General relativistic gravitational induction and causal temperatures

Abstract In this paper, we describe the process of general relativistic gravitational induction in spherically symmetric spacetimes by defining an energy momentum tensor for the induction process, which is divergence-free and hence conserved. The aforementioned tensor explicitly describes how the matter-free gravity, as measured by the geometrical Weyl curvature, interacts with the matter. This tensor is clearly different from the energy momentum tensor of the standard matter and we transparently show that in spherical symmetry, the Bianchi identities reduce to the conservation laws for these two such energy momentum tensors. Working with a semitetrad covariant formalism in spherically symmetric spacetimes, we then demonstrate the process of constructing a consistent causal thermodynamical picture for the free gravity and matter interaction via the general non-truncated Israel-Stewart heat transport equation. As an illustrative example, we consider the Lemaitre-Tolman-Bondi spacetime to highlight the relationship between the shear and the Weyl curvature in determining the inductive heat flux.