Temperature of free gravitational field: A geometrical perspective

In this paper, using a novel geometrical approach, we relate the concept of the thermodynamic temperature of the free gravitational field, to the non-affinity of real null geodesics in a Newman Penrose tetrad. This naturally links various temperature functions like Clifton, Ellis and Tavakol temperature, Hawking temperature, Unruh temperature etc., in their respective proper limits. Although our analysis is done within the realm of local rotational symmetry, we show that the result can be extended to other Petrov type D geometries, like the Kerr spacetime. We also obtain the geometrical and causal transport equations for this temperature function, in the form of a hyperbolic wave equation with a forcing term, sourced by Weyl curvature and matter. Finally, as a possible physical interpretation of the non-affinity, we relate the geometrical temperature with the gravitational red/blue shift of light rays travelling along null geodesics.