v 2 3 A ug 1 99 8 BRX TH 437 Equivalence of Hawking and Unruh Temperatures and Entropies Through Flat Space Embeddings

We present a unified description of temperature and entropy in spaces with either “true” or “accelerated observer” horizons: In their (higher dimensional) global embedding Minkowski geometries, the relevant detectors have constant accelerations aG; associated with their Rindler horizons are temperature aG/2π and entropy equal to 1/4 the horizon area. Both quantities agree with those calculated in the original curved spaces. As one example of this equivalence, we obtain the temperature and entropy of Schwarzschild geometry from its flat D=6 embedding. The relation between Hawking and Unruh effects has been extensively studied, and the emergence of temperature and entropy due to their respective “real” or “accelerated” horizons is well-understood. Recently, it was shown that the temperatures measured by accelerated detectors in de Sitter (dS) and Anti de Sitter (AdS) geometries can be obtained [1] from their corresponding constant (Rindler) accelerations in the appropriate global embedding Minkowski spacetimes (GEMS). Our purpose here is to generalize this method to provide a unified kinematical treatment of both effects in terms of accelerated motions in the GEMS; any Einstein geometry can be so embedded [2]. As a first illustration, we obtain the entropy of dS in this way. We will then derive, as our main example, the equivalence between temperature and entropy measured by the usual static detector in Schwarzschild geometry and their values as calculated from its Rindler-like motion in the (D=6) GEMS. A more complete discussion of this equivalence, and of its validity for other important geometries such as BTZ, Schwarzschild-AdS, Schwarzschild-dS and Riessner-Nordstrom, will be presented elsewhere. Recall first that in flat space, observers with constant acceleration of magnitude a, who follow a timelike Killing vector field ξ that encounters an event horizon, will thereby measure a temperature, 2πT = a. It is also well-known that the connection between surface gravity kH and temperature, kH = g 1/2 00 2πT (1) holds both in black hole spaces and for Rindler motions [3]; x is the timelike Killing vector of rest detectors and kH is defined as the horizon value of [− 1 2 (Dμξν) ]. In Rindler coordinates, the longitudinal interval is ds = Le(dτ 2 − dζ) and ζ=const detectors have