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mathematical roots of GeometroDynamics
1. http://en.wikipedia.org/wiki/Geometrodynamics
GeometroDynamics
from Wikipedia, the free encyclopedia
In theoretical physics, GeometroDynamics generally denotes a program of reformulation and
unification which was enthusiastically promoted by John Archibald Wheeler in the 1960s.
Einstein's GeometroDynamics
The term "GeometroDynamics" is rather loosely used as a synonym for General Relativity. More
properly, some authors use the phrase Einstein's GeometroDynamics to denote the initial value
formulation of General Relativity introduced by Arnowitt, Deser, and Misner (ADM) around 1960. In
this reformulation, space-times are sliced up into spatial hyperslices in a rather arbitrary fashion and the
vacuum Einstein field equation is reformulated as an evolution equation describing how -- given the
geometry of an initial hyperslice (i.e., the "initial value") -- the geometry evolves over "time".
This requires giving constraint equations which must be satisfied by the original hyperslice. It also
involves some "choice of gauge". Specifically, choices about how the coordinate system used to
describe the hyperslice geometry evolves.
Wheeler's GeometroDynamics
As described by Wheeler in the early 1960s, GeometroDynamics attempts to realize 3 catchy
slogans:
● mass without mass
● charge without charge
● field without field.
These slogans (due to Wheeler himself) -- which are discussed in more detail below -- capture the
general hope that GeometroDynamics would "do more with less".
Another way of summarizing the goals of Wheeler's original formulation of GeometroDynamics is
that Wheeler wished to lay the proper conceptual and mathematical foundation for Quantum Gravity and
also to unify Gravitation with ElectroMagnetism. (note: the strong and weak nuclear interactions were
not yet sufficiently well understood in 1960 to be included in the program). Wheeler's vision for
accomplishing these goals can be summarized as a program of reducing Physics to Geometry in an
1
even more fundamental way than had been accomplished by the ADM reformulation of General
Relativity.
Wheeler introduced the notion of geons -- gravitational wave packets confined to a compact region
of space-time and held together by the gravitational attraction of the (gravititational) field energy of the
wave itself. Wheeler was intrigued by the possibility that geons could affect test particles much like a
massive object. Hence the slogan "mass without mass".
Wheeler was also much intrigued by the fact that the (nonspinning) point-mass solution of General
Relativity -- the Schwarzschild vacuum -- has the nature of a wormhole. Similarly, in the case of a
charged particle, the geometry of the Reissner-Nordström electrovacuum solution suggests that the
symmetry between electric (which "end" in charges) and magnetic field lines (which never end) could be
restored if the electric field lines do not actually end but only go through a wormhole to some distant
location or even another branch of the Universe.
George Rainich had shown decades earlier that one can obtain the electromagnetic field tensor from
the electromagnetic contribution to the stress-energy tensor, which in General Relativity is directly
coupled to space-time curvature. Wheeler and Misner developed this into the so-called already unified
field theory which partially "unifies" Gravitation and ElectroMagnetism. This is very roughly the idea
behind the slogan "charge without charge".
Finally, in the ADM reformulation of General Relativity, Wheeler argued that the full Einstein field
equation can be recovered once the momentum constraint can be derived and suggested that this might
follow from geometrical considerations alone, making General Relativity something like a logical
necessity. Specifically, curvature (that is, the gravitational field as treated in General Relativity) might
arise as a kind of "averaging" over very complicated topological phenomena at very small scales -- the
so-called spacetime foam -- which would realize geometrical intuition suggested by Quantum Gravity.
This is roughly the idea behind the slogan "field without field".
These ideas were very imaginative and they captured the imagination of many physicists, even
though Wheeler himself quickly dashed some of the early hopes for his program. In particular, spin 1/2
fermions proved difficult to handle.
GeometroDynamics also attracted attention from philosophers intrigued by the suggestion that
GeometroDynamics might eventually realize mathematically some of the ideas of Descartes and Spinoza
concerning the nature of space.
Modern Notions of GeometroDynamics
More recently, Christopher Isham, Jeremy Butterfield, and their students have continued to develop
Quantum GeometroDynamics to take account of recent work toward a quantum theory of Gravity and
further developments in the very extensive mathematical theory of initial value formulations of General
Relativity. Some of Wheeler's original goals remain important for this work -- particularly the hope of
laying a solid foundation for Quantum Gravity. The philosophical program also continues to motivate
several prominent contributors.
References
● Anderson, E.. "Geometrodynamics: Spacetime or Space?". Retrieved on September 30, 2004.
This Ph.D. thesis offers a readable account of the long development of the notion of
"geometrodynamics".
2
● Butterfield, Jeremy (1999). The Arguments of Time. Oxford: Oxford University Press. ISBN 019-726207-4. This book focuses on the philosophical motivations and implications of the
modern geometrodynamics program.
● Prastaro, Agostino (1985). Geometrodynamics: Proceedings, 1985. Philadelphia: World Scientific.
ISBN 9971-978-63-6.
● Misner, Charles W; Thorne, Kip S., & Wheeler, John Archibald (1973). Gravitation. San
Francisco: W.H. Freeman. ISBN 0716703440. See chapter 43 for "superspace" and chapter
44 for "spacetime foam".
● Wheeler, John Archibald (1963). Geometrodynamics. New York: Academic Press. LLCN
62013645.
● Misner, C.; and Wheeler, J. A. (1957). "Classical Physics as Geometry". Ann. Phys. 2 (6): 525.
doi:10.1016/0003-4916(57)90049-0. online version (subscription required)
● J. Wheeler (1961). "Geometrodynamics and the Problem of Motion". Rev. Mod. Physics 44 (1):
63. doi:10.1103/RevModPhys.33.63. online version (subscription required)
● J. Wheeler (1957). "On the nature of Quantum GeometroDynamics". Ann. Phys. 2 (6): 604–614.
doi:10.1016/0003-4916(57)90050-7. online version (subscription required)
http://matpitka.blogspot.com/2012/02/progress-in-number-theoretic-vision.html#comments
(reprinted from Dr. Matti Pitkanen's physics blog)
At 10:00 PM, [email protected] said...
Wheeler's Geometrodynamics cannot be identified with General Relativity. The infinite-D space
of 3-metrics is the basic object and the dream is to quantize gravitation in this geometric framework
generalizing Einstein's geometrization program. This is one of the deep ideas of Wheeler.
At least 2 basic problems plague this approach.
The first problem is that one loses time. Space-times are what we want. How to make 3-D of
GeometroDynamics to 4-D of General Relativity. Semi-classical approximation to postulated path
integral over 4-geometries is the obvious approach but has formidable mathematical difficulties.
[The mathematical non-existence of the path integral is a quite general problem. QFT colleagues
have done their best to forget this. Pretend that there is no problem when problem is too difficult.
This has been the strategy of modern mainstream theoretical physics and guaranteed that nothing
new has emerged for 4 decades;-).]
The second problem is that one does not obtain fermions. Fermions are the problem of Classical
General Relativity too. Space-time need not allow spin structure at all so that one cannot talk about
spinor fields. This problem is much more general and also plagues string models and M-theory,
would have been excellent hint that space-times must be replaced with 4-surfaces and spinor
structure with induced spinor structure, has been put under the rug.
3-metrics are replaced with 3-surfaces in TGD framework. This solves both basic problems of
GeometroDynamics. WCW assigns to 3-surfaces space-time surfaces as analogs of Bohr orbits and
Classical Physics becomes part of Quantum Physics. The problems with fermions and spin are
3
circumvented via induced spinor structure. One fruit of labor is the geometrization of fermionic
statistics in terms of spinors of WCW.
Observer participancy is another very deep idea of Wheeler. The delayed choice experiment in
which one changes Geometric-Past is inspired by this idea.
The skeptic would react by saying that before we can talk about observer participancy, we must
have a physical definition for observer. We do not. The optimistic skeptic might try to imagine
what this definition might be on basis of existing and maybe some new ideas. The notion of self is
the TGD-inspired attempt to meet the challenge.
Evolution as a sequence of quantum jumps recreating the Universe repeatedly would
realize observer-participancy in the TGD Universe.
Most of us speak about cognitive, social, and cultural developments as something self evident.
But the theoretical physicist does not use these words. Very many words of Biology and
Neuroscience are absent from his vocabulary. Behavior, function, goal, homeostasis, punishments
and rewards, evolution -- everything relating to intentionality, goal directedness, values is absent.
The brutal reason is that the existing mathematical tools do not allow even attempt to define these
notions. New mathematics and new concepts are needed.
There is of course an easy way out. Self-deception which is as easy as cheating the innocent
laymen. Just say that "all that is" is nothing but a dance of quarks and consciousness is illusion and
life is nothing but complexity!
This is why I am talking about Physics as generalized number theory, p-adic physics, infinite
primes, hierarchy of Planck constants, etc. I am an observer wanting to participate the expansion of
our understanding about the white regions of the map;-).
http://matpitka.blogspot.com/2013/07/about-naturality-fine-tuning-and-recent.html#comments
topic: "About naturality, fine tuning, and the recent ego catastrophe in Theoretical Physics"
1. At 5:55 AM, Ulla said...
On Julian Barbours wikipedia page are 3 links:
● Anderson, Edward (2004) "Geometrodynamics: Spacetime or space?" Ph.D. thesis,
University of London. http://arxiv.org/abs/gr-qc/0409123
● (2007) "On the recovery of Geometrodynamics from two different sets of first principles,"
Stud. Hist. Philos. Mod. Phys. 38: 15. http://arxiv.org/abs/gr-qc/0511070
● Baierlein, R. F., D. H. Sharp, and John A. Wheeler (1962) "Three dimensional geometry as
the carrier of information about time," Phys. Rev. 126: 1864-1865. (Tthis I cannot find
for free on net).
2. At 9:24 PM, Matti Pitkanen said...
The basic problem of Wheeler's GeometroDynamics is that basic objects are 3-dimensional. One
should be able to get out 4-D space-time from this approach. Barboux is quite right in saying that
Time is lost in General Relativity based on Wheeler's GeometroDynamics.
Wheeler's
GeometroDynamics also has profound mathematical difficulties. For instance, it is difficult to get
fermions out of it and the geometry of super-space is poorly defined.
4
But why should Nature obey Wheeler's GeometroDynamics? Wheeler's notion of super space
must be modified so that it exists mathematically and is consistent with the basic space-time
symmetries (Poincare group) and geometric description of gravitation.
TGD's "World of Classical Worlds" allows us to achieve this elegantly. The news is that after
more than 2 decades this is still news!
http://matpitka.blogspot.com/2012/02/one-more-good-reason-for-p-adic.html#comments
(reprinted from Dr. Matti Pitkanen's physics blog)
At 3:02 PM, hamed said...
Dear Matti,
I want to know evolution of concepts from Classical physics to the GeometroDynamics of John
Wheeler (Superspace [WSP] ) and then to TGD (World of Classical Worlds [WCW] ). So I studied
correspondences between particle dynamic and GeometroDynamics in the book of gravitation by
Wheeler:
(1) Dynamic entity of Particle dynamics is particle and of GeometroDynamics is space (3dimensional space).
(2) Descriptors of momentary configuration of first is x,t(event) and of second is 3-Geometry.
(3) Classical history of first is x=x(t) and second is 4-Geometry.
(4) At first, every point on the world line gives a momentary configuration of particle. But at
second, every space like slice through 4-Geometry gives a momentary configuration of
space.
Where does WCW similar with superspace on subjects that listed above and where does is
different? Does ADM formalism any position in WCW like superspace?
Maybe if Wheeler had been equipped his superspace with Kähler geometry (and he had your
extraordinary ability of arguments;-)), he could like TGD assigns to each 3-surface a space-time. So
space-time should not mean in superspace without Kähler structure and the space-time is not
fundamental in this view.
As Wheeler wrote: “There is no space-time. There is no time. There is no before. There is no
after. The question of what happens 'next' is without meaning.”
For more clarification, I saw at The Labyrinth of Time by Lockwood: “Every such cosmological
model can be foliated in an infinity of different ways. Each foliation will consist of a sequence of 3geometries. (Wheeler refers to this freedom in the choice of a foliation as many-fingered time.)”
I mean Wheeler speak about some non-determinism of time? Then the time of superspace is
more like Subjective-Time(nondeterministic) of TGD rather than the Geometric-Time(deterministic)
of it?
5
At 8:25 PM, [email protected] said...
Dear Hamed,
It is fun to discuss and answer good questions. Here a brief comparison of WSP and WCW.
(A) In WSP, 3-geometry is the basic entity. In WCW, sub-manifold 3-geometry.
(B) There is no time or space-time in WSP context as Wheeler emphasizes. The theory should
assign in a natural manner to a given 3-geometry a 4-geometry to achieve this.
In WCW context, the definition of WCW Käehler function assigns to a given 3-surface a 4D space-time surface as preferred extremal of Käehler action. This makes Classical physics as
exact part of Quantum Theory and even that of WCW geometry. This new view about Bohr
orbits as something very real instead of being a fiction produced by stationary phase
approximation is the new philosophical viewpoint.
The notion of 3-surface generalizes and eventually one ends up with Zero Energy Ontology
(ZEO) and strong form of general coordinate invariance implying effective 2-dimensionality.
Partonic 2-surfaces and their 4-D tangent space data and CDs within CDs basically due to the
breaking of standard form of determinism.
On might argue that in strictly mathematical sense space-time becomes also in TGD
redundant locally (only locally due to the failure of strict determinism implying the need for
CDs within CDs). Space-time is, however, necessary for the interpretation of quantum
measurements which always assign to quantum events classical space-time correlates such as
frequencies. People talking about emergence often forget all about quantum measurement
theory. In any case, imbedding space and time are there and very relevant for the interpretation
of the theory!!
(C) WSP the classical history of 3-surface is a questionable concept since topology change is not
natural in this context. This view about 4-surface is very "Newtonian" and allows only X3xRtype space-time topologies.
In WCW with ZEO, one gives up completely the Newtonian view. Positive and Negative
energy parts of zero energy states as 3-surfaces at the 2 boundaries of CD can have totally
different topologies. Particle creation in a topological sense becomes possible and there is
space-time topology analogous to that of Feynman diagram.
Wheeler's many fingers would in this context become external lines of particle reaction. At
vertices incoming lines are join like lines of Feynman diagram so that space-time surface is not
anymore even 4-manifold. However, 3-surfaces are. Note that in string string world sheets are
2-manifolds but their time=constant sections are not always 1-manifolds.
(D) During the first year of TGD development, I tried ADM formalism but realized that canonical
quantization fails. The connection between time derivatives of imbedding space coordinates
and canonical momentum densities associated with action even in case of YM action is manyto-one and extremely non-linear so that one cannot do nothing in practice.
Standard quantization recipes would also yield horribly nonlinear poorly defined functional
differential equation (p→id/dx recipe generalized) so that a direct generalization from the tiny
6
hydrogen atom to the entire infinite Universe fails. Perhaps theoreticians were too ambitious at
this time;-)
For a long time, my heuristic guideline was that the non-locality of Kähler action for preferred
extremal as functional of 3-surface is the manner to get rid of local divergences. Reduction to ChernSimons terms brings, however, 3-D locality back but almost-topological character allows us to avoid
divergences now.
I have played now-and-then with the question whether Käehler geometry could be defined in terms
of preferred extremals of curvature scalar. What could these extremals be? How could one bring
complex and Käehler structures to the space of 3-geometries>
In the case of loop spaces, Kähler geometry is unique and there are excellent reasons to expect that
in higher dimensions even the target space containing 3-surfaces is more-or-less unique just from the
existence of the WCW geometry. What is the situation in WSP? Could it be that space-time dimension
is fixed?
In http://tgdtheory.com/public_html/articles/egtgd.pdf , I consider the analog for Wheeler's approach
in TGD based on Einstein-Maxwell action.
(1) Almost topological QFT property would mean that metric appears extremely implicitly in the
theory. It requires that the solutions of field equations involves are such that the action in
reduces to 3-D Chern-Simons term. The conformal invariance of Einstein-Maxwell equation
indeed implies that action reduces to boundary terms.
(2) Weak form of electric magnetic duality seems to make the 3-D contributions to action ChernSimons terms.
(3) One must also allow space-time regions of Euclidian signature and CP2 as solution of field
equation. Cosmological constant determining CP2-size scale (CP2 corresponds to nonvanishing cosmological constant) is needed in Euclidian regions. Average cosmological
constant could be quite small and reflect the fraction of Euclidian space-time volume.
(4) This would force a new view about black-hole interiors as Euclidian regions of space-time. CP2
was actually originally discovered as gravitational instanton.
(5) This is very near to TGD. Euclidian regions would carry electroweak and color quantum
numbers and Minkowskian regions classical quantum numbers. Could imbeddability to
M4xCP2 follow as a consistency condition?
Wheeler is quite right in what he says about absence of time in WSP. This is what general
coordinate invariance implies also in the 4-D view about General Relativity.
In the WCW approach based on Zero Energy Ontology, the position of space-time surface /CD in
imbedding space brings in Geometric-Time and one can speak about the positions of partonic 3surfaces.
Basically, I however think that "before" and "after" as we usually understand them apply to
Subjective-Time and not to Geometric(relativistic)-Time. One must identify Subjective-Time as
something different from Geometric-Time and this leads to quantum theories of Consciousness (and
things get even messier;-).
7
In quantum gravity based on Schrodinger equation in WSP, time could appear only as the analog of
time parameter in Hamiltonian unitary evolution. The basic equations of ADM, however, lead to
coordinate conditions saying that physical states are analogous to states with zero energy. Easy to guess
since Noether charges associated with diffeomorphisms of space-time vanish identically.
Again, one encounters the basic motivation of TGD. That is how to simultaneously realize Poincare
symmetries and geometrize gravitation. All the difficulties of GRT force to make the same question -what about sub-manifold gravity?
Perhaps the strongest objection that I have invented hitherto against sub-manifold gravity relates to
the superposition of fields. It is circumvented if it is replaced with superposition of effects of induced
fields (simultaneous topological condensation to several space-time sheets).
In path integral formalism, one makes a hypothesis that Schrodinger amplitude as a functional of 3geometry is sum over all space-time surfaces having 3-surface as boundary. Space-time would emerge
as the analog of point particle orbit. Classical space-times would correspond to extrema of Einstein's
action. There are technical problems due to the non-existence of path integral and one must make path
integral functional integral by making Euclidianization. A connection with the time of Special
Relativity is lost and achieved only in perturbative treatment of General Relativity in path integral
formalism around Minkowski space-time.
Spin 1/2 particles is a second problem of Wheeler's GeometroDynamics. In fact, spinor structure
does not even exist in generic space-time. (By the way, CP2 is one example of this.)
At 2:16 PM, hamed said...
Thank you very much for responding, Matti. Just a little question: why does any map from a
given 3-manifold X3 to M4xCP2 defines a surface?
At 12:02 AM, [email protected] said...
I am not sure whether I got the gist of the question. But this is the answer to the question as I
understood it and perhaps it is totally trivial to you.
The image of continuous map of X3 o H is the image of a 3-D geometric object and therefore a 3D surface.
Concrete example: hk = fk(x1,x2,x3) for k=1,...,8 (H-coordinates) when xi run over their value
range you get point set of H parameterized by 3 coordinates so that it is clearly 3-dimensional.
When H and X3 have p-adic coordinates, the situation is same. The image of X3 is parameterized
by 3 coordinates and in this sense 3-D.
Another basic manner to represent surfaces is as intersections of surfaces. The conditions
Hm(hk)=0, m=1,...5 for independent functions would give intersection of five(5) 7-D surfaces which
is generically a 3-D surface if non-trivial. In algebraic geometry, this is the manner to represent
surfaces.
8
In this representation, the expressions for induced metric and gauge potentials are not so easy to
deduce as using direct imbedding. One must in principle solve the imbedding space coordinates in
term of three of these functions locally.
[note: . The 30-year development/evolution of Topological GeometroDYnamics (TGD) physics
from Dr. Pitkänen's doctoral thesis at the University of Helsinki, Finland (~1982) to today is
archived at http://www.stealthskater.com/Pitkanen.htm#Evolution . The entire theory and
its many applications is at http://www.tgdtheory.com and selected files are also archived at
http://www.stealthskater.com/Pitkanen.htm in both Word(.doc) and Acrobat(.pdf) formats.]
http://www.viswiki.com/en/Geometrodynamics
General Relativity
Theory of Everything
Loop Quantum Gravity
Theoretical Physics
Quantum Gravity
Modified Newtonian Dynamics GeometroDynamics
John Archibald Wheeler
Le Sage's theory of gravitation
Kaluza-Klein theory
Supergravity
Stress-Energy tensor
Twistor theory
Brans-Dicke theory
String theory
Wormhole
M-theory
Classical Mechanics
Newton's Law of Universal Gravitation
Unified Field Theory
Superstring theory
Fermion
ArXiv
Schwarzschild metric
Tests of General Relativity
Einstein-Cartan theory
Mathematics of General Relativity
General Relativity http://www.viswiki.com/en/General_relativity
General Relativity or the General Theory of Relativity is the geometric theory of gravitation
published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It
unifies Special Relativity and Newton's law of universal gravitation and describes gravity as a geometric
property of space and time (or space-time). In particular, the curvature of space-time is directly related
to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are
present. The relation is specified by the Einstein field equations -- a system of partial differential
equations.
Many predictions of General Relativity differ significantly from those 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 dilation, the
gravitational redshift of light, and the gravitational time delay.
General Relativity's predictions have been confirmed in all observations and experiments to date.
Although it 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 fundamental being how General
Relativity can be reconciled with the laws of Quantum physics to produce a complete and self-consistent
theory of Quantum Gravity.
9
Einstein's theory has important astrophysical implications. It points towards the existence of black
holes (regions of space in which space and time are distorted in such a way that nothing, not even light,
can escape) as an end-state for massive stars. There is evidence that such stellar black holes as well as
more massive varieties of black hole are responsible for the intense radiation emitted by certain types of
astronomical objects such as active galactic nuclei or micro-quasars. The bending of light by gravity
can lead to the phenomenon of gravitational lensing where multiple images of the same distant
astronomical object are visible in the sky.
General Relativity also predicts the existence of gravitational waves which have since been
measured indirectly. A direct measurement is the aim of projects such as LIGO.
In addition, General Relativity is the basis of current cosmological models of a consistently
expanding Universe.
Theory of Everything http://www.viswiki.com/en/Theory_of_everything
The Theory of Everything (TOE) is a putative theory of theoretical physics that fully explains and
links together all known physical phenomena. Initially, the term was used with an ironic connotation to
refer to various over-generalized theories. For example, a great-grandfather of Ijon Tichy (a character
from a cycle of Stanisław Lem's science fiction stories of 1960s) was known to work on the "General
Theory of Everything". Physicist John Ellis claims[1] to have introduced the term into the technical
literature in an article in Nature in 1986 [2]. Over time, the term stuck in popularizations of Quantum
Physics to describe a theory that would unify or explain through a single model the theories of all
fundamental interactions of Nature.
There have been many theories of everything proposed by theoretical physicists over the last
century. But none have been confirmed experimentally. The primary problem in producing a TOE is
that the accepted theories of Quantum Mechanics and General Relativity are hard to combine.
Based on theoretical Holographic Principle arguments from the 1990s, many physicists believe that
11-dimensional M-theory (which is described in many sectors by matrix string theory and in many other
sectors by perturbative string theory) is the complete "Theory of Everything" although there is no
widespread consensus and M-theory is not a completed theory but rather an approach for producing one.
10
Loop Quantum Gravity http://www.viswiki.com/en/Loop_quantum_gravity
Simple spin network of the type used in Loop Quantum Gravity
Loop Quantum Gravity (LQG) --also known as Loop Gravity and Quantum Geometry -- is a
proposed quantum theory of space-time which attempts to reconcile the theories of Quantum Mechanics
and General Relativity. It suggests that space (i.e., the Universe) can be viewed as an extremely fine
fabric or network “weaved” of finite quantised loops (of excited gravitational fields) called spin
networks. When viewed over time, these spin networks are called "spin foam" which should not be
confused with "quantum foam". While some prefer String theory, other physicists consider Loop
Quantum Gravity to be a serious contender because it incorporates General Relativity and does not need
higher dimensions.
Loop Gravity preserves many of the important features of General Relativity while at the same time
employing quantization of both space and time at the Planck scale in the tradition of Quantum
Mechanics. The technique of loop quantization was developed for the nonperturbative quantization of
diffeomorphism-invariant gauge theory. Roughly, LQG tries to establish a quantum theory of Gravity in
which the very space where all other physical phenomena occur becomes quantized.
LQG is one of a family of theories called Canonical Quantum Gravity. The LQG theory also
includes matter and forces. But the theory does not address the problem of the unification of all physical
forces as other tentative quantum gravity theories do (for instance ,string theory).
Theoretical Physics http://www.viswiki.com/en/Theoretical_physics
Theoretical physics employs mathematical models and abstractions of physics in an attempt to
explain natural phenomena in a mathematical form. Its central core is mathematical physics though
other conceptual techniques are also used. The goal is to rationalize, explain, and predict physical
phenomena.
The advancement of Science depends in general on the interplay between experimental studies and
theory. In some cases, theoretical physics adheres to standards of mathematical rigor while giving little
weight to experiments and observations.
For example, while developing Special Relativity, Albert Einstein was concerned with the Lorentz
transformation which left Maxwell's equations invariant but was apparently uninterested in the
Michelson-Morley experiment on Earth's drift through a luminiferous ether. On the other hand, Einstein
was awarded the Nobel Prize for explaining the photoelectric effect -- previously an experimental result
lacking a theoretical formulation.
11
Quantum Gravity http://www.viswiki.com/en/Quantum_gravity
Quantum Gravity (QG) is the field of theoretical physics attempting to unify Quantum Mechanics
with General Relativity in a self-consistent manner or --more precisely -- to formulate a self-consistent
theory which reduces to ordinary quantum mechanics in the limit of weak gravity (potentials much less
than c2) and which reduces to "classical" General Relativity in the limit of large actions (action much
larger than reduced Planck's constant).
The theory must be able to predict the outcome of situations where both quantum effects and strongfield gravity are important (at the Planck scale, unless extra dimensional theories are correct).
Motivation for quantizing gravity comes from the remarkable success of the Quantum theories of the
other 3 fundamental interactions.
Although some quantum gravity theories such as string theory and other so-called "theories of
everything" also attempt to unify Gravity with the other fundamental forces, others such as Loop
Quantum Gravity make no such attempt at unification. They simply quantize the gravitational field
while keeping it separate from other force fields.
Observed physical phenomena in the early 21st Century can be described well by Quantum
Mechanics or General Relativity without needing both. This can be thought of as due to an extreme
separation of scales at which they are important. Quantum effects are usually important only for the
"very small". That is, objects no larger than ordinary molecules (for instance as of 2009, even viruses
have not been observed to undergo double-slit diffraction). General Relativistic effects, on the other
hand, show up only for the "very large" bodies such as collapsed stars. (Planets' gravitational fields, as
of 2009, are well-described by linearized gravity. So strong-field effects, any effects of gravity beyond
lowest nonvanishing order in phi/c2, have not been observed even in the gravitational fields of planets
and main sequence stars.)
Classical physics seems to be adequate over an enormous range of masses of objects from about
10−23 to 1030 kg. Thus there is a want of experimental evidence relating to Quantum Gravity. But the
"gap" spans 53 orders of magnitude.
Modified Newtonian Dynamics
http://www.viswiki.com/en/Modified_Newtonian_dynamics
In physics, Modified Newtonian dynamics (MOND) is a theory that proposes a modification of
Newton's Second Law of Dynamics (F = ma) to explain the galaxy rotation problem. When the uniform
velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of
gravity predicts that objects that are farther out will have lower velocities. For example, planets in the
Solar System orbit with velocities that decrease as their distance from the Sun increases. MOND theory
posits that acceleration is not linearly proportional to force at low values.
Expected (A) and observed (B) star velocities as a function of distance from the galactic center.
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MOND was proposed by Mordehai Milgrom in 1981 as an alternative way to model the observed
uniform velocity data in contrast to the more widely accepted theory of dark matter. The dark matter
theory suggests that each galaxy contains a halo of an as yet unidentified type of matter that provides an
overall mass distribution different from the observed distribution of normal matter. Milgrom noted that
Newton's Second Law for gravitational force has only been verified when gravitational acceleration is
large and suggested that for extremely low accelerations the theory may not hold.
John Archibald Wheeler is an eminent American theoretical physicist. One of the later collaborators
of Albert Einstein, he tried to achieve Einstein's vision of a unified field theory. He is also known for
having coined the terms black hole and wormhole and the phrase "It from Bit".
http://www.viswiki.com/en/John_Archibald_Wheeler
Le Sage's theory of gravitation http://www.viswiki.com/en/Le_Sage%27s_theory_of_gravitation
Le Sage's theory of gravitation is the most common name for the kinetic theory of gravity originally
proposed by Nicolas Fatio de Duillier in 1690 and later by Georges-Louis Le Sage in 1748. The theory
proposed a mechanical explanation for Newton's gravitational force in terms of streams of tiny unseen
particles (which Le Sage called ultra-mundane corpuscles) impacting on all material objects from all
directions.
According to this model, any 2 material bodies partially shield each other from the impinging
corpuscles, resulting in a net imbalance in the pressure exerted by the impacting corpuscles on the
bodies tending to drive the bodies together. This mechanical explanation for Gravity never gained
widespread acceptance although it continued to be studied occasionally by physicists until the beginning
of the 20th century, by which time it was generally considered to be conclusively discredited.
Kaluza-Klein theory http://www.viswiki.com/en/Kaluza%E2%80%93Klein_theory
In physics, Kaluza–Klein theory (or KK theory for short) is a model that seeks to unify the 2
fundamental forces of Gravitation and ElectroMagnetism. T he theory was first published in 1921 and
was proposed by the mathematician Theodor Kaluza who extended General Relativity to a 5dimensional space-time.
The space is compactified over the compact set C and after Kaluza-Klein decomposition, we have an
effective field theory over M.
The resulting equations can be separated out into further sets of equations -- one of which is
equivalent to Einstein field equations, another set equivalent to Maxwell's equations for the
electromagnetic field, and the final part an extra scalar field now termed the "radion".
Supergravity http://www.viswiki.com/en/Supergravity
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In theoretical physics, Supergravity (supergravity theory) is a field theory that combines the
principles of supersymmetry and General Relativity. Together, these imply that in supergravity, the
supersymmetry is a local symmetry (in contrast to non-gravitational supersymmetric theories such as the
Minimal Supersymmetric Standard Model (MSSM)).
Stress-Energy Tensor http://www.viswiki.com/en/Stress-energy_tensor
The stress-energy tensor (sometimes stress-energy-momentum tensor) is a tensor quantity in Physics
that describes the density and flux of energy and momentum in space-time, generalizing the stress tensor
of Newtonian physics.
It is an attribute of matter, radiation, and non-gravitational force fields. The stress-energy tensor is
the source of the gravitational field in the Einstein field equations of General Relativity just as mass is
the source of such a field in Newtonian gravity.
Twistor theory http://www.viswiki.com/en/Twistor_theory
The twistor theory -- originally developed by Roger Penrose in 1967 -- is the mathematical theory
which maps the geometric objects of the 4-dimensional space-time (Minkowski space) into the
geometric objects in the 4-dimensional complex space with the metric signature (2,2). The coordinates
in such a space are called "twistors."
The twistor theory was stimulated by a rationale indicating its particular usefulness in emergent
theories of quantum gravity.
The twistor approach appears to be especially natural for solving the equations of motion of massless
fields of arbitrary spin.
In 2003, Edward Witten used twistor theory to understand certain Yang-Mills amplitudes by relating
them to a certain string theory (the topological B model) embedded in twistor space. This field has
come to be known as twistor string theory.
Brans-Dicke theory http://www.viswiki.com/en/Brans-Dicke_theory
In theoretical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan-BransDicke theory) is a theoretical framework to explain gravitation. It is a well-known competitor of
Einstein's more popular theory of General Relativity.
It is an example of a scalar-tensor theory -- a gravitational theory in which the gravitational
interaction is mediated by a scalar field as well as the tensor field of General Relativity. The
gravitational constant G is not presumed to be constant but instead 1/G is replaced by a scalar field φ
which can vary from place-to-place and with time.
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The theory was developed in 1961 by Robert H. Dicke and Carl H. Brans[1] building upon, among
others, the earlier 1959 work of Pascual Jordan.
At present, both Brans-Dicke theory and General Relativity are generally held to be in agreement
with observation, although the experiments of the golden age of Reneral Relativity have considerably
constrained the allowed parameters of Brans-Dicke theory. Brans-Dicke theory represents a minority
viewpoint in Physics.
String theory http://www.viswiki.com/en/String_theory
String theory is a developing branch of theoretical physics that combines Quantum Mechanics and
General Relativity into a quantum theory of gravity [1]. The strings of string theory are 1-dimensional
oscillating lines. But they are no longer considered fundamental to the theory which can be formulated
in terms of points or surfaces too.
Since its birth as the dual resonance model which described the strongly interacting hadrons as
strings, the term "string theory" has changed to include any of a group of related superstring theories
which unite them. One shared property of all these theories is the Holographic Principle.
String theory itself comes in many different formulations -- each one with a different mathematical
structure and each best describing different physical circumstances. But the principles shared by these
approaches, their mutual logical consistency, and the fact that some of them easily include the Standard
Model of Particle Physics has led many physicists to believe that the theory is the correct fundamental
description of Nature.
In particular, string theory is the first candidate for the theory of everything (TOE) -- a way to
describe the known fundamental forces (gravitational, electromagnetic, weak and strong interactions)
and matter (quarks and leptons) in a mathematically-complete system.
Many detractors criticize string theory as it has not provided quantitative experimental predictions.
Like any other quantum theory of gravity, it is widely believed that testing the theory directly would
require prohibitively expensive feats of engineering. Whether there are stringent indirect tests of the
theory is unknown.
String theory is of interest to many physicists because it requires new mathematical and physical
ideas to mesh together its very different mathematical formulations. One of the most inclusive of these
is the 11-dimensional M-theory which requires space-time to have eleven dimensions [2] as opposed to
the usual 3 spatial dimensions and the fourth dimension of time.
The original string theories from the 1980s describe special cases of M-theory where the 11th
dimension is a very small circle or a line. If these formulations are considered as fundamental, then
string theory requires 10 dimensions. But the theory also describes universes like ours with 4
observable space-time dimensions as well as universes with up to 10 flat space dimensions and also
cases where the position in some of the dimensions is not described by a real number but by a
completely different type of mathematical quantity. So the notion of space-time dimension is not fixed
in string theory. It is best thought of as different in different circumstances [3].
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Interaction in the subatomic world: world lines of point-like
particles in the Standard Model or a world sheet swept up
by closed strings in string theory
Calabi-Yau manifold (3D projection)
String theories include objects more general than strings called "branes". The word brane (derived
from "membrane") refers to a variety of interrelated objects such as D-branes, black p-branes, and
Neveu-Schwarz 5-branes. These are extended objects that are charged sources for differential form
generalizations of the vector potential electromagnetic field.
These objects are related to one-another by a variety of dualities. Black hole-like black p-branes are
identified with D-branes which are endpoints for strings. This identification is called Gauge-gravity
duality. Research on this equivalence has led to new insights on Quantum ChromoDynamics -- the
fundamental theory of the strong nuclear force [4][5][6][7].
Wormhole http://www.viswiki.com/en/Wormhole
In Physics, a "wormhole" is a hypothetical topological feature of space-time that is fundamentally a
'shortcut' through space and time. Space-time can be viewed as a 2D surface (to simplify understanding)
that when 'folded' over, allows the formation of a wormhole bridge.
A wormhole has at least 2 mouths that are connected to a single throat or tube. If the wormhole is
traversable, matter can 'travel' from one mouth to the other by passing through the throat. While there is
no observational evidence for wormholes, space-times containing wormholes are known to be valid
solutions in General Relativity.
Embedded diagram of a Schwarzschild wormhole
The term wormhole was coined by the American theoretical physicist John Archibald Wheeler in
1957. However, the idea of wormholes had already been theorized in 1921 by the German
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mathematician Hermann Weyl in connection with his analysis of mass in terms of electromagnetic field
energy [1].
"This analysis forces one to consider situations...where there is a net flux of lines of
force through what topologists would call a handle of the multiply-connected space and
what physicists might perhaps be excused for more vividly terming a ‘wormhole’."
—John Wheeler in Annals of Physics
M-theory http://www.viswiki.com/en/M-theory
In theoretical physics, M-theory is a new limit of string theory in which 11 dimensions of space-time
may be identified. Because the dimensionality exceeds the dimensionality of 5 superstring theories in
10 dimensions, it was originally believed that the 11-dimensional theory is more fundamental and
unifies all string theories (and supersedes them).
However, in a more modern understanding, it is another, 6th possible description of physics of the
full theory that is still called "string theory". Though a full description of the theory is not yet known,
the low-entropy dynamics are known to be supergravity interacting with 2- and 5-dimensional
membranes.
This idea is the unique supersymmetric theory in 11 dimensions -- with its low-entropy matter
content and interactions fully determined -- can be obtained as the strong coupling limit of type IIA
string theory because a new dimension of space emerges as the coupling constant increases.
Drawing on the work of a number of string theorists (including Ashoke Sen, Chris Hull, Paul
Townsend, Michael Duff, and John Schwarz), Edward Witten of the Institute for Advanced Study
suggested its existence at a conference at USC in 1995 and used M-theory to explain a number of
previously observed dualities, sparking a flurry of new research in string theory called the second
superstring revolution.
In the early 1990s, it was shown that the various superstring theories were related by dualities which
allow physicists to relate the description of an object in one superstring theory to the description of a
different object in another superstring theory. These relationships imply that each of the superstring
theories is a different aspect of a single underlying theory -- proposed by Witten and named "M-theory".
Originally the letter 'M' in M-theory was taken from membrane -- a construct designed to generalize
the strings of string theory. However, as Witten was more skeptical about membranes than his
colleagues, he opted for "M-theory" rather than "Membrane theory". Witten has since stated that the
interpretation of the 'M' can be a matter of taste for the user of the word "M-theory" [1].
M-theory is not yet complete. However it can be applied in many situations (usually by exploiting
string theoretic dualities). The theory of electromagnetism was also in such a state in the mid-19th
century. There were separate theories for Electricity and Magnetism and -- although they were known
to be related -- the exact relationship was not clear until James Clerk Maxwell published his equations in
his 1864 paper "A Dynamical Theory of the Electromagnetic Field".
Witten has suggested that a general formulation of M-theory will probably require the development
of new mathematical language. However, some scientists have questioned the tangible successes of Mtheory given its current incompleteness and limited predictive power even after so many years of intense
research.
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In late 2007, Bagger, Lambert, and Gustavsson set off renewed interest in M-theory with the
discovery of a candidate Lagrangian description of coincident M2-branes based on a non-associative
generalization of Lie Algebra, Nambu 3-algebra, or Filippov 3-algebra. Practitioners hope the BaggerLambert-Gustavsson action (BLG action) will provide the long-sought microscopic description of Mtheory.
Classical Mechanics http://www.viswiki.com/en/Classical_mechanics
In the fields of Physics, Classical Mechanics is one of the 2 major sub-fields of study in the science
of Mechanics, which is concerned with the set of physical laws governing and mathematically
describing the motions of bodies and aggregates of bodies geometrically distributed within a certain
boundary under the action of a system of forces. The other sub-field is Quantum Mechanics.
Classical Mechanics is used for describing the motion of Macroscopic objects from projectiles to
parts of machinery as well as astronomical objects such as spacecraft, planets, stars, and galaxies. It
produces very accurate results within these domains, and is one of the oldest and largest subjects in
Science, Engineering, and Technology.
domain of validity for Classical Mechanics
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branches of Mechanics
Besides this, many related specialties exist dealing with gases, liquids, solids, and so on. Classical
Mechanics is enhanced by Special Relativity for objects moving with high velocity approaching the
speed-of-light; General Relativity is employed to handle gravitation at a deeper level; and Quantum
Mechanics handles the wave-particle duality of atoms and molecules.
The term "Classical Mechanics" was coined in the early 20th century to describe the system of
mathematical physics begun by Isaac Newton and many contemporary 17th century natural philosophers,
building upon the earlier astronomical theories of Johannes Kepler. Which in turn were based on the
precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo but before
the development of Quantum physics and Relativity.
Therefore, some sources exclude so-called "relativistic physics" from that category. However, a
number of modern sources do include Einstein's mechanics which in their view represents Classical
Mechanics in its most developed and most accurate form.
The initial stage in the development of Classical Mechanics is often referred to as Newtonian
mechanics and is associated with the physical concepts employed by and the mathematical methods
invented by Newton himself, in parallel with Leibniz and others. This is further described in the
following sections. More abstract and general methods include Lagrangian mechanics and Hamiltonian
mechanics. Much of the content of Classical Mechanics was created in the 18th and 19th centuries and
extends considerably beyond (particularly in its use of analytical mathematics) the work of Newton.
Newton's Law of Universal Gravitation
http://www.viswiki.com/en/Newton%27s_law_of_universal_gravitation
Newton's Law of Universal Gravitation is a general physical law derived from empirical
observations by what Newton called induction [1]. It describes the gravitational attraction between
bodies with mass. It is a part of Classical Mechanics and was first formulated in Newton's work
Philosophiae Naturalis Principia Mathematica (the Principia), first published on 5 July 5, 1687. In
modern language, it states the following:
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Every point mass attracts every other point mass by a force pointing along the line intersecting both
points. The force is directly proportional to the product of the 2 masses and inversely proportional to the
square of the distance between the point masses:
where:
F is the magnitude of the gravitational force between the two point masses,
G is the gravitational constant,
m1 is the mass of the first point mass,
m2 is the mass of the second point mass,
r is the distance between the two point masses.
Assuming SI units, F is measured in newtons (N); m1 and m2 in kilograms (kg); r in meters (m); and
the constant G is approximately equal to 6.673×10−11 N m2 kg−2. The value of the constant G was first
accurately determined from the results of the Cavendish experiment conducted by the British scientist
Henry Cavendish in 1798 (though Cavendish did not himself calculate a numerical value for G [2]).
This experiment was also the first test of Newton's theory of gravitation between masses in the
laboratory. It took place 111 years after the publication of Newton's Principia and 71 years after
Newton's death. So none of Newton's calculations could use the value of G. Instead he could only
calculate a force relative to another force.
Newton's law of gravitation resembles Coulomb's law of electrical forces which is used to calculate
the magnitude of electrical force between 2 charged bodies. Both are inverse-square laws in which force
is inversely proportional to the square of the distance between the bodies. Coulomb's Law has the
product of 2 charges in place of the product of the masses and the electrostatic constant in place of the
gravitational constant.
Newton's law has since been superseded by Einstein's theory of General Relativity. But it continues
to be used as an excellent approximation of the effects of Gravity. Relativity is only required when
there is a need for extreme precision or when dealing with gravitation for very massive objects.
Unified Field Theory http://www.viswiki.com/en/Unified_field_theory
In physics, a "unified field theory" is a type of field theory that allows all of the fundamental forces
between elementary particles to be written in terms of a single field. There is no accepted unified field
theory yet and this remains an open line of research. The term was coined by Albert Einstein who
attempted to unify the General Theory of Relativity with ElectroMagnetism.
A 'Theory of Everything' is closely related to 'Unified Field Theory' but differs by not requiring the
basis of nature to be fields and also attempts to explain all physical constants of Nature.
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This article describes Unified Field Theory as it is currently understood in connection with Quantum
Theory. Earlier attempts based on classical physics are described in the article on classical unified field
theories.
There may be no a priori reason why the correct description of nature has to be a unified field theory.
However, this goal has led to a great deal of progress in modern theoretical physics and continues to
motivate research. Unified field theory is only one possible approach to unification of Physics.
Superstring theory http://www.viswiki.com/en/Superstring_theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of Nature in
one theory by modeling them as vibrations of tiny supersymmetric strings. It is considered one of the
most promising candidate theories of Quantum gravity.
Superstring theory is a shorthand for supersymmetric string theory because unlike bosonic string
theory, it is the version of string theory that incorporates fermions and supersymmetry.
the 5 superstring interactions
Fermion http://www.viswiki.com/en/Fermion
In particle physics, fermions are particles which obey Fermi-Dirac statistics. They are named after
Enrico Fermi. In contrast to bosons which have Bose-Einstein statistics, only one fermion can occupy a
quantum state at a given time. This is the Pauli Exclusion Principle. Thus if more than one fermion
occupies the same place in space, the properties of each fermion (e.g., its spin) must be different from
the rest. Therefore fermions are usually associated with matter while bosons are often force carrier
particles (though the distinction between the two concepts is not clear cut in Quantum physics).
Fermions can be elementary (like the electron) or composite (like the proton). All observed
fermions have half-integer spin as opposed to bosons which have integer spin. This is in accordance
with the spin-statistics theorem which states that in any reasonable relativistic quantum field theory,
particles with integer spin are bosons while particles with half-integer spin are fermions.
In the Standard Model, there are 2 types of elementary fermions: quarks and leptons. In total, there
are 24 different fermions; 6 quarks and 6 leptons, each with a corresponding anti-particle:
12 quarks - 6 particles (u · d · s · c · b · t) with 6 corresponding antiparticles (u · d · s · c · b · t);
12 leptons - 6 particles (e− μ− τ− νe νμ ·ντ) with 6 corresponding antiparticles (e+ ·μ+ ·τ+ ·νe νμ ·ντ).
Composite fermions such as protons and neutrons are essential building blocks of matter. Weakly
interacting fermions can also display bosonic behavior as in superconductivity.
21
ArXiv http://www.viswiki.com/en/ArXiv
The arXiv (pronounced "archive" as if the "X" were the Greek letter Chi χ) is an archive for
electronic preprints of scientific papers in the fields of mathematics, physics, computer science,
quantitative biology, and statistics which can be accessed via the Internet.
In many fields of mathematics and physics, almost all scientific papers are placed on the arXiv. As
of October 3, 2008, arXiv.org passed the half-million article milestone with roughly 5,000 new e-prints
added every month [1].
Schwarzschild metric http://www.viswiki.com/en/Schwarzschild_metric
In Einstein's theory of General Relativity, the Schwarzschild solution (or the Schwarzschild vacuum)
describes the gravitational field outside a spherical, non-rotating mass such as a (non-rotating) star,
planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body
like the Earth or Sun. The cosmological constant is assumed to equal zero.
According to Birkhoff's theorem, the Schwarzschild solution is the most general spherically
symmetric, vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black
hole is a black hole that has no charge or angular momentum. A Schwarzschild black hole has a
Schwarzschild metric and cannot be distinguished from any other Schwarzschild black hole except by its
mass.
The Schwarzschild solution is named in honor of its discoverer Karl Schwarzschild who found the
solution in 1915, only about a month after the publication of Einstein's theory of General Relativity. It
was the first exact solution of the Einstein field equations other than the trivial flat space solution.
Schwarzschild had little time to think about his solution. He died shortly after his paper was published
as a result of a disease he contracted while serving in the German army during World War I.
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A plot of Flamm's paraboloid. It should not be confused with the unrelated concept of a gravity well.
The Schwarzschild black hole is characterized by a surrounding spherical surface called the "event
horizon" which is situated at the Schwarzschild radius (often called the "radius" of a black hole). Any
non-rotating and non-charged mass that is smaller than the Schwarzschild radius forms a black hole.
The solution of the Einstein field equations is valid for any mass M. So in principle (according to
General Relativity theory) a Schwarzschild black hole of any mass could exist if conditions became
sufficiently favorable to allow for its formation.
Tests of General Relativity http://www.viswiki.com/en/Tests_of_general_relativity
At its introduction in 1915, the General Theory of Relativity did not have a solid empirical
foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of
Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's
law of universal gravitation with special relativity. That light appeared to bend in gravitational fields in
line with the predictions of General Relativity was found in 1919. But it was not until a program of
precision tests was started in 1959 that the various predictions of General Relativity were tested to any
further degree of accuracy in the weak gravitational field limit, severely limiting possible deviations
from the theory.
Beginning in 1974, Hulse, Taylor, and others have studied the behavior of binary pulsars
experiencing much stronger gravitational fields than found in our Solar System. Both in the weak field
limit (as in our Solar System) and with the stronger fields present in systems of binary pulsars, the
predictions of General Relativity have been extremely well tested locally.
On the largest spatial scales such as galactic and cosmological scales, General Relativity has not yet
been subject to precision tests. Some have interpreted observations supporting the presence of dark
matter and dark energy as a failure of general relativity at large distances, small accelerations, or small
curvatures. The very strong gravitational fields that must be present close to black holes -- especially
those supermassive black holes which are thought to power active galactic nuclei and the more active
quasars -- belong to a field of intense active research.
Observations of these quasars and active galactic nuclei are difficult and the interpretation of the
observations are heavily dependent upon astrophysical models other than General Relativity or
competing fundamental theories of gravitation. But they are qualitatively consistent with the black hole
concept as modeled in General Relativity.
Einstein-Cartan theory http://www.viswiki.com/en/Einstein%E2%80%93Cartan_theory
Einstein–Cartan theory in theoretical physics extends General Relativity to correctly handle spin
angular momentum. As the master theory of Classical physics, General Relativity has one known flaw.
23
It cannot describe "spin-orbit coupling" -- i.e., exchange of intrinsic angular momentum (spin) and
orbital angular momentum. There is a qualitative proof showing that General Relativity must be
extended to Einstein–Cartan theory when matter with spin is present.
Experimental effects are too small to be observed at the present time because the spin tensor of
typical Macroscopic objects is often small and torsion is nonpropagating which means that torsion will
only appear within a massive body. In addition, only spinning objects couple to torsion.
The theory is named after Albert Einstein and Élie Cartan.
Mathematics of General Relativity http://www.viswiki.com/en/Mathematics_of_general_relativity
For a discussion of the minimal mathematics necessary to understand General Relativity, see
Introduction to mathematics of general relativity.
The mathematics of General Relativity refers to various mathematical structures and techniques that
are used in studying Albert Einstein's theory of General Relativity. The main tools used in this
geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of General Relativity.
Note: General Relativity articles using tensors will use the abstract index notation.
● Why tensors?
● Space-Time as a manifold
● Tensors in GR
● Tensor fields in GR
● Tensorial derivatives
● the Riemann curvature tensor
● the Energy-Momentum tensor
● the Einstein field equations
● the geodesic equations
● Langrangian formulation
● mathematical techniques for analyzing space-times
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