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, 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.