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

Recently, a relativistic gravitation theory has been proposed [J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004)] that gives the modified Newtonian dynamics 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, I consider the strong gravity regime of TeVeS. I study spherically symmetric, static, and vacuum spacetimes relevant for a nonrotating 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 nonvanishing radial component. I show that in the first branch of solutions the {beta} and {gamma} parametrized post-Newtonian (PPN) coefficients in TeVeS are identical to these of general relativity, 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, I derive analytic expressions for the physical metric and discuss their implications. Applying these solutions to the case of black holes, itmore » 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.« less