Hawking Radiation from Four-dimensional Schwarzschild Black Holes in M-theory

Recently a method has been developed for relating four dimensional Schwarzschild black holes in M-theory to near-extremal black holes in string theory with four charges, using suitably defined “boosts” and T-dualities. We show that this method can be extended to obtain the emission rate of low energy massless scalars for the four dimensional Schwarzschild hole from the microscopic picture of radiation from the near extremal hole. Typeset using REVTEX 1 ] In the past couple of years there has been considerable progress in understanding the statistical basis of the thermodynamics of near-extremal five and four dimensional black holes in string theory [1]. Further, a simple effective model gives the correct low energy Hawking radiation from such holes. A natural question to ask is: Can we get the emission from neutral (i.e. Schwarzschild) holes as well? Recently there has been some progress in understanding Schwarzschild black holes in M theory. The basic idea is to use 11-dimensional Lorentz invariance properties to relate Schwarzschild holes boosted along x to string theory states carrying Ramond-Ramond charges [2]. In [3] a concrete map was found which relates Schwarzschild strings and black p-branes. It was found that the precise relationship is not through a genuine boost in the compact direction, but through a boost in the covering space. The boosted coordinate is then re-compactfied on a circle of radius which is related to the original radius by Lorentz contraction. This is not an exact symmetry of the theory, but provides a concrete map at the classical level. T-dualities can generate other charges from the momentum charge, and a combination of boosts and T-dualities may be used to relate Schwarzschild holes with other known black holes in string theory carrying charges, as in [4]. Specific maps which relate a five-dimensional Schwarzschild black hole with the standard five dimensional black hole in string theory with three large charges were given in [3]. Using similar steps one can map the four dimensional Schwarzschild hole to three sets of near-extremal 5-branes of M-theory, intersecting in a common line along x, and carrying momentum along this direction [5]. This is a description of a four dimensional black hole with four charges in M theory [6] The entropy of the latter near extremal hole is known to follow from a microscopic calculation [7,8], and thus we get the entropy of the neutral hole [5]. In the microscopic picture we can also get the emission from the above near extremal model [9] , so one wonders if the maps allow us to predict the emission from neutral holes. We show in this paper that, using a general relation between absorption cross-sections derived in [3], such is indeed the case, at leading order in the energy if we assume during the calculation 2 that the radius of x is large enough. The important issue turns out to be the way the maps act on the emitted quantum while they transform the black hole it turns out that the low energy scalars emitted from the neutral hole indeed map to quanta whose emission we can compute from the microscopics of the near extremal hole. We will consider 11-dimensional M-theory compactified on a space T p × S. The torus has sides Li, i = 1, · · · 6 and the circle, which will be taken to be along the x 11 direction, has a radius R. In this space a “Schwarzschild string” is a product of a Schwarzschild black hole in the noncompact (10−p) dimensions and flat space along the (p+1) compact dimensions. (This is thus a (p + 1) black brane. We call this a string since it extends along x.) The 11-dimensional metric is ds11 = −(1− ( r0 r ))dt + dr (1− ( r0 r )n) (1) +rdΩn+1 + dz 2 + p ∑ i=1 (dx), where n = 7− p and r = ∑9 i=p+1(x ). When the Schwarzschild radius is smaller than the radius R, a second object the periodic Schwarzschild black hole which is a periodic version of a (11− p) dimensional Schwarzschild hole with x as one of the transverse directions is entropically favorable. This is the subject of [2], [10][12]. A proposal for the microscopic description from this phase has been given in terms of a gas of D0 branes [11,12]. The map which relates the Schwarzschild string to a charged hole consists of a boost in the covering space in which x is noncompact, but the other p directions are still compact z′ = z coshα + t sinhα, t′ = t coshα + z sinhα. (2) The boosted coordinate z has to be then compactified on a radius R which is related to R by a standard Lorentz contraction