Thermodynamical interpretation of the geometrical variables associated with null surfaces

The emergent gravity paradigm interprets gravitational field equations as describing the thermodynamic limit of the underlying statistical mechanics of microscopic degrees of freedom of the spacetime. The connection is established by attributing a heat density Ts to the null surfaces where T is the appropriate Davies-Unruh temperature and s is the entropy density. The field equations can be obtained from a thermodynamic variational principle which extremizes the total heat density of all null surfaces. The explicit form of s determines the nature of the theory. We explore the consequences of this paradigm for an arbitrary null surface and highlight the thermodynamic significance of various geometrical quantities. In particular, we show that: (a) A conserved current, associated with the time development vector in a natural fashion, has direct thermodynamic interpretation in all Lanczos-Lovelock models of gravity. (b) One can generalize the notion of gravitational momentum, introduced in arXiv 1506.03814 to all Lanczos-Lovelock models of gravity such that the conservation of the total momentum leads to the relevant field equations. (c) The thermodynamic variational principle which leads to the field equations of gravity can also be expressed in terms of the gravitational momentum in all Lanczos-Lovelock models. (d) Three different projections of gravitational momentum related to an arbitrary null surface in the spacetime lead to three different equations, all of which have thermodynamic interpretation. The first one reduces to a Navier-Stokes equation for the transverse drift velocity. The second can be written as a thermodynamic identity TdS = dE + P dV. The third describes the time evolution of the null surface in terms of suitably defined surface and bulk degrees of freedom. The implications are discussed.