# Proper temperature of Schwarzschild AdS black hole revisited

## TL;DR

This paper revisits the temperature measurement of Schwarzschild AdS black holes, resolving previous issues with imaginary temperatures by applying an extended Tolman procedure, ensuring a real temperature everywhere outside the horizon.

## Contribution

It introduces an extended Tolman method accounting for trace anomalies to accurately determine the free-fall temperature of Schwarzschild AdS black holes.

## Key findings

- Free-fall temperature is real everywhere outside the horizon.
- Temperature vanishes at the horizon and infinity.
- Resolves previous issues with imaginary temperature values.

## Abstract

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.