# Birkhoff's theorem with cosmological constant

#### pervect

Staff Emeritus
Science Advisor
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.

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