  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.
