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What does symmetry in GR mean and where does it lead?

  1. Mar 1, 2012 #1
    Steve Carlip says this [back in 2007], here,:


    "...Until fairly recently, no one had a clear idea of the microscopic states responsible for black hole entropy. Today, we suffer the opposite problem: we have many explanations, each describing a different set of states but all agreeing on the final numbers. .......

    [1] One attractive possibility is that a hidden symmetry of classical general relativity controls the thermodynamic properties of black holes. ......

    [2] The uncertainty principle prevents us from simply saying,"A black hole is present." [Instead, we must find a way to impose constraints strong enough to ensure the presence of a black hole,but weak enough to be allowed by quantum mechanics.] . ......

    [3]The key point is that the horizon constraints break the fundamental symmetry of general relativity, general covariance (technically, diffeomorphism invariance). [As a result, states that would normally be considered equivalent, differing only by a "gauge" transformation, are now physically distinct.]

    [The brackets[] are not Carlips and are included for perspective.]

    Is this area of research active: 'hidden symmetries in GR???' and has it produced anything interesting....Is it known by maybe a different term today?? Any suggested reading sources??

    And what in layman's terms do 1,2,3 mean???

    Wiki had nothing I could find under "Symmetry in GR". I just got Taylor and Wheelers' "Exploring Black Holes, Introduction to GR" an hour ago in the mail and there is nothing in the index there about symmetry in GR.

    edit: Wikipedia makes mention:
    "In general relativity, the symmetric stress-energy tensor acts as the source of spacetime curvature... "

    thank you.
    Last edited: Mar 1, 2012
  2. jcsd
  3. Mar 1, 2012 #2


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    I hope you aren't expecting a short answer. Algebraic symmetries are at the heart of GR and are used to classify spacetimes into 'families' according to the structure of the Lie group formed by all possible solutions of Killings equation. Effectively this means that we can think in families of solutions that share certain core properties. Spaces of constant curvature for instance have stringent conditions on the curvature tensor, which are also called symmetries.

    I think what Carlip means in [1] is that the different ensembles of states that apparently describe the same thing, are not actually different because they share some algebraic property ( a symmetry) and are actually the same thing ( this might be tautologous ... ).

    Fully understanding symmetry requires a close acquaintance with group theory, every physicists nightmare.:wink:
  4. Mar 9, 2012 #3
    I was especially interested in Carlip's comment #2 about uncertainty.....
    There have been some long discussions in these forums and I thought I had a reasonable
    understanding, but what is meant by

    is really baffling....

    These were, then, discovered after the formulation of GR??.....they were likely not formal input restrictions which Einstein recognized and which guided him beforehand????

    When Einstein was deciding what mathematics to use for his new theory, I seem to recall he was aware of 'stress, strain' and these are described by symmetric tensors.....we now know.....was such 'symmetry' understood/known in the 1920's???
    Last edited: Mar 9, 2012
  5. Mar 9, 2012 #4
    Mine too. Do you know of any good book on Lie groups, Lie algebras, representation theory, tensor products, direct sums, etc? Annoyingly advanced physics books act as if this stuff is common knowledge (which it probably is, to a lot of readers).
  6. Mar 10, 2012 #5


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    I'm guessing, but I'd say Einstein was not aware of the richness of GR right away. He wanted a theory with geodesic motion because he understood that free-fall is what distinguishes gravity. Later, Killing found the GR equivalent of Noether's theorem which enables space-times to be classified according to their invariants.

    I cannot recommend a specific book but there must be many now aimed at physicists. I can write down everything I know about groups in a few sentences so I'm not the person to answer this.

    I do have a copy of 'The Theory of Groups and Quantim Mechanics' by Hermann Weyl, which kicked off a lot of this, but I wouldn't recommend it to any non-geniuses like me.
  7. Mar 10, 2012 #6
    that's what I've found to be in the literature so far....and seems convincing considering Minkowski introduced Einstein to the 'merger' of space and time.....and that it took others to solve Einstein's own equations.....
  8. Mar 10, 2012 #7
    Hmm? "and that it took others to solve Einstein's own equations....."

    What do you mean by that Naty?

    His equations was his own as far as I can see? And worked in the presentation of them he gave. That you can continue to manipulate them and get more or less far fetched outcomes that he hadn't considered, as 'time travels' for example, doesn't minimize what he did, to me.

    He was one of a kind.
  9. Mar 10, 2012 #8


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    Some related topics:
    Symmetry (physics)
    Spacetime symmetries

    (It's always easier to find the answer to a question if you already know the answer! :smile:)
  10. Mar 10, 2012 #9


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    Googling off Carlip's essay suggests these papers.

    Carlip, http://arxiv.org/abs/hep-th/9812013
    "Restricted to a black hole horizon, the "gauge" algebra of surface deformations in general relativity contains a Virasoro subalgebra with a calculable central charge. The fields in any quantum theory of gravity must transform accordingly, i.e., they must admit a conformal field theory description."

    Solodukhin, http://arxiv.org/abs/hep-th/9812056
    "The existence of black hole horizon is considered as a boundary condition to be imposed on the fluctuating metrics. ... The diffeomorphisms preserving this condition act in (arbitrary small) vicinity of the horizon and form the group of conformal transformations ..."

    Maldacena, Strominger, http://arxiv.org/abs/hep-th/9702015
    "Fundamental string theory has been used to show that low energy excitations of certain black holes are described by a two dimensional conformal field theory. This picture has been found to be extremely robust. In this paper it is argued that many essential features of the low energy effective theory can be inferred directly from a semiclassical analysis of the general Kerr-Newman solution of supersymmetric four-dimensional Einstein-Maxwell gravity, without using string theory."

    There seems to be some differences between the various views, as reviewed by Joan Simon. The Kerr/CFT picture seems to apply to extremal black holes, Carlip proposes using Killing horizons, while Solodukhin proposes using apparent horizons.

    As for whether the horizon absolutely exists or only approximately exists in quantum gravity, there are ideas like Mathur's fuzzball conjecture in which the horizon only exists for macroscopic objects.
    Last edited: Mar 10, 2012
  11. Mar 10, 2012 #10
    Einstein as far as I know did not find any solutions to the Einstein equations. Even the simplest known solution, that of a nonspinning point mass, was found by Schwarzchild.
  12. Mar 10, 2012 #11
    "The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor) with the energy and momentum within that spacetime (expressed by the stress–energy tensor).

    Similar to the way that electromagnetic fields are determined using charges and currents via Maxwell's equations, the EFE are used to determine the spacetime geometry resulting from the presence of mass-energy and linear momentum, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation.

    As well as obeying local energy-momentum conservation, the EFE reduce to Newton's law of gravitation where the gravitational field is weak and velocities are much less than the speed of light.

    Solution techniques for the EFE include simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as flat spacetime with a small deviation, leading to the linearised EFE. These equations are used to study phenomena such as gravitational waves."

    From http://en.wikipedia.org/wiki/Einstein_field_equations.

    and this one http://www.btinternet.com/~j.doyle/SR/Emc2/Derive.htm is for SR.

    One thing though, he (Jim Doyle) uses the older definition of 'relativistic mass' in his description, although Einstein later found 'invariant mass' to be the preferred concept. But you can use both.
    Last edited: Mar 10, 2012
  13. Mar 10, 2012 #12
    Um, I'm not sure why your quoting all this. The point is not that Einstein didn't come up with the Einstein equations, but rather that he didn't find solutions to them himself.
  14. Mar 10, 2012 #13
    You mean that SR and GR wasn't his?

    Seems we're defining 'solutions' quite differently here?
    Maybe you have another word, that will clear it up :)
    Last edited: Mar 10, 2012
  15. Mar 10, 2012 #14
    Okay, then we're agreed on that :) The other solutions you're speaking of I would define as manipulating the equations he gave for SR and GR, taking them to their limits more or less. And who knows, maybe there still are new ways to use them?
  16. Mar 10, 2012 #15
    I mean solution in the mathematical sense of solution of an equation. Einstein came up with but did not solve the field equations of general relativity.
  17. Mar 10, 2012 #16
    Are you sure that all derivations of his equations are known now? There can be no new derivations in the future? What Einstein did was to introduce SR and GR. To prove it mathematically he was forced to both learn new math as well as invent some of it. He also was the one introducing the concept of symmetries that we all use nowadays.

    But I see how you thought, and we used 'solutions' differently. To me SR and GR represent the solutions for Einsteins equations and all further derivations coming from them is manipulating the concepts he introduced of SpaceTime.

    As long as we're discussing SpaceTime that is. Quantum mechanics is a different coin, or side of it at least.
  18. Mar 10, 2012 #17
    No, only a very small number of exact mathematical solutions are known for the Einstein equations. Most solutions are not expressible using elementary functions.
  19. Mar 10, 2012 #18
    Okay, and some of them are very far from what we observe, as closed timelike curves, and 'wormholes' as the Einstein-Rosen Bridge. Even though Einstein and Rosen created it in 'an attempt to to explain fundamental particles like electrons as space-tunnels threaded by electric lines of force.' As for if you can manipulate the field equations further I don't know, but I would really like too :)
  20. Mar 12, 2012 #19
    Here is a candidate for hidden symmetry in GR :

    [tex]R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R=G_{\mu \nu}=g_{\mu \nu}\Lambda-\Pi_{\mu \nu}[/tex]

    Personally, I think it leads to a link between gravity, quantum theory and the accelerating expansion. Just my opinion though.

    If you would like to debate this, please go to the forum http://www.bautforum.com/showthread.php/129309-Negative-Mass-Interpretation-of-General-Relativity

    (mods: Please let me know if the link is not allowed, I will delete.)
    Last edited: Mar 12, 2012
  21. Mar 12, 2012 #20


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    There are few solutions to the equations that are physically realistic. For instance there are more than 100 perfect fluid solutions but only a few are physically acceptable. It is a problem, maybe, with the EFE that there are many solutions that are so weird they describe implausible alternative universes. Vacuum solutions are much rarer.

    Messenger, what is ∏ in the equation ?
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