Generalized Birkhoff theorem and its applications in mimetic gravity

Abstract

There is undetermined potential function $V(\varphi )$ 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 $\beta (r)$ if another metric function $\alpha (r)$ possesses a pole of first order within mimetic gravity. The zero point theorem may be regarded as the generalization of Birkhoff theorem $\alpha \beta =1$ in GR. As a corollary, we show that there is a modified black hole solution for any given $V(\varphi )$, 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.

arXiv comment11 pages

References in corpus (14)

• The Relation between Physical and Gravitational Geometry
• Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations
• On the recently proposed Mimetic Dark Matter
• Fitting galactic rotation curves with conformal gravity and a global quadratic potential
• Impact of a global quadratic potential on galactic rotation curves
• Nonsingular Black Hole
• Covariant Horava-like and mimetic Horndeski gravity: cosmological solutions and perturbations
• Spherically symmetric static vacuum solutions in Mimetic gravity
• Black hole solutions in de Rham-Gabadadze-Tolley massive gravity
• Dark Energy Oscillations in Mimetic $F(R)$ Gravity
• Ghost Free Mimetic Massive Gravity
• Vaidya solution and its generalization in de Rham-Gabadadze-Tolley massive gravity
• Beyond dRGT as Mimetic Massive
• Dynamical spacetimes in conformal gravity

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