# Taking the temperature of a black hole

TL;DR: In this paper, the global embedding of a black hole spacetime into a higher dimensional flat spacetime is used to define a local temperature for observers in free fall outside a static black hole.

Abstract: We use the global embedding of a black hole spacetime into a higher dimensional flat spacetime to define a local temperature for observers in free fall outside a static black hole. The local free-fall temperature remains finite at the event horizon and in asymptotically flat spacetime it approaches the Hawking temperature at spatial infinity. Freely falling observers outside an AdS black hole do not see any high-temperature thermal radiation even if the Hawking temperature of such black holes can be arbitrarily high.

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TL;DR: In this paper, the Hartle-Hawking-like states of quantum field theories on asymptotically AdS black hole backgrounds were studied with particular regard to the phase structure of interacting theories.

Abstract: We study Hartle–Hawking-like states of quantum field theories on asymptotically AdS black hole backgrounds, with particular regard to the phase structure of interacting theories. By a suitable analytic continuation we show that the equilibrium dynamics of field theories on large asymptotically AdS black holes can be related to the low-temperature states of the same field theory on the AdS soliton (or pure AdS) background. This allows us to gain insight into Hartle–Hawking-like states on large-radius Schwarzschild- or rotating-AdS black holes. Furthermore, we exploit the AdS/CFT correspondence to explore the physics of strongly coupled large N theories on asymptotically AdS black holes. In particular, we exhibit a plausibly complete set of phases for the M2-brane world-volume superconformal field theory on a BTZ black hole background. Our analysis partially resolves puzzles previously raised in connection with Hawking radiation on large AdS black holes.

128 citations

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TL;DR: Compared the response function of an Unruh-DeWitt detector for different space-times and different vacua is compared and it is found that both detectors register thermal radiation, but for a given, equivalent acceleration the fixed detector in the Schwarzschild space-time measures a higher temperature.

Abstract: We compare the response function of an Unruh-DeWitt detector for different space-times and different vacua and show that there is a detailed violation of the equivalence principle. In particular comparing the response of an accelerating detector to a detector at rest in a Schwarzschild space-time we find that both detectors register thermal radiation, but for a given, equivalent acceleration the fixed detector in the Schwarzschild space-time measures a higher temperature. This allows one to locally distinguish the two cases. As one approaches the horizon the two temperatures have the same limit so that the equivalence principle is restored at the horizon.

85 citations

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TL;DR: In this article, a new type of global embedding of curved space is introduced to present a unified description of the Hawking and Unruh effects, which simplifies as well as generalises the conventional embedding approach.

55 citations

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TL;DR: In this article, the occupation number of particles whose frequencies are measured in the proper time of an infalling observer in two crucially different space-time foliations: Schwarzschild and Eddington-Finkelstein is calculated.

Abstract: We investigate an important question of Hawking-like radiation as seen by an infalling observer during gravitational collapse. Using the functional Schrodinger formalism we are able to probe the time dependent regime which is out of the reach of the standard approximations like the Bogolyubov method. We calculate the occupation number of particles whose frequencies are measured in the proper time of an infalling observer in two crucially different space-time foliations: Schwarzschild and Eddington-Finkelstein. We demonstrate that the distribution in Schwarzschild reference frame is not quite thermal, though it becomes thermal once the horizon is crossed. We approximately fit the temperature and find that the local temperature increases as the horizon is approached, and diverges exactly at the horizon. In Eddington-Finkelstein reference frame the temperature at the horizon is finite, since the observer in that frame is not accelerated. These results are in agreement with what is generically expected in the absence of backreaction. We also discuss some subtleties related to the physical interpretation of the infinite local temperature in Schwarzschild reference frame.

37 citations

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TL;DR: In this article, the effect of velocity and acceleration on the temperature felt by particles and strings in backgrounds relevant in holographic models was analyzed and a global Minkowski embedding for the planar AdS black hole was found.

Abstract: We analyse the effect of velocity and acceleration on the temperature felt by particles and strings in backgrounds relevant in holographic models. First, we compare accelerated strings and strings at finite temperature. We find that for fixed Unruh temperature felt by the string endpoints, the screening length is smaller for the accelerated Wilson loop than for the static one in a thermal background of the same temperature; hence acceleration provides a ``more efficient'' mechanism for melting of mesons. Secondly, we show that the velocity-dependence of the screening length of the colour force, previously obtained from a moving Wilson loop in a finite temperature background, is not specific for the string, but is a consequence of the generic fact that an observer which moves with constant velocity in a black hole background measures a velocity-dependent temperature. Finally, we analyse accelerated particles and strings in the AdS black hole background, and show that these feel a temperature which increases as a function of time. As a byproduct of our analysis we find a global Minkowski embedding for the planar AdS black hole.

34 citations

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TL;DR: In this article, it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature, which leads to a slow decrease in the mass of the black hole and to its eventual disappearance.

Abstract: In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature\(\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K\) where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.

10,923 citations

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TL;DR: In this paper, the authors examined various aspects of black-hole evaporation and proposed a technique for replacing the collapse by boundary conditions on the past horizon, which retains the essential features of the collapse while eliminating some of the difficulties.

Abstract: This paper examines various aspects of black-hole evaporation. A two-dimensional model is investigated where it is shown that using fermion-boson cancellation on the stress-energy tensor reduces the energy outflow to zero, while other noncovariant techniques give the Hawking result. A technique for replacing the collapse by boundary conditions on the past horizon is developed which retains the essential features of the collapse while eliminating some of the difficulties. This set of boundary conditions is also suggested as the most natural set for a preexistent black hole. The behavior of particle detectors under acceleration is investigated where it is shown that an accelerated detector even in flat spacetime will detect particles in the vacuum. The similarity of this case with the behavior of a detector near the black hole is brought out, and it is shown that a geodesic detector near the horizon will not see the Hawking flux of particles. Finally, the work of Berger, Chitre, Nutku, and Moncrief on scalar geons is corrected, and the spherically symmetric coupled scalar-gravitation Hamiltonian is presented in the hope that someone can apply it to the problem of black-hole evaporation.

4,344 citations

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TL;DR: The correspondence between supergravity and string theory on AdS space and boundary conformal eld theory relates the thermodynamics of N = 4 super Yang-Mills theory in four dimensions to the thermodynamic properties of Schwarzschild black holes in Anti-de Sitter space as mentioned in this paper.

Abstract: The correspondence between supergravity (and string theory) on AdS space and boundary conformal eld theory relates the thermodynamics of N = 4 super Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in Anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic connement, and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large AdS Schwarzschild black hole must scale \holographically" with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large N gauge theories in four dimensions without supersymmetry.

4,209 citations

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TL;DR: This review includes discussion of classical black hole thermodynamics, Hawking radiation from black holes, the generalized second law, and the issue of entropy bounds.

Abstract: We review the present status of black hole thermodynamics. Our review includes discussion of classical black hole thermodynamics, Hawking radiation from black holes, the generalized second law, and the issue of entropy bounds. A brief survey also is given of approaches to the calculation of black hole entropy. We conclude with a discussion of some unresolved open issues.

729 citations

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CERN

^{1}TL;DR: An analytic manifold of the Schwarzschild line element was found in this article, which is the complete analytic extension of the real 3-dimensional rotation group and the one-dimensional translation group.

Abstract: An analytic manifold is found, the most important properties of which are that it is complete and that it contains the manifold of the Schwarzschild line element. It is thus the complete analytic extension of the latter. The manifold is represented as a Riemannian surface in a six-dimensional pseudo-Euclidean space. The subspace $d\ensuremath{\phi}=d\ensuremath{\vartheta}=0$ is visualized as a two-dimensional Riemannian surface in a 3-dimensional hyperplane in the six-dimensional space. Although the manifold admits groups of motion isomorphic to the real 3-dimensional rotation group and the one-dimensional translation group, it is impossible to introduce a global time-coordinate in such a way that the latter is realized as translations in time. Hence in any global set of coordinates the gravitational field is nonstationary, although it can be made stationary for $rg1$ to any desired approximation. The question of what happens to small test bodies reaching the Schwarzschild critical radius is discussed.

214 citations