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Now that an equivalence of Navier Stokes and Einstein equations has been established, and it is known solutions to Einstein-Maxwell-Boltzmann exist and are unique, and it is known that Einstein equations in a wave guage with electromagnetic source with a lorentz gauge also have unique solutions that exist, what are the next steps? What are the good references on research for existence and uniqueness for einsten solutions without electromagnetic sources? Also, what does it mean physically that there are unique solutions to Einstein Field equations when electromagnetic sources are included?

For those struggling with the concept of existance and uniqueness, I think this post provides some good insight.

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The Navier-Einstein paper is a link in p+1 to p+2 dimensions. The two references to EMB go to the same paper which proves its result for 3 and 4 dimensions only. So one issue is whether one is interested in p-dimensional theory or only "classical" GR with D=4. I suppose the Answers might bifurcate on the answer to this. –  Roy Simpson Jan 28 '11 at 12:29
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General Relativity has a well-defined Initial Value Problem. There are existence and uniqueness theorems for all space-times with suitable initial data. You can find all this in the chapter on the IVP in Wald –  user346 Jan 28 '11 at 13:13
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Dear Humble, I am extremely grateful to you for the link to the paper because I had to miss it in some way, and it is cool, but could you try to make the question more specific? If you're asking how to write the most famous paper about similar topics that will be celebrated in 2012, and it surely looks like you're asking exactly that, well, no one knows, and even if someone happened to know, he would probably not be telling you the stuff in advance on a random server. Also, the new paper you linked has no electromagnetism in it, so your mixing of "Maxwell" into the question just adds chaos. –  Luboš Motl Jan 28 '11 at 14:33
    
@lumo: your right, I think I am being a little ambitious in asking this question, although it would be exciting if someone ponied up an answer. I guess my thinking is that even if the new paper has no maxwell equations in it, it seems that if exact solutions exist to problems with electromagnetism, then there should be an analog, or possible equivanlence in some equations similar to Navier Stokes. I guess I would welcome any suggestions on how I would ask a question that is better balanced and still interesting enough for this forum. –  Humble Jan 28 '11 at 21:58
    
@Roy Simpson: I have to confess my mind was only tracking in D=4 GR, it sometimes takes me time to really digest all the details and understand these things better. –  Humble Jan 28 '11 at 22:04

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The formal classification of solutions to Einstein field equations is the Penrose-Petrov-Pirani scheme. This classifies solutions according to eigen-Killing vectors of the Weyl tensor. These solutions range from the type D solutions for black holes to the type N solutions for gravity waves. In between there are type I II and III solutions for Robinson-Trautman spacetimes. These turn out to have a relationship to each other, where type D solutions are near source terms and type N solutions are far field. The other solutions are in between. This is a gravitational analogue of the near and far field solutions for Mazwell’s equations. A source on these solutions is in:

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge: Cambridge University Press. (2003)

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Someone clearly has a personal issue with you @Lawrence, otherwise so many of your perfectly good answers would not be getting down-voted without explanation. It sucks. –  user346 Feb 8 '11 at 17:39

Classically, there are initial configurations in general relativity without closed timelike curves which can be shown to evolve into future states with closed timelike curves. The boundary between both regions is the Cauchy horizon, and evolution across the Cauchy horizon is nondeterministic.

For anti de Sitter space without any conformal boundary conditions, we also have nondeterministic evolution, even far away from conformal infinity.

Also, classically, anything can come out of a white hole singularity.


Probably the best bet for dealing with questions about existence and uniqueness is to work with the ADM time-slicing Hamiltonian approach.

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