A gravitational energy–momentum and the thermodynamic description of gravity

Abstract

A proposal for the gravitational energy-momentum tensor, known in the literature as the square root of Bel-Robinson tensor, is analyzed in detail. Being constructed exclusively from the Weyl part of the Riemann tensor, such tensor encapsulates the geometric properties of free gravitational fields in terms of optical scalars of null congruences: making use of the general decomposition of any energymomentum tensor, we explore the thermodynamic interpretation of such geometric quantities. While the matter energy-momentum is identically conserved due to Einstein's field equations, the SQBR is not necessarily conserved and dissipative terms could arise in its vacuum continuity equation. We discuss the possible physical interpretations of such mathematical properties.

Canonical Noether and the energy–momentum non-uniqueness problem in linearized gravity

“…Due to Magnano and Sokolowski's no-go result [1], one can consider energy-momentum tensors in higher derivative gravity in order to obtain a spin-2 gauge invariant expression (invariant under linearized diffeomorphisms), such as the variants of the Bel-Robinson tensor [48,49], or the linearized Gauss-Bonnet gravity energy-momentum tensor [50][51][52], which are both invariant under the spin-2 gauge transformation (linearized diffeomorphisms). However, since these models require higher derivative actions, they are not connected to spin-2 Fierz-Pauli theory via standard Lagrangian based energy-momentum derivations such as the Noether method or the Hilbert (metric) method.…”

Newman–Penrose scalars and black hole equations of state

“…where c is an arbitrary constant introduced in order to compare with different possible conventions as in Ref. [46]. Different choices for the arbitrary function H have appeared in the literature, basically depending on the vanishing of the covariant divergence of t ab , u a ∇ b t ab .…”

“…[47,48] and, on the other hand, H = 0 has been chosen in Ref. [46] in order to secure a traceless t ab and, therefore, a massless carrier of the gravitational field. A fluid-like interpretation of the SQBR tensor can be specified once a timelike congruence u a , with u a u a = 1, has been chosen.…”

“…In the frame adapted to the principal null directions (the comoving frame), and choosing H = 0, these thermodynamic quantities read [46] µ g = c |Ψ 2 |,…”

“…where the triad {x a , y b , z c } is a set of orthonormal basis vectors (see [46] for details). Interestingly, the EoS…”