Covariant approach to the thermodynamic structure of a generic null surface

Kinematics and dynamics of null hypersurfaces in the Einstein-Cartan spacetime and related thermodynamic interpretation

“…The idea was originally introduced in [30,76] based on an infinitesimal virtual displacement along the auxiliary null vector field. Since the original analysis was a non-covariant one, we here followed the spirit of our earlier work [50] in order to provide a covariant formalism for a torsion-full spacetime.…”

“…In [50], while analysing the thermodynamic interpretation provided to a generic null surface (under virtual displacement) in general spacetimes without any torsion, the authors described the notion of a "gravitational pressure" defined as P = − 1 8π G ab k a l b . Hence for a generic null surface in Einstein gravity…”

“…Contrary to this, the pressure term (leading to the work function) is entirely from quantities defined in the manifold (M, g, ∇). Infact, it has been explicitly shown [50] that at least for Einstein gravity (with zero torsion), the covaraint expression of the energy (6.11) reduces to expressions of energy for well known spacetimes. For example, the computation of (6.11) for the Scharzschild metric gives us the mass term.…”

“…A completely covariant approach to the thermodynamic structure provided by G ab k a l b on the null surface was given in [50] and was also applied in the case of Scalar-Tensor theory of gravity [51]. It was pointed out in [30] that G ab k a l b is more natural a projection component than G ab l a l b for the thermodynamic interpretation.…”

“…The thermodynamic interpretation for Ĝab l a l b has been provided in [46] and hence will not be pursued here. In absense of torsion, it can be shown [50] that G ab k a l b is related to the dynamics of the ingoing expansion scalar as opposed to the outgoing one in spacetimes without torsion. This can be achieved by taking the trace of the evolution equation of the transversal deformation rate tensor [75].…”