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Modern Cosmology Part II From Special to General Relativity Max Camenzind ZAH Heidelberg Em CamSoft D Neckargem und, Germany M.Camenzindlsw.uniheidelberg.de Abstract Modern Cosmology is based on Einsteins view of gravity which is an extension of Special Relativity developped by Einstein in . Special Relativity SR is the physical theory of measurement in inertial frames of reference proposed in by Albert Einstein after the considerable and independent contributions of Hendrik Lorentz, Henri Poincare and others in the paper On the Electrodynamics of Moving Bodies. It generalizes Galileos principle of relativity that all uniform motion is relative, and that there is no absolute and well dened state of rest no privileged reference frames from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. Special Relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source . General Relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in November . It is the current description of gravitation in modern physics. It generalises special relativity and Newtons law of universal gravita tion, providing a unied description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the fourmomentum massenergy and linear momentum of whatever matter and radiation are present. The rela tion is specied by the Einstein eld equations, a system of partial differential equations. Many predictions of General Relativity differ signicantly fromthose of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time di lation, the gravitational redshift of light, and the gravitational time delay. General relativitys predictions have been conrmed in all observations and experiments to date. Although Gen eral Relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most funda mental being how General Relativity can be reconciled with the laws of quantum physics to produce a complete and selfconsistent theory of quantum gravity. Version October , Copyright C by Max Camenzind Table of Contents Modern Cosmology Part II From Special to General Relativity Max Camenzind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... II From Special to General Relativity Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MichelsonMorley Experiment and the Aetherwind . . . . . . . . . . . . . Postulates of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . PseudoRotations in D . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minkowski Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Concept of Minkowski SpaceTime . . . . . . . . . . . . . . . . . . . . . . . SpaceTime and Lorentz Transformations . . . . . . . . . . . . . . . . . . Vectors and Tensors in Minkowski SpaceTime . . . . . . . . . . . . . . . Causal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . Forces in D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newtonian Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . EnergyMomentum Tensor of Perfect Fluids . . . . . . . . . . . . . . . . Relativistic Plasma Equations . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Hydrodynamics as a Conservative System c . . . . . . Electromagnetism in Minkowski SpaceTime . . . . . . . . . . . . . . . . . . . . General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... Einsteins Principles of Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . Einstein Equivalence Principle EEP . . . . . . . . . . . . . . . . . . . . The Strong Equivalence Principle SEP . . . . . . . . . . . . . . . . . . Einsteins Vision of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Concept of SpaceTime . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity is an Afne Connection on SpaceTime . . . . . . . . . . . . . . . Calculus on Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . Torsion and Curvature of SpaceTime . . . . . . . . . . . . . . . . . . . . . Curvature and Einsteins Equations . . . . . . . . . . . . . . . . . . . . Is General Relativity the Correct Theory of Gravity . . . . . . . . . . . . . . . . Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . PostKeplerian Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . MCamenzind . On Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . PlanckLength and Limits of General Relativity . . . . . . . . . . . . . Alternative Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . BransDicke Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fR Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . Aberration Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Denition of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . TOVEquations for Compact Objects . . . . . . . . . . . . . . . . . . . . Curvature in a Spatially Flat Universe . . . . . . . . . . . . . . . . . . . . Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merging of two Black Holes at Cosmological Distances . . . . . . . . . . Gravitational Waves from Compact Binary Systems . . . . . . . . . . . . A Calculus for Differentiable Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christoffel Symbols and Covariant Derivative . . . . . . . . . . . . . . . . . . . Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ricci and Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weyl Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gradient, Divergence and LaplaceBeltrami Operator . . . . . . . . . . . . . . . Differential Forms on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . pForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hodge Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirac Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Perturbations of Minkowski Space and the Nature of Gravitational Waves . . . . . . . . . . . . . Linearized Gravity and Gauge Transformations . . . . . . . . . . . . . . . . . . . On Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Einsteins Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transverse Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Detection of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . Resonant Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laser Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SpaceBorne Interferometers . . . . . . . . . . . . . . . . . . . . . . . . Part II From Special to General Relativity Chapter Special Relativity FIGURE . Special Relativity is the physical theory of measurement in inertial frames of reference proposed in by Albert Einstein at the age of . Special relativity SR also known as the special theory of relativity is the physical theory of measurement in inertial frames of reference proposed in by Albert Einstein after the considerable and independent contributions of Hendrik Lorentz, Henri Poincar e and others in the paper On the Electrodynamics of Moving Bodies. It generalizes Galileos principle of relativity that all uniform motion is relative, and that there is no absolute and welldened state of rest no privileged reference frames from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. Special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source. The principle of relativity, which states that there is no preferred inertial reference frame, dates back to Galileo, and was incorporated into Newtonian Physics. However, in the late th century, the existence of electromagnetic waves led physicists to suggest that the universe was lled with a substance known as aether, which would act as the medium through which these waves, or vibrations travelled. The aether was thought to constitute an absolute reference frame against which speeds could be measured, and could be considered xed and motionless. Aether supposedly had some wonderful properties it was sufciently elastic that it could support electro magnetic waves, and those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, including the MichelsonMorley experi ment, indicated that the Earth was always stationary relative to the aether something that was difcult to explain, since the Earth is in orbit around the Sun. Einsteins solution was to discard the notion of an aether and an absolute state of rest. Special relativity is formulated so as to not assume that any particular frame of reference is special rather, in relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in a vacuum is always measured to be c, even when measured by multiple systems that are moving at different but constant velocities. The theory of special relativity is the combination of two ideas and their seemingly weird consequences. Chapter The laws of physics are the same wherever you are. This means that an experiment carried out in a moving train will give the same results as when it is performed in a lab. Furthermore, if there were no windows on the train and it was moving at a constant speed, there is no experiment that you could do to see whether or not it was actually moving. The speed of light is the same for everyone. The speed of light being the same wherever you are might not seem strange, but think about how we normally experience speeds. A ball thrown on a moving train will have a greater speed than a ball thrown with the same force by someone standing on the platform. This is because the speed of the train is added to that of the ball to give its total speed. But this isnt the case with light. If you measure the speed of the light produced by torches on a moving train and a stationary platform, you will get the same speed the speed of the train doesnt matter. When you measure the speed of light, it doesnt matter if you are moving or stationary, or if the source of the light is moving the speed is always the same ,, metres per second. But the only way that the laws of physics and the speed of light can always be the same is for something else to change. Special relativity shows that measurements of distance and time depend on how fast you are travelling a result that goes against our everyday experiences. If you measured the length of a baguette and the time it took you to eat it, there would be no difference whether you were on a moving train or standing on a platform but that is only because the speed of the train is so small. As speeds increase towards the speed of light, the socalled relativistic effects of time dilation clocks running slow and length contraction objects getting shorter become more and more obvious. But the most famous part of special relativity is the equation E mc , where E is energy, m is mass and c is the speed of light. The equation stems, in part, from the relationship between energy and momentum that Einstein developed to ensure that the speed of light was the same for everyone no matter what they were doing. The equation tells us that energy and mass can be changed from one to the other that they are equivalent. Space and Time The Universe has at least three spatial and one temporal time dimension. It was long thought that the spatial and temporal dimensions were different in nature and independent of one another. However, according to the special theory of relativity, spatial and temporal separations are inter convertible within limits by changing ones motion. In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being threedimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. According to certain Euclidean space perceptions, the universe has three dimensions of space and one dimen sion of time. By combining space and time into a single manifold, physicists have signicantly simplied a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels. In mathematics, a covariant metric tensor g is a nonsingular symmetric tensor eld of rank that is used to measure distance in a space. In other words, given a smooth manifold, we make a choice of , tensor on the manifolds tangent spaces. At a given point in the manifold, this tensor takes a pair of vectors in the tangent space to that point, and gives a real number. If it is positive, this is just an inner product on each tangent space, which is required to vary smoothly from point to point. The concept of spacetime combines space and time to a single abstract space, for which a unied coordinate system is chosen. Typically three spatial dimensions length, width, height, and one temporal dimension time are required. Dimensions are independent components of a coordinate grid needed to locate a point in a certain dened space. For example, on the globe Special Relativity the latitude and longitude are two independent coordinates which together uniquely determine a location. In spacetime, a coordinate grid that spans the dimensions locates events rather than just points in space, i.e. time is added as another dimension to the coordinate grid. This way the coordinates specify where and when events occur. However, the unied nature of spacetime and the freedom of coordinate choice it allows imply that to express the temporal coordinate in one coordinate system requires both temporal and spatial coordinates in another coordinate system. Unlike in normal spatial coordinates, there are still restrictions for how measurements can be made spatially and temporally see Spacetime intervals. Until the beginning of the th century, time was believed to be independent of motion, pro gressing at a xed rate in all reference frames however, later experiments revealed that time slowed down at higher speeds with such slowing called time dilation explained in the theory of Special Relativity. Many experiments have conrmed time dilation, such as atomic clocks onboard a Space Shuttle running faster than synchronized Earthbound inertial clocks and the relativistic decay of muons from cosmic ray showers. The duration of time can therefore vary for various events and various reference frames. When dimensions are understood as mere com ponents of the grid system, rather than physical attributes of space, it is easier to understand the alternate dimensional views as being simply the result of coordinate transformations. Spacetimes are the arenas in which all physical events take place an event is a point in spacetime specied by its time and place. For example, the motion of planets around the Sun may be described in a particular type of spacetime, or the motion of light around a rotating star may be described in another type of spacetime. The basic elements of spacetime are events. In any given spacetime, an event is a unique position at a unique time. Because events are spacetime points, an example of an event in classical relativistic physics is t, x, y, z, the location of an elementary pointlike particle at a particular time. A spacetime itself can be viewed as the union of all events in the same way that a line is the union of all of its points, organized into a manifold a locally at metric space. An understanding of calculus and differential equations is necessary for the understanding of nonrelativistic physics. In order to understand Special Relativity one also needs an understanding of tensor calculus. To understand the general theory of relativity, one needs a basic introduction to the mathematics of curved spacetime that includes a treatment of curvilinear coordinates, non tensors, curved space, parallel transport, Christoffel symbols, geodesics, covariant differentiation, the curvature tensor, Bianchi identity, and the Ricci tensor. Basics of Special Relativity This Section is devoted to the consequences of Einsteins principle of Special Relativity, which states that all the fundamental laws of physics are the same for all uniformly moving nonaccelerating observers. In particular, all of them measure precisely the same value for the speed of light in vacuum, no matter what their relative velocities. Before Einstein wrote, several principles of relativity had been proposed, but Einstein was the rst to state it clearly and hammer out all the counterintuitive consequences. This theory has a wide range of consequences which have been experimentally veried, in cluding counterintuitive ones such as length contraction, time dilation and relativity of simul taneity, contradicting the classical notion that the duration of the time interval between two events is equal for all observers. On the other hand, it introduces the spacetime interval, which is in variant. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of matter and energy, as expressed in the massenergy equivalence formula E mc , where c is the speed of light in a vacuum. The predictions of special relativity agree well with Newtonian mechanics in their common realmof applicability, specically in experiments in which all velocities are small compared with the speed of light. Special Relativity reveals that c is not just the velocity of a certain phenomenon, namely the propagation of electromagnetic radiation Chapter light, but rather a fundamental feature of the way space and time are unied as spacetime. One of the consequences of the theory is that it is impossible for any particle that has rest mass to be accelerated to the speed of light. The theory is termed special because it applies the principle of relativity only to inertial ref erence frames, i.e. frames of reference in uniform relative motion with respect to each other. Einstein developed general relativity to apply the principle more generally, that is, to any frame so as to handle general coordinate transformations, and that theory includes the effects of gravity. From the theory of general relativity it follows that special relativity will still apply locally i.e., to rst order, and hence to any relativistic situation where gravity is not a signicant factor. Inertial frames should be identied with nonrotating Cartesian coordinate systems constructed around any free falling trajectory as a time axis. . MichelsonMorley Experiment and the Aetherwind In the late nineteenth century, most physicists were convinced, contra Newton , that light is a wave and not a particle phenomenon. They were convinced by interference experiments whose results can be explained classically only in the context of wave optics. The fact that light is a wave implied, to the physicists of the nineteenth century, that there must be a medium in which the waves propagat there must be something to wave and the speed of light should be measured relative to this medium, called the aether. The Earth orbits the Sun, so it cannot be at rest with respect to the medium, at least not on every day of the year, and probably not on any day. The motion of the Earth through the aether can be measured with a simple experiment that compares the speed of light in perpendicular directions. This is known as the Michelson Morley experiment and its surprising result was a crucial hint for Einstein and his contemporaries in developing Special Relativity. FIGURE . The Earth travels a tremendous distance in its orbit around the Sun, at a speed of around km/s. The Sun itself is travelling about the Galactic Center at even greater speeds, and there are other motions at higher levels of the structure of the Universe. Since the Earth is in motion, it was expected that the ow of aether across the Earth should produce a detectable aether wind. Special Relativity Michelson and Morley designed in an experiment, employing an interferometer and a halfsilvered mirror, that was accurate enough to detect aether ow. The mirror system reected the light back into the interferometer. If there were an aether drift, it would produce a phase shift and a change in the interference that would be detected. However, no phase shift was ever found. The negative outcome of the MichelsonMorley experiment left the whole concept of aether without a reason to exist. Worse still, it created the perplexing situation that light evidently behaved like a wave, yet without any detectable medium through which wave activity might propagate. FIGURE . The Michelson interferometer produces interference fringes by splitting a beam of monochromatic light so that one beam strikes a xed mirror and the other a movable mirror. When the reected beams are brought back together, an interference pattern results, which should depend on the direction of the aether wind. . Postulates of Special Relativity The rst principle of relativity ever proposed is attributed to Galileo, although he probably did not formulate it precisely. Galileos principle of relativity says that sailors on a uniformly moving boat cannot, by performing onboard experiments, determine the boats speed. They can determine the speed by looking at the relative movement of the shore, by dragging something in the water, or by measuring the strength of the wind, but there is no way they can determine it without observing the world outside the boat. A sailor locked in a windowless room cannot even tell whether the ship is sailing or docked. This is a principle of relativity, because it states that there are no observational consequences of absolute motion. One can only measure ones velocity relative to something else. As physicists we are empiricists we reject as meaningless any concept which has no observ able consequences, so we conclude that there is no such thing as absolute motion. Objects have velocities only with respect to one another. Any statement of an objects speed must be made with respect to something else. Our language is misleading, because we often give speeds with no reference object. When Kepler rst introduced a heliocentric model of the Solar System, it was resisted on the grounds of common sense. If the Earth is orbiting the Sun, why cant we feel the motion Relativity provides the answer there are no local, observational consequences to our motion. Now that the Earths motion is generally accepted, it has become the best evidence we have for Galilean relativity. On a daytoday basis we are not aware of the motion of the Earth around the Sun, despite the fact that its orbital speed is a whopping km/s. We are also not aware of the Suns km/s motion around the center of the Galaxy, or the roughly km/s motion of the local group of galaxies which includes the Milky Way relative to the rest frame of the cosmic background radiation. We have become aware of these motions only by observing extraterrestrial references in the above cases, the Sun, the Galaxy, and the cosmic background radiation. Our Chapter everyday experience is consistent with a stationary Earth. Einsteins principle of relativity says, roughly, that every physical law and fundamental phys ical constant including, in particular, the speed of light in vacuum is the same for all non accelerating observers. This principle was motivated by electromagnetic theory and in fact the eld of special relativity was launched by a paper entitled in English translation on the electro dynamics of moving bodies Einstein . Einsteins principle is not different from Galileos, except that it explicitly states that electromagnetic experiments such as measurement of the speed of light will not tell the sailor in the windowless room whether or not the boat is moving, any more than uid dynamical or gravitational experiments. Since Galileo was thinking of exper iments involving bowls of soup and cannonballs dropped from towers, Einsteins principle is effectively a generalization of Galileos. Einstein discerned two fundamental propositions that seemed to be the most assured, regard less of the exact validity of the then known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light and the independence of physical laws es pecially the constancy of the speed of light from the choice of inertial system. In his initial presentation of special relativity in he expressed these postulates as The Principle of Relativity The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other. The Principle of Invariant Light Speed ... light is always propagated in empty space with a denite velocity speed c which is independent of the state of motion of the emitting body. That is, light in vacuum propagates with the speed c a xed constant, independent of direction in at least one system of inertial coordinates the stationary system, regardless of the state of motion of the light source. Following Einsteins original presentation of Special Relativity in , many different sets of postulates have been proposed in various alternative derivations. However, the most common set of postulates remains those employed by Einstein in his original paper. . Lorentz Transformations Einstein has said that all of the consequences of special relativity can be derived from examination of the Lorentz transformations. Relativity theory depends on reference frames. The term reference frame as used here is an observational perspective in space at rest, or in uniform motion, from which a position can be measured along spatial axes. In addition, a reference frame has the ability to determine mea surements of the time of events using a clock any reference device with uniform periodicity. An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame it is a point in spacetime. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired. For example, the explosion of a a supernova may be considered to be an event. We can completely specify an event by its four spacetime coordinates The time of occurrence and its dimensional spatial location dene a reference point. Lets call this reference frame S. Since there is no absolute reference frame in relativity theory, a concept of moving doesnt strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore S and S are not comoving. Special Relativity Lets dene the event to have spacetime coordinates t, x, y, z in system S and t ,x ,y ,z in S see Fig. .. Then the Lorentz transformation species that these coordinates are related in the following way t t vx/c x x vt y y z z v c is the Lorentz factor and c is the speed of light in a vacuum. A quantity invariant under Lorentz transformations is known as a Lorentz scalar. FIGURE . Two observers S and S move in xdirection with speed v, each using their own Cartesian coordinate system to measure space and time intervals. The coordinate systems are oriented so that the xaxis and the x axis are collinear, the yaxis is parallel to the yaxis, as are the zaxis and the zaxis. The relative velocity between the two observers is v along the common xaxis. The inverse transformation is then simply given as tt vx /c . xx vt . yy . zz .. .. On the Derivation In most textbooks, the Lorentz transformation is derived from the two postulates the equivalence of all inertial reference frames and the invariance of the speed of light. However, the most gen eral transformation of space and time coordinates can be derived using only the equivalence of all inertial reference frames and the symmetries of space and time. The general transformation Chapter depends on one free parameter with the dimensionality of speed, which can be then identied with the speed of light c. This derivation uses the group property of the Lorentz transforma tions, which means that a combination of two Lorentz transformations also belongs to the class Lorentz transformations. In the following we shortly discuss the rst way to derive the Lorentz transformations. The Lorentz transformation is a linear transformation. Thus ct Act Bx . x Dx E ct . with four unknown functions of v. The origin of the reference frame S has the coordinate x and moves with velocity v relative to the reference frame S, so that x vt. For x we have dx/dt v and for x we nd dx /dt v. Thus v dx dt E D c,v dx dt E C c,. and hence D A and E v A/c A with v/c. Three unknowns A, B and E are left. These coefcients now follow from the invariance of the speed of light ct x ct x Act Bx Ax Ect .. This implies A E ct A B x AE Bxct . . Thereore, B E and A E A A .. and A .. This leads to the solutions A. BE. DA. E.. .. General Lorentz Transformation The Lorentz transformation given above is for the particular case in which the velocity v of S with respect to S is parallel to the xaxis. We now give the Lorentz transformation in the general case. Suppose the velocity of S with respect to S is v. Denote the spacetime coordinates of an event in S by t, r. For a boost in an arbitrary direction with velocity v, it is convenient to decompose the spatial vector r into components perpendicular and parallel to the velocity v rr r . Then only the component r in the direction of v is warped by the gamma factor t t v r/c . r r r vt . . These transformation laws can be written in matrix form t r v T /c v/c I v v T /v t r . where I is the identity matrix and v T denotes the transpose of v of a row vector. Special Relativity . PseudoRotations in D Proper Lorentz transformations x x form a group with det and . First there are the conventional rotations, such as a rotation in the x y plane cos sin sin cos . There are also Lorentz boosts, which may be thought of as rotations between space and time directions also called pseudorotations. An example is given by cosh sinh sinh cosh . The boost parameter , unlike the rotation angle, is dened from to . There are also discrete transformations which reverse the time direction or one or more of the spatial directions. When these are excluded we have the proper Lorentz group, SO, . A general transformation can be obtained by multiplying the individual transformations the explicit expression for this sixparameter matrix three boosts, three rotations is not sufciently pretty or useful to bother writing down. In general Lorentz transformations will not commute, so the Lorentz group is non abelian. The set of both translations and Lorentz transformations is a tenparameter nonabelian group, the Poincar e group. The boosts correspond to changing coordinates by moving to a frame which travels at a con stant velocity, but lets see it more explicitly. For the transformation given by ., the trans formed coordinates t and x will be given by ct ct cosh x sinh . x ct sinh x cosh . . From this we see that the point dened by x is moving with a velocity v c x ct sinh cosh tanh . . To translate into more pedestrian notation, we can replace tanh v/c and the relations cosh . sinh . to obtain the wellknown classical expressions for the Lorentz transformations. For the transformation of an arbitrary vector a under a Lorentz transformation with veloc ity v we decompose the vector a into a component parallel to the unit vector n in the direction of v and a component perpendicular to this direction aa na ,a a n. . The transformation is then given as a a cosh a sinh a a . a a sinh a cosh a a . a a .. Chapter . Physical Predictions These transformations, and hence Special Relativity, lead to different physical predictions than Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counterintuitive Time dilation the time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers reference frames e.g., the twin paradox which concerns a twin who ies off in a spaceship traveling near the speed of light and returns to discover that his or her twin sibling has aged much more. Consider the interval between two ticks of a clock moving at the speed v in xdirection, T t t . An observer sitting in a system S x x sees the timeinterval Tt t t vx /c t vx /c t t T .. Relativity of simultaneity two events happening in two different locations that occur si multaneously in the reference frame of one inertial observer, may occur nonsimultaneously in the reference frame of another inertial observer lack of absolute simultaneity. Lorentz contraction the dimensions e.g., length of an object as measured by one ob server may be smaller than the results of measurements of the same object made by another observer e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage. L x x x x L .. Therefore, from the system S the lenght appears as L /. Composition of velocities velocities and speeds do not simply add, for example if a rocket is moving at / the speed of light relative to an observer, and the rocket res a missile at / of the speed of light relative to the rocket, the missile does not exceed the speed of light relative to the observer. In this example, the observer would see the missile travel with a speed of / the speed of light. If the observer in S sees an object moving along the x axis at velocity u, then the observer in the S system, a frame of reference moving at velocity v in the x direction with respect to S, will see the object moving with velocity u where u x u x v vu x /c . u y u y vu x /c . u z u z vu x /c .. The inverse transformation is given by u x u x v vu x /c . u y u y vu x /c . u z u z vu x /c .. Special Relativity .. On the Derivation Let an object be moving with velocities u and u with respect to inertial frames S and S, repsectively. The frame S is itself moving with velocity v along the xaxis. The we get u x dx dt dx/dt dt/dt u x v u x /c u x v u x /c . u y dy dt dy/dt dt/dt u y u x /c . u z dz dt u z u x /c .. Similarly, we can write u x dx dt dx /dt dt /dt u x v u x /c u x v u x /c . u y dy dt dy /dt dt /dt u y u x /c . u z dz dt dz /dt dt /dt u z u x /c .. So, the velocity perpendicular to the xaxis only suffers from timedilation. For small velocities, v/c , this gives the famous Galilean transformation u u x v. If one of the velocities is the speed of light, e.g. u x c, then u x cv v/c c.. The velocity of light is indeed an unsurmountable speed limit. Example If a radioactive nucleus travels in the Lab with a speed of .c is emitting an electron with velocity .c, then with respect to the Lab the velocity of the electron is not .c, but only .c. The velocity addition theorem has been veried by a number of experiments. .. Remark The velocityaddition theorem can easily be treated with pseudorotations. Since Lorentz transformations form a group, performing two Lorentz transformations in the xdirection produces also a Lorentz tranformation. Using pseudorotations we nd ct ct cosh x sinh . x ct sinh x cosh . ct ct cosh x sinh ct cosh x sinh . x ct sinh x cosh ct sinh xcosh, . with . With tanh v/c and the wellknown theorem for the hyperbolic tangent tanh tanh tanh tanh tanh . we obtain for the combined velocity v v v v v /c .. Chapter Inertia and momentum as an objects speed approaches the speed of light from an ob servers point of view, its mass appears to increase thereby making it more and more dif cult to accelerate it from within the observers frame of reference. Equivalence of mass and energy E mc The energy content of an object at rest with mass m equals mc . Conservation of energy implies that in any reaction a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies. . Minkowski Diagrams The Minkowski diagram was developed in by Hermann Minkowski and provides an illustra tion of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations. The Minkowski diagram is a spacetime diagram with usually only one space dimension. It is a superposition of the coordinate systems for two observers moving rel ative to each other with constant velocity. Its main purpose is to allow for the space and time coordinates x and t used by one observer to read off immediately the corresponding x and t used by the other and vice versa. From this onetoone correspondence between the coordinates the absence of contradictions in many apparently paradoxical statements of the theory of relativity becomes obvious. Also the role of the speed of light as an unconquerable limit results graphically from the properties of space and time. The shape of the diagram follows immediately and without any calculation from the postulates of special relativity, and shows the close relationship between space and time discovered with the theory of relativity. If ct instead of t is assigned on the time FIGURE . Minkowski diagram for the translation of the space and time coordinates x and t of a rst observer into those of a second observer blue moving relative to the rst one with of the speed of light c. Each point in the diagram represents a certain position in space and time. Such a position is called an event whether or not anything happens at that position. axes, the angle between both path axes results to be identical with that between both time axes. This follows from the second postulate of the special relativity, saying that the speed of light is Special Relativity the same for all observers, regardless of their relative motion. is given by tan v c .. Relativistic time dilation means that a clock moving relative to an observer is running slower and nally also the time itself in this system this is important for understanding e.g. the GPS system. This can be read immediately from the adjoining Minkowski diagram Fig. . The observer at A is assumed to move from the origin O towards A and the clock from O to B. For this observer at A all events happening simultaneously in this moment are located on a straight line parallel to its path axis passing A and B. Due to OB lt OA he concludes that the time passed on the clock moving relative to him is smaller than that passed on his own clock since they were together at O. FIGURE . Time dilation Both observers consider the clock of the other as running slower. For the speed of a photon passing A both observers measure the same value even though they move relative to each other. A second observer having moved together with the clock from O to B will argue that the other clock has reached only C until this moment and therefore this clock runs slower. The reason for these apparently paradoxical statements is the different determination of the events happening synchronously at different locations. Due to the principle of relativity the question of who is right has no answer and does not make sense. .. Speed of Light Another postulate of special relativity is the constancy of the speed of light. It says that any observer in an inertial reference frame measuring the speed of light relative to himself obtains the same value regardless of his own motion and that of the light source. This statement seems to be paradox, but it follows immediately from the differential equation yielding this, and the Minkowski diagram agrees. It explains also the result of the MichelsonMorley experiment which was considered to be a mystery before the theory of relativity was discovered, when photons were thought to be waves through an undetectable medium. For world lines of photons passing the origin in different directions x ct and x ct holds. That means any position on such a world line corresponds with steps on x and ctaxis of equal absolute value. From the rule for reading off coordinates in coordinate system with tilted Chapter FIGURE . Minkowski diagram for coordinate systems. For the speeds relative to the system in black v .c and v .c holds. Any observer in an inertial reference frame measuring the speed of light relative to himself obtains the same value regardless of his own motion and that of the light source. axes follows that the two world lines are the angle bisectors of the x and ctaxis. The Minkowski diagram shows that they are angle bisectors of the x and ct axis as well. That means both observers measure the same speed c for both photons. In principle further coordinate systems corresponding to observers with arbitrary velocities can be added in this Minkowski diagram. For all these systems both photon world lines represent the angle bisectors of the axes. The more the relative speed approaches the speed of light the more the axes approach the corresponding angle bisector. The path axis is always more at and the time axis more steep than the photon world lines. The scales on both axes are always identical, but usually different from those of the other coordinate systems. The Concept of Minkowski SpaceTime In the mathematician Hermann Minkowski explored a way of visualizing these processes that proved to be especially well suited to disentangling relativistic effects. This was their rep resentation in spacetime. Quite puzzling relativistic effects could be comprehended with ease within the spacetime representation and work in the theory of relativity started to be transformed into work on the geometry of spacetime. . SpaceTime and Lorentz Transformations We build a spacetime by taking instantaneous snapshots of space at successive instants of time and stacking them up. It is easiest to imagine this if we start with a two dimensional space. The snapshots taken at different times are then stacked up to give us a three dimensional spacetime. In this spacetime, a small body at rest will be represented by a vertical line. To see why it is vertical, recall that it has to intersect each instantaneous space at the same spot. A vertical line will do this. If it is moving, it will intersect each instantaneous space at a different spot a moving body is presented by a line inclined to the vertical. A standard convention is to represent trajectories of light signals by lines at degrees to the vertical. SR uses a at dimensional Minkowski space, which is an example of a spacetime. This Special Relativity space, however, is very similar to the standard dimensional Euclidean space. The differential of distance ds in cartesian D space is dened as ds dx dx dx ,. where dx , dx , dx are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes ds dx dx dx c dt .. This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry of our spacetime, very similar to rotational symmetry of Euclidean space. Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. Basically, SR can be stated in terms of the invariance of spacetime interval between any two events as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry the Poincar e group of Minkowski spacetime. If we reduce the spatial dimensions to , so that we can represent the physics in a D space ds dx dx c dt ,. we see that the null geodesics lie along a dualcone dened by the equation ds dx dx c dt . or simply dx dx c dt ,. which is the equation of a circle of radius c dt. Having recognised the fourdimensional nature of spacetime, we are driven to employ the Minkowski metric, , given in components valid in any inertial reference frame as . which is equal to its reciprocal, , in those frames. Then we recognize that coordinate transformations between inertial reference frames are given by the Lorentz transformation matrix . For the special case of motion along the xaxis, we have . which is simply the matrix of a boost like a rotation between the x and ct coordinates. Where indicates the row and indicates the column. Also, and are dened as v c , .. More generally, a transformation from one inertial frame ignoring translations for simplicity to another must satisfy , T ,. Chapter where there is an implied summation of and from to on the righthand side in accordance with the Einstein summation convention. The Poincar e group is the most general group of trans formations which preserves the Minkowski metric and this is the physical symmetry underlying Special Relativity. Taking the determinant of gives us det .. Lorentz transformations with det are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with det are called improper Lorentz transformations and consist of discrete space and time reections combined with spatial rotations and boosts. They dont form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation. In , Henri Poincar e was the rst to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz. Later in the same year, Einstein derived the Lorentz transformation under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame, obtaining results that were algebraically equivalent to Larmors and Lorentzs , , but with a different interpretation. . Vectors and Tensors in Minkowski SpaceTime To probe the structure of Minkowski space in more detail, it is necessary to introduce the con cepts of vectors and tensors. We will start with vectors, which should be familiar. Of course, in spacetime vectors are fourdimensional, and are often referred to as fourvectors. This turns out to make quite a bit of difference for example, there is no such thing as a cross product between two fourvectors. A scalar is a single quantity function whose value does not change under Lorentz transfor mations. We already have introduced the concept of a vector for quantities such as dx ,f or p . They generally transofrm under a Lorentz transormation as V V V .. Such a quantity is called a contravariant vector, to distinguish it from a covariant one U U U ,. where .. From this, it follows that the scalar product is invariant U V U V U V ,. i.e. the expression V V . denes a mapping of contravariant vectors to covariant ones. Linear Lorentz transformations forma subset of general coordinate transformations in Minkowski space x x x ,x ,x ,x . A vector is said to be contravariant if it transforsm as A x x A .. A vector B is said to be covariant if it transforms as B x x B .. Special Relativity As a consequence, the product B A B A is invariant under these tranformations. Although any vector can be written in a contravariant or covariant form, there are some vectors which appear more naturally contravariant such as dx others covariant, such as /x . This gradient is covariant /x /x .. Therefore, the divergence V /x is Lorentzinvariant or a scalar quantity, and therefore similarly, the dAlembertian /x /x c t . is also Lorentzinvariant. This demonstrates that the wave equation is invariant under Lorentz transformations, as it should be. .. Tensors of Higher Rank Vectors are in a way tensors of rst rank. Similarly, tensors of higher rank are dened by means of their transformation properties T T T .. In particular, the energymomentum tensor will be a second rank symmetric tensor, i.e. T T . A particular example of a tensor of higher rank is the totally antisymmetric LeviCivita tensor , even permutation of , odd permutation of , otherwise . The transformed tensor satises ,. since the left hand side is simply the determinant of . The LeviCivita tensor also satises .. In Relativity, all proper physical quantities must be given in tensorial form. So to trans form from one frame to another, we use the general tensor transformation law T i ,i ,...,i p j ,j ,...,j q i i i i i p i p j j j j j q j q T i ,i ,...,i p j ,j ,...,j q . Here j k j k is the reciprocal matrix of j k j k . . Causal Structure In Fig. the interval AB is timelike i.e., there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter or information to travel from A to B, so there can be a causal relationship with A the cause and B the effect. The interval AC in the diagram is spacelike i.e., there is a frame of reference in which events A and C occur simultaneously, separated only in space. However there are also frames in which A precedes C as shown and frames in which C precedes A. If it were possible for a causeandeffect relationship to exist between events A and C, then paradoxes of causality would Chapter FIGURE . The light cones in Minkowski space are at. The timeaxis runs vertically, the spatial axes horizontally. A light cone is the path that a ash of light, emanating from a single event A localized to a single point in space and a single moment in time and traveling in all directions, would take through spacetime. If we imagine the light conned to a twodimensional plane, the light from the ash spreads out in a circle after the event A occurs. result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself wont give rise to a paradox, one can show that faster than light signals can be sent back into ones own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously. Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel faster than light in a vacuum. However, some things can still move faster than light. For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly. Even without considerations of causality, there are other strong reasons, why fasterthanlight travel is forbidden by Special Relativity. For example, if a constant force is applied to an object for a limitless amount of time, then integrating F dp/dt gives a momentum that grows without bound, but this is simply because p mv approaches innity as v approaches c. To an observer who is not accelerating, it appears as though the objects inertia is increasing, so as to produce a smaller acceleration in response to the same force. This behavior is in fact observed in particle accelerators LHC e.g.. . Velocity and Acceleration Recognising other physical quantities as tensors also simplies their transformation laws. First note that the velocity fourvector U is given by U dx d c v x v y v z . Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity fourvector of one particle from one frame to another. U also has an invariant form U U U c .. Special Relativity FIGURE . The light cones in Minkowski space are at. The timeaxis runs vertically, the spatial axes horizontally. At each event we nd a forward and backward light cone. Photons and other massless particles move along the light cone, while the trajectories of normal particles are conned to the interior of the light cones. A detector can only measure photons which come in from the backward light cone. So all velocity fourvectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity at the least, you are always moving forward through time. The acceleration vector is given by a dU /d. Given this, differentiating the above equation by produces a U .. So in relativity, the acceleration fourvector and the velocity fourvector are orthogonal, acceler ation is always spacelike. .. Energy and Momentum Similarly, momentum and energy combine into a covariant vector p m U E/c p x p y p z ,. where m is the invariant mass. The invariant magnitude of the momentum vector is p p p E/c p .. We can work out what this invariant is by rst arguing that, since it is a scalar, it doesnt mat ter which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero. p E rest /c mc .. We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero. Chapter The rest energy is related to the mass according to the celebrated equation discussed above E rest mc .. Note that the mass of systems measured in their center of momentum frame where total momen tum is zero is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames. . Forces in D To use Newtons third law of motion, both forces must be dened as the rate of change of mo mentum with respect to the same time coordinate. That is, it requires the D force dened above. Unfortunately, there is no tensor in D which contains the components of the D force vector among its components. If a particle is not traveling at c, one can transform the D force from the particles comoving reference frame into the observers reference frame. This yields a vector called the fourforce. It is the rate of change of the above energy momentum fourvector with respect to proper time. The covariant version of the fourforce is F dp d dE/c/d dp x /d dp y /d dp z /d ,. where is the proper time. In the rest frame of the object, the time component of the four force is zero, unless the in variant mass of the object is changing this requires a nonclosed system in which energy/mass is being directly added or removed from the object in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the fourforce are not equal to the components of the threeforce, because the threeforce is dened by the rate of change of momentum with respect to coordinate time, i.e. dp dt , while the fourforce is dened by the rate of change of momentum with respect to proper time, i.e. dp d . In a continuous medium, the D density of force combines with the density of power to form a covariant vector. The spatial part is the result of dividing the force on a small cell in space by the volume of that cell. The time component is /c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism. Relativistic Hydrodynamics In physics and astrophysics, uid dynamics is a subdiscipline of uid mechanics that deals with uid owthe natural science of uids liquids and gases in motion. It has several subdisciplines itself, including aerodynamics the study of air and other gases in motion and hydrodynamics the study of liquids in motion. . Newtonian Euler Equations The foundational axioms of uid dynamics are the conservation laws, specically, conservation of mass, conservation of linear momentum also known as Newtons Second Law of Motion, and conservation of energy also known as First Law of Thermodynamics. The basic variables are the mass density , the velocity v, pressure P and internal energy density e. The corresponding equations are the conservation of mass, momenta and total energy with internal energy e and Special Relativity specic enthalpy h see e.g. LandauLifshitz VI t v. t vvvP. t v e v v h . G,. These laws can easily be expressed as true conservation laws in Cartesian coordinates, but not in curvilinear coordinates see Fig. . FIGURE . The conservative formulation of the Euler equations consists of equations, which can be combined into one vectorial equation for the state vector U , v, E T . In addition, we need an equation of state for the pressure P. Hydrodynamic instabilities play a major role in determining the efciency and performance of inertial connement fusion implosions. In laserdriven implosions, highperformance cap sules require high aspect ratios the ratio of the radius to the shell thickness. These capsules are susceptible to hydrodynamic instabilities of the RayleighTaylor, RichtmyerMeshkov, and KelvinHelmholtz varieties, which can in principle severely degrade capsule performance. .. Example RayleighTaylor Instability The RayleighTaylor instability, or RT instability after Lord Rayleigh and G. I. Taylor, is an instability of an interface between two uids of different densities, which occurs when the lighter uid is pushing the heavier uid. This is the case with an interstellar cloud and shock system. The equivalent situation occurs when gravity is acting on two uids of different density with the dense uid above a uid of lesser density such as water balancing on light oil. As the instability develops, downwardmoving irregularities dimples are quickly magnied into sets of interpenetrating RayleighTaylor ngers Fig. . Therefore the RayleighTaylor instability is sometimes qualied to be a ngering instability. The upwardmoving, lighter material is shaped like mushroom caps. Chapter FIGURE . Hydrodynamical simulation of the RayleighTaylor instability in Newtonian uid dynamics pseudocolors for the density distribution yellow is the light uid blue the heavy uid. Gravity g acts in vertical direction downwards. Time progresses from left to right. This shows that the boundary between the heavy uid and the light uid is heavily unstable, leading to a kind of mushroom structure and vortices. Finally, the entire boundary will become turbulent. RayleighTaylor instabilities develop behind the supernova blast wave on a time scale of a few hours. The importance of the RayleighTaylor RT instability and turbulence in accelerating a thermonuclear ame in Type Ia supernovae SNe Ia is well recognized. Flame instabilities play a dominant role in accelerating the burning front to a large fraction of the speed of sound in a Type Ia supernova. The KelvinHelmholtz instabilities accompanying the RT instability in SNe Ia drives most of the turbulence in the star, and, as the ame wrinkles, it will interact with the turbulence generated on larger scales. . EnergyMomentum Tensor of Perfect Fluids Many applications in relativistic Astrophysics are based on a hydrodynamical description of mat ter the internal structure of white dwarfs and neutron stars is based on the hydrostatic approx imation, and accretion onto compact objects in general requires a timedependent treatment of gas dynamics. We can dene a perfect uid such that in local comoving coordinates the uid is isotropic. In Minkowskian spacetime, the energymomentum tensor of the uid is given by T tt ,T xx T yy T zz P,. where is the total proper energy density and P the pressure. When each uid element has a spatial velocity v i with respect to some xed lab frame, the expression of the energymomentum tensor is obtained via a Lorentz boost T Pu u P .. Here, u is the uid velocity, satisfying u u c . The equations for conservation of energy and momentum can be written as T , in Minkowski spacetime. In order to extend Special Relativity this expression to curved spacetime we only need to replace the Minkowskian metric by the general Lorentz metric of the spacetime and partial derivatives with covariant ones. Thus, in a general curved spacetime, the stress energy tensor for a perfect uid plasma is given by T Pu u Pg .. In the strong gravity regime, pressure and stresses are typically so large that we cannot assume that the uid is incompressible. In addition, the pressure contributions to the stress tensor can be of the same order as those from the energy density for relativistic uids. This makes relativistic plasmas behave very differently from the type of plasmas that we encounter in daily life, where the stress energy tensors are dominated by their rest mass density. . Relativistic Plasma Equations The general relativistic hydrodynamic equations consist of the local conservation laws of the stressenergy tensor T the Bianchi identities and of the matter current density the continuity equation T . J ,. where J is the masscurrent J u .. In distinction to the energy density , we denote the rest mass energy density as . The above expression for the stressenergy tensor can be extended to a nonperfect plasma as follows see e.g. MTW T u u Ph q u q u ,. where h is the spatial projection tensor h g u u . In addition, and are the shear and bulk viscosities. The expansion , describing the divergence or convergence of the uid world lines, is dened as u . The symmetric, tracefree, spatial shear tensor is dened by u h u h h .. Finally, q is the heat energy ux vector, which is spacelike, u q . In order to close the system, the equations of motion and the continuity equation must be sup plemented with an equation of state EOS relating some fundamental thermodynamical quanti ties. In general, the EOS takes the form P P , . Traditionally, most of the approaches for numerical integrations of the general relativistic hydrodynamic equations have adopted spacelike foliations of the spacetime, within the formulation. . Relativistic Hydrodynamics as a Conservative System c In the framework of special relativity, the motion of an ideal uid is governed by particle number conservation and energymomentum conservation. In the lab frame of reference, these two con servation equations can be written in closed divergence form, similar to the Newtonian equations U t F i x i .. The vedimensional state vector U D, S i , T , i , , , consists of the relativistic den sity D, the momentum density vector S and the total energy density with pressure P. The Chapter transformation between the rest frame quantities , the specic enthalpy h, pressure P and veloc ity v are given by DW . S W hv . W h P D E D, . where the Lorentz factor is traditionally designated as W / v , h e/ P/ is the relativistic specic enthalpy, and E is the total energy density Bernoulli energy. The corresponding ux vectors are given by F i Dv i ,S j v i P i j , Pv i .. The state of the relativistic plasma is therefore given either in terms of the vedimensional state vector U UP, or in terms of the primitive variables P , v ,v ,v ,P T . While the expression for the state vector U in terms of the primitive variables P is trivial, the inverse relation involves the calculation of the Lorentz factor D/W . v S/E P . P DW hE.. The Lorentz factor can be expressed in terms of the pressure W P S EP .. For given D, S and E, one can derive from the above relations an implicit expression for P fP DhP, WP E P , . where / denotes the specic proper volume, which is related to the enthalpy variation dh s dP . . This equation must be solved for all grid points in order to recover the pressure from the values of the state vector U. Electromagnetism in Minkowski SpaceTime Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that nite propagationspeed of the E and B elds required certain behaviors on charged particles. The general study of moving charges forms the LinardWiechert potential, which is a step towards special relativity. The Lorentz transformation of the electric eld of a moving charge into a nonmoving ob servers reference frame results in the appearance of a mathematical term commonly called the magnetic eld. Conversely, the magnetic eld generated by a moving charge disappears and be comes a purely electrostatic eld in a comoving frame of reference. Maxwells equations are thus simply an empirical t to special relativistic effects in a classical model of the Universe. As electric and magnetic elds are reference frame dependent and thus intertwined, one speaks of electromagnetic elds. Special relativity provides the transformation rules for how an electro magnetic eld in one inertial frame appears in another inertial frame. Special Relativity Maxwells equations in the Dformare already consistent with the physical content of Special Relativity. But we must rewrite them to make them manifestly invariant. The charge density and current density J x ,J y ,J z are unied into the currentcharge vector J c J x J y J z .. The law of charge conservation, t J , becomes J .. The electric eld E x ,E y ,E z and the magnetic induction B x ,B y ,B z are now unied into the rank antisymmetric covariant electromagnetic eld tensor, called Faraday tensor F E x /c E y /c E z /c E x /c B z B y E y /c B z B x E z /c B y B x .. The density, f , of the Lorentz force, f EJ B, exerted on matter by the electromagnetic eld becomes f F J .. Faradays law of induction, E B t , and Gausss law for magnetism, B , combine to form F F F .. Although there appear to be equations here, it actually reduces to just four independent equa tions. Using the antisymmetry of the electromagnetic eld, one can either reduce to an identity or render redundant all the equations except for those with , , either ,, or ,, or ,, or ,,. This equation is nothing than the vanishing of the exterior derivative of the Faraday form F, dF see calculus on manifolds. The electric displacement D x ,D y ,D z and the magnetic eld H x ,H y ,H z are now unied into the rank antisymmetric contravariant electromagnetic displacement tensor D D x cD y cD z c D x cH z H y D y cH z H x D z cH y H x .. Ampres law, H J D t , and Gausss law, D , combine to form D J .. In a vacuum, the constitutive equations are D F .. Antisymmetry reduces these equations to just six independent equations. Because it is usual to dene F by F F ,. Chapter the constitutive equations may, in a vacuum, be combined with Ampres law to get F J .. The energy density of the electromagnetic eld combines with Poynting vector and the Maxwell stress tensor to form the D electromagnetic stressenergy tensor. It is the ux density of the momentum vector and as a rank mixed tensor it is T F D F D ,. where is the Kronecker delta. When upper index is lowered with , it becomes symmetric and is part of the source of the gravitational eld. The conservation of linear momentum and energy by the electromagnetic eld is expressed by f T ,. where f is again the density of the Lorentz force. This equation can be deduced from the equa tions above with considerable effort. Chapter General Relativity General Relativity is the currently accepted theory of gravitation having been introduced by Ein stein in , replacing the Newtonian theory. It plays a major role in astrophysics in situations involving strong gravitational elds, for example the study of neutron stars, black holes, and gravitational lensing. The theory also predicts the existence of gravitational radiation, which manifests itself by the transfer of energy due to a changing gravitational eld, for example that of a binary pulsar. General Relativity therefore also provides the theoretical foundation for the subject of Cosmology, in which one studies the structure and evolution of the Universe on the largest possible scales. The nal steps to the theory of General Relativity were taken by Einstein and Hilbert at almost the same time. Both had recognised aws in Einsteins October work and a correspondence between the two men took place in November . How much they learnt from each other is hard to measure, but the fact that they both discovered the same nal form of the gravitational eld equations within days of each other must indicate that their exchange of ideas was helpful. On the th November Einstein made a discovery about which he wrote For a few days I was beside myself with joyous excitement. The problem involved the advance of the perihelion of the planet Mercury. Le Verrier, in , had noted that the perihelion the point where the planet is closest to the sun advanced by per century more than could be accounted for from other causes. Many possible solutions were proposed, Venus was heavier than was thought, there was another planet inside Mercurys orbit, the sun was more oblate than observed, Mercury had a moon and, really the only one not ruled out by experiment, that Newtons inverse square law was incorrect. This last possibility would replace the /d by /d p , where p for some very small number. By the advance was more accurately known, per century. From Einstein had realised the importance of astronomical observations to his theories and he had worked with Freundlich to make measurements of Mercurys orbit required to conrm the general theory of relativity. Freundlich conrmed per century in a paper of . Einstein applied his theory of gravitation and discovered that the advance of per century was exactly accounted for without any need to postulate invisible moons or any other special hypothesis. Of course Einsteins November paper still does not have the correct eld equations, but this did not affect the particular calculation regarding Mercury. Freundlich attempted other tests of general relativity based on gravitational redshift, but they were inconclusive. Also in the November paper Einstein discovered that the bending of light was out by a factor of in his work, giving . arcsec. In fact after many failed attempts due to cloud, war, incompetence etc. to measure the deection, two British expeditions in were to conrm Einsteins prediction by obtaining . . arcsec and . . arcsec. On November Einstein submitted his paper The eld equations of gravitation which give the correct eld equations for general relativity. The calculation of bending of light and the advance of Mercurys perihelion remained as he had calculated it one week earlier. Five days before Einstein submitted his November paper Hilbert had submitted a paper The foundations of physics which also contained the correct eld equations for gravitation. Hilberts paper contains some important contributions to relativity not found in Einsteins work. Hilbert Chapter applied the variational principle to gravitation and attributed one of the main theorems concerning identities that arise to Emmy Noether who was in G ottingen in . No proof of the theorem is given. Hilberts paper contains the hope that his work will lead to the unication of gravitation and electromagnetism. Immediately after Einsteins paper giving the correct eld equations, Karl Schwarzschild found in a mathematical solution to the equations which corresponds to the gravitational eld of a massive compact object. At the time this was purely theoretical work but, of course, work on neutron stars, pulsars and black holes relied entirely on Schwarzschilds solutions and has made this part of the most important work going on in astronomy today. The starting point for the application of Einsteins theory to cosmology is what is termed cosmological principle sometimes also called the Copernican principle Viewed on sufciently large distance scales, there are no preferred directions or preferred places in the Universe. Stated simply, this principle means that averaged over large enough distances, one part of the Universe looks approximately like any other part. In this sense, the Earth is not a preferred location in the Universe the physical laws tested in our labs should apply to all positions in the Universe. In this Section, we shortly describe the essential elements of Einsteins theory of gravity and derive the most general form of isotropic world models. Einsteins Principles of Equivalence The principle of equivalence has historically played an important role in the development of grav itation theory. Newton regarded this principle as such a cornerstone of mechanics that he devoted the opening paragraph of the Principia to it. In , Einstein used the principle as a basic ele ment of General Relativity. We now regard the principle of equivalence as the foundation, not of Newtonian gravity or of GR, but of the broader idea that spacetime is curved. One elementary equivalence principle is the kind Newton had in mind when he stated that the property of a body called mass is proportional to the weight, and is known as the weak equivalence principle WEP. An alternative statement of WEP is that the trajectory of a freely falling body one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces is independent of its internal structure and composition. In the simplest case of drop ping two different bodies in a gravitational eld, WEP states that the bodies fall with the same acceleration this is often termed the Universality of Free Fall. . Einstein Equivalence Principle EEP A more powerful and farreaching equivalence principle is known as the Einstein equivalence principle EEP. It states that . WEP is valid. . The outcome of any local nongravitational experiment is independent of the velocity of the freelyfalling reference frame in which it is performed. . The outcome of any local nongravitational experiment is independent of where and when in the universe it is performed. The second piece of EEP is called local Lorentz invariance LLI, and the third piece is called local position invariance LPI. For example, a measurement of the electric force between two charged bodies is a local non gravitational experiment a measurement of the gravitational force between two bodies Cavendish experiment is not. General Relativity The Einstein equivalence principle is the heart and soul of gravitational theory, for it is pos sible to argue convincingly that if EEP is valid, then gravitation must be a curved spacetime phenomenon, in other words, the effects of gravity must be equivalent to the effects of living in a curved spacetime. As a consequence of this argument, the only theories of gravity that can embody EEP are those that satisfy the postulates of metric theories of gravity, which are . Spacetime is endowed with a symmetric metric. . The trajectories of freely falling bodies are geodesics of that metric. . In local freely falling reference frames, the nongravitational laws of physics are those written in the language of special relativity. The argument that leads to this conclusion simply notes that, if EEP is valid, then in local freely falling frames, the laws governing experiments must be independent of the velocity of the frame local Lorentz invariance, with constant values for the various atomic constants in order to be independent of location. The only laws we know of that fulll this are those that are com patible with special relativity, such as Maxwells equations of electromagnetism. Furthermore, in local freely falling frames, test bodies appear to be unaccelerated, in other words they move on straight lines but such locally straight lines simply correspond to geodesics in a curved spacetime. General Relativity is a metric theory of gravity, but then so are many others, including the BransDicke theory. Neither, in this narrow sense, is superstring theory, which, while based fun damentally on a spacetime metric, introduces additional elds dilatons, moduli that can couple to material stressenergy in a way that can lead to violations, say, of WEP. Therefore, the notion of curved spacetime is a very general and fundamental one, and therefore it is important to test the various aspects of the Einstein Equivalence Principle thoroughly. A direct test of WEP is the comparison of the acceleration of two laboratorysized bodies of different composition in an external gravitational eld. If the principle were violated, then the accelerations of different bodies would differ. The simplest way to quantify such possible violations of WEP in a form suitable for comparison with experiment is to suppose that for a body with inertial mass m I , the passive gravitational mass m P is no longer equal to m I , so that in a gravitational eld g, the acceleration is given by m I am P g.. Now the inertial mass of a typical laboratory body is made up of several types of massenergy rest energy, electromagnetic energy, weakinteraction energy, and so on. If one of these forms of energy contributes to m P differently than it does to m I , a violation of WEP would result. One could then write m P m I A A E A /c ,. where E A is the internal energy of the body generated by interaction A, and A is a dimensionless parameter that measures the strength of the violation of WEP induced by that interaction, and c is the speed of light. A measurement or limit on the fractional difference in acceleration a and a measured between two bodies then yields a quantity called the E otv os ratio dened as a a a a A A E A m I, c E A m I, c .. Many highprecision E otv ostype experiments have been performed, from the pendulum ex periments of Newton, Bessel and Potter, to the classic torsionbalance measurements of E otv os, Chapter FIGURE . Torsion balance with which the EotWash group at the University of Washington is looking for departures from Newtonian gravity at submillimeter separations. The pendulum shown silvery is suspended by a torsion ber above a uniformly rotating attractor. The gap between them can be as small as mm. Ten holes in the pendulum and holes in the attractor of theminvisible in the attractors lower plate serve as negative test masses. Their deployment is such that only a shortrange gravitational anomaly would produce signicant torque pulses as the attractor rotates. Pendulum twists are monitored by a laser beam and mirrors. Dicke, Braginsky and their collaborators. In the modern torsionbalance experiments, two objects of different composition are connected by a rod or placed on a tray and suspended in a horizontal orientation by a ne wire Fig. . If the gravitational acceleration of the bodies differs, there will be a torque induced on the suspension wire, related to the angle between the wire and the direc tion of the gravitational acceleration g. If the entire apparatus is rotated about some direction with angular velocity , the torque will be modulated with period /. In the experiments of E otv os and his collaborators, the wire and g were not quite parallel because of the centripetal acceler ation on the apparatus due to the Earths rotation the apparatus was rotated about the direction of the wire. In the Dicke and Braginsky experiments, g was that of the Sun, and the rotation of the Earth provided the modulation of the torque at a period of hr. Beginning in the late s, numerous experiments were carried out primarily to search for a fth force, but their null results also constituted tests of WEP. In the freefall Galileo experiment performed at the University of Colorado, the relative freefall acceleration of two bodies made of uranium and copper was measured using a laser interferometric technique. The E otWash experiments carried out at the University of Washington used a sophisticated torsion balance tray to compare the accelerations of various materials toward local topography on Earth, movable laboratory masses, the Sun and the galaxy, and have recently reached levels of . The resulting upper limits on are summarized in Figure . . The Strong Equivalence Principle SEP In any metric theory of gravity, matter and nongravitational elds respond only to the spacetime metric g. In principle, however, there could exist other gravitational elds besides the metric, such as scalar elds, vector elds, and so on. If, by our strict denition of metric theory, matter does not couple to these elds, what can their role in gravitation theory be Their role must be that of mediating the manner in which matter and nongravitational elds generate gravitational elds and produce the metric once determined, however, the metric alone acts back on the matter General Relativity FIGURE . Selected tests of the weak principle, showing bounds on , which measures fractional difference in acceleration of different materials or bodies. The freefall and E otWash experiments were originally performed to search for a fth force. The shaded band shows current bounds on for gravitating bodies from lunar laser ranging LURE. The STEP experiment would reach an accuracy of . Credits C. Will in the manner prescribed by EEP. What distinguishes one metric theory from another, therefore, is the number and kind of gravitational elds it contains in addition to the metric, and the equations that determine the structure and evolution of these elds. From this viewpoint, one can divide all metric theories of gravity into two fundamental classes purely dynamical and priorgeometric . By purely dynamical metric theory we mean any metric theory whose gravitational elds have their structure and evolution determined by coupled partial differential eld equations. In other words, the behavior of each eld is inuenced to some extent by a coupling to at least one of the other elds in the theory. By prior geometric theory, we mean any metric theory that contains absolute elements, elds or equations whose structure and evolution are given a priori, and are independent of the structure and evolution of the other elds of the theory. These absolute elements typically include at background metrics , cosmic time coordinates t, and algebraic relationships among otherwise dynamical elds. General Relativity is a purely dynamical theory, since it contains only one gravitational eld, the metric itself, and its structure and evolution are governed by partial differential equations Einsteins equations. BransDicke theory and its generalizations are purely dynamical theories, too the eld equation for the metric involves the scalar eld as well as the matter as source, and that for the scalar eld involves the metric. Chapter By discussing metric theories of gravity from this broad point of view, it is possible to draw some general conclusions about the nature of gravity in different metric theories, conclusions that are reminiscent of the Einstein equivalence principle, but that are subsumed under the name strong equivalence principle. Consider a local, freely falling frame in any metric theory of gravity. Let this frame be small enough that inhomogeneities in the external gravitational elds can be neglected throughout its volume. On the other hand, let the frame be large enough to encompass a system of gravitating matter and its associated gravitational elds. The system could be a star, a black hole, the solar system or a Cavendish experiment. Call this frame a quasilocal Lorentz frame. To determine the behavior of the system, we must calculate the metric. The computation proceeds in two stages. First we determine the external behavior of the metric and gravitational elds, thereby establishing boundary values for the elds generated by the local system, at a boundary of the quasilocal frame far from the local system. Second, we solve for the elds generated by the local system. But because the metric is coupled directly or indirectly to the other elds of the theory, its structure and evolution will be inuenced by those elds, and in particular by the boundary values taken on by those elds far from the local system. This will be true, even if we work in a coordinate system in which the asymptotic form of g in the boundary region between the local system and the external world is that of the Minkowski metric. Thus the gravitational environment, in which the local gravitating system resides, can inuence the metric generated by the local system via the boundary values of the auxiliary elds. Consequently, the results of local gravitational experiments may depend on the location and velocity of the frame relative to the external environment. Of course, local nongravitational experiments are unaffected, since the gravitational elds they generate are assumed to be negligible, and since those experiments couple only to the metric, whose form can always be made locally Minkowskian at a given spacetime event. Local gravitational experiments might include Cavendish experiments, measurement of the acceleration of massive selfgravitating bodies, studies of the structure of stars and planets, or analyses of the periods of gravitational clocks. We can now make several statements about different kinds of metric theories. A theory which contains only the metric g yields local gravitational physics which is in dependent of the location and velocity of the local system. This follows from the fact that the only eld coupling the local system to the environment is g, and it is always possible to nd a coordinate system in which g takes the Minkowski form at the boundary between the local system and the external environment. Thus the asymptotic values of g are con stants independent of location, and are asymptotically Lorentz invariant, thus independent of velocity. General Relativity is an example of such a theory. A theory, which contains the metric g and dynamical scalar elds, yields local gravitational physics, which may depend on the location of the frame but which is independent of the ve locity of the frame. This follows from the asymptotic Lorentz invariance of the Minkowski metric and of the scalar elds, but now the asymptotic values of the scalar elds may de pend on the location of the frame. An example is BransDicke theory, where the asymptotic scalar eld determines the effective value of the gravitational constant, which can thus vary as the scalar eld varies. On the other hand, a form of velocity dependence in local physics can enter indirectly if the asymptotic values of the scalar eld vary with time cosmologi cally. Then the rate of variation of the gravitational constant could depend on the velocity of the frame. A theory which contains the metric g and additional dynamical vector or tensor elds or priorgeometric elds yields local gravitational physics which may have both location and velocitydependent effects. General Relativity These ideas can be summarized in the strong equivalence principle SEP, which states that . WEP is valid for selfgravitating bodies as well as for test bodies. . The outcome of any local test experiment is independent of the velocity of the freely falling apparatus. . The outcome of any local test experiment is independent of where and when in the Universe it is performed. The distinction between SEP and EEP is the inclusion of bodies with selfgravitational inter actions planets, stars and of experiments involving gravitational forces Cavendish experiments, gravimeter measurements. Note that SEP contains EEP as the special case in which local gravi tational forces are ignored. Einsteins Vision of Gravity The general theory of relativity took seven years of work by Einstein, the nal two to three being years of intense and exhausting labor. No one else was even close to Einsteins ideas. Had he not worked on them, they would most probably not have emerged then. We may not even have them today. In some ways, Einsteins theory is conservative. It is the last classical eld theory in the sense that classical can mean nonquantum. In another sense, it is anything but conservative. The theory is quite different from any theory before or after. It treats a force by means of geometry and eventually leads to startling notions black holes, other universes and the bridges to them and even the possibility of time travel. All other theories of forces have been readily swept into quantum theory. General relativity has resisted and the problem of bringing general relativity and quantum theory together remains one of the most difcult, outstanding puzzles of modern physics. The seven years of work divides loosely into two phases. The earlier phase of his work was governed by powerful physical intuitions that seemed as much rationally as instinctively based. He felt a compelling need to generalize the principle of relativity from inertial motion to accelerated motion. He was transxed by the ability of acceleration to mimic gravity and by the idea that inertia is a gravitational effect. As Einstein struggled to incorporate these ideas into a new physical theory, he was drawn to use the mathematics of curvature as a means of formulating the new theory. Newtons celebrated theory of gravitation presumed instantaneous action at a distance. The sun now exerts a gravitational force on the earth now with a magnitude set by Newtons inverse square law. The key part was the now. If the sun were to move slightly, the resulting alteration in the force it exerts on the earth would be felt by us instantaneously according to Newtonian theory. That means that Newtons theory depends upon a notion of absolute simultaneity. A change there is felt here at the same moment. However Einsteins theory had banished absolute simultaneity from physics. Different observers would judge different pairs of events to be simultaneous. Newtons theory had to be adjusted to accommodate this new relativity. Modern cosmology begins with Hubbles observation that the universe of galaxies is expand ing. A theoretical basis for this observation has been given by Einsteins theory of gravity, more than years earlier. In modern terms, Einsteins theory of gravity is a gauge theory with the Lorentz group as the gauge group which is operating in the tangent space. Gravity is therefore modeled by means of an afne connection of a Lorentzian manifold. In this Section, we sum marize all the elements necessary to understand the geometry of the Friedmann Universe and of A modern introduction into General Relativity can be found in the textbooks by Carroll and Hobson et al. . The latter one does include the basic concepts for Cosmology Friedmann models and Ination. A more mathematically oriented treatment is given by Straumann . This textbook also includes a complete overview for modern differential geometry of Riemannian manifolds theory of tensor elds, afne connections, curvature and pforms. Chapter its generalisations, such as the perturbed Friedmann Universe or Brane Cosmology. However, it is not the purpose of this Section to introduce all concepts in sufcient depth, for this attend a lecture on General Relativity. FIGURE . In Albert Einstein published the fundamental paper uber die relativistische Theorie der Gravitation. . The Concept of SpaceTime Special relativity showed that the absolute space and time of Newtonian physics could be only an approximation to their true nature. However, the special theory of relativity is incapable of explaining gravity because SR assumes the existence of inertial frames it does not explain how inertial frames are to be determined. Machs principle, which states that the distribution of matter determines space and time, suggests that matter is related to the denition of inertial frames, but Mach never elucidated any means by which this might happen. General relativity attacks this problem and in so doing, discovers that gravity is related to geometry. The equivalence principle is the fundamental basis for the general theory of relativity. The strict equivalence between gravity and inertial acceleration means that freefalling frames are completely equivalent to inertial frames. In general relativity, GR it is spacetime geometry that determines freefalling inertial, geodesic worldlines, telling matter how to move. Matter, in turn, tells spacetime how to curve. Geometry is related to matter and energy through Einsteins equation. The metric equation provides a general formalism for the spacetime interval in general geometries, not just the Minkowski at spacetime of special relativity SR. Matter and energy determine inertial frames, but within an inertial frame there is no inuence by any outside matter. Thus Machs principle is present more in spirit than in actuality in the general theory of relativity. .. The Concept of a Metric To introduce the concept of a metric, let us consider Euclidean dimensional space with Carte sian coordinates x, y. A parametrized cureve xt, yt begins at t and ends at t . The length General Relativity of the curve is given by s ds dx dy t t x y dt . . Here, ds dx dy is the line element. The square of the line element, also called the metric, is then given as ds dx dy .. FIGURE . sphere is a manifold which needs to be covered by more than one chart. For this representation, we also can use polar coordinates r, with the expression for the metric ds dr r d .. In a similar manner, in dimensional Euclidean space, the metric is given by ds dx dy dz ,. in Cartesian coordinates, and ds dr r d sin d . in spherical coordinates. .. SpaceTime as a Manifold A manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specic dimension, called the dimension of the manifold. Thus a line and a circle are one dimensional manifolds, a plane and sphere the surface of a ball are twodimensional manifolds, and so on. More formally, every point of an ndimensional manifold has a neighborhood homeo morphic to the ndimensional space R n . Although manifolds resemble Euclidean spaces near each point locally, the global struc ture of a manifold may be more complicated. For example, any point on the usual twodimensional surface of a sphere Fig. is surrounded by a circular region that can be attened to a circular region of the plane, as in a geographical map. However, the sphere differs from the plane in the Chapter FIGURE . Manifolds might have quite complicated structure. large in the language of topology, they are not homeomorphic. The structure of a manifold is encoded by a collection of charts that form an atlas, in analogy with an atlas consisting of charts of the surface of the Earth. For most applications a special kind of topological manifold, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can dene directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of an ndimensional differentiable manifold has a tangent space. This is an ndimensional Euclidean space consisting of the tangent vectors of the curves through the point. FIGURE . The tangent plane at a point x in the manifold is generated by all tangent vectors of curves t on the manifold passing through the point x. To measure distances and angles on manifolds, the manifold must be Riemannian. A Rie mannian manifold is a differentiable manifold in which each tangent space is equipped with an inner product lt ..., ... gt or metric in a manner which varies smoothly from point to point. Given two tangent vectors u and v, the inner product lt u, v gt gives a real number. The dot or General Relativity scalar product is a typical example of an inner product. This allows one to dene various notions such as length, angles, areas or volumes, curvature, gradients of functions and divergence of vector elds. Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family of positive denite inner products g p T p MT p M R, p M . such that, for all differentiable vector elds X, Y on M, pg p Xp, Y p . denes a smooth function M R. In a systemof local coordinates on the manifold M given by n realvalued functions x ,x ,...,x n , the vector elds x ,..., x n . give a basis of tangent vectors at each point of M. Relative to this coordinate system, the compo nents of the metric tensor are, at each point p, g ij pg p x i p , x j p .. Equivalently, the metric tensor can be written in terms of the dual basis dx , . . . , dx n of the cotangent space as g i,j g ij dx i dx j .. Endowed with this metric, the differentiable manifold M, g is a Riemannian manifold. The concept of spacetime combines space and time to a single abstract space, for which a unied coordinate system is chosen. Typically three spatial dimensions length, width, height, and one temporal dimension time are required. Dimensions are independent components of a coordinate grid needed to locate a point in a certain dened space. For example, on the globe the latitude and longitude are two independent coordinates which together uniquely determine a location. In spacetime, a coordinate grid that spans the dimensions locates events rather than just points in space, i.e. time is added as another dimension to the coordinate grid. This way the coordinates specify where and when events occur. However, the unied nature of spacetime and the freedom of coordinate choice it allows imply that to express the temporal coordinate in one coordinate system requires both temporal and spatial coordinates in another coordinate system. Unlike in normal spatial coordinates, there are still restrictions for how measurements can be made spatially and temporally. .. Einstein I Minkowski space M has to be generalized to a general curved pseudoRiemannian manifold M, g with metric tensor eld g such that SpaceTime is locally still Minkowskian, i.e. the tangent space T p MM . The notion of an event is fundamental in relativity. An event is characterized by its position x and its time t. An event p is given by the coordinates t, x, y, z in the dimensional Space Time. Already in Special Relativity, time and space appear as an entity. Two neighboring events t, x, y, z and t dt, x dx, y dy, z dz have then a distance ds which is determined by the metric of Minkowski space, x ct, ds c dt dx dy dz dx dx .. Chapter This distance is invariant against Lorentz transformations. We often say the spacetime of SR is at, since the resulting curvature vanishes. A suitable tool to picturize a spacetime is to use spacetime diagrams Fig. . The timeaxis is running vertically, and space is running horizontally. In Minkowski space, the lightcones have a constant openening angle of degrees. In a curved spacetime this may change. Gravity cannot be included into Special Relativity despite many desperate attempts to do this years ago. Einstein postulated therefore that Minkowski space is only realized locally in a D pseudoRiemannian manifold M, g. This means strictly speaking, Minkowski space M is the tangent space T p MM at each event p of the spacetime M. Einstein had the idea that the effects of gravity are expressed in terms of a generalized Minkowski metric element of the form ds , g x dx dx ,. which gives the distance between neighboring events in M. The ensemble of all events parametrized by local coordinates t, x i is called the SpaceTime. The metric tensor g is a secondrank sym metric tensor, which in general depends on the events. ds now measures the proper time of timelike curves x s/c g dx d dx d d. . .. Examples of Simple SpaceTimes The Schwarzschild spacetime as the expression of the gravitational eld of nonrotating stars ds expr c dt expr dr r d sin d .. Due to spherical symmetry, it only has two independent functions r and r, which only depend on the spherical radius r. and are the usual angles on the sphere r const, r is a measure for the surface of the sphere, given by r , and t is a measure for time observed at spatial innity, r . Schwarzschild found in the solution for this ansatz exp exp GM c r .. The gravitational eld of rapidly rotating stellar objects ds c dt d dt exp r dr exp d . already has independent functions depending now on the radius r and on , but not on and t, r, etc. In addition to Schwarzschild, this line element contains an off diagonal term g , related to the angular momentum of the star. This metric is the starting point for rapidly rotating neutron stars and Black Holes the Kerr metric. For Black Holes the metric coefcients are given by simple polynomials the miracle of the Kerr solution . sin . aMr . e ,e .. General Relativity For this one uses the following polynomials in r and cos r Mr a . r a cos . r a a sin .. This solution is uniquely given by two parameters the mass M of the source and the Kerr parameter a, which is related to the angular momentum of the source, J H aM. In physical units, the mass is given in terms of the gravitational radius GM/c , and similarly for the angular momentum, a is in units of GM/c . This metric is asymptotically at and approaches the Schwarzschild metric in the limit a . It can be shown that the above ansatz for the Kerr metric indeed satises Einsteins vacuum solutions, but this is a very complicated procedure. Flat Cosmological Spacetimes Cosmological spacetimes are given by the metric ds c dt R td ,. where d is the metric of a space of constant curvature sphere, an Euclidean space E or a hyperbolic space. The essential degree of freedom is the expansion factor Rt which scales all spatial lengths see later on. The simplest example is the stretching of Minkowski space called a at Universe ds c dt R t dx dy dz ,. often written in spherical coordinates t, r, , as ds c dt R t dr r d sin d .. The expansion factor Rt is a solution of the Friedmann equation for the expansion veloc ity H R/R H G c .. All of the above spacetimes have some high degree of symmetries. . Gravity is an Afne Connection on SpaceTime In order to compare tangent spaces at neighboring events, one needs a connection on the manifold M. As in Riemannian geometry, this connection is required to be metric, so that the correspond ing Christoffel symbols are uniquely given by derivatives of the metric elements. This is a basic postulate of Einsteins theory of gravity one could construct more general theories of gravity which include torsion. Physically speaking, we associate observers e a , a , , , , i.e. an orthonormal tetrad or Vierbein eld, satisfying ge a ,e b ab ,. where is the at Minkowskian metric with signature , or . An observer is a global orthonormal basis eld in the tangent space of each event p, where e is timelike and e i i , , are spacelike. One could also construct null tetrads in order to dene the geometry In the following, the convention for indices is as follows greek indices are related to local coordinate systems, latin indices a, b, c, ... mark observer elds, latin indices i, k, l, ... specify spatial components. Chapter of the spacetime. The dual elements of e a is a basis of the cotangent space T p , denoted by a , satisfying a e b a b . They dene the metric g ab a b . The denition of these observer elds is not unique, since any observer derived by means of a local Lorentz transformation is also an observer e a x b a xe b x ,x T x.. These are Lorentz transformations operating in the tangent space of each event. As an example we consider static observers in the Schwarzschild metric .. Such an observer is given by the following tetrad e exp t ,e r exp r ,e r ,e r sin .. It is then clear that they satisfy ge a ,e b ab , where ab is the Minkowski metric. The dual basis is a basis of oneforms a with a e b a b exp dt , r exp dr , rd, r sin d. . We now consider a satellite which is orbiting the central star in the equatorial plane of the Schwarzschild spacetime with velocity u given by uu t t , gu, u . . u /u t d/dt is the angular velocity of the satellite Keplerian e.g. as measured by xed stars. The Lorentz transformation between the static observer e a and the satellite observer e a is then given by a boost transformation with velocity V r sin / and Lorentz factor S / V , where GM/r c . is the redshift factor between a static observer at radius r and innity. This provides us the Lorentz transformation e S e Ve . e r e r . e e . e S Ve e .. The trajectory of this observer, with tangent vector e is now a helical path in spacetime. .. The Concept of a Connection A connection is now dened as a linear mapping between the tangent space at the event x and the tangent space at a neigboring event displaced by dx. It is sufcient to dene this mapping for an arbitrary basis e a of the tangent space e a e b c ab e c c b e a e c ,. with the additional properties for any function f and any vector eld X fX e b f X e b . X fe b f X e b X.fe b .. Thus, the oneforms dened as b a b ca c . are called connection oneforms. They are identical with the Christoffel symbols, if the basis in the tangent space is the natural basis implied by the coordinate system e e e .. General Relativity Remember that the rst index in the Christoffel symbols is a oneform index, the second is a matrix index. From the duality between tangent and cotangent space we nd then X a a b X b . for any vector eld X. From this denition we nd the covariant derivative for any vector eld XX a e a Xe a dX a a b X b ,. or in components with respect to an orthonormal basis a X b e a X b , b c e a X c .. Similarly, for a oneform a a we have b d b a a b ,. or in components a b e a b, c b e a c .. When we use the coordinate basis of the chosen chart, the covariant derivatives of vector elds and oneforms are given in the wellknown form X X , X . , .. The covariant derivative for vector elds and oneforms can now be extended to arbitrary tensor elds, in general, by requiring that the operation of satises the Leibniz rule when acting on tensor products S T S TS T. . In this sense, the covariant derivative of a tensor eld T is given as follows T T , T T ,. and similarly for a tensor eld A by means of A A , A A .. .. Parallel Transport and Geodesics A connection on a vector bundle here the tangent space species then the notion of parallel transport along curves in the manifold. Let be a curve on the manifold, and X a vector eld dened on M. A vector eld is called autoparallel along , if X.. In coordinates, we have XX , dx ds ,. and therefore dX ds dx ds X .. Chapter The vector eld is autoparallel if dX ds dx ds X .. For any curve s and X in the tangent space T M we nd a unique vector eld Xs given along s with the initial condition X X . This operation is called the parallel displacement of a vector eld along a curve s. A curve is called a geodesic, if the tangent eld is autoparallel along s. According to the above analysis, this means d x ds dx ds dx ds .. Geodesics are the trajectories of freely falling bodies in the gravitational eld given by the afne connection. A satellite e.g. will move on geodetic curves in the gravitational eld of the Earth, planets move on the geodetic curves in the solar gravitational eld. .. Gravity is a Metric Connection So far, the concept of a metric and the concept of the connection are independent of each other. Each Riemannian manifold, however, carries a particular connection which is uniquely associated with the metric. For this, we say An afne connection is said to be a metric connection if the parallel transport along any smooth curve in the manifold preserves the inner product. One can then prove that this statement is equivalent to g , which is equivalent to the Ricci identity X.gY, Z g X Y, Z gY, X Z.. With the condition that torsion vanishes, X Y Y X X, Y , . we can write the above equation as X.gY, Z g Y X, Z gX, Y , Z gY, X Z.. X, Y denotes the Lie bracket for vector elds see next Section. With cyclic permutation of the vector elds we obtain Y.gZ, X g Z Y, X gY, Z, X gY, X Z. Z.gX, Y g X Z, Y gZ, X, Y gX, Z Y.. Now we add the rst and third equation and subtract the second one to get g Z Y, X X.gY, Z Y.gZ, X Z.gX, Y gZ, X, Y gY, Z, X gX, Y , Z . . For the fundamental vector elds X k ,Y j and Z i the Lie bracket vanishes, i , j and g i , j g ij , which means that g i j , k m ij g mk k g ji j g ik i g kj . General Relativity or g mk m ij g jk,i g ik,j g ij,k .. With the inverse metric g ij , we now get the famous expression for the LeviCivita connection m ij g mk g ki,j g kj,i g ij,k .. This afne connection is therefore uniquely determined by derivatives of the metric tensor and is therefore called metric connection, or LeviCivita connection. For any pseudoRiemannian manifold there is then a unique afne connection such that it is i torsionfree and ii metric. This particular connection is usually called the LeviCivita con nection, or pseudoRiemannian connection. .. Einstein II It is now one of the fundamental postulates of Einsteins theory of gravity that gravity is related to the LeviCivita connection of the Lorentzian manifold. This means in particular that there is no torsion associated with gravity. .. Strong Principle of Equivalence Since the Lorentz connection transforms inhomogeneously as x xd x. for any Lorentz transformation between local observers, a a b x b , we always can nd locally an observer system such that p , i.e. the connection can be transformed away just locally, but not globally. This is not the case for the curvature The weak principle of equivalence states that effects of gravitation can be transformed away locally by suitably accelerated frames of reference by going to local Minkowskian coordinates. We can formulate, however, a much stronger requirement, the socalled .. Einstein III The strong principle of equivalence holds, which states that any physical interaction other than gravitation behaves in a local inertial frame as if gravitation were absent. E.g. Maxwells equations will have their familiar forms as in SR. The strong principle of equivalence allows us to extend any physical law that is expressed in a covariant way to curved SpaceTime. Ordinary derivatives are just replaced by covariant ones. .. Example Christoffel Symbols in Schwarzschild The Schwarzschild metric is given in Schwarzschild coordinates t, r, , , here, we use natural units G c, g M r M r r r sin . with its inverse, g M r M r /r /r sin .. Chapter Fist, we calculate the various partial derivatives of the Schwarzschild metric t g , spacetime is static . r g M r M rM r r sin . g r sin cos . g , spacetime is axisymmetric . . With the above denition of the Christoffel symbols, we get the following expressions given as symmetric matrices t M rrM M rrM . r M r M r M rMr Mr M r sin . /r /r sin cos . /r / tan /r / tan . . With these expressions, we can calculate the equations of motion for the acceleration with respect General Relativity to Schwarzschild time t from the geodesics equation d x ds dx ds dx ds . d ds dx dt dt ds dx dt dx dt dt ds . d ds dx dt dt ds dx dt d t ds dx dt dx dt dt ds . d x dt dt ds dx dt d t ds dx dt dx dt dt ds . d x dt dt ds dx dt dx dt dt ds dx dt d t ds . d x dt dx dt dx dt dx dt t dx ds dx ds ds dt . d x dt dx dt dx dt dx dt t dx dt dx dt .. The Newtonian Limit For the spatial components, we get in terms of velocities v k dx k /dt the acceleration dv i dt i tt i tk v k i km v k v m v i t tk v k v i t km v k v m .. The leading term in v/c is given by the force i tt , which only has a radial part. Therefore, the pseudoNewtonian gravitational force is f r i tt GM r GM/c r.. The last correction is due to the existence of the horizon near a Black Hole. .. Example Relativistic Hydrodynamics The hydrodynamical equations just follow from the energymomentum tensor T for perfect uids by replacing partial derivatives in terms of covariant ones T .. This is supplemtented the mass conservation J . for the mass current J u with restmass density . Since the energymomentum tensor is a symmetric tensor, equation . can be written in terms of the Christoffel symbols T T , T .. All the coordinate and gravity effects are hidden in the Christoffel symbols. . Calculus on Differentiable Manifolds Many of the techniques from multivariate calculus also apply to differentiable manifolds. One can dene the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing the total derivative of a function Chapter the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function dened on a Euclidean space, at least locally. There are, however, important differences in the calculus of vector elds and tensor elds in general. In brief, the directional derivative of a vector eld is not welldened, or at least not dened in a straightforward manner. Several generalizations of the derivative of a vector eld or tensor eld do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are The Lie derivative which is uniquely dened by the differential structure, but fails to satisfy some of the usual features of directional differentiation. An afne connection which is not uniquely dened, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an afne connection is not unique, it is an additional piece of data which must be specied on the manifold. Ideas from integral calculus also carry over to differential manifolds. These are naturally expressed in the language of exterior calculus and differential forms. The fundamental theorems of integral calculus in several variables namely Greens theorem, the divergence theorem, and Stokes theorem generalize to a theorem also called Stokes theorem relating the exterior derivative and integration over submanifolds. .. The Lie Derivative A Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor elds over a manifold M. The vector space of all Lie derivatives on M forms an innite dimensional Lie algebra with respect to the Lie bracket dened by A, B L A BL B A. . The Lie derivatives are represented by vector elds, as innitesimal generators of ows active diffeomorphisms on M. Looking at it the other way round, the group of diffeomorphisms of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory. For a geometrical interpretation of the Lie derivative, see Fig. . FIGURE . The geometry of Lie derivatives. General Relativity A vector eld, expressed in terms of this selected set of basis vectors, is written as XX a x a .. One denes the Lie bracket X, Y of a pair of vector elds as X, Y XY a YX a x a X b Y a x b Y b X a x b x a ,.. The second denition is intrinsic in that it does not rely on the use of coordinates. Since a vector eld can be identied with a rstorder differential operator on functions, the Lie bracket of two vector elds can be dened as follows. If X and Y are two vector elds, then the Lie bracket of X and Y is also a vector eld, denoted by X, Y , dened by the equation X, Y f XY f Y Xf . . Using a local coordinate expression for X and Y, one can prove that this is equivalent to the previous denition of the Lie bracket. .. Covariant Derivative The covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Historically, at the turn of the th century, the covariant derivative was introduced by Grego rio RicciCurbastro and Tullio LeviCivita in the theory of Riemannian and pseudoRiemannian geometry. Ricci and LeviCivita following ideas of Elwin Bruno Christoffel observed that the Christoffel symbols used to dene the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector elds on a manifold. This new derivative the LeviCivita connection was covariant, in the sense that it satised Riemanns requirement that objects in geometry should be independent of their description in a particular coordinate system. The denition has been given in the Section on afne connections. .. Diferential Forms Differential forms are extremely helpful concepts in direct calculation. A zeroform is a scalar function. The oneforms a dened above are the basis elements of the cotangent space, its com ponents are the components of covariant vectors. A general oneform A can always be written as AA dx A a a . The vector potential of classical electrodynamics is the standard example. A new operation introduced when one works with forms is called the wedge product. If x and y are coordinates, then dx and dy are oneforms, and dx dy dy dx is called a twoform. An example of a pform is A p A ... dx dx ... dx ,. where A ... is a completely antisymmetric tensor with p indices. In fact, the set of pforms in a ndimensional manifold is a vector space p of dimension n/pn p see Table . In dimensions we have one zeroform, oneforms basis in the cotangent space, forms the Farady tensor e.g., forms currents and only one form volumeform. Formally, the wedge product of a pform with a qform is given by the alternating operator A.. The exterior derivative d takes a pform into a p form, e.g. a oneform dd dx , dx dx , , dx dx .. Chapter Forms dim dim TABLE . Number of linearly independent pforms for D and D . In general, for a pform A given by AA ... dx dx ... dx . the exterior derivative is given by its local expression dA dA ... dx dx ... dx A ... x dx dx dx ... dx .. With this explicit denition, one can show dA B dA B p A dB . ddA . for pform A and a qform B. One can also dene an antiderivation i which makes a p form out of a pform dened as i V V , ..., V p V, V , ..., V p ,. i.e. just by contraction with the rst index. With the Farady tensor F we can e.g. build the one form E i V F, in components E V F . This operation is called the inner product of V with . Applying both operations, the inner product and the exterior derivative, leaves the degree of a pform invariant L X di X i X d. is equivalent to the Lie derivative on pforms. The Lie derivative L is given by its action on functions L X f X.f dfX , . its action on vector elds L X Y X, Y X Y , Y X , e ,. and the Leibniz rule for the compatibility with higher rank tensors L X STL X STSL X T.. From the last property, we can derive for a oneform and a vector eld Y L X YL X YL X YL X Y.. Writing out in components, we have X X , L X Y L X Y ,. or making use of the Liebracket L X Y X , Y Y , X Y , Y X , X , X , Y .. Since this last equation is valid for any vector eld Y , we conclude L X X , X , .. General Relativity . Torsion and Curvature of SpaceTime A physical theory of gravity also requires some dynamical evolution for the connection. This is usually formulated in terms of the curvature associated with the connection. The calculation of the Riemann tensor is therefore one of the major tasks when dealing with specic spacetimes. For this purpose, we denote by XM the space of all smooth vector elds on the manifold M. Conventionally, the torsion elds T are dened as bilinear mappings T XM XM XM on the set of all vector elds on the manifold TX, Y X Y Y X X, Y . . Curvature is dened as a trilinear mapping R XM XM XM XM RX, Y Z X Y Z Y X Z X,Y Z.. The components R a bcd of this vector eld denes the Riemann tensor which has four indices. These quantities obviously satisfy the antisymmetry conditions TX, Y TY, X , RX, Y RY, X , . as well as TfX, gY fg TX, Y . RfX, gY hZ fghRX, Y Z . for any functions f, g and h. Local expressions for the Riemann tensor The components of the curvature tensor are then given by inner product between tangent and cotangent space R i jkl lt dx i ,R k , l j gtlt dx i , k l l k j gt lt dx i , k s lj s l s kj s gt . . From this we get the famous expression for any Riemannian manifold of dimension n R i jkm k i mj m i kj i ks s mj i ms s kj ,. or in particular for spacetimes of dimensions in local coordinates R .. The above expression reminds us of the denition of the Faraday form, dened in terms of the vector potential A , as follows F A A .. The nonlinear parts are missing, since electromagnetism is given by an Abelian group U. The Faraday tensor is in fact the corresponding curvature. In a nonAbelian gauge theory, we nd the same expression as above, except that the gauge potemtial A is now an element of the Lie algebra of the gauge group, SUn e.g., i.e. it is an antisymmetric matrix. The corresponding curvature is now given by F A A A ,A .. The bracket ..., ... is the Liebracket commutator of the corresponding Liealgebra. This is the reason for using the above sign convention in the denition of the Riemann tensor. The second pair of indices in the Riemann tensor , is related to the form character, the rst pair to the Lie algebra of the Lorentz group. With this in mind it is quite easy to remember the ordering of the indices. Chapter .. Cartans Structure Equations Since torsion TX, Y and curvature RX, Y are antisymmetric tensors, they naturally dene corresponding twoforms TX, Y T a X, Y e a . RX, Y e b a b X, Y e a .. The exterior derivatives of the basic oneforms a and of the connection forms satisfy Cartans structure equations T a d a a b b . a b d a b a d d b .. The wedge operator denotes the exterior products for pforms. The form is the curvature form which gives, when expressed locally, a b R a bcd c d . the components of the Riemann tensor R a bcd in orthonormal coordinates. Similarly, we have torsion twoforms T a T a bc b c .. The proof of Cartans equations is given in footnote. Local expressions In local coordinates, a metric connection is expressed in terms of the Christof fel symbols g g , g , g , . such that the connection form is given in a local coordinate basis as dx ,. and therefore d , dx dx , , dx dx .. For the proof of Cartans structure equations, we use the above denition of torsion. Written as oneforms, this means T a X, Y e a X Y Y X X, Y X b Ye b Y b Xe b a X, Y e a X. a Y Y. a X a X, Y e a a Y b a X a X b a Y e a d a X, Y e a a b b X, Y e a .. The proof of the second structure equation is similar. Written as a twoform, this means a c X, Y e a X Y e c Y X e c b c X, Y e b X b c Ye b Y b c Xe b a c X, Y e a X. a c Y Y. a c X a c X, Y e a b c Y a b X b c X a b Y e a d a c X, Y e a a b b c X, Y e a .. General Relativity Also, dx dx dx dx . Accordingly, Cartans second structure equation is equivalent to the conventional denition . of the Riemann tensor in local coordinates R .. . Curvature and Einsteins Equations Another consequence of the afne connection is an additional symmetry of the Riemann tensor gRX, Y Z, U gRX, Y U, Z . gRX, Y Z, U gRZ, UX, Y . . The Riemann tensor is the fundamental entity for the construction of the eld dynamics. It satis es the following essential symmetries which are important for the concrete calculation R a bcd R a bdc . R abcd R bacd . R abcd R cdab .. The rst property results from the fact that curvature is a twoform, the second one that curvature is an element of the Lie algebra of the Lorentz group, and the third one gives a fundamental relation between spacetime indices and intrinsic indices metric condition. This last property follows from the cyclic identity for a torsionfree connection RX, Y Z RZ, XY RY, ZX , . or in components R abcd R adbc R acdb .. Making use of the antisymmetry in the rst and second pair of indices, we nd R abcd R adbc R acdb R dabc R cadb R dcab R dbca R cbad R cdba R cdab R bdca R bcad R cdab R badc R cdab R abcd .. Hence R abcd R cdab .. In total, the Riemann tensor has components, while the last symmetry reduces it to inde pendent components. The Riemann tensor of D spacetimes has independent components. Astrophysical spacetimes have usually many symmetries such that the total number of indepen dent components is drastically reduced. In comparison, the metric tensor only has independent components, i.e. only half of the components of the Riemann tensor are due to the metric, or the Ricci tensor R ab , while the other components are hidden in the Weyl tensor dened as follows C abcd R abcd g ac R bd g bd R ac g bc R ad g ad R bc Rg ac g bd g ad g bc .. Chapter The Weyl tensor has the same symmetries as the curvature tensor, but is tracefree, g bd C abcd . Facit In four dimensions, the Riemann tensor has independent components, in two di mensions just one component, and in three dimensions components. In D, the metric tensor only has independent components, but there is more information in the Riemann tensor components are in the Ricci tensor, the other components are hidden in the Weyl tensor. The Riemann tensor is nowused to construct the Ricci tensor, R bd R a bad . For the Schwarzschild spacetime . e.g. we get the following expressions for the Ricci tensor R R r r R R . R rr R rr R rr R rr . R R R r r R . R R R r r R . with all other components vanishing. With the Ricci scalar R R a a as the trace of the Ricci tensor we now can construct the Einstein tensor G ab R ab Rg ab .. The Einstein tensor is symmetric, G ab G ba , and divergencefree, G a ba due to the Bianchi identity. .. Einstein IV Einstein postulated that the tensor G ab couples to the matter content of the Universe G ab G c T ab ,. where T ab is the symmetric energymomentum tensor of all matter in the Universe particles, baryons, galaxies, photons, neutrinos and quantum elds. As a consequence of the above prop erties, the divergence of the energymomentum tensor vanishes identically T ab b .. .. The Hilbert Action Einsteins equations can be derived from the action called Hilbert action A G R gd L matter , gd x. where L matter is the Lagrangian density for matter depending on some variables denoted collec tively as , c , since for any domain D of spacetime D R gd x D G g gd x. . The variation of this action with respect to will lead to the equation of motion for matter, L matter / , while the variation of the action with respect to the metric tensor g leads to Einsteins equations R Rg G L matter g GT .. Here, T L matter /g is the energymomentum tensor of matter elds. see any textbook on General Relativity General Relativity .. Einsteins Equations with Cosmological Constant Let us now consider a new matter action L matter L matter /G, where is a real constant. The equation of motion for the matter does not change under this transformation, since is constant. But the action now picks up an extra term proportional to , which can be written in two different ways, A G R gd x L matter , G gd x G R gd x L matter , gd x. and Einsteins equations get modied. This simple manipulation has many backdrops in theoret ical Physics. It can be interpreted in different manners The rst interpretation is based on the rst line of the above equations, it treats as a shift in the matter Lagrangian, which in turn will lead to a shift in the matter Hamiltonian. This could be thought of as a shift in the zero point energy of the matter system. Such a constant shift in energy does not affect the dynamics of matter, while gravity picks up an extra contribution in the form of a new term Q in the energymomentum tensor R R GT Q ,Q G .. The second line in Eq . can be interpreted as a gravitational eld, described by the Lagrangian of the formL grav /GR, interacting with matter. In this interpreta tion, gravity is described by two constants, the Newtons constant G and the cosmological constant . It is then natural to modify the left hand side of Einsteins equations in the form of R R GT .. In this interpretation, the spacetime is curved even in the absence of matter, T , since the left hand side does not admit at spacetimes as solutions. It is even possible to consider a situation where both effects can occur. If gravitational theories are in fact described by the Lagrangian of the form R , then there is an intrinsic cosmological constant in nature, just as there is a Newtonian constant G in nature. If the matter Lagrangian contains energy densities which change due to the dynamics, then L matter can pick up constant shifts during dynamical evolution. For this we consider a scalar eld with the Lagrangian L / V,. which has the energymomentum tensor T V .. For eld congurations which are constant e.g. at the minimum of the potential V , this contributes an energymomentum tensor T V min , which has exactly the same form as a cosmological constant. It is then the combination of these two effects of very different nature which is relevant and the source will be T e V min /G g .. min can change during the dynamical evolution, leading to a timedependent cosmologi cal constant. Chapter The termQ in Einsteins equations behaves very pecularly compared to the energymomentum tensor of normal matter. Q is in the form of an energymomentum tensor of an ideal uid with energy density and pressure P . Obviously, either the pressure or the energy density of this uid must be negative. Such an equation of state, P , also has another important implication in GR. The relative acceleration between two geodesics, g, satises in GR the following equation gGP.. The source of this relative acceleration between geodesics is P and not alone. This shows, as long as P gt , gravity remains attractive, while P lt leads to repulsive forces. A positive cosmological constant therefore leads to repulsive gravity. .. Gravity as a Gauge Theory We now have the means to compare the formalism of connections and curvature in Riemannian geometry to that of gauge theories in particle physics. In both situations, the elds of interest live in vector spaces which are assigned to each point in spacetime. In Riemannian geometry the vector spaces include the tangent space, the cotangent space, and the higher tensor spaces constructed from these. In gauge theories, on the other hand, we are concerned with internal vector spaces. The distinction is that the tangent space and its relatives are intimately associated with the manifold itself, and were naturally dened once the manifold was set up an internal vector space can be of any dimension we like, and has to be dened as an independent addition to the manifold. In math lingo, the union of the base manifold with the internal vector spaces dened at each point is a ber bundle, and each copy of the vector space is called the ber in perfect accord with our denition of the tangent bundle. Nongravitational interactions are described nowadays in terms of gauge theories. In this sense, Maxwells theory is a U gauge theory resulting from local phase transformations on quantum elds, x expix x. The vector potentials A x are the connection coefcients, and the Faraday tensor F /F dx dx is the corresponding curvature. Strong interaction is a SU gauge theory, where the internal space is dened by the color space each quark can carry a specic color. The gauge elds A b a are then the local expressions of a connection oneform A dx with values in the Liealgebra of SU. In this case, we have connection elds A x, , ..., , corresponding to the gluon elds of strong interaction. In this sense, the gauge transformations for gravity are the local Lorentz transformations op erating between different observers at the same events in spacetime. We have connection elds A x, , ..., , i.e. in total connection coefcients. Note, however, that the dynamics proposed by Einstein is different from the YangMills dynamics of nonAbelian gauge theories. Is General Relativity the Correct Theory of Gravity Most of the tests for Einsteins theory of gravity have been done for stellar objects, such as the Sun or neutron stars. In good aprroximation, stars are spherical objects and the gravitational eld is given in terms of spherically symmetric metric elements. Das einfachste metrische Feld wird von einemkugelsymmetrischen Stern erzeugt. In diesem Falle reduziert die hohe Symmetrie Kugelsymmetrie die m oglichen metrischen Koefzienten auf zwei wesentliche Funktionen The metric produced by the Sun is to a good approximation spherically symmetric and can be expressed in any metric theory by two metric components g r und g rr r ds r dt r dr r d sin d .. A detailed analysis of all these tests is not the topic of the present lecture, see e.g. any lecture on GR or on Relativistic Astrophysics, and Clifford Will . General Relativity The two parameters and are known as Robertson parameters. Each metric theory of gravity predicts certain values for the Robertson parameters. Einsteins equations e.g. determine these parameters uniquely via differential equations ,.. The above Ansatz corresponds to a postNewtonian expansion in the metric, and the solar system experiments nowallowto determine these two parameters. The Newtonian limit is xed, therefore there is no change in chosing the rst order for g . Remark The Schwarzschild metric was the rst solution of Einsteins equations, published in by Karl Schwarzschild. The gravitational eld of the Sun, and other nonrotating stars is described by the Schwarzschild solution, though the deviations from at space are incredibly small, even near the surface of the Sun. These deviations from at Minkowski space are of the order of R S /R . For neutron stars these deviations are considerably higher, since a neutron star has a radius of km, therefore R S /R /. FIGURE . The light cones are strongly deformed near the surface of a Black Hole light cones are pointing towards the center of the Black Hole so that no light can escape from the Schwarzschild surface, given by r R S . . Gravitational Redshift Let us consider an at r const, const and const und stellen uns die Frage, wie die Zeit seiner Uhr sich at coordinate time t correct time at spatial innity. d ds/c measures then the proper time of the Observer in a local inertial frame d GM c r dt . . With respect to innity, a clock seems to be slower in the gravitational eld. This also means that photons are redshifted in the gravitational eld of a star. The frequency ratio of the photon B A d A d B g A g B . Chapter determines the ratio of frequencies of a photon at different position A and B in the gravitational eld. This is the case e.g. for the emission of a spectral line with wavelength emitted at the surface of a compact star observed at some location B B B GM c R .. The spectral lines of a compact star are therefore redshifted by z B GM c R GM c R .. This is known as gravitational redshift. The approximation is only valid for noncompact ob jecst. For White Dwarfs a redshift of z has been measured Sirius B, for neutron stars we would expect a redshift of z . only one example known, and for Black Holes, the redshift is by denition z . A nonrotating Black Hole has a radius R R S Fig. . FIGURE . Perihelion advance in a body system. For the planet Mercury, the perihelion ad vance results in arcsec per century. In more compact binary system, the periastron advance can be considerably higher. . PostKeplerian Effects Apart from gravitational redshift, three other general relativistic effects are observable in the solar system and are nowadays of principal importance for the calculation of ephemerids of planets The perihelion precession for the Mercury orbit by arcsec per century Fig. . Each planet shows a perihelion precession, which however gets smaller for high semimajor axis a GM M P c e a .. General Relativity Light deection on the solar surface in the metric . is given by GM c R ... This formula directly shows that half of the light deection is produced by the Newtonian potential , the other half comes from the curved space. By measuring the light deec tion the solar surface very accurately, one is able to constrain the value of the parameter Fig. . Light deection will be an important effect e.g. for the GAIA mission to measure star positions from the Lagrange point L. The Shapiro timedelay for signals propagating in the solar system. This effect is due to longer propagation of signals in the space curved by the Sun compared to a propagation far away from the solar surface. FIGURE . In a gravitational lense, the gravitational eld of a galaxy e.g. deects the photon paths so that multiple images can occur. This gure also shows that photons propagate longer in a curved spacetime. General Relativity predicts the bending of light by gravity, gravitational time dilation and length contraction, gravitational redshifts and blueshifts, the precession of Mercurys orbit, and the existence of gravitational radiation. All these effects have been measured, although gravita tional radiation has been observed only indirectly via the decay of the orbits of binary pulsars. The LIGO project is an attempt to build a giant MichelsonMorley type of interferometer to detect gravitational radiation directly. Two interferometers have been built, each one with perpendicular lightcarrying vacuum pipes kilometers long. The relativistic periastron shift and Shapiro timedelay are essential effects used in astronomy to determine the exact pulse arrival times for radio pulses emitted by pulsars in compact binary systems see Camenzind . . On Gravitational Waves Gravitational waves are ripples in the fabric of space and time produced by violent events in the distant universe, for example by the collision of two black holes or by the cores of supernova Chapter FIGURE . Photon trajectories are strongly affected by the gravitational eld of a rotating Black Hole. The Black Hole is bombarded by laser photons along the equatorial plane. Photons with low impact parameters are captured by the horizon. explosions. Gravitational waves are emitted by accelerating masses much as electromagnetic waves are produced by accelerating charges. These ripples in the spacetime fabric travel to Earth at the speed of light, bringing with them information about their violent origins and about the nature of gravity. Albert Einstein predicted the existence of these gravitational waves in in his general theory of relativity, but only since the s has technology become powerful enough to permit detecting them and harnessing them for science. Although they have not yet been detected di rectly, the inuence of gravitational waves on a binary pulsar two neutron stars orbiting each other has been measured accurately and is in good agreement with the predictions. Scientists therefore have great condence that gravitational waves exist. Joseph Taylor and Russel Hulse were awarded the Nobel Prize in Physics for their discovery of the binary pulsar which shows a decay of the binary orbit due to the emission of gravitational waves. In contrast to Newtonian gravity, timedependent processes lead in GR to the emission of gravitational waves, just as in Electrodynamics. These waves propagate at the speed of light through the underlying spacetime. They represent a kind of ripples in the space. The characteris tics of gravitational waves are given in details in Appendix B. The Laser Interferometer GravitationalWave Observatory LIGO is a facility dedicated to the detection of cosmic gravitational waves and the harnessing of these waves for scientic re search. It consists of two widely separated installations within the United States, operated in unison as a single observatory. When it reaches maturity, this observatory will be open for use by the national community and will become part of a planned worldwide network of gravitational wave observatories. Burst sources are the most likely to have large amplitudes at higher frequency therefore, they are the best candidates for detection by resonant mass detectors. Burst sources must be very violent events. One candidate is the gravitational collapse of a massive star to form a neutron star. The strength of emission depends on the degree of nonsphericity in the collapse and also on the speed of the collapse. A perfectly spherical collapse will produce no waves, whereas a highly antisymmetric collapse will produce strong waves. The burst of gravitational waves will cover a large frequency bandwidth, however the newly created neutron star is expected to have quadrupole modes that resonate on the order of kHz, creating gravitational waves at that frequency. Supernovae are thought to occur at a rate of about one per years in our galaxy and at a rate of several per year at a distance out to the center of the Virgo cluster. General Relativity FIGURE . Constraints on the Robertson parameter as determined by Shapiro effect and light deection measurements. Graphics courtesy C. Will . PlanckLength and Limits of General Relativity Physics in the th century was dominated by two great revolutions in the way we think about the nature of the universe at the most fundamental level quantum theory and relativity theory. Currently, they are physicists best understanding of the gears and wheels behind how everything works however, each has limitations and it remains an unnished revolution. General Relativity is a completely classical theory, its quantum form is unknown. Under general relativity theory, which is a classical not a quantum theory, the force of gravity is propagated by gravitational waves, which transmit the force of gravity at the speed of light. Under a quantumtheory of gravity, we focus on the quanta of gravity the gravitons, the elementary force particles that transmit gravity through a process of graviton exchange. Gravitons, never experimentally observed, are particles of zero mass which travel at the speed of light and have a quantum spin of . Quantum theory is our understanding of how things work at the ultramicroscopic scale of atoms and subatomic particles. Indeed, quantum theory was developed, in large part, to under stand how it is possible for atoms to exist in our universe. That process of discovery revealed laws of nature completely alien to the ways of thinking we all develop, based on our daytoday experiences with the world. For instance, it was discovered that a single particle could behave as if it were in two places at once, and that a pair of particles, even a great distance apart, could behave in some ways as a single entity. Quantum gravity is of interest both to permit gravity to be unied with the other three forces, as well as to unify general relativity with quantum physics. Quantum gravity is at the heart of physics Theory of Everything. Simple dimensional arguments show that the physical phenomena where quantum gravita tional effects becomes relevant as those characterized by the length scale L Planck G/c Chapter m, called the Planck length. Here is the Planck constant that governs the scale of the quantum effects, G is the Newton constant that governs the strength of the gravitational force, and c is the speed of light, that governs the scale of the relativistic effects. The Planck length is extremely small. To have an idea, the Planck length is as many times smaller than an atom, as an atom is smaller than the solar system. Current technology is not yet capable of observing physical effects at scales that are so small although several recent suggestions of how it could be possible to do so have appeared. Because of this, we have no direct experimental guidance for building a quantum theory of gravity. This is not by itself a complete impediment, because general relativity and quantum mechanics are themselves strong guidances for constructing the theory several major advances in the history of physics have been obtained in the absence of new experiments, from the effort of merging two empirically supported but apparently contradictory theories. Examples are Newtons merge of Keplers and Galileos theories, Maxwells merge of electric and magnetic theory, or Einsteins derivation of special relativity from the apparent contradiction between electromagnetism and mechanics. However, until genuine quantum grav itational phenomena are directly or indirectly observed, we cannot conrm or falsify any of the current tentative theories. If we could measure the geometry of space and time at the Planck scale, we should be able to see quantum gravitational effects. Many arguments indicate that the Planck length may appear as a limit to the innite divisibility of space, that is, as a minimal length. Intuitively, any attempt to measure smaller distances would result in the concentration of too much energy in too small a space, with the result of forming a microblackhole, effectively subtracting the region from observation. A minimal length would complete the tern of fundamental scales in Nature, together with the speed of light, which is the maximal velocity of a body, and the Planck constant, which is the minimal amount of action exchanged between two systems. .. Planck Units All of the quantities that have Planck attached to their name can ultimately be understood from the concept of the Planck mass. The Planck mass, roughly speaking, is the mass a point particle would need to have for its classical Schwarzschild radius the size of its event horizon to be the same size as its quantummechanical Compton wavelength or the spread of its wavefunction. The signicance of this mass is that it is the energy scale at which the quantum properties of the object remember, this is a point particle are as important as the general relativity properties of the object. Therefore it is likely to be the mass scale at which quantum gravity effects start to matter. Thus, the Planck length is the typical quantum size of a particle with a mass equal to the Planck mass. The Planck time is then just the Planck length divided by the speed of light. With the three main constants of physics, G, c and , one can form the following units Planck mass M P Gravitational radius of this mass is equal to Compton wavelength GM P /c /M P c,. or M P c G . g,E P M P c . GeV. . Planck length L P L P M P c G c . m. . General Relativity Planck time t P t P L P c . s.. Planck temperature T P T P E P k B . K. . Planck density p P M P L P c G g cm .. The Planck time comes from a eld of mathematical physics known as dimensional analysis, which studies units of measurement and physical constants. The Planck time is the unique com bination of the gravitational constant G, the relativity constant c, and the quantum constant , to produce a constant with units of time. For processes that occur in a time t less than one Planck time, the dimensionless quantity t P /t is large. Dimensional analysis suggests that the effects of both quantum mechanics and gravity will be important under these circumstances, requiring a theory of quantum gravity. Unfortunately, all of our scientic experiments and human experience happens over billions of billions of billions of Planck times, which makes it hard to directly probe the events happening at the Planck scale. As of , the smallest time interval that was directly measured was on the order of attoseconds s, or about . Planck times. Before a time classied as a Planck time, all of the four fundamental forces are presumed to have been unied into one force. All matter, energy, space and time are presumed to have exploded outward from the original singularity. Nothing is known of this period. In the era around one Planck time, it is projected by present modeling of the fundamental forces that the gravity force begins to differentiate from the other three forces. This is the rst of the spontaneous symmetry breaking which lead to the four observed types of interactions in the present universe. .. Loop Quantum Gravity and String Theory Loop quantum gravity LQG, also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity. Loop quantum gravity suggests that space can be viewed as an extremely ne fabric or network woven of nite quantised loops of excited gravitational elds called spin net works. When viewed over time, these spin networks are called spin foam, which should not be confused with quantum foam. A major quantum gravity contender with string theory, loop quan tum gravity incorporates general relativity without requiring string theorys higher dimensions. In , Abhay Ashtekar reformulated Einsteins eld equations of general relativity, using what have come to be known as Ashtekar variables, a particular avor of EinsteinCartan the ory with a complex connection. In , Carlo Rovelli and Lee Smolin used this formalism to introduce the loop representation of quantum general relativity, which was soon developed by Ashtekar, Rovelli, Smolin and many others. In the Ashtekar formulation, the fundamental ob jects are a rule for parallel transport technically, a connection and a coordinate frame called a vierbein at each point. Because the Ashtekar formulation was backgroundindependent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit spatial diffeomorphism invariance of the vacuum state plays an essential role in the regulariza tion of the Wilson loop states. String theorists are working very hard to create a theory of everything out of models in which strings are the elementary entity of physics. Todays extradimensional string theories have proven Chapter remarkably robust in creating accurate models of physics particles and interactions. String the ories provide a model for all four of physics forces and for the elementary particles of matter as well. Among string theories vibrational string patterns is a pattern that exactly produces the properties of the graviton. Thus, string theory is a quantum theory that incorporates gravity. But string theories are backgrounddependent. The background is spacetime, and in string theories all of the forces including gravity operate against the background of spacetime. This seems to present a conceptual stumbling block in the way of using string theories to recon cile general relativitys gravity with quantum physics other forces, because general relativity is a backgroundindependent theory. Under general relativity, the force of gravity shapes spacetime is spacetime. Quantum effects in gravity are expected to inuence the following areas of physics .. i Early Universe According to the currently standard cosmological model, the Universe was very dense and hot in the past. Extrapolating back the model, we encounter a singular point of innite density, tem perature and curvature, conventionally denoted the Big Bang. However, this nal extrapolation is certainly incorrect, because quantum gravitational phenomena become dominant when the uni verse is very dense and hot, and these effects are not included in the usual model. A quantum theory of gravity is needed in order to take these effects into account and study the early instants in the life of our universe. Some current theories of gravity in particular loop quantum gravity, indicate that the singular BigBang point is never reached, and the current expansion of the uni verse might have been preceded by a collapsing phase, and a Big Bounce. One of the major hopes of observing traces of quantum gravitational phenomena is in this cosmological context, as traces of early universe phenomena left in the cosmic background radiation currently under intense observation, or in the background gravitational wave radiation, which is expected to be observed in the next decade. .. ii Black Holes Quantum gravity should play a role in several aspects of blackhole physics. First, it should give a complete understanding of the thermal radiation that black holes are expected to produce, rst computed by Stephen Hawking. Second, Hawkings analysis shows that black holes carry enormous entropy about to the power for a solar mass black hole. What is the statistical mechanical origin of this number which is enormous even by the standards of thermodynamics Third, quantumgravity is expected to replace the innite singularity that general relativity predicts at the center of black holes with a more physically reasonable picture. Finally, the theory should explain what happens at the end of the Hawking evaporation of a black hole. .. iii Astrophysical Effects Several astrophysical quantumgravitational effects have been suggested. None has been observed so far, but different calculations suggest that they might be observed in the near future. An example is a small dependence of the speed of light on the color of the light, caused by the Planckscale granularity of space. The effect is very small because of the smallness of the Planck scale, but it might become detectable if it is cumulated over a very long integalactic path traveled by the light. Observations for testing this prediction are ongoing. Alternative Theories of Gravity Einsteins eld equations are not unique. Various alternative theories have been created in the last years. We discuss a few aspects of these theories. . BransDicke Theory The key ideas of this theory are General Relativity matter, reprepresented by the energymomentum tensor, and a coupling constant x a scalar eld the scalar eld xes the value of G the gravitational eld equations relate the curvature to the energymentum tensor of the scalar eld and matter. The coupled equations for the scalar eld and the metric in this theory are T M . R Rg c T M T .. In the limit , is not affected by the matter distribution, and can therefore be set to a constant /G. In this case, T vanishes, and hence the BransDicke theory reduces to Einsteins equations. The BransDicke theory is an interesting construction, because it shows that is possible to construct theories that are consistent with the Einstein principle of equivalence. Einsteins theory is beautiful, but it is not unique. Tests of theories are therefore very important, see e.g. Will . A reasonably conservative limit follows from experiments gt , Einsteins theory is most probably the correct theory, at least in the solar system. . fR Gravity fR gravity is a type of modied gravity theory proposed as an alternative to Einsteins General Relativity. Although it is an active eld of research in Cosmology, there are known problems with the theory. It has the potential, in principle, to explain the accelerated expansion of the Universe without adding unknown forms of dark energy or dark matter. In fR gravity, one seeks to generalize the Lagrangian of the EinsteinHilbert action Sg R gd x. to S f g fR gd x. where G, g is the determinant of the metric tensor g g and fR is some function of the Ricci Curvature. An interesting feature of these theories is the fact that the gravitational constant is time and scale dependent. To see this, add a small scalar perturbation to the metric in the conformal Newtonian gauge ds dt ij dx i dx j ,. where and are the Newtonian potentials and use the eld equations to rst order. After some lengthy calculations, one can dene a Poisson equation in the Fourier space and attribute the extra terms that appear on the right hand side to an effective gravitational constant G e . Doing so, we get the gravitational potential valid in subhorizon scales k a H G eff a k m ,. where m is a perturbation in the matter density and G e is G eff F k a R m k a R m . Chapter and m RF ,R F .. This class of theories, when linearized, exhibits three polarization modes for the gravita tional waves, of which two correspond to the massless graviton helicities and the third scalar is coming from the fact that if we take into account a conformal transformation, the fourth order theory fR becomes General Relativity plus a scalar eld. Sign Conventions There is unfortunately no accepted system of sign conventions in GR. Different textbooks use different sign conventions. Let us write S, , , . T M Pu u SP g . R S . G S G c T . R SSR .. In this text, we have used the natural convention S , S , and S . This is the same convention as in MTW . Hobson et al. use the conventions S , S , and S. Exercises . Lorentz Transformations . Aberration Formula Calculate the light aberration within Special Relativity hint use the velocity addition theorem. . Denition of Curvature What is the geometric meaning of a connection on a manifold How is the curvature tensor dened What is the geometric meaning of the curvature tensor What are the symmetries of the Riemann tensor How many independent components has the Riemann tensor in and dimensions . TOVEquations for Compact Objects Write down the TolmanOppenheimerVolkoff TOV equations for the structure of a nonrotating neutron star. Which relativistic effects modify the Newtonian hydrostatic equilibrium . Curvature in a Spatially Flat Universe Consider a spatially at expanding universe stretching Minkowski space ds dt a t ij dx i dx j . with an expansion factor at. With the observer eld dt , i at dx i . this metric can still be written in Minkowski form, ds ab a b . General Relativity . Use the rst Cartan structure equation d a a b b . to derive the connection forms a b for this metric. . Use the second Cartan structure equation a b d a b a c c b R a bcd c d . to derive the curvature forms i and i j , i, j , , and from this the Riemann tensor R a bcd . . Calculate from these expressions the Ricci tensors R ,R i ,R ij , as well as the Ricci scalar, and from there Einsteins eld equations. . Gauge Transformations Prove that the gauge transformations h h . leave the Riemann tensor invariant. This is analogous to gauge transformations in electromag netism which leave invariant the Faraday tensor F . . Merging of two Black Holes at Cosmological Distances Give an estimate for the gravitational wave amplitude expected from the merging of two Black Holes with masses of one million solar masses at a distance of Gpc. What are the typical wavelength and frequency of such waves Compare with the sensitivity diagram for LISA. . Gravitational Waves from Compact Binary Systems Compute the loss of energy due to emission of gravitational waves of a binary system consisting of two neutron stars with masses M and M , binary period P, semimajor axis a and eccentricity e lt dE dt gt G M M M M a c e / e e .. For the motion of two point masses M and M in the orbital plane x, y we have the fol lowing relations a GM M E ,E GM M a . e EL M M G M M . r ae e cos . r M M M r. r M M M r.. the following calculations can also simply be done by using Christoffel symbols. Chapter Since t, this leads to a timedependent moment of inertia I xx M x M x M M M M r cos . I yy M M M M r sin . I xy M M M M r sin cos . II xx I yy M M M M r .. From this we get the energy loss due to the quadrupole formula dE dt G c I xx I yy I xy I .. The time derivative of r r e sin GM M ae . and follows from angular momentum conservation L M M /M r GM M a e r .. After somewhat lengthy calculations, you obtain the energy loss averaged over one revolution lt dE dt gt P b P b dE dt dt P b dE dt d. . This leads to the basic formula .. Compute from this formula over the total energy E of the binary system the timechange of the semimajor axis a of the pulsar orbit a T m m m m c a e / e e . We use masses in units of M ,m the mass of the pulsar m the mass of the compagnion and the fundamental constant T GM c .s.. This equation can be solved with the ansatz a ta , t t / . with the initial value a , . Determine the crashtime t for a typical binary system. General Relativity References Camenzind, M. , Compact Objects in Astrophysics White Dwarfs, Neutron Stars and Black Holes, SpringerVerlag, Berlin Sean M. Carroll , Spacetime Geometry An Introduction to General Relativity, Addison Wesly M.P. Hobson, G. Efstathiou and A. Lasenby , General Relativity An Introduc tion for Physicists, Cambbridge University Press Misner, C., Thorne, Kip. S. and Wheeler, J. , Gravitation, Freeman Kawamura, M., Oohara, K., Nakamura, T. , GR Numerical Simulations on Coalescing Binary Neutron Stars and GaugeInvariant Wave Extraction, astro ph/ Nelemans, G. et al. , The gravitational wave signal from the Galactic disk pop ulation of binaries containing two compact objects, astroph/. Peters, P.C. , Phys. Rev. , Peters, P.C., Matthews, J. , Phys. Rev. , . Will, Clifford , The Confrontation between General Relativity and Experiment, Living Reviews in Relativity, lrr Appendix Appendix A Calculus for Differentiable Riemannian Manifolds In this Appendix we summarize the most important formulae for calculus on Riemannian mani folds. Christoffel Symbols and Covariant Derivative In a smooth coordinate chart x i , i , . . . , n, the Christoffel symbols are given by m ij g km x i g kj x j g ik x k g ij .. Here g ij is the inverse matrix to the metric tensor g ij . In other words, i j g ik g kj . and thus n i i g i i g ij g ij . is the dimension of the manifold. Christoffel symbols satisfy the symmetry relation i jk i kj . which is equivalent to the torsionfreeness of the LeviCivita connection. The contracting rela tions on the Christoffel symbols are given by i ki g im g im x k g g x k log g x k . and g k i k g gg ik x k . where g is the absolute value of the determinant of the metric tensor g ik . These are useful when dealing with divergences and Laplacians see below. The covariant derivative of a vector eld with components v i is given by v i j j v i v i x j i jk v k . and similarly the covariant derivative of a ,tensor eld a form with components v i is given by v ij j v i v i x j k ij v k .. For a ,tensor eld with components v ij this becomes v ij k k v ij v ij x k i k v j j k v i . Appendix A and likewise for tensors with more indices. The covariant derivative of a function scalar is just its usual differential i i ,i x i .. Because the LeviCivita connection is metriccompatible, the covariant derivatives of metrics vanish, k g ij k g ij .. The geodesic Xt starting at the origin with initial speed v i has Taylor expansion in the chart Xt i tv i t i jk v j v k Ot .. Riemann Curvature Tensor If one denes the curvature operator as RU, V W U V W V U W U,V W and the coordinate components of the ,Riemann curvature tensor by RU, V W R ijk W i U j V k , then these components are given by R ijk x j ik x k ij n s js s ik ks s ij ,. where n denotes the dimension of the manifold. Lowering indices with R ijk g s R s ijk one gets R ikm g im x k x g k x i x m g i x k x m g km x i x g np n k p im n km p i .. The symmetries of the tensor are R ikm R mik and R ikm R kim R ikm .. It is symmetric in the exchange of the rst and last pair of indices, and antisymmetric in the ipping of a pair. The cyclic permutation sum sometimes called rst Bianchi identity is R ikm R imk R imk .. The second Bianchi identity is m R n ik R n imk k R n im ,. that is, R n ikm R n imk R n imk ,. which amounts to a cyclic permutation sum of the last three indices, leaving the rst two xed. . Ricci and Scalar Curvature Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the Riemann tensor, but contain less information. The Ricci curvature tensor is essentially the unique nontrivial way of contracting the Riemann tensor R ij R ij g m R ijm g m R imj ij x i x j ij m m m i jm .. The Ricci tensor R ij is symmetric. By the contracting relations on the Christoffel symbols, we have R ik ik x m i km k x i log g .. Calculus for Differentiable Riemannian Manifolds The scalar curvature is the trace of the Ricci curvature, Rg ij R ij g ij g m R ijm .. The gradient of the scalar curvature follows from the Bianchi identity R m m R, . that is, R m R m .. . Einstein Tensor The Einstein tensor Gab is dened in terms of the Ricci tensor R ab and the Ricci scalar R, G ab R ab g ab R, . where g is the metric tensor. The Einstein tensor is symmetric, with a vanishing divergence, which is due to the Bianchi identity, a G ab G ab a .. . Weyl Tensor The Weyl tensor is given by C ikm R ikm n R i g km R im g k R k g im R km g i nn Rg i g km g im g k ,. where n denotes the dimension of the Riemannian manifold. The Weyl tensor is tracefree, C m amk . Gradient, Divergence and LaplaceBeltrami Operator The gradient of a function is obtained by raising the index of the differential i , that is i i g ik k g ik ,k g ik k g ik x k .. The divergence of a vector eld with components V m is m V m V m x m V k log g x k g V m g x m .. The LaplaceBeltrami operator acting on a function f is given by the divergence of the gradient f i i f det g x j g jk det g f x k .. The divergence of an antisymmetric tensor eld of type , simplies to k A ik g A ik g x k .. For a symmetric tensor T ab of type , energymomentum tensor e.g. one gets k T ik g T ik g x k i km T km .. Appendix A Differential Forms on Manifolds In this section, we collect the main denitions and formulas useful to deal with differential forms. This language has become pretty common in the recent Literature and sometimes confuses stu dents used to the index notation. We should say that in a theory like GR, where the coordinates do not have any physical meaning, differential forms represent the most natural mathematical formalism, even though, in some problems, the index notation is preferable. So, often one has the necessity to switch from one formalism to the other, going from forms to indexes and vice versa. This has induced me to collect some formulas in few pages, with the hope they will be useful to the readers as they have been for me. Unfortunately, many different notations are used in the Literature, anyone valid and moti vated by precise choices. The denitions and formulas below are in accordance with a notation commonly used in Physics and refers to an arbitrary number of dimensions unless differently specied and to any signature of the ndimensional manifold. . pForms Let M be an ndimensional manifold with signature s. Denote the n p dimensional space of pforms on the cotangent bundle as p T M n . Let e a e a dx be a form transforming under the vectorial representation of the local symmetry group SOn s, s. The canonical basis for p T M n is naturally induced by the local basis e a a , through the wedge product antisymmetric tensor product. Specically, a basis for pforms in n dimensions is given by the collection of all the possible linearly independent pforms which can be formed by wedging the n vectors e a . For example, the natural basis for forms in four dimensions is made up of the forms given below e e e e ,e e e e ,e e e e ,e e e e .. Any pform p TM n can be expanded on the canonical basis according to the following denition p a a p e a e a p ,. where the square brackets denote antisymmetrization. An nform can be naturally integrated on the ndimensional manifold M, by considering that it contains the natural volume element according to the following denition e a e a n e a e a n n dx dx n s a a n dV , . where dV g dx dx n denotes the volume element and a ...a n the totally antisymmetric symbol, with the condition that a ...a n for a i lt a i . . Inner Product We now introduce the inner or scalar product between differential pforms and vectors v dened on the tangent bundle TM. Let p T M n and v v a e a TM, where the vector elds e a e a are a local basis on TM. By denition we have lt e a ,e b gt a b . The following prescription allows to evaluate any differential form in particular directions represented by vector Namely, s corresponds to the number of minus signs in the metric. Calculus for Differentiable Riemannian Manifolds elds, obtaining a p form, according to the following prescription v p a a p v b lt e a e a p ,e b gt p p i pi a i b a a i a p v b e a e a i e a i e a p p p ba a p v b e a e a p ,. where in the last line we moved the index saturated with the components of the vector v on the left by using the antisymmetry of the indexes of and renamed the others. By using the above formula we can extract the components of a pform by evaluating it on p vectors of the local basis, i.e. e a ,,e a p p p a a p , p T M n ,. namely, the pform is a smooth map that at any point x M associates an antisymmetric tensor of type , p. . Wedge Product Let us nowintroduce the exterior or wedge product between two generic differential forms. The wedge product is a map p T M n q T M n pq T M n p q n dened as pq a a p b b q e a e a p e b e b q pq pq pq a a p a p a pq e a e a p e a p e a pq ,. so that the components of the resulting p qform are e a ,,e a pq pq pq a a p a p a pq .. . Hodge Dual Another useful operator is the socalled Hodge dual, usually denoted by the symbol . The Hodge dual is a map p T M n np T M n , acting on the canonical basis according to the following prescription e a e a p np a a p a p a n e a p e a n np pnp a a p a p a n e a p e a n .. Notice that the above denition slightly differs from the standard one, but it results particularly convenient for a reason that will be clear soon. In fact, an interesting consequence of the denition above is that the wedge product of a pform with its Hodge dual generates the volume element according to the following formula e a e a p e b e b p p a a p b b p dV , . where dV is the natural volume element on the ndimensional manifold. It is worth noting that no dependence on the signature or dimensions appear in the formula above, so that it will be par ticularly convenient to rewrite actions in terms of differential forms. Notice, also, that operating twice with the Hodge dual one obtains the initial form apart for a possible sign factor, i.e. e a e a p spnp e a e a p .. Appendix A By using the denitions . and ., we can easily extract the expression of the dual of a generic pform. Specically, p a a p e a e a p np p pnp a a p a a p a p a n e a p e a n .. In other words, the dual of a generic pform is the n pform of components e a ,e a np p pnp a a p a a p a p a n . So let and p TM n be two pform, we have by the formula above that p a a p a a p dV . . Hence, apart for the factor /p, the wedge product in . corresponds to the scalar product between the components of the two pforms multiplied by the natural volume element. This can be rewritten as p e a ,,e a p ,e a ,,e a p dV , . where the symbol . . . , . . . denotes the internal product. We remark that wedging the pform with the canonical basis the factorial of p disappears. . Exterior Derivative The exterior derivative operator d is a map from p T M n to p T M n dened as p T M n d p p b a a p e b e a e a p , p T M n ,. where, as usual, we contained in parentheses the components of the resulting pform. By the denition given above we can immediately extract an important property of the exterior deriva tives, i.e. d d , namely the composition of two derivative operators is the vanishing operator. Moreover, assuming that p TM n and q TM n , it is very easy to show the follow ing formula dd p d.. In general, the presence of a local symmetry requires the denition of a covariant derivative. In this framework a local SOs, n s symmetry is present the Lorentz group e.g., therefore we have to dene a new exterior derivative operator acting on sos, n s Liealgebravalued p forms which generates sos, n s valued p forms. Namely, the derivative operator has to transform in the adjoint representation of the local symmetry group. In this respect, let us introduce a sos, n s Liealgebravalued connection form a b and dene the new derivative operator d as d d.. We claim that the above derivative operator has exactly the property required, as can be easily demonstrated. In order to operatively dene the covariant derivative operator, we rstly specify its action on the basis form e a , we have d e a de a a b e b ,. Calculus for Differentiable Riemannian Manifolds which, as can be easily recognized, is the denition of the torsion form T a . Specically, T a d e a de a a b e b .. So, in the presence of torsion the covariant exterior derivative operator fails in annihilating the ba sis element. Sometimes, the equation d e a T a is referred as rst Cartan structure equation. It is worth noting that the composition of two covariant exterior derivative does not trivially vanish, rather we have d d e a a b e b ,. which allows to extract the following expression for the curvature form a b d a b a c c b ,. known as the second Cartan structure equation. It is worth remarking that if a b a b a c d c b ,. namely the connection is a pure gauge, ab ba being a representation of the local symmetry group a Lorentz transformation e.g.. Then, one can demonstrate that by assuming a b de a iff T a ,. which implies that e a dx a , where x a are functions of the original set of coordinates. Moreover, we have e a x a , so that the components of the local basis simply represent the soldering forms in at space between two local arbitrary accelerated reference frames, with the origin placed at the same point of the tangent bundle. Two useful identities can be easily derived from the above denitions, i.e. d a b , .a d T a b a b e b , .b respectively known as second and rst Bianchi identity. We now refer to a specic case we assume that n and s , which means that we are referring to a dimensional pseudoRiemannian manifold M , which is locally isomorphic to Minkowski spacetime with signature , , , , so the local symmetry group is SO, . In this framework the HilbertPalatini action for General Relativity can be rewritten as S H e, e a e b ab .. Remembering denition . and formula . we can easily write S H e, e a e b ab ab cd e a e b e c e d d xdeteR , . where we used dV dete d x. This is exactly the Hilbert action. Appendix A . Dirac Action An analog procedure allows to rewrite also the Dirac action in the formalismof differential forms. For this we need to dene the action of the exterior covariant derivative on a spinor eld, which is a form complex function transforming under the spinor representation of the SO, local group. We do not enter in the details about the construction of the spinor bundle, we only say that the exterior covariant derivative operator acts on the spinor elds and according to the following rules Dd i ab ab , .a Dd i ab ab , .b where ab i a , b . are the generators of the Lorentz group. Now, the Dirac action for a spinor eld of mass m can be written as S , i e a a DD a i me a .. Remembering that, according to our notation, e a e b b a dV , the demonstration follows immediately. This form is needed e.g. for the discussion of spinor elds representing quarks and leptons in the early universe. Appendix B Perturbations of Minkowski Space and the Nature of Gravitational Waves When we rst derived Einsteins equations, we checked that we were on the right track by con sidering the Newtonian limit. This amounted to the requirements that the gravitational eld be weak, that it be static no time derivatives, and that test particles be moving slowly. In this sec tion we will consider a less restrictive situation, in which the eld is still weak, but it can vary with time, and there are no restrictions on the motion of test particles. This will allow us to discuss phenomena which are absent or ambiguous in the Newtonian theory, such as gravitational radiation where the eld varies with time. Linearized Gravity and Gauge Transformations The weakness of the gravitational eld is once again expressed as our ability to decompose the metric into the at Minkowski metric plus a small perturbation, g h ,h .. Under this condition, the inverse metric is simply given by g h ,. where h h . We can raise and lower indices just by using the at Minkowski metric . For this reason we may consider h as a symmetric tensor of second rank dened on Minkowski space. We want to nd the equation of motion obeyed by the perturbations h, which come by exam ining Einsteins equations to rst order. We begin with the Christoffel symbols, which are given by h h h .. Since the connection coefcients are rst order quantities, the only contribution to the Riemann tensor will come from the derivatives of the s, not the terms. Lowering an index for conve nience, we obtain R h h h h .. The Ricci tensor is obtained by contracting over and , giving R h h hh .. Her we have dened the trace of the perturbation, h h , and the dAlembertian operator in Minkowski space t x y z . Finally, we obtain the Ricci scalar R h h. . Appendix B Putting all this together, we obtain the Einstein tensor G R R . h h hh h h. The linearized eld equations are then G GT ,. where T is the energymomentum tensor calculated in zeroth order from h. We do not include higherorder corrections to the energymomentum tensor, because the amount of energy and momentum must itself be small for the weakeld limit to apply. In other words, the lowest nonvanishing order in T is automatically of the same order of magnitude as the perturbation. Notice that the conservation law to lowest order is simply T . We will most often be concerned with the vacuum equations, which as usual are just R , where R is given by .. . On Gauge Invariance With the linearized eld equations in hand, we are almost prepared to set about solving them. First, however, we should deal with the important issue of gauge invariance. This issue arises because the demand that g h does not completely specify the coordinate system on spacetime there may be other coordinate systems, in which the metric can still be written as the Minkowski metric plus a small perturbation, but the perturbation will be different. Thus, the decomposition of the metric into a at background plus a perturbation is not unique. The notion that the linearized theory can be thought of as one governing the behavior of tensor elds on a at background can be formalized in terms of a background spacetime M ,a physical spacetime M p , and a diffeomorphism M M p . As manifolds M and M p are the same since they are diffeomorphic, but we imagine that they possess some different tensor elds on M we have dened the at Minkowski metric , while on M p we have some metric g which obeys Einsteins equations. We imagine that M is equipped with coordinates x and M p is equipped with coordinates y, although these will not play a prominent role. The diffeomorphism allows us to move tensors back and forth between the background and physical spacetimes. Since we would like to construct our linearized theory as one taking place on the at background spacetime, we are interested in the pullback g of the physical metric. We can dene the perturbation as the difference between the pulledback physical metric and the at one h g .. From this denition, there is no reason for the components of h to be small however, if the gravitational elds on M p are weak, then for some diffeomorphisms we will have h . We therefore limit our attention only to those diffeomorphisms for which this is true. Then the fact that g obeys Einsteins equations on the physical spacetime means that h will obey the linearized equations on the background spacetime since , as a diffeomorphism, can be used to pull back Einsteins equations themselves. In this language, the issue of gauge invariance is simply the fact that there are a large number of permissible diffeomorphisms between M and M p where permissible means that the per turbation is small. Consider a vector eld x on the background spacetime. This vector eld generates a oneparameter family of diffeomorphisms M M . For sufciently small, if is a diffeomorphism for which the perturbation dened by h is small than so will be, although the perturbation will have a different value. Specically, we can dene a family of perturbations parameterized by h g g .. Perturbations of Minkowski Space and the Nature of Gravitational Waves The second equality is based on the fact that the pullback under a composition is given by the composition of the pullbacks in the opposite order, which follows from the fact that the pullback itself moves things in the opposite direction from the original map. Plugging in the relation for h, we nd h h h ,. since the pullback of the sum of two tensors is the sum of the pullbacks. Now we use our assump tion that is small in this case h will be equal to h to lowest order, while the other two terms give us a Lie derivative h h h L .. Since the background metric is at, we therefore nd h h .. This formula represents the change of the metric perturbation under an innitesimal diffeomor phism along the vector eld this is called a gauge transformation in linearized theory. The innitesimal diffeomorphisms provide a different representation of the same physical situation, while maintaining our requirement that the perturbation be small. Therefore, the above result tells us what kind of metric perturbations denote physically equivalent spacetimes those related to each other by , for some vector eld . The invariance of our theory un der such transformations is analogous to traditional gauge invariance of electromagnetism under A A . The analogy is different from the previous analogy we drew with electromag netism, relating local Lorenz transformations in the orthonormalframe formalism to changes of basis in an internal vector bundle. In electromagnetism the invariance comes about because the eld strength F A A is left unchanged by gauge transformations similarly, we nd that the transformation . changes the linearized Riemann tensor by R .. Our abstract derivation of the appropriate gauge transformation for the metric perturbation is veried by the fact that it leaves the curvature and hence the physical spacetime unchanged. Gauge invariance can also be understood from the slightly more lowbrow, but considerably more direct route of innitesimal coordinate transformations. Our diffeomorphism can be thought of as changing coordinates from x to x . The minus sign, which is unconven tional, comes from the fact that the new metric is pulled back from a small distance forward along the integral curves, which is equivalent to replacing the coordinates by those a small dis tance backward along the curves. Following through the usual rules for transforming tensors under coordinate transformations, you can derive precisely . although you have to cheat somewhat by equating components of tensors in two different coordinate systems. Degrees of Freedom The metric perturbation h is a symmetric , tensor on Minkowski spacetime. This means, under spatial rotations the component is a scalar, the i component form a threevector, and the ij components form a twoindex symmetric spatial tensor. Each spatial tensor can be decom posed into a trace and a tracefree part in group representations this corresponds to irreducible This can easily be proved from equation . Appendix B representations of the rotational group SO. We therefore write h as h . h i w i . h ij ij s ij .. denotes the trace of h ij , and s ij is traceless ij h ij . s ij h ij kl h kl ij .. The entire metric can thus be written as ds dt w i dt dx i dx i dt ij s ij dx i dx j .. Here we have not yet chosen a gauge, we just have conveniently decomposed the metric pertur bations into two scalar modes, one vector mode and a tensor mode, adding to independent components of the perturbation h . To get a feeling for the physical interpretation of these modes, we consider the motion of test particles as described by the geodesic equation. For this we need the Christoffel symbols . i i w i . j j . i j j w i i w j h ij . jk j w k k w j h ij . i jk j h ki k h ji i h jk .. Here we use h ij s ij ij . We decompose the momentum p dx /d, where /m for massive particles, in terms of the energy E and the threevelocity v i dx i /dt p dt d E,p i Ev i .. Then we write the geodesic equation as though a force would act on the particles dp dt p p E .. For we get the energy evolution dE dt E k v k j w k k w j h jk v j v k .. The spatial components i become E dp i dt i w i i w j j w i h ij v j j h ki k h ji i h jk v j v k .. This ansatz can easily be generalized to cosmological spacetimes in order to describe general perturbations evolving under the expansion of the Universe. Perturbations of Minkowski Space and the Nature of Gravitational Waves For a physical interpretation we introduce the gravitoelectric and gravitomagnetic eld in terms of scalar and vector potentials, where the vector w acts as a vector potential G i i w i . H i w i ijk j w k .. Then we can write E dp i dt G i v H i h ij v j j h ki k h ji i h jk v j v k .. The rst two terms on the right hand side describe how the test particle responds to the scalar and vector perturbations and w i in a way reminiscent of the Lorenz force in electromagnetism. We also nd couplings to the spatial perturbations h ij of linear and quadratic order in the velocity. . Einsteins Equations We can now decompose the Riemann tensor in our variables R jl j l j w l h jl . R jkl j k w l k h lk . R ijkl j k h li i k h lj .. To obtain the Ricci tensor we contract with the at metric R k w k . R j w j j k w k j k s k j . R ij i j i w j ij s ij k i s k j ,. where ij i j is the at Laplacian. Finally, we can calculate the Einstein tensor G k l s kl . G j w j j k w k j k s k j . G ij ij i j ij k w k i w j ij s ij k i s k j ij k m s km .. With this decomposition we see that in fact equations are just constraint equations and do not present true dynamical evolution equations. To see this we start with the rst equation which can be written as GT k m s km .. This is an equation for with no time derivatives. If we know what are T and s ij are doing all the time, the potential is uniquely determined by boundary conditions. is therefore not a propagating degree of freedom, it will be determined by the energymomentum tensor and the strain. Next we consider the i equation, which we write as jk j k w k GT j j k s k j .. We use the notation j w k j w k k w j . j w k j w k k w j .. Appendix B This is an equation for the vector eld w j which also does not contain time derivatives. Finally, the ij equation is ij i j GT ij ij i j ij ij k w k i w j s ij k i s k j ij k m s jm .. Once again, there are not time derivatives acting on , which is therefore determined from the other elds. The only propagating degree of freedom in Einsteins theory are those in the strain ten sor s ij . In terms of elds, which depend on the behaviour under spatial rotations we may classiy the scalars and as spin, the vector w i as spin and the strain tensor as spin degrees of freedom. Only the spin degree of freedom is a true dynamical mode in General Relativity. . Transverse Gauge The different metric components of h will transform under a general gauge transformation generated by a vector eld as . w i w i i i . i i . s ij s ij i j k k ij .. rst we consider the transverse gauge. This is closely related to the Coulomb gauge of electro magnetism, i A i . Similarly, we x the strain by means of i s ij ,. by choosing j to satisfy j j i i i s ij .. The value of is still undetermined. We can choose this term for the condition i w i by means of i w i i i .. With htis gauge, Einsteins equations become G GT . G j w j j GT j . G ij ij i j i w j ij s ij GT ij .. Gravitational Wave Solutions Let us consider now the transverse gauge, by neglecting source terms, T . Then the equation is .. For suitable boundary conditions we can achieve everywhere. The i component is then w i ,. Perturbations of Minkowski Space and the Nature of Gravitational Waves which again implies w i . We turn next to the trace of the ij component with the above values ,. which also implies . We are then left with the tracefree part of the ij equation s ij ,. which becomes a wave equation for the traceless strain tensor. It is convenient to express the metric tensor in this transverse traceless gauge h TT s ij . This quantity is purely spatial, traceless and transverse, i.e. h TT . h TT . h TT .. In analogy to electromagnetism, plane waves are solutions of this equation h TT A expik x , . where A is a constant symmetric , tensor, which is purely spatial and traceless A , A .. The constant kvector is the wave vector with k k . The plane wave . is a solution of the linearized equation, provided the wave vector is null. This means that gravitational waves propagate with the speed of light. Any superposition of plane waves is also a solution. The condition of transversality means that h TT ik A expik x . or that k A .. We now consider a wave travelling in the zdirection, i.e. k ,,,k ,,,.. In this case, the transversality requires that A . The only nonzero components are therefore A ,A ,A ,A . But A is traceless and symmetric, i.e. of the form A A A A A . For a plane wave travelling in the zdirection, the two amplitudes A and A completely char acterize the wave. Appendix B For getting a feeling what happens if a wave passes by, we consider the motion of test parti cles in the presence of the gravitational eld represented by the wave. For this we consider the relative motion of nearby particles with velocities described by the vector eld U . Nearby geodesics are then given in terms of a separation vector X , which satises the equation of geodesic deviation see Appendix A D d X R U U X .. The velocity is simply given by U ,,,. Therefore, we only need to compute the Riemann tensor R which is given by R h TT h TT h TT h TT .. But with h TT , we get simply R h TT .. For slowly moving particles we have in lowest order t x , so the equation of geodesic deviation becomes t X h TT, X .. For our plane wave this means that only X and X will be affected the test particles are only disturbed in directions perpendicular to the wave vector. This is similar to electromagnetism, where the electric and magnetic elds in a plane wave. Our plane wave is characterized by two amplitudes which are denoted for convenience as follows h A . h A ,. so that the amplitude tensor has the form A h h h h .. Let us consider the effects exerted by h for h . Then we have the two equations X X h expik x . X X h expik x . . These can be solve dimmediately in lowest order as X h expik x X . X h expik x X .. Perturbations of Minkowski Space and the Nature of Gravitational Waves FIGURE . The mode of gravitational waves. The phases shown are , /, , /, . FIGURE . The mode of gravitational waves. Thus particles initially separated in the xdirection will oscillate in the xdirection, and likewise for those in the ydirection. If we start with a ring of stationary particles in the x y plane, they will bounce back and forth in the shape of a , as the wave passes by Fig. . The equivalent analysis for the case where h , but h would yield the solutions X X X h expik x . X X X h expik x . . In this case, the circle of particles would bounce back and forth in the shape of a , as shown in Fig. . These two quantities measure therefore two independent modes of linear polarisation of a gravitational wave, known as the plus and cross polarisations. Out of these two modes we also can construct right and lefthanded circularly polarized modes by dening h R h ih . h L h ih .. The effect of a pure h R wave would be to rotate the particles in a righthanded sense, as shown in Fig. , and similarly for the lefthanded modes. Remark In a general theory of gravity we nd three independent polarisation modes for trans verse gravitational waves. In addition to the above elliptic modes of General Relativity, also a scalar mode can appear which just represents a radial oscillation of the particles of a ring. This scalar mode is excluded in General Relativity, but appears e.g. in the BransDicke theory. These modes can be represented by the possible oscillations for a loop of a string. These give rise to three massles degees of freedom a spin particle the dilaton eld in the notation of Sect. . and a massless spin particle the graviton. Quantized strings inevitably give rise to gravity. The extra spin scalar mode reects the fact that string theory actually predicts a scalartensor theory Appendix B FIGURE . The circular mode of gravitational waves. of gravity, rather than ordinary General Relativity. A massles scalar is however not observed in reality, so some mechanism must be at work to give a mass to the scalar at low energies. The Detection of Gravitational Waves The physical effect of a passing gravitational wave is to slightly perturb the relative positions of freely falling masses. If two test masses are separated by a distance L, the change in the distance is roughly L L h. . Let us build a gravitational wave observatory with test bodies separated by some distance of order of kilometers. Then to detect a wave amplitude of the order of h would require a sensitivity of L h L km cm. . This is to compare to the size of atoms, a cm. This means that a gravitational wave observatory will have to be sensitive to changes in distances much smaller than the size of the constituent atoms out of which the masses have to be made. The original proposal was to use resonant rigid bodies to measure the strains exerted by the waves called Weber detectors. The basic modes of an elastic body excited by a gravitational wave are quadrupolar oscillations, whereby the surface remains constant. For this we construct a quadrupole consisting of four elastic springs which are coupled to a rectangular plate. The use of elastic springs enables us to consider resonance effects. For a weakly damped coupling such a detector can absorb energy from the gravitational wave, and this energy can be measured. A freely moving particle suffers in a gravitational wave of the form h Aexpit and h a change in position by y yAexpit. This corresponds to a force K m yAexpit. For elastically damped motion of a spring we have the equation of motion y y y K/m. . The solution is y yA i expit . . The energy absorbed from the wave corresponds to the work done by the force K within one wave period T, divided by the period. For the quadrupolar apparatus this gives E T T ReKRe y dt A y m .. In fact, resonant wave detectors, such as bars or spheres, do not measure the absorbed energy, but the strain exerted by the wave. Perturbations of Minkowski Space and the Nature of Gravitational Waves Laser interferometers provide a way to overcome this difculty. A laser with a typical wave length cm is detected at a beamsplitter, which sends the photons down to evacuated tubes of length L Fig. . At the ends of the cavities are test masses, represented by mirrors which are suspended from pendulums. A wealth of detectors has been built in the last years Resonant Bar Detectors . Nautilus Rome, Italy . Explorer CERN, Switzerland . Auriga Lengaro, Italy . Niobe Perth, Australia . Allegro Louisiana, USA . IGEC International Gravitational Events Collaboration Spherical Detectors . MiniGRAIL Leiden, The Netherlands . Sfera Rome, Italy . Graviton Sao Paulo, Brazil . TIGA Louisiana, USA Laser Interferometers . LIGO Livingston / Hanford, USA Advanced LIGO in . . VIRGO France / Italy . TAMA Japan . Geo Hannover, Germany . AIGO Australia Space Laser Interferometer . LISA JPL, NASA, ESA, to be launched in . . Resonant Detectors For more than years, gravitational waves have eluded conrmed experimental detection. The pioneering proposal to detect gravitational waves was made by Joseph Weber in the early s. He proposed using a large piezoelectric crystal to detect the oscillating strain produced by an oscillating gravitational eld. By Weber had constructed the rst resonantmass gravitational wave antenna. It was a large, roomtemperature aluminum bar that was vibrationally isolated in a vacuum chamber. Quartz strain gauges were used to monitor the bars fundamental mode of vibration. By Weber had achieved strain sensitivities of a few parts in and had constructed several more gravitational wave detectors. He soon announced that he had observed coincidences between them. These results generated great excitement in the eld and other groups began constructing gravitational wave detectors. In the end, however, Webers ndings could not be conrmed by other groups who built similar detectors. By the early s other groups were involved in building advanced gravitational wave detec tors. These groups made a number of signicant improvements over Webers original design. One improvement was to lower the temperature of the bar to liquid helium temperatures Kelvin. Appendix B FIGURE . Webers bar antenna. The second was a better suspension of the bar with increased vibration isolation. A third was the use of a resonant transducer and low noise amplier to observe the motion of the bar. The small resonator not only amplied the displacement but attenuated large amplitude vibrations at low frequencies. Today there are three detectors of this type being operated the LSU ALLEGRO detector, the Rome EXPLORER detector, and the Australian NIOBE detector. The best current antennas, such as the LSU ALLEGRO detector, are sensitive enough to detect a gravitational collapse in our galaxy, if the energy converted to gravitational waves is a few percent of a solar mass. However, the conventional wisdom is that we need to look at least orders of magnitude further in distance, out to the Virgo Cluster, to have an assured event rate of several per year. This requires improving the energy resolution of the detector by orders of magnitude. . Spherical Detectors The LSU group e.g. proposes using a special arrangement of attached resonators is proposed, which are termed Truncated Icosahedral Gravitational Wave Antenna, or TIGA. They have constructed a small truncated icosahedron to test a model for a spherical resonant mass gravita tional wave antenna. This shape was machined from an Al cylindrical bar and is cm in diameter. The rst quadrupole resonances were near Hz. It was suspended from its center of mass. They observed the motion of the prototypes surface using accelerometers attached to its surface in the symmetric truncated icosahedral arrangement. They have tested a rst order direction nding algorithm, which uses xed linear combinations of six accelerometer responses to rst infer the relative amplitudes of the quadrupole modes and from these the location of the impulse. Although a complete investigation of the practicality of a spherical gravitational wave antenna has not been completed, the LSUwork has generated great excitement in the eld of resonant mass detectors. Several groups have begun exploring the possibility of constructing large spherical antennas. These include GRAVITON in Brazil, GRAIL in the Netherlands, ELSA in Italy, and TIGA in the United States. Two collaborations to build such antennas have also been formed the US Gravity Wave Coop and the international OMEGA collaboration. A spherical gravitational wave detector can be equally sensitive to a wave from any direction, Perturbations of Minkowski Space and the Nature of Gravitational Waves and also able to measure its direction and polarization. A special arrangement of attached resonators is proposed, which is termed a Truncated Icosahedral Gravitational Wave Antenna, or TIGA. An analytic solution to the equations of motion is found for this case. We nd that direct deconvolution of the gravitational tensor components can be accomplished with a specied set FIGURE . Quadrupole spherical detector MiniGRAIL in Leiden. of linear combinations of the resonator outputs, which we call the mode channels. This group has developed one simple noise model for this system and calculate the resulting strain noise spectrum. They conclude that the angleaveraged energy sensitivity will be times better than for the typical equivalent bartype antenna with the same noise temperature. The MiniGRAIL detector is a cryogenic cm diameter spherical gravitational wave an tenna made of CuAl alloy with a mass of kg, a resonance frequency of Hz and a bandwidth around Hz, possibly higher Fig. . The quantumlimited strain sensitivity dL/L would be . The antenna will operate at a temperature of mK. Two other sim ilar detectors will also be built, one in Rome and one in Sao Paulo already nanced, which will strongly increase the chances of detection by looking at coincidences. The sources are for instance, nonaxisymmetric instabilities in rotating single and binary neutron stars, small black hole or neutronstar mergers. When a gravitational wave passes by, the spheroidal quadrupole modes of the sphere will be excited. The amplitude will be of the order of meters. . Laser Interferometers The Laser Interferometer GravitationalWave Observatory LIGO is a facility dedicated to the de tection of cosmic gravitational waves and the measurement of these waves for scientic research. It consists of two widely separated installations within the United States, operated in unison as a single observatory. This observatory is available for use by the world scientic community, and is a vital member in a developing global network of gravitational wave observatories. Gravitational waves are ripples in the fabric of spacetime. When they pass through LIGOs Appendix B Lshaped detector they will decrease the distance between the test masses in one arm of the L, while increasing it in the other Fig. . These changes are minute just centimeters, or one hundredmillionth the diameter of a hydrogen atom over the kilometer length of the arm. Such tiny changes can be detected only by isolating the test masses from all other disturbances, such as seismic vibrations of the Earth and gas molecules in the air. The measurement is performed by bouncing highpower laser light beams back and forth between the test masses in each arm, and then interfering the two arms beams with each other. The slight changes in testmass distances throw the two arms laser beams out of phase with each other, thereby disturbing their interference and revealing the form of the passing gravitational wave. Laser interferometers are gigantic L FIGURE . A schematic design of a gravitational wave interferometer. The mirrors represent the freely falling test masses. FIGURE . Gravity wave experiments based on laser interferometric techniques. shaped instruments of kilometer size arms, built at on the Earths surface. Laser beams are bounced back and forth along the two arms, being reected by mirrors at the ends. These mirrors Perturbations of Minkowski Space and the Nature of Gravitational Waves are suspended by wires. Each can move slightly in the direction of the arm, as if it were a free mass. The reected beams are recombined and their interference pattern monitored by a photodetector. A gravitational wave passing through the interferometer causes displacements of the mirrors and a shift in the interference pattern. The amplitude of the displacement will be extremely small in comparison with the arms length. The magnitude of the relative displacement is like the width of a persons hair in comparison with the distance from the Sun to nearby stars. At least two detectors located at widely separated sites are essential for the certain detection of gravitational waves. Regional phenomena such as microearthquakes, acoustic noise, and laser uctuations can cause disturbances that simulate a gravitational wave event. This may happen locally at one site, but such disturbances are unlikely to happen simultaneously at two widely separated sites. FIGURE . LIGO sensitivity achieved in various science runs S, S and S, compared to the design sensitivity solid line. First coincident observations are expected to take place in when the GEO and LIGO detectors come online. The expected initial sensitivity as a function of frequency is plotted in Figure , together with the performance of upgraded detectors. The sensitivity is given in terms of the s noise background within a bandwidth of Hz i.e., linear amplitude spectral density at the output of the interferometer. The output signal of the interferometer is a timedependent signal xt. The power spectrum of this signal gives us a statistics of the portion of the signal power falling within given frequency bins. The power spectrum can be generated from Fourier transforms P f lim T x T .. The power spectrum is the Fourier transform of the autocorrelation function lt xtxt gt P f expit d . . Since the wave amplitude h is dimensionless, one often plots the quantity hf P f as a function of the frequency f. hf has therefore the dimension / Hz. If a signal is truly Appendix B FIGURE . Advanced EUROsensitivity to be achieved in , compared to the design sensitivity of LIGO II. Tracks of various merger sources are also shown. random, we will never observe any long term correlation, i.e. no power concentration in the long frequency region. At low frequencies the performance will be limited by seismic noise, at medium frequen cies by thermally induced motions of the optical components and at high frequencies by photoelectron shot noise. This sensitivity of initial instruments is sufcient to detect a rare su pernova originating in our Galaxy or coalescence of a binary consisting of two stellar mass black holes at a distance of Mpc. To start serious gravitational wave astronomical observations, a careful upgrading of the existing technology has to take place Fig. . This includes kWtype lasers to reduce the shot noise level, possibly purely diffractive optics to avoid problems with absorbed light inside optical components, new materials for mirror substrates and cooling of the main optics to reduce internal thermal noise. The uctuating radiation pressure of the illuminating light requires mirror masses of up to tons. There are three more or less well dened plans for future upgraded detectors advanced LIGO, the Japanese Large Scale Cryogenic Gravitational Wave Telescope LCGT, and EURO a third generation gravitational wave interferometer in Europe. The performance of EURO, the most ambitious future detector, is described at frequencies above a few hundred Hz by the standard quantum limit, where shot noise and radiation pressure noise are balanced at lower frequencies there is the Newtonian gravity gradient noise. The sensitivity of EURO as shown in Figure represents the limits possibly reached when different topologies, like recycling parameters, are chosen for optimum sensitivity at each particular frequency. A network of upgraded interferome ters and enhanced bar detectors will be able to register signals from stellar mass black holes from cosmological distances, quakes in neutron star cores and hence an understanding of the state of matter at very high densities in our Galaxy, supernovae and coalescing neutron star binaries at a General Relativity redshift of z , etc. This is truly an attractive scenario for gravitational wave astronomy. . SpaceBorne Interferometers At low frequencies below a few Hz the performance of groundbased detectors is limited by gravitational gradient noise, as caused, for instance, by motions inside the Earths crust or in the atmosphere. Measurement and subtraction of this disturbance can only work to a certain extent. To enter this very interesting frequency range it is necessary to go into space, as it is planned with the Laser Interferometric Space Antenna LISA. LISA is a cornerstone mission of ESA, and included in NASAs Structure and Evolution of the Universe Roadmap. The scheduled launch is around . The technology will be tested in the precursor mission LISA Pathnder in . In LISA, three spacecrafts are arranged in an equilateral triangle of side km, trailing the Earth by degrees in a heliocentric orbit cf. Figure . Each of the three crafts follows its own elliptic orbit slightly out of the ecliptic. Over the course of a year, the triangle seems to rotate about its centreofmass, maintaining the relative distances constant to within a percent, without any active corrections. Under the inuence of gravitational waves the relative distances between the craft change. These are, therefore, continuously registered with laser interferometry. To avoid the noise caused by the uctuating solarwind and radiation pressure, the distance is measured between free ying test masses, each shielded by its surrounding spacecraft by use of the socalled dragfree technique. The Laser Interferometer Space Antenna LISA is a mission to measure gravitational waves from various black hole sources, compact binary stars, and a stochastic background of gravitational waves from the very early Universe. Each spacecraft carries two freeying masses with associated sensors, two identical tele scopes and two optical benches to measure the relative displacement of the spacecraft. The proof masses will be freeying. The noisereduction system will detect their movement relative to the spacecraft and actuate the Field Effect Electric Propulsion thrusters at micronewton levels to make sure that the spacecraft follows the masses. The lasers will operate as a Michelson interferometer to detect and measure relative movements of each spacecraft generated by the gravitational waves. The three spacecraft will y in a quasiequilateral triangle formation in a heliocentric orbit at . million km Earths distance from the Sun, located degrees million km behind the Earth in its orbit. After months commissioning and up to months for transfer to operational orbit, LISA is estimated to stay in orbit for two years. The sensitive frequency range of LISA is between . mHz and Hz Figure . For LISA there are guaranteed sources Galactic compact binaries of period in the relevant range will be observed with a signalto noise ratio of up to . But much more fascinating are the signals to be expected from a variety of less wellknown origin namely, events involving supermassive black holes that are believed to exist at the centre of every galaxy in the Universe. It is not clear how such black holes formed. It is possible that a midsized black hole forms simultaneously with the formation of the galaxy and then grows in size by accreting matter in the form of ordinary stars and black holes found in their vicinity. If this is so, then a small black hole or a neutron star falling into a supermassive black hole emits gravitational waves. As the body slowly spirals into the hole, both its orbit and spin are expected to precess, more violently as the body approaches the black hole, and it samples the geometry of spacetime as it tumbles round. The dynamics of the body, as well as the nature of the spacetime in which the body whirls around will be encoded in the waves we can potentially observe with LISA. In the early history of their formation galaxies are believed to have interacted strongly with one another leading to their mutual collision and merger. Such mergers should also involve the coalescence of the associated black holes. The waves emitted in the process will be visible at a very high signaltonoise ratio, wherever in the Universe the source might be. Thus, LISA should give us a complete census of the supermassive black hole population in the Universe. Finally, and most importantly, it is hoped that LISA, or one of its successors, will shed light on the conditions that prevailed, when the Universe was born. Nothing could be more exciting. Appendix B FIGURE . Top A schematic diagram of the LISA spacecraft in formation as they orbit around the Sun. The spacecraft are separated from each other by million km and trail behind the Earth at a distance of million km equivalent to degrees. Bottom The sensitivity of the LISA interferometer. At frequencies f below f lt mHz, the double white dwarf population in the Galactic disk forms an unresolved background for LISA. Above this limit, some few thousand double white dwarfs and a few tens of neutron star binaries will be resolved. A few of them are indicated by stars. The most prominent sources will be binaries consisting of supermassive Black Holes at cosmological distances. List of Figures Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aether drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MichelsonMorley experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . Frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time dilation in Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . Speed of light in Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . Light cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RayleighTaylor instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EotWash experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WEP Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Einsteins Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tangent plane of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light cone structure near a Black Hole . . . . . . . . . . . . . . . . . . . . . . . Perihelion advance in a body system . . . . . . . . . . . . . . . . . . . . . . . Gravitational lense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photon trajectories around a Black Hole . . . . . . . . . . . . . . . . . . . . . . Robertson parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plus mode of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . Plus mode of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . Circular mode of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . Weber antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MiniGRAIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laser interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity wave experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGO sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EURO sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISA in orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures List of Tables Number of linearly independent pforms for D and D . . . . . . . . . .