Birkhoff's theorem with cosmological constant

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Birkhoff's theorem states that any vacuum solution of Einstein's equations must be static and asymptotically flat, leading to the conclusion that the gravitational field inside a spherical shell of matter is zero, regardless of the shell's expansion. The discussion raises the question of how the introduction of a cosmological constant affects this conclusion. Specifically, it explores whether the gravitational field inside a spherical shell remains zero when considering a non-zero cosmological constant in the universe. The implications of this scenario on gravitational dynamics and the nature of vacuum solutions are examined. Understanding these effects is crucial for further developments in cosmological models.
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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?
 
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