Spherically Symmetric, Static Spacetimes in a Tensor-Vector-Scalar Theory

Recently, a relativistic gravitation theory has been proposed by Bekenstein cite{dg:Bek} 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. Here, I consider the strong gravity regime of Tensor-Vector-Scalar (TeVeS) and study spherically symmetric, static and vacuum spacetimes. 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. The Parameterized Post-Newtonian (PPN) coefficients are calculated for these cases. For the first branch of solutions, I derive exact analytic expressions for the physical metric and apply them to the case of black holes. Implications are discussed.