A simple proof of Birkhoff's theorem for cosmological constant

Abstract

We provide a simple, unified proof of Birkhoff's theorem for the vacuum and cosmological constant case, emphasizing its local nature. We discuss its implications for the maximal analytic extensions of Schwarzschild, Schwarzschild(-anti)-de Sitter and Nariai spacetimes. In particular, we note that the maximal analytic extensions of extremal and over-extremal Schwarzschild-de Sitter spacetimes exhibit no static region. Hence the common belief that Birkhoff's theorem implies staticity is false for the case of positive cosmological constant. Instead, the correct point of view is that generalized Birkhoff's theorems are local uniqueness theorems whose corollary is that locally spherically symmetric solutions of Einstein's equations exhibit an additional local Killing vector field.PACS numbers: 04.20.Cv, 04.20.Gz Birkhoff's theorem, that any spherically symmetric vacuum solution of Einstein's equations is locally isometric to a region in Schwarzschild spacetime, is not only a classic contribution to general relativity but is also an important tool in gravitational physics and cosmology. First proven for the vacuum Einstein equations, this theorem is traditionally credited to Birkhoff [1]; however it was recently rediscovered by Deser, when providing an alternate proof, that this theorem was published two years earlier by Jebsen [2,3,4]. (An english translation of Jebsen's paper with an introduction by Deser was published in General Relativity and Gravitation in 2005 [5, 6].) In older references, this theorem is credited as independently discovered by not only Birkhoff and Jebsen but also Alexandrow [7] and Eiesland [8].1 The multiple provenance of this theorem is a testament to its significance in general relativity. It is natural to consider its extension to other situations in gravitational physics. Birkhoff's theorem readily generalizes to the inclusion of electromagnetic fields [10], cosmological constant [8], with explicit exhibition of both the Schwarzschild-de Sitter and Nariai solutions in [11,12] (also in [13] without citation of previous references), and to other symmetry groups [14,15].2 Generalized Birkhoff's theorems have also been proven in lower and higher dimensions [16,17,18,19,20,21], certain alternate theories of gravity [22,23,24,25,26,27,28] including Lovelock gravity [29,30,31,32], and shown not to hold on Randall-Sundrum branes [33] and in some modified theories of gravity [34,35].Although Birkhoff's theorem is a classic result, many current textbooks and review articles on general relativity no longer provide a proof or even a careful statement of the theorem. Frequently it is cited as proving that the spherically symmetric vacuum solution is static. This is clearly not the case as recognized in many (but not all) proofs of the theorem (See, for example, the comment on Jebsen's proof in [36]). The interior region of the maximal analytic extension of Schwarzschild spacetime is not static; instead, it exhibits a spacelike Killing vector field. Of course, other regions of this spacetime do exhi...

Embedding Black Holes and Other Inhomogeneities in the Universe in Various Theories of Gravity: A Short Review

“…The Jebsen-Birkhoff theorem familiar from GR textbooks and stating the uniqueness of the Schwarzschild solution under the assumptions of vacuum, spherical symmetry, and asymptotic flatness can easily be generalized to show that the Schwarzschild-de Sitter/Kottler geometry is the unique spherically symmetric solution of Eq. (3.37) with Λ > 0 [127,119,50]. The Schwarzschild-de Sitter/Kottler line element is…”

Black hole non-modal linear stability: the Schwarzschild (A)dS cases

“…Alternatively, by Birkhoff's theorem in a cosmological background [28], the only possible spherically symmetric perturbation of a Schwarzschild-(A)dS black hole amounts to a change of the black hole mass, and this does not affect the unperturbed value Q − = 0. Since J 2 commutes with the wave operator in (93), it follows from (101) and Lemma 4.…”

IDEAL characterization of higher dimensional spherically symmetric black holes

“…The gluing is done with the help of the tortoise coordinate r * from (49). Penrose conformal diagrams for the extensions can be found in [18,25].…”

“…Proposition 6 ( [18,26,25,1]). Consider a pseudo-Riemannian geometry (M,ḡ) of dimension n = 2 + m that locally, say atx ∈M, has the form (M, g) × r (S, Ω) of a 2 + m-warped product, withḡ…”