Uniqueness theorems for black holes

[Max Green]

I am under the impression, there is no unique solutions to Einstein's field equations for a cosmological constant, or for higher dimensional spacetimes. Has anybody got a counter example for a solution including the cosmo constant to show there are multiple solutions, for example, i know of the de sitter spacetimes (\ads) , but I am not sure of any others. Is the FLRW metric an exact solution for a cosmological constant? I am studying specifically black holes.

 

[martinbn]

Of course there are infinitely many solution of the Einstein field equations with cosmological constant.

 

[Max Green]

Could you give me an example of a few of these solutions? Other than de sitter/anti solution

 

[PeterDonis]

I am under the impression, there is no unique solutions to Einstein's field equations for a cosmological constant

Where are you getting this impression from? And what do you mean by "unique solutions"? Obviously there are infinitely many de Sitter spacetimes, since there are infinitely many possible values for the cosmological constant. Do those count as different solutions, or do all of them count as just one? Also, if the cosmological constant is zero or negative, the global character of the solution is significantly different; does that count as a different solution?

Is the FLRW metric an exact solution for a cosmological constant?

The FLRW metric with a cosmological constant (and no other matter content) is just de Sitter spacetime (more precisely, de Sitter if the constant is positive, or anti-de Sitter if the constant is negative).

 

[Max Green]

I mean, for asymptotically flat 4-d spacetimes, we have unique solutions, ie. kerr metric for rotational source with no charge, in which we only require the mass and angular momentum to find a unique black hole solution. I'm wanting to apply this for EFE with a cosmological constant, why are there multiple solutions instead of a unique one such as kerr metric, kerr newman, schwarzschild in the vacuum flat case.

 

[PeterDonis]

for asymptotically flat 4-d spacetimes

I'm wanting to apply this for EFE with a cosmological constant

There are no asymptotically flat solutions with a nonzero cosmological constant. So what you are trying to do doesn't make sense.

 

[PeterDonis]

why are there multiple solutions instead of a unique one such as kerr metric, kerr newman, schwarzschild in the vacuum flat case

What "multiple solutions" are you referring to?

 

[Max Green]

What "multiple solutions" are you referring to?

Ok to rephrase this, why is there no uniqueness theorem for EFE with a cosmological constant..can you give a counter example? or a simple reason as to why we don't have uniqueness theorems like birkhoff's or robinsons/israels but for a cosmo constant

 

[PeterDonis]

why is there no uniqueness theorem for EFE with a cosmological constant

If you mean a solution to the EFE with a cosmological constant and no other stress-energy present, the solutions are known: de Sitter spacetime for a positive cosmological constant, and anti-de Sitter spacetime for a negative cosmological constant.

I'm still not sure what you mean by "unique solutions" or why you think this is somehow an issue.

 

[Max Green]

so is there a uniqueness theorem for the EFE with a cosmological constant? if there is only one solution (de sitter spacetime) for positive constant, that is a unique solution for a cosmological constant? are there other metrics that can describe a spacetime with cosmo constant, T=0, other than de sitter?

 

[PeterDonis]

so is there a uniqueness theorem for the EFE with a cosmological constant?

I don't know if there is a named theorem. See below.

if there is only one solution (de sitter spacetime) for positive constant, that is a unique solution for a cosmological constant?

You tell me; you're the one who keeps using the term "unique solution" without explaining what you mean by it, despite being asked.

are there other metrics that can describe a spacetime with cosmo constant, T=0, other than de sitter?

More precisely, as I've said, de Sitter for a positive cosmological constant, or anti-de Sitter for a negative cosmological constant. There are also combinations of the Schwarzschild solution with both of those: Schwarzschild-de Sitter and Schwarzschild-anti de Sitter. Those are the only solutions I'm aware of with a nonzero cosmological constant and zero stress-energy tensor.

 

[Max Green]

Haha calm down mate only asking, you’re clearly not hearing what I’m saying. Don’t worry about it, I’ll get help from somebody qualified

 

[PeterDonis]

you’re clearly not hearing what I’m saying

Well, I've answered every question you asked except for the one about "unique solution", which, as I said, is a term you keep using without explaining what you mean by it. You're welcome to ask further questions if you need to--but you'll have to do so in a new thread, see below.

I’ll get help from somebody qualified

This, however, just got you this thread closed and a warning. People here respond to your questions out of the goodness of their hearts. Please keep that in mind when responding to them.