Distribution function of the Atoms of Spacetime and the Nature of Gravity

The fact that the equations of motion for matter remain invariant when a constant is added to the Lagrangian suggests postulating that the field equations of gravity should also respect this symmetry. This principle implies that: (1) the metric cannot be varied in any extremum principle to obtain the field equations; and (2) the stress-tensor of matter should appear in the variational principle through the combination $T_{ab}n^an^b$ where $n_a$ is an auxiliary null vector field, which could be varied to get the field equations. This procedure selects naturally the Lanczos-Lovelock models of gravity in $D$-dimensions and Einstein's theory in $D=4$. Identifying $n_a$ with the normals to the null surfaces in the spacetime leads to a thermodynamic interpretation for gravity, in the macroscopic limit. Several geometrical variables and the equation describing the spacetime evolution acquire a thermodynamic interpretation. Extending these ideas one level deeper, we can obtain this variational principle from a distribution function for the ``atoms of spacetime'', which counts the number of microscopic degrees of freedom of the geometry. This is based on the curious fact that the renormalized spacetime endows each event with zero volume, but finite area!