The tetralogy of Birkhoff theorems

Abstract

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. Hence, Birkhoff-type theorems exist only for those physical situations which are reducible to two dimensions.

26 pages, updated references, minor text changes, accepted by Gen. Relat. Grav

References in corpus (22)

• A new cubic theory of gravity in five dimensions: Black hole, Birkhoff's theorem and C-function
• Fourth order gravity: equations, history, and applications to cosmology
• General class of wormhole geometries in conformal Weyl gravity
• Spherically Symmetric Solutions to Fourth-Order Theories of Gravity
• Static Solutions for 4th order gravity
• The Cauchy problem for metric-affine f(R)-gravity in presence of perfect-fluid matter
• A general solution in the Newtonian limit of f(R)- gravity
• The Newtonian limit of metric gravity theories with quadratic Lagrangians
• Gauss-Bonnet black holes with non-constant curvature horizons
• Ramifications of Lineland
• General solutions of Einstein's spherically symmetric gravitational equations with junction conditions
• Static wormholes in vacuum for conformal gravity
• Quantum Dilaton Gravity in Two Dimensions with Fermionic Matter
• The strong equivalence principle from gravitational gauge structure
• Spontaneous breaking of conformal invariance, solitons and gravitational waves in theories of conformally invariant gravitation
• Duality in 2-dimensional dilaton gravity
• Shock waves and Birkhoff's theorem in Lovelock gravity
• Four-dimensional couplings among BF and massless Rarita-Schwinger theories: a BRST cohomological approach
• On the validity of the 5-dimensional Birkhoff theorem: The tale of an exceptional case
• On the generalized Freedman-Townsend model
• Time (in)dependence in general relativity
• Birkhoff's theorem and perturbations in $f(R)$ theories

Cited by in corpus (2)

• Exact radial solution in 2+1 gravity with a real scalar field
• Symmetry Inheritance and Jebsen-Birkhoff Theorem