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

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 study the time evolution of 2-point functions and entanglement entropy in strongly anisotropic, inhomogeneous and time-dependent N = 4 $$ \mathcal{N}=4 $$ super Yang-Mills theory in the large N and large ’t Hooft coupling limit using AdS/CFT. On the gravity side this amounts to calculating the length of geodesics and area of extremal surfaces in the dynamical background of two colliding gravitational shockwaves, which we do numerically. We discriminate between three classes of initial conditions corresponding to wide, intermediate and narrow shocks, and show that they exhibit different phenomenology with respect to the nonlocal observables that we determine. Our results permit to use (holographic) entanglement entropy as an order parameter to distinguish between the two phases of the cross-over from the transparency to the full-stopping scenario in dynamical Yang-Mills plasma formation, which is frequently used as a toy model for heavy ion collisions. The time evolution of entanglement entropy allows to discern four regimes: highly efficient initial growth of entanglement, linear growth, (post) collisional drama and late time (polynomial) fall off. Surprisingly, we found that 2-point functions can be sensitive to the geometry inside the black hole apparent horizon, while we did not find such cases for the entanglement entropy.

Austria

Christian Ecker

Daniel Grumiller

Philipp Stanzer

Stefan Stricker

United States

Wilke Schee

Exploring nonlocal observables in shock wave collisions

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.

Steven B. Giddings

Hawking radiation, the Stefan–Boltzmann law, and unitarization

We determine holographically 2-point correlators of gauge invariant operators with large conformal weights and entanglement entropy of strips for a time-dependent anisotropic 5-dimensional asymptotically anti-de Sitter spacetime. At the early stage of evolution where geodesics and extremal surfaces can extend beyond the apparent horizon all observables vary substantially from their thermal value, but thermalize rapidly. At late times we recover quasi-normal ringing of correlators and holographic entanglement entropy around their thermal values, as expected on general grounds. We check the behaviour of holographic entanglement entropy and correlators as function of the separation length of the strip and find agreement with the exact expressions derived in the small and large temperature limits.

Evolution of holographic entanglement entropy in an anisotropic system

We study the dynamics of gravitationally collapsing massive shells in AdS spacetime, and show in detail how one can determine extremal surfaces traversing them. The results are used to solve the time evolution of the holographic entanglement entropy in a strongly coupled dual conformal gauge theory, which is is seen to exhibit a regime of linear growth independent of the shape of the boundary entangling region and the equation of state of the shell. Our exact results are finally compared to those of two commonly used approximation schemes, the Vaidya metric and the quasistatic limit, whose respective regions of validity we quantitatively determine.

United Kingdom

Ville Keränen

Germany

Hiromichi Nishimura

Netherlands

Olli Taanila

Finland

Aleksi Vuorinen

Gravitational collapse of thin shells: time evolution of the holographic entanglement entropy