TY  - JOUR
A1  - Schmidt, Hans-Jürgen
T1  - The tetralogy of Birkhoff theorems
JF  - General relativity and gravitation
N2  - 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 can be expressed by the Schwarzschild metric; for , 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 2-D 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 2-D.
KW  - Birkhoff theorem
KW  - Einstein space
KW  - Isometry group
Y1  - 2013
U6  - https://doi.org/10.1007/s10714-012-1478-5
SN  - 0001-7701
VL  - 45
IS  - 2
SP  - 395
EP  - 410
PB  - Springer
CY  - New York
ER  -