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Holography in Action

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 Lanczos-Lovelock action into a new pair of surface and bulk terms, such that the surface term becomes Wald entropy on a horizon and the bulk term is the energy density (which is the ADM Hamiltonian density for Einstein gravity). We show that this new pair also obeys a holographic relationship and give a thermodynamic interpretation to this relation in this context. Since the bulk and surface terms, in this decomposition, are related to energy and en