Generalized Birkhoff theorem and its applications in mimetic gravity

Abstract

There is undetermined potential function V (φ) in the action of mimetic gravity which should be resolved through physical means. In general relativity(GR), the static spherically symmetric(SSS) solution to the Einstein equation is a benchmark and its deformation also plays a crucial role in mimetic gravity. The equation of motion is provided with high nonlinearity, but we can reduce primal nonlinearity to a frequent Riccati form in the SSS case of mimetic gravity. In other words, we obtain an expression of solution to the functional differential equation of motion with any potential function. Remarkably, we proved rigorously that there is a zero point of first order for the metric function β(r) if another metric function α(r) possesses a pole of first order within mimetic gravity. The zero point theorem may be regarded as the generalization of Birkhoff theorem αβ = 1 in GR. As a corollary, we show that there is a modified black hole solution for any given V (φ), which can pass the test of solar system. As another corollary, the zero point theorem provides a dynamical mechanism for the maximum size of galaxies. Especially, there are two analytic solutions which provide good fits to the rotation curves of galaxies without the demand for particle dark matter.

Effective Quantum Dust Collapse via Surface Matching

“…To avoid confusion with in-/outside the horizon or black hole interior /exterior, we refer to the two regions as matter and vacuum regions. 3 In alternative theories of gravity there are results in supporting the validity of Birkhoff's theorem [78] (for f (R) gravity), [79] (generalisation in mimetic gravity), [80] (Dilaton gravity), but there is also evidence against it [81,82].…”

Mimetic Black Holes

“…Some attempts have been made in this direction in the Refs. [32,33,34,35,36,37]. But the difficult part is that in order to have a static black hole solution we need to have an event horizon.…”