We propose a new formula for the entropy of a dynamical black hole—valid to leading order for perturbations off of a stationary black hole background—in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in <math><mi>n</mi></math> dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. In particular, in general relativity, our formula for the entropy of a dynamical black hole differs from the standard Bekenstein-Hawking formula <math><mi>A</mi><mo>/</mo><mn>4</mn></math> by a term involving the integral of the expansion of the null generators of the horizon. We show that, to leading perturbative order, our dynamical entropy in general relativity is equal to <math><mn>1</mn><mo>/</mo><mn>4</mn></math> of the area of the apparent horizon. Our formula for entropy in a general theory of gravity is obtained from the requirement that a “local physical process version” of the first law of black hole thermodynamics hold for perturbations of a stationary black hole. It follows immediately that for first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the leading-order change in entropy occurs at second order in perturbation theory, and the second law is obeyed at leading order if and only if the modified canonical energy flux is positive (as is the case in general relativity but presumably would not hold in more general theories of gravity). Our formula for the entropy of a dynamical black hole differs from a formula proposed independently by Dong and by Wall. We obtain the general relationship between their formula and ours. We then consider the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole. We show that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using our notion of black hole entropy but using a modified notion of von Neumann entropy for matter. On the other hand, the generalized second law for the Dong-Wall entropy is equivalent to an integrated version of QNEC, using the unmodified von Neumann entropy for the entropy of matter.

Germany

Stefan Hollands

United States

Robert M. Wald

Victor G. Zhang

Entropy of dynamical black holes

In this paper, the existence of a massive dark photon, associated with a new gauge group is considered. The dark photon can be kinetically mixed with the photon. To study some applications, the thermo field dynamics formalism is used. Exploring the topological structure of this approach, the influence of dark photons on the Stefan–Boltzmann law and the Casimir effect at zero and finite temperature is calculated.

Brazil

V. Prata

A. Santos

Canada

Faqir Khanna

Stefan–Boltzmann law and Casimir effect for dark photons

We investigate the second law of black hole mechanics in gravitational theories with higher derivative terms in the action. Wall has described a method for defining an entropy that satisfies the second law to linear order in perturbations around a stationary black hole. We show that this can be extended to define an entropy that satisfies the second law to quadratic order in perturbations, provided that one treats the higher derivative terms in the sense of effective field theory. We also address some outstanding issues with Wall’s method, in particular its gauge invariance and its relation to the Iyer-Wald entropy.

Italy

Áron Kovács

United Kingdom

Harvey Reall

The second law of black hole mechanics in effective field theory

The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.

S.C. Ulhoa

R.G.G. Amorim

J.S. da Cruz Filho

A.F. Santos

F.C. Khanna

Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

Gravitons are described by the propagator in teleparallel gravity in nearly flat space-time. Finite temperature is introduced by using Thermofield Dynamics formalism. The gravitational Casimir effect and Stefan-Boltzmann law are calculated as a function of temperature. Then an equation of state for gravitons is determined.

T.F. Furtado

On Gravitational Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

We use the scalar field constructed in phase space to analyze the analogous Stefan-Boltzmann law and Casimir effect, both of them at finite temperature. The temperature is introduced by Thermo Field Dynamics (TFD) formalism and the quantities are analyzed once projected in the space of coordinates. We show that using the framework of phase space it is possible to introduce a thermal energy which is related to temperature as it vanishes when the temperature tends to zero. In fact given such a correlation the formalism of TFD is equivalent when project is in momenta space when compared to coordinates space.

Stefan-Boltzmann Law and Casimir Effect for the Scalar Field in Phase Space at Finite Temperature

We formulate an “action principle” for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings “In memoriam Rudolf Haag” in September 2016 and the “Wolfhart Zimmermann memorial symposium” in May 2017.

Action principle for OPE

It has recently been suggested (S.B. Giddings (2016) [2] ) that the Hawking black-hole radiation spectrum originates from an effective quantum “atmosphere' which extends well outside the black-hole horizon. In particular, comparing the Hawking radiation power of a (3+1) -dimensional Schwarzschild black hole of horizon radius rH with the familiar Stefan–Boltzmann radiation power of a (3+1) -dimensional flat space perfect blackbody emitter, Giddings concluded that the source of the Hawking semi-classical black-hole radiation is a quantum region outside the Schwarzschild black-hole horizon whose effective radius rA is characterized by the relation Δr≡rA−rH∼rH . It is of considerable physical interest to test the general validity of Giddings's intriguing conclusion. To this end, we study the Hawking radiation of (D+1) -dimensional Schwarzschild black holes. We find that the dimensionless radii rA/rH which characterize the black-hole quantum atmospheres, as determined from the Hawking black-hole radiation power and the (D+1) -dimensional Stefan–Boltzmann radiation law, are a decreasing function of the number D+1 of spacetime dimensions. In particular, it is shown that radiating (D+1) -dimensional Schwarzschild black holes are characterized by the relation (rA−rH)/rH≪1 in the large D≫1 regime. Our results therefore suggest that, at least in some physical cases, the Hawking emission spectrum originates from quantum excitations very near the black-hole horizon.

Israel

Shahar Hod

Hawking radiation and the Stefan–Boltzmann law: The effective radius of the black-hole quantum atmosphere

Where does Hawking radiation originate? A common picture is that it arises from excitations very near or at the horizon, and this viewpoint has supported the “firewall” argument and arguments for a key role for the UV-dependent entanglement entropy in describing the quantum mechanics of black holes. However, closer investigation of both the total emission rate and the stress tensor of Hawking radiation supports the statement that its source is a near-horizon quantum region, or “atmosphere,” whose radial extent is set by the horizon radius scale. This is potentially important, since Hawking radiation needs to be modified to restore unitarity, and a natural assumption is that the scales relevant to such modifications are comparable to those governing the Hawking radiation. Moreover, related discussion suggests a resolution to questions regarding extra energy flux in “nonviolent” scenarios, that does not spoil black hole thermodynamics as governed by the Bekenstein–Hawking entropy.