Jebsen-Birkhoff theorem in alternative gravity

Abstract

We discuss the validity, or lack thereof, of the Jebsen-Birkhoff theorem in scalar-tensor theories by generalizing it and regarding the Brans-Dicke-like scalar as effective matter. Both the Jordan and Einstein frames are discussed and an apparent contradiction between static spherical solutions of scalar-tensor gravity and Hawking's theorem on Brans-Dicke black holes is clarified. The results are applied to metric and Palatini f (R) gravity.

Birkhoff’s theorem in f(T) gravity

“…It has been confirmed to be valid in the Palatini formalism of f (R) gravity, while it no longer holds in the metric formalism generally [7,19,20]. Similar to the origination of f (R) theories, the f (T ) gravity is extended from the teleparallel gravity which is equivalent to the theory of general relativity.…”

“…(27) only present in the form of (A ′ /A), just as in the case of Eq. (20). Consequently, the relation (21) is preserved, and the static metric so (25) is obtained again.…”

“…The f (T ) models investigated by several authors before take variant forms, most of which are more complex than that we just considered in (16) form. It is impracticable to express the field equations of all these cases so specifically as in the (20). Nonetheless, without being concerned with the particular forms of f (T ) models, we give the two field equations in vacuum E…”

Is There Unification in the 21st Century?

“…The failure of Birkhoff's theorem in higher derivative gravity theories means that spherically symmetric solution is time-dependent and dynamical. A similar failure of Birkhoff's theorem in a generalization of Einstein's gravity called f (R) theory, in which the action is a nonlinear function of the scalar curvature R, also shows that spherically symmetric solutions are time-dependent [7]. As a result, black holes in these theories are dynamical.…”

Higgs shifts from electron–positron annihilations near neutron stars

“…the fact that the Schwarzschild solution is the unique spherically symmetric vacuum solution, does not hold in metric f (R) gravity, and R = 0 even if T = 0 (see [69] for a dedicated study of possible counterexamples). This means that a putative observed Higgs shift will be differently interpreted in various approaches, as due to the Higgs-Kretschmann coupling in ordinary general relativity, of as a Ricci-Higgs coupling in higher dimensional theories [70,71], f (R) theories [72,73], or in Hořava gravity [74][75][76]. Further observables will therefore be necessary to disentangle the various theoretical scenarios.…”