Abstract
It is often argued that {\it all the information of a gravitational theory is encoded in the surface term of the action}; which means one can find several physical quantities just from the surface term without incorporating the bulk part of the action. This has been observed in various instances; e.g. derivation of the Einstein's equations, surface term calculated on the horizon leads to entropy, etc. Here I investigate the role of it in the context of entropy function and entropy of extremal near horizon black holes. Considering only the Gibbons-Hawking-York (GHY) surface term to define an entropy function for the extremal near horizon black hole solution, it is observed that the extremization of such function leads to the exact value of the horizon entropy. This analysis again supports the previous claim that there exists a ``{\it holographic}'' nature in the gravitational action -- surface term contains the information of the bulk.
Highlights
Introduction and motivationIn this paper, I shall develop the entropy function formalism alone from the surface term of the gravitational action in the presence of the matter action of the theory
The Noether charge corresponding to the GHY term, calculated on the horizon, leads to the entropy [8,9]
The calculation of entropy for an extremal near horizon black hole reduces to a set of algebraic equations which are given by the relations, presented in (18)
Summary
I shall develop the entropy function formalism alone from the surface term of the gravitational action in the presence of the matter action of the theory. The Noether charge corresponding to the GHY term, calculated on the horizon (similar to the Wald charge associated with the EH action), leads to the entropy [8,9]. This has been even tested successfully in the context of Virasoro algebra and the Cardy formula [8,10]. Considering the above facts, it is important to investigate if the entropy function formalism can be developed purely from the surface or the boundary term of the action. The final section will summarize the results and conclude
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