Boundary conditions, energies and gravitational heat in general relativity (a classical analysis)

Abstract

The variation of the energy for a gravitational system is directly defined from the Hamiltonian field equations of General Relativity. When the variation of the energy is written in a covariant form it splits into two (covariant) contributions: one of them is the Komar energy, while the other is the so-called covariant ADM correction term. When specific boundary conditions are analyzed one sees that the Komar energy is related to the gravitational heat while the ADM correction term plays the role of the Helmholtz free energy. These properties allow to establish, inside a classical geometric framework, a formal analogy between gravitation and the laws governing the evolution of a thermodynamical system. The analogy applies to stationary spacetimes admitting multiple causal horizons as well as to AdS Taub-bolt solutions.

Action growth for black holes in modified gravity

“…where M is a four-dimentional space-time manifold with boundary ∂M, g is the determinant of the metric tensor gµν (x µ ), and F (R) is a function of the Ricci scalar R. As in GR, in order to deal with a proper well posed variational problem for the metric tensor [28], one needs to subtract to the Lagrangian a suitable boundary term. In the so called Jordan frame (JF), one has to work with the following action [29,30,31,32,33,34,35,36],…”

“…As in GR, in order to deal with a proper well posed variational problem for the metric tensor [28], one needs to subtract to the Lagrangian a suitable boundary term. In the so called Jordan frame (JF), one has to work with the following action [29,30,31,32,33,34,35,36],…”

The Relative Energy of Homogeneous and Isotropic Universes From Variational Principles

“…To this research effort two of the present authors gave a number of contributions (see for example Refs. [11,13,15,16,17]) mainly proposing a pure variational route to the definition of the relative conserved currents between two solutions of the field equations interpreted as the amount of conserved current needed to pass from one solution to the other. The mathematical framework introduced to this aim goes well beyond General Relativity and extends in fact to any gauge-natural Lagrangian field theory (see for example Ref.…”

Renormalized entropies in a de Sitter spacetime

“…The regulated entropy is related to the parameters that appear in the renormalized action for the coupled gravitational-quantum scalar field. The essence of DLM's model is to cancel the divergence of entropy, by introducing an counterterm/background in the Lagrangian [9,10]. Especially in [9], a background subtraction term is necessary to obtain a finite result.…”

“…The essence of DLM's model is to cancel the divergence of entropy, by introducing an counterterm/background in the Lagrangian [9,10]. Especially in [9], a background subtraction term is necessary to obtain a finite result. For example, a constant of integration Q 0 absorbs the divergent entropy coming from the Misner string.…”