Black holes in the tensor-vector-scalar theory of gravity and their thermodynamics

Tensor-vector-scalar (TeVeS) theory, a relativistic theory of gravity, was designed to provide a basis for the modified Newtonian dynamics. Since TeVeS differs from general relativity (e.g., it has two metrics, an Einstein metric and a physical metric), black hole solutions of it would be valuable for a number of endeavors ranging from astrophysical modeling to investigations into the interrelation between gravity and thermodynamics. Giannios has recently found a TeVeS analogue of the Schwarzschild black hole solution. We proceed further with the program by analytically solving the TeVeS equations for a static spherically symmetric and asymptotically flat system of electromagnetic and gravity fields. We show that one solution is provided by the Reissner-Nordström metric as physical metric, the TeVeS vector field pointing in the time direction, and a TeVeS scalar field positive everywhere (the last feature protects from superluminal propagation of disturbances in the fields). We work out black hole thermodynamics in TeVeS using the physical metric; black hole entropy, temperature, and electric potential turn out to be identical to those in general relativity. We find it inconsistent to base thermodynamics on the Einstein metric. In light of this, we reconsider the Dubovsky-Sibiryakov scenario for violating the second law of thermodynamics in theories with Lorentz symmetry violation.

Figure 1

Our solution for $\varphi -{\varphi }_c$ as a function of $r$ for several values of $M/r_h$: the higher $M/r_h$, the lower the curve. Both axes are in arbitrary units.