Method of Complex Paths and General Covariance of Hawking Radiation

Abstract

We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation. This also shows that there is no correspondence between the particles detected by the model detector and the particle spectrum obtained by the quantum field theoretic analysis -a result known in other contexts as well. * 571 Mod. Phys. Lett. A 2001.16:571-578. Downloaded from www.worldscientific.com by GEORGE WASHINGTON UNIVERSITY on 02/05/15. For personal use only.

Quantum tunneling from three-dimensional black holes

“…Using this approach we calculate the imaginary part of the classical action for this classically forbidden process of emission across the horizon. In this semi-classical method the probabilities of crossing the horizon from inside to outside, and from outside to inside, are given by [7,8] P emission ∝ exp −2 ImI = exp −2 (ImW + + ImK) , (3.10)…”

“…If we compare this equation (3.13) with Γ = exp (−βω), which is Boltzmann factor for particle of energy ω, and β is the inverse temperature of the horizon [7,8], we can derive the Hawking temperature for black holes. Comparing equation (3.13) with the Boltzmann factor of energy, we can find the Hawking temperature (taking = 1) as…”

“…These radiations have also been viewed as quantum tunneling of particles from the horizons of black holes [3,4,5]. Many well known black holes have been researched for these radiations [6][7][8][9][10][11][12]. In one of the procedures the wave equation governing the emission of particles is solved in the background of the black hole spacetimes by using complex path integration techniques and WKB approximation.…”

Temperature and entropy of Schwarzschild–de Sitter space-time

“…[23] (see also Refs. [19]), the spacetime structure of spherically symmetric space-times can be understood via R and T regions. If at the given event in the coordinate system (3), the inequality…”

“…In this paper, we extend the method of complex paths to space-times with multiple horizons and obtain the spectrum of particles produced in these space-times. (The method of complex paths has proved to be useful in obtaining the temperature associated with a quantum field propagating in a spherically symmetric coordinate spacetimes with single horizon [18][19][20][21].) We show that the temperature of radiation in these space-times is proportional to the effective surface gravity -inverse harmonic sum of surface gravity of each horizon.…”