Final state conjecture in general relativity?

[21joanna12]

Hi all,

I am really curious about the final state conjecture in general relativity, but I don't really understand it... There seems to be a really good explanation provided by Willie Wong here:

http://math.stackexchange.com/questions/50521/open-problems-in-general-relativity

however it is still a bit beyond me. I was wondering if anyone could explain it in layman terms? So far, I get that there are two possibilities for the future of our universe- contract into a singularity or expand to form mainly empty space. First thing I am stuck on is what Minkowski space has to do with a sparse universe... I am guessing it is about setting the state of the universe (I think in the stress-energy tensor?) to being 'very sparse' and then when you see how it evolves with time, you can solve the field equations to find that the metric tensor approaches Minkowski space? I thought we already exist in Minkowski space, but it seems to me like this is what the above article is suggesting...

Thank you for any help :)

EDIT: Am I correct in saying that Minkowski space is flat space is flat spacetime and so the fact that when you run the Einstein field equations forward in time the metric tensor is either asymptotic to Minkowski space or forms a singularity means that in the future our spacetime will become eiither more flat and keep on expanding forever or there will be a big crunch? If so, II still don't see why this would be a problem for general relativity?

 

[WannabeNewton]

Hi Joanna. We do not currently live in Minkowski space-time and Willie Wong is not saying that we are in any way. We live in a curved space-time. What he/she is saying is if we start with a solution to the Einstein equations consisting of a universe that is already sparse then the solution will asymptotically approach the Minkowski solution. This is because, as Christodoulou et al proved in an extremely famous ~500 page book, Minkowski space-time is a stable solution to the Einstein equations. This means that small perturbations of Minkowski space-time, e.g. a sparse universe, will eventually (asymptotically) settle down to the stationary Minkowski space-time.

If on the other hand we expect the universe to settle down asymptotically to a stationary black hole solution to the Einstein equations then one has to prove quite generally the black hole uniqueness theorem for the very clear reason given by Willie Wong i.e. the asymptotic stationary state has to be unique. Finally, for the exact same reasons as for the Minkowski solution, one must show that the family of stationary black hole solutions, e.g. Kerr-Newman, are stable.

A few years ago there was held the Kerr conference to celebrate the 50 year anniversary of Kerr's discovery of the solution to the Einstein equations of a stationary axisymmetric rotating space-time. In it Mihalis Dafermos gave a very self-contained and clear overview of the stability of Kerr. I would recommend watching it:

 

[bcrowell]

Googling turned up this statement of the final state conjecture:

...generic asymptotically flat initial data sets have maximal future developments, namely solutions of the Einstein vacuum equations, which look, asymptotically, in any finite region of space, as a Kerr black hole solution.

-- http://www.researchperspectives.org/topicmaps/grant.php?id=1065710

Willie Wong's statement of the problem doesn't seem right, since he talks about the "universe," but the universe isn't asymptotically flat.

The FSC seems pretty physically plausible to me. We don't expect to have gravitational states that look like an uncollapsed static equilibrium, for basically the same reason that such a thing can't happen in classical electromagnetism: if a field is such as to give a restoring force to a particle no matter which direction it moves from equilibrium, then that field would violate Gauss's law. But if the equilibrium is dynamic, then we expect orbiting bodies to gradually radiate away their energy in the form of gravitational waves. (Again, this is the same idea as in classical electromagnetism.)

I would think that the FSC would be false unless you assumed some energy conditions, since stuff could probably fly apart before it got a chance to form black holes.

I imagine that the FSC also tells us that in some sense the generalized entropy not only increases, as required by the generalized second law of thermodynamics, but also increases to something like the possible value it could possibly have. But this is probably tricky to state exactly, since the entropy might be infinite.