What does Birkhoff's theorem really tell us?

Abstract

Birkhoff's theorem is a classic result that characterizes locally spherically symmetric solutions of the Einstein equations. In this paper, we illustrate the consequences of its local nature for the cases of vacuum and positive cosmological constant. We construct several examples of initial data for spherically symmetric spacetimes on Cauchy surfaces of different topology than R × S 2 , that of the maximal analytic extension of Schwarzschild and Schwarzschild-de Sitter spacetimes. The spacetimes formed from the evolution of these initial data sets also have very different physical properties; in particular they need not contain a static region or be asymptotically flat or asymptotically de Sitter. We also present locally spherically symmetric initial data sets for de Sitter spacetimes that are not covered by the maximal analytic extension of de Sitter spacetime itself. Finally we illustrate the utility of Birkhoff's theorem in identifying the spacetimes associated with two spherically symmetric initial data sets; one proven to exist but not explicitly exhibited and one which has negative ADM mass.

Spacetime foam: a review

“…Such generalizations come in two distinct strands. Morrow-Jones, Witt, and Schleich have carried out detailed studies of locally spherically symmetric initial data [81][82][83]. At first sight, the requirement of spherical symmetry seems very restrictive: it is widely thought that for a vacuum spacetime, spherical symmetry leads inevitably to Schwarzschild or Schwarzschild-(anti) de Sitter spacetimes.…”

Spacetime foam: a review

“…Such generalizations come in two distinct strands. Morrow-Jones, Witt, and Schleich have carried out detailed studies of locally spherically symmetric initial data [77][78][79]. At first sight, the requirement of spherical symmetry seems very restrictive: it is widely thought that for a vacuum spacetime, spherical symmetry leads inevitably to Schwarzschild or Schwarzschild-(anti) de Sitter spacetimes.…”

“…At first sight, the requirement of spherical symmetry seems very restrictive: it is widely thought that for a vacuum spacetime, spherical symmetry leads inevitably to Schwarzschild or Schwarzschild-(anti) de Sitter spacetimes. But while Birkhoff's theorem implies that a spherically symmetric vacuum spacetime must be locally isometric to some region of Schwarzschild, local patches can be sewn together to form spacetimes that look drastically different [79]. In particular, one can form connected sums of large classes of three-manifolds with such data, including arbitrary quotients of S 3 , R 3 , and H 3 , with a construction that is almost completely explicit [77].…”

Midisuperspace foam and the cosmological constant

“…The setting for locally spherically symmetric midisuperspace has been studied extensively by Morrow-Jones, Witt, and Schleich [5][6][7]. At first sight, this symmetry requirement may seem too strong: it is widely believed that spherical symmetry with Λ > 0 leads inevitably to Schwarzschild-de Sitter space.…”

“…At first sight, this symmetry requirement may seem too strong: it is widely believed that spherical symmetry with Λ > 0 leads inevitably to Schwarzschild-de Sitter space. But while Birkhoff's theorem implies that a spherically symmetric vacuum spacetime must be locally isometric to some region of Schwarzschild-de Sitter space, local patches can be sewn together to form a spacetime that looks drastically different [7]. In particular, we will be able to construct explicit initial data containing both expanding and contracting regions.…”