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Digging around in the archives, I came across the following pair of excellent questions from way back in July 2004:
Let me reformulate these questions, in three groups:
First: is there some reasonable way of "counting" the distinct source-free solutions to the Maxwell field equations? How about the vacuum solutions of the Einstein field equation? What about vacuum solutions in (the field theoretic reformulation of) Newtonian gravitation, or the Brans-Dicke theory of gravitation, the Nordstrom theory of gravitation, or the source-free solutions in other classical field theories in physics? How do these "counts" compare? Do the "degrees of freedom" implied by our counting suggest useful ways of reformulating our theories?
Second: is there some reasonable way of "counting" the distinct Lorentzian four-manifolds, taking account of our freedom to change coordinates? What about "counting" the number of one-forms on a four-dimensional smooth manifold? Or antisymmetric two-forms? Symmetric rank two tensor fields? How do these "counts" compare with each other, and with the counts from the first group? (We would expect, for example, that there should be "more" Lorentzian four-manifolds than there are vacuum solutions of the EFE or Brans-Dicke field equations, simply because said solutions are Lorentzian manifolds which are required to satisfy some additional conditions.) Does our counting suggest any "irredundant" representations of Lorentzian manifolds?
To avoid possible confusion, let me stress that "counting" the number of distinct Lorentzian four-manifolds is not the same thing at all as counting the number of linearly independent components of the metric tensor.
Third: we know how to compute the Riemann tensor if we are given the metric tensor, by a kind of differentiation. Going the other way, we'd expect to be able to find the metric tensor from the Riemann tensor. Suppose you'd never seen the standard line element expressing the Schwarzschild vacuum solution to the EFE. If I gave you the (20? 21?) linearly independent components of the Riemann tensor, could you in principle deduce the metric tensor? Would the answer be, in a suitable sense taking account of gauge freedom, unique? Put in other words, do curvature, connection, and metric describe completely equivalent geometric information? (Compare: do the field three-vectors and potential four-vector in Maxwell's theory contain completely equivalent information?)
I don't plan to answer these questions in PF since I have promised to do that in another forum, but I will try to remember to provide a link here. But they are good questions and I invite forum members to try to think of good answers. We just had a thread on a classroom discussion for (as I guessed) junior high schoolers; these questions would make good discussion questions for a solid graduate level course in gtr! (That's another reason why I plan to give a link to a discussion elsewhere; these questions are probably a bit too sophisticated for discussion in PF.)
(Incidently, I consider it somewhat scandalous that AFAIK no gtr textbooks even mention these problems, which are clearly fundamental. But I soften that judgement by recognizing the near-impossibility of cramming even the basics into a reasonable one year course, much less a one-semester course, even at the fast paced graduate level. Still it seems useful to sometimes point out that even MTW attemped to cover only the first two thirds or so to an adequate introduction to a serious study of gtr. There outta be a book, called simply "Track Three"!)
To avoid possible confusion, I should stress that good answers are known. To avoid violating the constitutional proscription on cruel and unusual punishment, in case anyone gets hooked, then stuck, I should cite one of my favorite papers, "Counting solutions of Einstein's equations", by S. T. C. Siklos, Class. Quantum Grav. 13 (1996): 1931-1948, which addresses the first two questions. For some hints on the third group, try http://mathworld.wolfram.com/MetricEquivalenceProblem.html
(Warning! Counting invariants does not suffice, in the case of Lorentzian manifolds; the standard counterexample is this: all invariants to all orders vanish for all pp-waves, but there are zillions of distinct pp-wave spacetimes.)
It is delightful to end by noting that when Einstein said, "a theory should be as simple as possible, but no simpler", he had in mind a very simple and elementary notion for comparing the "richness" of physical theories, on which Siklos's paper is based! Einstein's idea seems to have arisen in the context of comparing the "richness" of Nordstrom's theory with Newtonian gravitation and Maxwell's theory, c. 1913. What landmark prediction did he narrowly miss deducing from "counting" alone?
kurious said:
Let me reformulate these questions, in three groups:
First: is there some reasonable way of "counting" the distinct source-free solutions to the Maxwell field equations? How about the vacuum solutions of the Einstein field equation? What about vacuum solutions in (the field theoretic reformulation of) Newtonian gravitation, or the Brans-Dicke theory of gravitation, the Nordstrom theory of gravitation, or the source-free solutions in other classical field theories in physics? How do these "counts" compare? Do the "degrees of freedom" implied by our counting suggest useful ways of reformulating our theories?
Second: is there some reasonable way of "counting" the distinct Lorentzian four-manifolds, taking account of our freedom to change coordinates? What about "counting" the number of one-forms on a four-dimensional smooth manifold? Or antisymmetric two-forms? Symmetric rank two tensor fields? How do these "counts" compare with each other, and with the counts from the first group? (We would expect, for example, that there should be "more" Lorentzian four-manifolds than there are vacuum solutions of the EFE or Brans-Dicke field equations, simply because said solutions are Lorentzian manifolds which are required to satisfy some additional conditions.) Does our counting suggest any "irredundant" representations of Lorentzian manifolds?
To avoid possible confusion, let me stress that "counting" the number of distinct Lorentzian four-manifolds is not the same thing at all as counting the number of linearly independent components of the metric tensor.
Third: we know how to compute the Riemann tensor if we are given the metric tensor, by a kind of differentiation. Going the other way, we'd expect to be able to find the metric tensor from the Riemann tensor. Suppose you'd never seen the standard line element expressing the Schwarzschild vacuum solution to the EFE. If I gave you the (20? 21?) linearly independent components of the Riemann tensor, could you in principle deduce the metric tensor? Would the answer be, in a suitable sense taking account of gauge freedom, unique? Put in other words, do curvature, connection, and metric describe completely equivalent geometric information? (Compare: do the field three-vectors and potential four-vector in Maxwell's theory contain completely equivalent information?)
I don't plan to answer these questions in PF since I have promised to do that in another forum, but I will try to remember to provide a link here. But they are good questions and I invite forum members to try to think of good answers. We just had a thread on a classroom discussion for (as I guessed) junior high schoolers; these questions would make good discussion questions for a solid graduate level course in gtr! (That's another reason why I plan to give a link to a discussion elsewhere; these questions are probably a bit too sophisticated for discussion in PF.)
(Incidently, I consider it somewhat scandalous that AFAIK no gtr textbooks even mention these problems, which are clearly fundamental. But I soften that judgement by recognizing the near-impossibility of cramming even the basics into a reasonable one year course, much less a one-semester course, even at the fast paced graduate level. Still it seems useful to sometimes point out that even MTW attemped to cover only the first two thirds or so to an adequate introduction to a serious study of gtr. There outta be a book, called simply "Track Three"!)
To avoid possible confusion, I should stress that good answers are known. To avoid violating the constitutional proscription on cruel and unusual punishment, in case anyone gets hooked, then stuck, I should cite one of my favorite papers, "Counting solutions of Einstein's equations", by S. T. C. Siklos, Class. Quantum Grav. 13 (1996): 1931-1948, which addresses the first two questions. For some hints on the third group, try http://mathworld.wolfram.com/MetricEquivalenceProblem.html
(Warning! Counting invariants does not suffice, in the case of Lorentzian manifolds; the standard counterexample is this: all invariants to all orders vanish for all pp-waves, but there are zillions of distinct pp-wave spacetimes.)
It is delightful to end by noting that when Einstein said, "a theory should be as simple as possible, but no simpler", he had in mind a very simple and elementary notion for comparing the "richness" of physical theories, on which Siklos's paper is based! Einstein's idea seems to have arisen in the context of comparing the "richness" of Nordstrom's theory with Newtonian gravitation and Maxwell's theory, c. 1913. What landmark prediction did he narrowly miss deducing from "counting" alone?
