Abstract
This chapter explains that due to co-homological properties, the entropy can be geometrically calculated as an integral at space infinity; hence, the geometrical properties of the horizons become irrelevant. In particular, there is no need for the horizon to be Killing. This traditional assumption is related to the wish of reducing the entropy to be an integral at the horizon with the consequent need to control the limit behavior at the horizon itself, although these features turn out to be unnecessary and produce too stringent requirements that are not justified by phenomenology. The traditional physical quantities at singularities can be recovered by (artificially) splitting the domain of integration into the union of a number of regions surrounding the singularities themselves. The chapter reviews the basic frameworks to define entropy of gravitational systems and discusses the relation among different perspectives. Particular attention is paid to macroscopic prescriptions in order to clarify the relation between gravitational entropy and the geometric properties of space-times. The chapter also discusses the fundamental contributions that geometry and calculus of variations offer in characterizing the entropy of black-hole solutions and of general singular solutions in general relativity.
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