Holography in action

The Einstein-Hilbert action and its natural generalizations to higher dimensions (like the Lanczos-Lovelock action) have certain peculiar features. All of them can be separated into a bulk and a surface term, with a specific (“holographic”) relationship between the two, so that either term can be used to extract information about the other. Further, the surface term leads to entropy of the horizons on shell. It has been argued in the past that these features are impossible to understand in the conventional approach but find a natural explanation if we consider gravity as an emergent phenomenon. We provide further support for this point of view in this paper. We describe an alternative decomposition of the Einstein-Hilbert action and the Lanczos-Lovelock action into a new pair of surface and bulk terms, such that the surface term becomes the Wald entropy on a horizon and the bulk term is the energy density (which is the Arnowitt-Deser-Misner Hamiltonian density for Einstein gravity). We show that this new pair also obeys a holographic relationship, and we give a thermodynamic interpretation of this relation in this context. Since the bulk and surface terms, in this decomposition, are related to the energy and entropy, the holographic condition can be thought of as analogous to inverting the expression for entropy given as a function of energy $S=S(E,V)$ to obtain the energy $E=E(S,V)$ in terms of the entropy in a normal thermodynamic system. Thus the holographic nature of the action allows us to relate the descriptions of the same system in terms of two different thermodynamic potentials. Some further possible generalizations and implications are discussed.