Spherically symmetric, static spacetimes in a tensor-vector-scalar theory

TL;DR

This paper investigates spherically symmetric, static vacuum solutions in TeVeS, a relativistic theory of gravity that reproduces MOND, revealing two solution branches with distinct physical properties and causality issues near black holes.

Contribution

It identifies and analyzes two solution branches in TeVeS for static, spherically symmetric spacetimes, highlighting their differences from general relativity and their implications for black hole physics.

Findings

First branch matches GR PPN parameters, second does not.

Solutions near black holes can violate causality.

Some solutions exhibit negative energy density.

Abstract

Recently, a relativistic gravitation theory has been proposed [J. D. Bekenstein, Phys. Rev. D {\bf 70}, 083509 (2004)] that gives the Modified Newtonian Dynamics (or MOND) in the weak acceleration regime. The theory is based on three dynamic gravitational fields and succeeds in explaining a large part of extragalactic and gravitational lensing phenomenology without invoking dark matter. In this work we consider the strong gravity regime of TeVeS. We study spherically symmetric, static and vacuum spacetimes relevant for a non-rotating black hole or the exterior of a star. Two branches of solutions are identified: in the first the vector field is aligned with the time direction while in the second the vector field has a non-vanishing radial component. We show that in the first branch of solutions the \beta and \gamma PPN coefficients in TeVeS are identical to these of general relativity (GR) while in the second the \beta PPN coefficient differs from unity violating observational determinations of it (for the choice of the free function $F$ of the theory made in Bekenstein's paper). For the first branch of solutions, we derive analytic expressions for the physical metric and discuss their implications. Applying these solutions to the case of black holes, it is shown that they violate causality (since they allow for superluminal propagation of metric, vector and scalar waves) in the vicinity of the event horizon and/or that they are characterized by negative energy density carried by the fields.