Abstract
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!
Highlights
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 respect this symmetry
The results described so far suggest that the dynamics of spacetime is the thermodynamic limit of the statistical mechanics of microscopic degrees of freedom, which we shall call the atoms of spacetime
It is based on the desire to have a strong physical principle to describe the dynamics of gravity, viz. that the field equations should be invariant under the shift Tba → Tba + δba
Summary
While the difference between a hot body and a cold one was known even to the cavemen, physicists struggled for centuries to understand the nature of heat [1]. Boltzmann introduced a paradigm shift in which matter was treated as discrete at small scales and the thermal phenomena were related to the (suitably averaged) mechanical attributes of these discrete degrees of freedom. This provides a deeper level of description of spacetime, such that the thermodynamic variational principle, mentioned in Item (2) above, can be obtained from it.
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