# Effective Tolman temperature induced by trace anomaly

## Abstract

Despite the finiteness of stress tensor for a scalar field on the four-dimensional Schwarzschild black hole in the Israel–Hartle–Hawking vacuum, the Tolman temperature in thermal equilibrium is certainly divergent on the horizon due to the infinite blue-shift of the Hawking temperature. The origin of this conflict is due to the fact that the conventional Tolman temperature was based on the assumption of a traceless stress tensor, which is, however, incompatible with the presence of the trace anomaly responsible for the Hawking radiation. Here, we present an effective Tolman temperature which is compatible with the presence of the trace anomaly by using the modified Stefan–Boltzmann law. Eventually, the effective Tolman temperature turns out to be finite everywhere outside the horizon, and so an infinite blue-shift of the Hawking temperature at the event horizon does not appear any more. In particular, it is vanishing on the horizon, so that the equivalence principle is exactly recovered at the horizon.

## 1 Introduction

On the other hand, the renormalized stress tensor for a conformal scalar field could be finite on the background of the Schwarzschild black hole [7]. At infinity, the proper energy density \(\rho \) is positive finite, which is consistent with the Stefan–Boltzmann law as \(\rho =\sigma T^4_\mathrm{H}\), where \(\sigma =\pi ^2/30\). If one considered a motion of an inertial observer [7, 8, 9, 10], the negative proper energy density could be found near the horizon in various vacua and its role was also discussed in connection with the information loss paradox [10]. However, it might be interesting to note that the local temperature (1) is infinite at the horizon, although the proper energy density at the horizon \(r_\mathrm{H}\) is negative finite as \(\rho (r_\mathrm{H}) =-12 \sigma T^4_\mathrm{H}\) as seen in Ref. [7].

Now, it appears to be puzzling in that the Tolman temperature at the horizon is positively divergent despite the negative finite energy density there. More worse, the energy density happens to vanish at a certain point outside the horizon [7], but the local temperature (1) is positive finite at that point. In these regards, the Tolman temperature runs contrary to the finite renormalized stress tensor, which certainly requires that the Stefan–Boltzmann law to relate the stress tensor to the proper temperature should be appropriately modified in such a way that they are compatible each other.

To resolve the above conflict between the finiteness of the renormalized stress tensor and the divergent behavior of the proper temperature, it is worth noting that the usual Tolman temperature rests upon the traceless stress tensor; however, the trace of the renormalized stress tensor is actually not traceless because of the trace anomaly. So we should find a modified Stefan–Boltzmann law in order to get the proper temperature commensurate with the finite renormalized stress tensor. In fact, this was successfully realized in the two-dimensional case where the stress tensor was perfect fluid [11]. In this work, we would like to extend the above issue to the case of the four-dimensional more realistic Schwarzschild black hole, where the renormalized stress tensor is no more isotropic.

Using the exact thermal stress tensor calculated in Ref. [7], we solve the covariant conservation law and the equation for the trace anomaly, and then obtain the proper quantities such as the proper energy density and pressures written explicitly in terms of the trace anomaly in Sect. 2. In Sect. 3, we derive the effective Tolman temperature from the modified Stefan–Boltzmann law based on thermodynamic analysis. It shows that the effective Tolman temperature exactly reproduces the Hawking temperature at infinity, but it has a maximum at a finite distance outside the horizon and eventually it is vanishing rather than divergent on the horizon. Finally, a conclusion and a discussion are given in Sect. 4.

## 2 Proper quantities in terms of trace anomaly

## 3 Effective Tolman temperature

## 4 Conclusion and discussion

It has been widely believed that the Tolman temperature is divergent at the horizon due to the infinite blue-shift of the Hawking radiation. However, the usual Stefan–Boltzmann law assuming the traceless stress tensor should be consistently modified in order to discuss the case where the stress tensor is no longer traceless in the process of the Hawking radiation. From the modified Stefan–Boltzmann law, we obtained the effective Tolman temperature without the red-shift factor related to the origin of the divergence at the horizon, so that it is finite everywhere outside the black hole horizon.

The intriguing behavior of the effective Tolman temperature on the horizon may be understood by the Unruh effect [36]. The static metric (2) near the horizon can be written by the Rindler metric for a large black hole whose curvature scale is negligible. The Unruh temperature is divergent due to the infinite acceleration of the frame where the fixed detector is very close to the horizon. So the Unruh temperature is equivalent to the locally fiducial temperature for the Schwarzschild black hole [33]. Conversely speaking, based on the equivalence principle, the Unruh temperature measured by the geodesic detector should vanish on the horizon since the proper acceleration of the geodesic detector vanishes. In this regard, it appears natural to conclude that the freely falling observer from rest does not see any excited particles on the horizon in thermal equilibrium and thus the effective Tolman temperature vanishes at the horizon.

On the other hand, AMPS argument is that the firewall on the horizon should be defined in an evaporating black hole rather than the black hole in thermal equilibrium [37]. The firewall is certainly characterized by the divergent proper temperature in that the average frequency \(\omega \) of an excited particle with a thermal bath can be identified with the proper temperature as \( \omega \sim T\). Using the advantage of the effective Tolman temperature, we find the reason why the firewall could not exist in thermal equilibrium: the fact that the red-shift factor responsible for the divergence at the horizon could be canceled out.

## Notes

### Acknowledgements

We would like to thank Jeong-Hyuck Park and Edwin J. Son for exciting discussions. W. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (2017R1A2B2006159).

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