Abstract
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.
Highlights
The proper temperature of the gravitating system of a perfect fluid in thermodynamic equilibrium has been defined by the well-known Tolman temperature [1,2]
We cannot conclude that the difference between them comes from the dimensionality, since we can exactly get the same free-fall temperature as TBT for the two-dimensional Schwarzschild black hole (20) by using a slight different higher-dimensional embedding method [23]
We consider the new expression for the Stefan-Boltzmann law such as p = αT 2 and ρ = αT 2 − Tμμ, TBT can be obtained from Eq (25); this does not satisfy the relation (13) which comes from the first law of thermodynamics
Summary
The proper temperature of the gravitating system of a perfect fluid in thermodynamic equilibrium has been defined by the well-known Tolman temperature [1,2]. In the Hartle-Hawking-Israel state, the energy density and pressure are finite at the horizon even though the Tolman temperature is infinite at the horizon [14,15]. The energy density at the horizon is negative in the Hartle-Hawking-Israel state, so that it seems to be nontrivial task to relate the negative energy density to the positive temperature if the conventional Stefan-Boltzmann law is just assumed. In these respects, it raises some natural questions.
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