A Birkhoff's theorem with cosmological constant

[pervect]

Birkhoff's theorem says that any vacuum solution of Einstein's equations must be static, and asymptotically flat.

One of the consequences of Birkhoff's theorem is that the gravitational field inside any spherical shell of matter is zero, even if the shell is expanding.

But what happens if we allow a cosmological constant? Can we still say that the field inside a spherical shell of matter (including expanding shells) is zero if we assume that the universe has a non-zero cosmological constant?

Some context might help explain why I am asking this question. I am addressing the question of the effect (if any) of the cosmological expansion on the orbits of the Solar system. I want to justify ignoring the gravitational effect of the homogeneous part of the universe on the solar system via Birkhoff's theorem.

I'm a bit unclear about the applicability of Birkhoff's theorem to the case with the cosmological constant, unfortunately - and the universe in the latest models does have a cosmological constant.

Ultimately I want to reconcile the approach taken in
http://xxx.lanl.gov/abs/astro-ph/9803097

mentioned in Ned Wright's cosmology FAQ:
http://www.astro.ucla.edu/~wright/cosmology_faq.html#SS

which predicts a very, very small cosmological effect to the arguments presented in

http://arxiv.org/abs/gr-qc/0602002

which predict no effect at all. I wish to argue that what is important is the total mass contained within a sphere of radius R of the sun, and that the bulk of the expanding universe does not contribute at all to any solar system expansion.

To do this successfully, I need to know if Birkhoff's theorem does work in the presence of a cosmological constant.

 

[Greg Bernhardt]

@pervect did you ever find out more about this?

 

[pervect]

@pervect did you ever find out more about this?

No, I never did

 

[PeterDonis]

Birkhoff's theorem says that any vacuum solution of Einstein's equations must be static, and asymptotically flat.

More precisely, that any spherically symmetric vacuum solution must be isometric to the Schwarzschild geometry, which means that it has a 4th Killing vector field in addition to the 3 due to spherical symmetry. Note that the 4th Killing vector field is not necessarily timelike everywhere, so the solution is not necessarily static, and that the solution does not have to be asymptotically flat. For example, consider a black hole surrounded by a vacuum region surrounded by a homogeneous, isotropic FRW region that extends outward indefinitely. Such a solution will not be static inside the black hole's horizon, and it will not be asymptotically flat because FRW spacetime isn't.

what happens if we allow a cosmological constant?

There is a generalization of Birkhoff's theorem for this case which says that any spherically symmetric solution with a cosmological constant but no other stress-energy present is isometric to Schwarzschild-de Sitter spacetime. See, for example, these papers:

https://arxiv.org/pdf/0908.4110.pdf

https://arxiv.org/pdf/0910.5194.pdf

(Note that the second of these gives examples which are not asymptotically flat, similar to the one I described above. Actually, Schwarzschild-de Sitter spacetime itself is not asymptotically flat because de Sitter spacetime isn't.)

Can we still say that the field inside a spherical shell of matter (including expanding shells) is zero if we assume that the universe has a non-zero cosmological constant?

Not if "zero field" means "Minkowski spacetime", since for the case you describe the spacetime geometry inside the shell will be de Sitter or anti-de Sitter (depending on the sign of the cosmological constant), not Minkowski.

 

[PeterDonis]

the bulk of the expanding universe does not contribute at all to any solar system expansion

The presence of a cosmological constant doesn't contribute at all to expansion of the solar system, but it does change the parameters of the stable orbits in the solar system by a miniscule amount.