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?

 

[Valerie Park]

The answer is yes. Birkhoff's theorem still holds, with the addition of a term related to the cosmological constant. The gravitational field inside a spherical shell of matter is still zero, even if the shell is expanding and there is a non-zero cosmological constant.

 

[sir-pinski]

Birkhoff's theorem with a cosmological constant is a generalization of the original theorem that still holds true. It states that any spherically symmetric vacuum solution of Einstein's equations with a cosmological constant must be static and asymptotically de Sitter. This means that the universe described by the solution will have a constant, non-zero value of the cosmological constant and will expand at a constant rate.

In this case, the theorem still holds true in the sense that the gravitational field inside a spherical shell of matter will be zero, even if the shell is expanding. This is because the presence of the cosmological constant does not affect the spherical symmetry of the solution, and therefore the field inside the shell is still described by the same vacuum solution.

However, the presence of the cosmological constant does have an impact on the overall geometry of the universe. It introduces a repulsive force that counteracts the attractive force of gravity, causing the expansion of the universe to accelerate. This means that the shells of matter will expand at a faster rate than they would in the absence of a cosmological constant.

In summary, Birkhoff's theorem with a cosmological constant still holds true and tells us that the gravitational field inside a spherical shell of matter is zero. However, the cosmological constant does introduce a new dynamic to the universe, causing it to expand at an accelerated rate.