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xxxxxxxx 30 Scientific American, November 2010 Photograph/Illustration by Artist Name T. Padmanabhan is a theoretical physicist at IUCAA, Pune. His work on gravity has won international acclaim, including the First Award from the Gravity Research Foundation, USA. He was a Sackler Distinguished Astronomer at Cambridge and is currently the President of the Cosmology Commission of the IAU. Grappling With Gravity p h ys i c s The underlying description of gravity may lie in a microstructure made up of “atoms of spacetime” By Thanu Padmanabhan T he first hint that gravity is an odd-man-out among the fundamental forces came when Einstein realized that the ideas of Newtonian gravity need to be modified to make them compatible with the special theory of relativity. The special theory of relativity demands that no influence can propagate with a speed greater than that of light. So, if the Sun disappears at some instant, its gravitational influence on Earth can disappear only after a delay of at least eight minutes because light takes about eight minutes to reach the Earth from the Sun. On the other hand, according to Newton’s theory of gravity, the gravitational force will disappear instantaneously. The attempt to correct such inconsistencies be- tween Newtonian gravity and special relativity led Einstein to the general theory of relativity which interprets gravity as a manifestation of the curvature of spacetime. In this approach, verified by numerous experiments, gravity is described in terms of what is called the spacetime metric, which is a mathematical object that essentially allows one to compute all the geometrical properties of the spacetime, as well as the influence of gravity on other physical systems. Einstein also introduced a set of equations which allow us to determine this metric from the distribution of matter and energy which acts as the source for gravity. Every massive body curves the spacetime around it and other bodies move in this curved geometry along ‘straight lines’ which are defined to be the shortest paths between two events. The no- in brief The outstanding problem in physics is to combine Einstein’s description of gravity with quantum theory. Such a theory is needed for the mathematical consistency of theoretical physics. Gravity is today understood as the manifestation of the curvature of spacetime. So, it is not really a force, unlike electricity or magnetism. A conceptual rather than technical breakthrough may be needed. Instead of the conventional view that gravity is a fundamental interaction, it could be that a microstructure—the “atoms of space time”—give rise to it. The idea chimes with thermodynamics, where if you can heat something, it has a microstructure. Recent work suggests this may be more than an analogy—it could be a physical reality. Photo illustration by Rajat Baran w w w. s c i a m . c o . i n S C I E N T I F I C A M E R I C A N I n d i a 31 T h e D av i e s - U n r u h E f f e c t Turning Up the Heat Paul Davies and Bill Unruh independently discovered, in the mid-seventies, a curious result which arises when you combine the principles of general relativity and quantum theory. They showed that, if you run with an acceleration carrying a thermometer, it will show a reading different from the one when you are at rest. This result plays a crucial role in the emergent perspective of gravity. One usually thinks of the temperature of a system as an absolute entity, independent of who is measuring it. Further, the temperature of the vacuum, one would have thought, is zero for all observers. Not so. The vacuum will appear as a zero temperature state for an observer at rest; but one who is accelerating through the vacuum will attribute to it a nonzero temperature. Why does this happen? Vacuum, in the language of quantum theory, is not just empty space. It contains a bristling crowd of virtual particles characterizing the vacuum fluctuations of the quantum field. For example, the vacuum state harbors virtual photons, which are the quanta of electromagnetic radiation. These are directly responsible for some of the subtlest effects in quantum electrodynamics measured in the laboratory. But the pattern of vacuum fluctuations is such that a detector for photons will not be excited by these fluctuations as long as the detector is at rest or when it is moving with a uniform velocity. But, as Unruh found, it is a different story if the detector is accelerating. The vacuum exhibits a different pattern of fluctuations as seen by the accelerated detector and the detector will now get excited. In fact, the work of Davies and Unruh shows that an accelerated detector will see a spectrum of photons which is identical to what an unaccelerated detector will detect when immersed in a hot oven at some temperature. Such an oven—and in fact, every hot object at some temperature—emits thermal radiation of a specific pattern which depends only on the temperature. The pattern of vacuum fluctuations as seen by the accelerated observer mimics this radiation. The temperature perceived by the detector is proportional to the acceleration. Unfortunately, it is very tiny; an acceleration of 1g produces a temperature of about 10−65 Kelvin, which is immeasurably small. But, in principle, a person accelerating through the vacuum carrying a glass of water will find that the water heats up. tion of a straight line on a curved surface (say, on the surface of sphere) is quite different from our intuitive notion of a straight line based on the ideas of flat surfaces. Such a motion appears like the one produced by a force between the two bodies which we call the gravitational force. Thus, as we understand it today, gravity is the manifestation of the curvature of spacetime. In this sense, gravity clearly stands apart and has a geometric interpretation unlike, say, electricity or magnetism. Einstein’s ideas work fine as long as you are in the domain of classical physics. But the exact laws of physics appear to be quantum mechanical rather than classical. At microscopic scales when we are dealing with molecules, atoms, nuclei, elementary particles or when studying physics at sufficiently high energies—like in the earliest moments in the evolution of the universe—we need to use the laws of quantum theory. When one incorporates the principles of quantum theory into the known classical laws of, say, electricity and magnetism, we end up getting a more fundamental description known as quantum electrodynamics. This theory— verified by experiments to enormous accuracy—is a successful prototype showing how to combine the principles of quantum theory with a known classical law for force. Similarly, we need to combine Einstein’s description of gravity, given in terms of a curved spacetime, with the principles of quantum theory, thereby obtaining a quantum theory of gravity. One can, in fact, show that a situation in which gravity is described by classical laws while the matter is described by quantum laws will be mathematically inconsistent. Hence quantum gravity is needed for the mathematical consistency of theoretical physics. The trouble is that a significant amount of effort during the past half century has not resulted in any conclusive model for quantum gravity. A NEW PARADIGM recent research suggests an intriguing possibility. It appears that we need yet another fundamental revision in our point of view regarding gravity: Gravity could just be an emergent phe- 32 S C I E N T I F I C A M E R I C A N I n d i a —T.P.. nomenon. What is an emergent phenomenon? Simple examples of emergent phenomena in physics include elasticity or gas dynamics. The laws of elasticity or gas dynamics, expressed as mathematical equations, were known for centuries. They are quite adequate for understanding, say, the elastic vibrations of a steel wire used in a suspension bridge or the flow of air past an airplane wing—without ever referring to the individual atoms which make up the steel wire or air. We say that gas dynamics or elasticity is an emergent phenomenon while the really fundamental description is in terms of the atoms making up the matter. This has implications when you move from the classical to the quantum domain. We cannot obtain a fundamental quantum description of matter by, say, combining the laws of elasticity with the principles of quantum theory; instead, we first describe the elastic solid as made of billions of atoms and then provide a quantum mechanical description of these atoms. If gravity is an emergent phenomenon, then Einstein’s equations describing the dynamics of the spacetime have a status similar to the laws of elasticity or gas dynamics. By analogy, we should not try to quantize gravity by combining Einstein’s theory directly with the principles of quantum theory. Instead, we must identify the “atoms of spacetime” and provide a quantum description of this structure. This idea that gravity is an emergent phenomenon has a long history. It was first introduced by the Russian scientist, A. Sakharov, in 1968. Another intriguing connection, suggesting a similarity between the surface properties of black holes and fluid mechanics was investigated by T. Damour, Kip Thorne, W. Price and their collaborators in the eighties. An attempt to obtain Einstein’s equations from a thermodynamic perspective was made by T. Jacobson in 1995. Other approaches, very similar in spirit, were developed by G. Volovik and Bei-Lok Hu. My collaborators and I started on a concrete programme to explore this idea in 2002. More recently, E. Verlinde has tried to reas- J a n u a r y 2 0 11 semble many of these ideas from the perspective of string theory. This article will mostly concentrate on the efforts by my collaborators and I. If the emergent picture is correct, most of the previous attempts to quantize Einstein’s equations were in the wrong direction and similar to someone attempting to discover atomic physics by quantizing the laws of elasticity. We will need a major paradigm shift to make progress. EMERGENT PHENOMENA let us now translate these ideas to gravity. A natural length scale in quantum gravity (called the Planck length, see box on right) can be built from three fundamental constants in physics: Newton’s gravitational constant, Planck’s constant relevant in quantum theory and the speed of light which plays a crucial role in relativity. The description of spacetime in terms of its metric and other geometrical properties can be thought of as an emergent phenomenon valid at scales large compared to the Planck length. Variables like metric, curvature, etc. used to describe spacetime in Einstein’s theory are analogous to density, velocity, etc. in fluid mechanics and will have no significance in the microscopic description. In that case, quantization of the metric itself is not of any use in unraveling the microscopic structure of spacetime, any more than quantizing the density and velocity of a fluid will help us to understand molecular dynamics. Just as the proper description of molecules of a fluid requires introduction of new degrees of freedom and a theoretical formalism based on quantum mechanics, the microscopic description of spacetime will require the introduction of new degrees of freedom (“atoms of spacetime”) and a different theoretical formalism. Since we do not know what these are, one needs to proceed indirectly, in the same way as nineteenth century physicists, especially the German scientist Ludwig Boltzmann (1844-1906), did before the atomic nature of matter was directly observed. Boltzmann realized that the temperature and heat content of a fluid arises due to the random motion of discrete microscopic structures which must exist in the fluid. These new degrees of freedom, which we now know are related to the actual molecules—but were hypothetical objects at the time of Boltzmann—make the fluid capable of storing energy internally and exchanging it with the surroundings. If the fluid is treated as a continuum all the way down to the smallest scales, without any constituent atoms, then it is not possible for it to exhibit thermal phenomena. Given the fact that fluids can be heated up and can have a temperature, Boltzmann’s genius was in inferring the existence of the atomic nature of matter, before direct observations revealed these underlying discrete degrees of freedom. So Boltzmann said, “If you can heat it, it has microstructure.” Amazingly enough, we can heat up spacetime and thus apply Boltzmann’s logic to it. HOT SPACETIMES one of the intriguing consequences of Einstein’s theory is the existence of observers who cannot access information from some region of space and time. Just as standing at a seashore you cannot see beyond the horizon, a spacetime horizon prevents these observers from getting information from the other side. The most well-known example is in the case of black holes in which the sur- w w w. s c i a m . c o . i n planck length Getting A Measure of Gravity A natural length scale in quantum gravity, called the Planck length, can be built from three fundamental constants in physics—Newton’s gravitational constant, Planck’s constant relevant in quantum theory and the speed of light which plays a crucial role in relativity. This length has a very tiny value of 10-33 cm. (This notation stands for the number obtained by dividing 1 by 10 repeatedly 33 times; that is, 10−1 = 1/10 = 0.1; 10−2 = 1/(10 × 10) = 0.01 etc. In decimal notation 10−33 will have 32 zeros followed by 1 after the decimal point. Similarly, the notation 108, for example, stands for the number obtained by multiplying 10 eight times.) To see how tiny the Planck length is, recall that the size of the visible universe is about 1028 cm which is about 28 factors of ten away from the everyday scale of few centimeters. So, in the macroscopic sector, these 28 powers of ten contain all of the cosmic phenomena. If we now move in the microscopic direction we need to travel farther—to 33 factors of ten—before we reach the Planck length at 10−33 cm. Thus Planck length is farther away in the microscopic direction than the edge of the universe is in the macroscopic direction. It is this tiny scale which plays the role similar to intermolecular separation in gas dynamics. In the case of gases, the intermolecular separation is only about 6 factors of ten smaller than everyday scales but in gravity we have to go much, much deeper. face of the black hole prevents the signals from the inside to reach the outside observer. However, what is not often emphasized in popular literature, is that horizons are ubiquitous in Einstein’s description. The simplest context in which a horizon comes into play is in the case of an observer who is moving along a given direction with an acceleration. It turns out that such an observer has a horizon and cannot receive information from the region beyond it, in a manner very similar to the situation in the case of a black hole. Research in the seventies by J. Bekenstein, S. Hawking, P. Davies and W. Unruh led to a tantalizing discovery: observers with horizons perceive the spacetimes to be hot, endowed with thermodynamical properties. For example, it can be shown that an observer accelerating through empty space—who sees a horizon —will feel as though he is sitting inside a microwave oven set at a particular temperature proportional to his acceleration. (See Turning Up the Heat). This effect is real in the sense that such an accelerating observer can use this temperature he notices to heat up, say, a glass of water. In addition to temperature one can also attribute another thermodynamic variable called entropy to the horizons. Though it may be less familiar to some of you than the temperature, it is probably the most crucial entity in thermodynamics. Roughly speaking, the entropy measures our lack of information about the state of the microscopic degrees of the system. When we study the properties of a gas, say, we cannot measure the positions and speeds of each of the molecules in the gas. Instead, we describe the state of the gas in terms of some macroscopic observables like its volume, temperature—which essentially is a measure of the average kinetic energy of the molecules—and its pressure. There are a large number of possible distributions of molecules inside the gas with different speeds which will all result in the same average S C I E N T I F I C A M E R I C A N I n d i a 33 properties like pressure, volume and temperature. An analogy might be useful here: Suppose you are told that on tossing a coin 1,000 times, it turned up heads 480 times. This kind of macroscopic information is consistent with several different microscopic distributions of heads and tails in each individual toss, just keeping the total number of heads at 480. In such a case, we say that there exist several possible microstates, consistent with what we know about the macrostate. This lack of precise information about the microstates (in this case, the precise result of each toss; in the case of a gas, the precise position and speed of each molecule) is what is encoded in the entropy. The more microstates are possible with some macroscopic specification, the larger is the entropy. This connection between lack of information about a system and its entropy turns out to be very general. Because of this connection, one can also attribute an entropy to the horizons. In the case of gas, the lack of information is due to our inability to keep track of billions and billions of molecules and our resorting to an average description. In the case of observers with a horizon, it is the blocking of information by the horizon which leads to the entropy. In a way, the entropy associated with a horizon turns out to be more important in the theoretical developments compared to the temperature of the horizons. The description of gravity based on the equations proposed by Einstein is very natural when space has three dimensions. Physicists have, in recent years, toyed with the idea that the space could actually have more than three dimensions with the extra dimensions being very tiny and compact and hence not detectable in low energy experiments. The natural generalization of Einstein’s theory to higher dimensions leads to a class of models known as Lanczos-Lovelock models of gravity. All these models share the fact that observers who perceive a horizon will attribute thermodynamical variables to the horizon. It turns out that the temperature attributed by an observer to a particular horizon does not depend on which theory one is considering; i.e., whether it is Einstein’s theory or Lanczos-Lovelock theory. But the entropy associated with the horizon does contain information about the underlying theory. The entropy happens to be just proportional to the area of the horizon in Einstein’s theory but is given by a much more complicated relation—first pointed out by Bob Wald in 1993—in other theories. This is what makes entropy more important than temperature. Roughly speaking, this is due to the fact that entropy actually contains information about the microstates of the system. These facts about the horizon were known for several decades but most people thought of them as rather tantalizing curiosities. But from the perspective of emergent gravity, they acquire a much deeper significance. To see this, recall Boltzmann’s dictum: “If you can heat it, it has microstructure.” It is obvious that spacetimes can be hot from the perspective of observers who perceive a horizon. Clearly, there must be microscopic degrees of freedom in the spacetime which are responsible for this thermal behaviour. The The connection between gravity and thermodynamics which started out as an analogy in the seventies has now become a physical reality 34 S C I E N T I F I C A M E R I C A N I n d i a connection between thermodynamics and gravity is not a mathematical curiosity but a physical reality that needs to be taken into account properly. I started exploring this angle in 2002 and this program has led to interesting insights. A THERMODYNAMIC PARADIGM FOR GRAVITY The starting point was to explore the connection between the dynamical equations which describe gravity and the thermodynamics of horizons. In a normal thermodynamical system, say, a gas in a box, changes in the physical parameters like the volume will lead to changes in heat content and internal energy. The heat content can be related to the entropy and one can write down a relation between the changes in entropy, volume and internal energy of the gas. This is usually called the first law of thermodynamics. I discovered in 2002 that exactly the same relation holds for horizons in Einstein’s theory. Moving the horizon by a small amount turns out to be analogous to changing the volume of a gas and Einstein’s equations which describe the spacetimes— with horizons in slightly different positions— reduce exactly to the first law of thermodynamics (See The Thermodynamics Behind Einstein’s Equations). Since 2002, my group as well as several other groups, have investigated the situation in a wide class of models more general than Einstein’s theory. In all the cases, it has been found that the gravitational field equations of the theory become identical to a thermodynamic law. This suggests that the concept of atoms of spacetime is quite real and has direct physical content. Around 2004, I realized that, if this idea is correct, one should be able to directly determine the density of these microscopic degrees of freedom from macroscopic dynamics itself. Once again, this is closely related to another key fact about the thermodynamics of gases which had intrigued people before the atomic nature of matter was understood. It turns out that, at a given temperature, each microscopic degree of freedom can store an amount of energy proportional to the temperature. This principle is called equipartition of energy since it says that the total available energy is divided equally among all the available degrees of freedom. (There are exceptions to this rule but they are not relevant to our discussion.) More importantly, the equipartition law gives a direct link between microscopic and macroscopic physics of matter. If you know the total thermal energy of a body, and its temperature, you can immediately calculate the number of microscopic degrees of freedom of the body. If you take one mole of gas and keep it at what chemists call standard temperature and pressure, it will contain a definite number of degrees of freedom. This number is called Avogadro’s number, named after the nineteenth century Italian scientist. The importance of these ideas lies in the fact that they were developed before one had direct evidence about the atomic structure of matter. Even without any direct observational evidence for atoms or molecules, nineteenth century physicists could count the number of molecules in some amount of gas without knowing quite precisely what they were counting. We are in a similar situation as regards atoms of spacetime. We now know that a spacetime, just like a material body, can be hot. Following Boltzmann, it seems natural to attribute this heat content to microscopic degrees of freedom. But does the law of equipartition hold? If so, can we use it to estimate the number of microscopic degrees of J a n u a r y 2 0 11 degrees of freedom The Thermodynamics Behind Einstein’s Equations There is a remarkable connection between the first law of thermodynamics and the laws describing gravity in a wide class of theories. The connection can be explained by an analogy. Suppose, some amount of gas is confined to a box, the volume of which can be changed by moving the piston. If one lets the piston move outward due to the pressure of the gas, one can extract some mechanical work from the gas as well as change the internal energy of the gas. The gas can also exchange heat with the surPiston in a roundings that can be expressed in terms displaced locationof the entropy change Piston of the gas. The first law of thermodynamics relates the changes in these three quantities: entropy, internal energy and mechanical work. Let us move on from a box of gas to a spacetime with a horizon. The location of the horizon plays a role analogous to the position of the piston. While you cannot push around a horizon, you can certainly Gas made of molecules consider two different spacetimes with the Piston in a displaced location Piston Gas made of molecules Horizon in a displaced location HORIZON Black hole freedom present in a given region of spacetime? Remarkably enough, both of these can be achieved. I did this calculation for Einstein’s theory in 2004 and extended it to a more general class of theories last year. These results show that the laws governing gravity in a wide class of models imply the equipartition of energy among the microscopic degrees of freedom. What is more, this analysis allows one to compute the number density of the atoms of spacetime. When you carry out the analysis you also discover one crucial difference between the gaseous system and gravity. In the former, the relevant degrees of freedom reside in the bulk volume of the gas. In the case of gravity, the relevant degrees of freedom corresponding to a region of space reside on the boundary of that space and not in the bulk. (Physicists call such systems holographic.) In Einstein’s theory, the surface density of these degrees of freedom is a constant and hence the total number of degrees of freedom increases in proportion with the surface area. In more general theories, the degrees of freedom still reside on the surface but their density is not a constant. In fact, given the mathematical expression for entropy density, one can reconstruct the full theory. It is now obvious that the proper description of gravity should start from the entropy density of spacetime—or equivalently, the density of atoms of spacetime, which is the same thing. This is again in perfect analogy with the thermodynamic description of matter. One of the principles of thermodynamics tells you that, in equilibrium, the entropy must be maximum. If the entropy of a system, w w w. s c i a m . c o . i n horizons at two slightly different locations. This disHorizon in a displacedagain location placement of the horizon causes changes in the properties of the spacetime which—in turn—is governed by the equation describing the gravity. Remarkably enough, one can prove that this equation reduces to a form identical to the equation in HORIZO N the case of gas with a piston. This result was first obtained by me in 2002 in the simplest context of horizons in Einstein’s theory. Further work by different groups has now established that this result is true for a wide class of theories of Black hole gravity much more general than Einstein’s theory. In the general context, the temperature associated with the horizon is independent of the theory one is studying but the entropy depends crucially on the theory. Remarkably enough, the thermodynamic description picks out the correct expression for entropy in each theory thereby showing that the entropy density associated with a horizon contains the necessary information to reconstruct the underlying theory. This, in essence, reduces the description of gravity to a thermodynamic one based on entropy density contributed by the microscopic degrees of freedom. —T.P. say a gas, is given in terms of other relevant variables then one can figure out the equilibrium properties of the system by just maximizing it. Aseem Paranjape and I found in 2007 that exactly the same principle holds for gravity. We could write down a mathematical expression for the entropy content of a region of a spacetime and by maximizing it, obtain the equations governing the dynamics of the spacetime in a wide class of models of gravity including Einstein’s theory. A spacetime obeys Einstein’s equations because the atoms of spacetime maximize the entropy—just as a gas obeys gas laws because the atoms of gas maximize its entropy. The connection between gravity and thermodynamics which started out as an analogy in the seventies has now become a physical reality. The situation has an exact parallel with the days of Boltzmann when we had no direct evidence for the atomic nature of matter but thermodynamics held the key. We have no direct evidence for the atoms of spacetime but, thanks to the thermodynamic description, we can say a lot about them. n more to explore Thermodynamical Aspects of Gravity: New insights, T. Padmanabhan Reports in Progress of Physics, 73, 046901 (2010) [available online at arxiv.org/pdf/arXiv:0911.5004] Black Holes and Time Warps: Einstein’s Outrageous Legacy, Kip S. Thorne , W. W. Norton, 1995. A Journey into Gravity and Spacetime, J.A. Wheeler, Scientific American Library, W. H. Freeman, 1999. S C I E N T I F I C A M E R I C A N I n d i a 35