Structure of the gravitational action and its relation with horizon thermodynamics and emergent gravity paradigm

If gravity is an emergent phenomenon, as suggested by several recent results, then the structure of the action principle for gravity should encode this fact. With this motivation we study several features of the Einstein-Hilbert action and establish direct connections with horizon thermodynamics. We begin by introducing the concept of holographically conjugate variables in terms of which the surface term in the action has a specific relationship with the bulk term. In addition to $g_{ab}$ and its conjugate momentum $\sqrt{-g}{M}^{cab}$, this procedure allows us to (re)discover and motivate strongly the use of $f^{ab}=\sqrt{-g}{g}^{ab}$ and its conjugate momentum $N_{ab}^c$. The gravitational action can then be interpreted as a momentum-space action for these variables. We also show that many expressions in classical gravity simplify considerably in this approach. For example, the field equations can be written in the form ${\partial }_c{f}^{ab}=\partial {\mathcal{H}}_g/\partial N_{ab}^c$, ${\partial }_c{N}_{ab}^c=-\partial {\mathcal{H}}_g/\partial f^{ab}$ (analogous to Hamilton’s equations) for a suitable Hamiltonian ${\mathcal{H}}_g$, if we use these variables. More importantly, the variation of the surface term, evaluated on any null surface which acts a local Rindler horizon can be given a direct thermodynamic interpretation. The term involving the variation of the dynamical variable leads to $T\delta S$ while the term involving the variation of the conjugate momentum leads to $S\delta T$. We have found this correspondence only for the choice of variables $(g_{ab},\sqrt{-g}{M}^{cab})$ or $(f^{ab},N_{ab}^c)$. We use this result to provide a direct thermodynamical interpretation of the boundary condition in the action principle, when it is formulated in a spacetime region bounded by the null surfaces. We analyze these features from several different perspectives and provide a detailed description, which offers insights about the nature of classical gravity and emergent paradigm.