A Quantal Tolman Temperature

The conventional Tolman temperature based on the assumption of the traceless condition of energy-momentum tensor for matter fields is infinite at the horizon if Hawking radiation is involved. However, we note that the temperature associated with Hawking radiation is of relevance to the trace anomaly, which means that the traceless condition should be released. So, a trace anomaly-induced Stefan-Boltzmann law is newly derived by employing the first law of thermodynamics and the property of the temperature independence of the trace anomaly. Then, the Tolman temperature is quantum-mechanically generalized according to the anomaly-induced Stefan-Boltzmann law. In an exactly soluble model, we show that the Tolman factor does not appear in the generalized Tolman temperature which is eventually finite everywhere, in particular, vanishing at the horizon. It turns out that the equivalence principle survives at the horizon with the help of the quantum principle, and some puzzles related to the Tolman temperature are also resolved.Comment: 5 pages, 1 figure, version to appear in EPJ

Entropy and temperatures of Nariai black hole

The statistical entropy of the Nariai black hole in a thermal equilibrium is calculated by using the brick-wall method. Even if the temperature depends on the choice of the time-like Killing vector, the entropy can be written by the ordinary area law which agrees with the Wald entropy. We discuss some physical consequences of this result and the properties of the temperatures.Comment: 12 pages, no figur

The role of the cosmological constant as a pressure in the (2+1)-dimensional black string

It has been claimed that the cosmological constant in AdS black holes such as the BTZ black hole plays the role of the thermodynamic variable of a pressure in the thermodynamic first law and the Smarr relation from the scaling law of the Christodoulou-Ruffini formula. However, the dual solution of the BTZ black hole is the black string which is asymptotically flat despite the presence of the cosmological constant, and so the explicit form of the pressure with the role of the cosmological constant is unclear in the black string since the pressure is subject to the choice of the energy-momentum tensor. Thus, we show that if the pressure of the black string is still assumed to be proportional to the cosmological constant similar to the case of the BTZ black hole, then the thermodynamic first law is consistent with the Smarr relation from the Christodoulou-Ruffini formula, and the thermodynamic quantities for the pressure are well-behaved under the dual transformation.Comment: 15 pages, version to appear in PL

Proper temperature of Schwarzschild AdS black hole revisited

The Unruh temperature calculated from the global embedding of the Schwarzschild AdS spacetime into Minkowski spacetime was identified with the local temperature measured by a free-fall observer; however, it would be imaginary in a certain region outside the event horizon. So, the temperature was assumed to be zero of no thermal radiation for that region. In this paper, we revisit this issue in the exactly soluble two-dimensional Schwarzschild AdS black hole and present an alternative resolution to this problem by using the Tolman's procedure. However, the process is not straightforward in the sense that one should extend the original procedure to rest upon the traceless energy-momentum tensor in such a way that it could encompass the case of the non-vanishing trace of energy-momentum tensor in the presence of the trace anomaly. Consequently, we show that the free-fall temperature turns out to be real everywhere outside the event horizon without any imaginary value, in particular, it vanishes both at the horizon and at the asymptotic infinity.Comment: 12 pages, 1 figure, version published in PL