general relativity, Birkhoff–Jebsen's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically In general relativity, Birkhoff–Jebsen's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution (i.e. the spacetime outside of a spherical, nonrotating, gravitating body) must be given by the Schwarzschild metric. The converse of the theorem is true and is called Israel's theorem. The conver In general relativity, Birkhoff–Jebsen's theorem states that any…

We classify the existent Birkhoff-type theorems into four classes: First, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star/ therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff's theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein's vacuum field equation with Lambda = 0 can be expressed by the Schwarzschild metric/ for Lambda unequal 0, it is the Schwarzschild-de Sitter metric instead. Fourth, in differential geometry, any statement of the type: every member of a family of pseudo-Riemannian space-times has more isometries than expected from the original metric ansatz, carries the name Birkhoff-type theorem. Within the fourth of these classes we present some new results with further values of dimension and signature of the related spaces/ including them are some counterexamples: families of space-times where no Birkhoff-type theorem is valid. These counterexamples further confirm the conjecture, that the Birkhoff-type theorems have their origin in the property, that the two eigenvalues of the Ricci tensor of two-dimensional pseudo-Riemannian spaces always coincide, a property not having an analogy in higher dimensions. Hen

We attempt to answer whether Birkhoff's theorem (BT) is valid in the Einstein-Aether (EA) theory. The BT states that any spherically symmetric solution of the vacuum field equations must be static, unique, and asymptotically flat. For a general spherically symmetric metric with metric functions A(r,t) & B(r,t), and aether components a(r,t) & b(r,t), we prove the conditions for the staticity of spacetime using two different methods. We point out that BT is valid in EA theory only for special values of c1+c3, c1+c4, and c2, where we can show that all these special cases are asymptotically flat. In particular, when the aether has only a temporal component i.e., b(r,t)=0, the c14≠0 case gives us spherically symmetric static solutions with singularities without Killing nor universal horizons, at least for special values of c14. However, when we have an aether vector with temporal and radial components, we prove that the staticity and the flatness at infinity hold for only a special metric and a particular combination of the aether parameters. These solutions have universal horizons.

AbstractBrikhoff's theorem states that if the geometry of a given region of space-time is first spherically symmetric and secondly a solution to the Einstein empty space equations, that then that geometry is a piece of the Schwarzschild geometry. Here we show that Birkhoff's theorem holds on differentiable manifolds in general.