Spin-2 Amplitudes in Black-Hole Evaporation

Quantum amplitudes for $s=2$ gravitational-wave perturbations of Einstein/scalar collapse to a black hole are treated by analogy with $s=1$ Maxwell perturbations. The spin-2 perturbations split into parts with odd and even parity. We use the Regge-Wheeler gauge; at a certain point we make a gauge transformation to an asymptotically-flat gauge, such that the metric perturbations have the expected falloff behaviour at large radii. By analogy with $s=1$, for $s=2$ natural 'coordinate' variables are given by the magnetic part $H_{ij} (i,j=1,2,3)$ of the Weyl tensor, which can be taken as boundary data on a final space-like hypersurface $\Sigma_F$. For simplicity, we take the data on the initial surface $\Sigma_I$ to be exactly spherically-symmetric. The (large) Lorentzian proper-time interval between $\Sigma_I$ and $\Sigma_F$, measured at spatial infinity, is denoted by $T$. We follow Feynman's $+i\epsilon$ prescription and rotate $T$ into the complex: $T\to{\mid}T{\mid} \exp(-i\theta)$, for $0<\theta\leq\pi/2$. The corresponding complexified {\it classical} boundary-value problem is expected to be well-posed. The Lorentzian quantum amplitude is recovered by taking the limit as $\theta\to 0_+$. For boundary data well below the Planck scale, and for a locally supersymmetric theory, this involves only the semi-classical amplitude $\exp(iS^{(2)}_{\rm class}$, where $S^{(2)}_{\rm class}$ denotes the second-variation classical action. The relations between the $s=1$ and $s=2$ natural boundary data, involving supersymmetry, are investigated using 2-component spinor language in terms of the Maxwell field strength $\phi_{AB}=\phi_{(AB)}$ and the Weyl spinor $\Psi_{ABCD}=\Psi_{(ABCD)}$

Black hole evaporation in a spherically symmetric non-commutative space-time

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in noncommutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side. Relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, we have considered from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes has been shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F have been derived which are compatible with the adiabatic approximation.Comment: 8 pages, Latex file with IOP macros, prepared for the QFEXT07 Conference, Leipzig, September 200

Gravitational amplitudes in black-hole evaporation: the effect of non-commutative geometry

Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space-time and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in noncommutative geometry have shown that, in general relativity, the effects of noncommutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy-momentum tensor as a source on the right-hand side. The present paper, relying on the recently obtained noncommutativity effect on a static, spherically symmetric metric, considers from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes is shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F are derived which are compatible with the adiabatic approximation here exploited. Approximate formulae for the particle emission rate are also obtained within this framework.Comment: 14 pages, 2 figures, Latex macros. In the final version, section 5 has been amended, the presentation has been improved, and References 21-24 have been added. Last misprints amended in Section 5 and Ref. 2

Relic Radiation from an Evaporating Black Hole

We present a non-string-theoretic calculation of the microcanonical entropy of relic integer-spin Hawking radiation -- at fixed total energy $E$. The only conserved macroscopic quantity is the total energy $E$ (the total energy of the relic radiation). Data for a boundary-value approach, with massless, integer-spin perturbations, are set on initial and final space-like hypersurfaces. In the resulting 1-dimensional statistical-mechanics problem, the real part of the (complex) time separation at spatial infinity, $T = {\mid}T{\mid}\exp(-i\delta), \delta >0$, is the variable conjugate to the total energy. We count the number of weak-field configurations on the final space-like hypersurface with energy $E$. One recovers the Cardy formula and the Bekenstein-Hawking entropy, if Re(T) is of the order of the black-hole life- time, leading to a statistical interpretation of black-hole entropy. The microcanonical entropy includes a logarithmic correction to the black-hole area law, which is {\it universal} (independent of black-hole parameters). Here, the discreteness of the energy levels is crucial. This approach is compared with that of string theory for the transition to the fundamental-string r\'egime in the final stages of evaporation. The squared coupling, $g^2$, regulating the transition to a highly-excited string state and {\it vice versa}, can be related to the angle, $\delta$, of complex-time rotation above. The strong-coupling r\'egime corresponds to a Euclidean black hole, while the physical limit of a Lorentzian space-time (as $\delta \to 0_+$) corresponds to the weak-coupling r\'egime. This resembles the transition to a highly-excited string-like state which subsequently decays into massless particles, thereby avoiding the naked singularity.Comment: To appear in International Journal of Modern Physics

Black Hole Evaporation in a Noncommutative Charged Vaidya Model

The aim of this paper is to study the black hole evaporation and Hawking radiation for a noncommutative charged Vaidya black hole. For this purpose, we determine spherically symmetric charged Vaidya model and then formulate a noncommutative Reissner-Nordstr$\ddot{o}$m-like solution of this model which leads to an exact $(t-r)$ dependent metric. The behavior of temporal component of this metric and the corresponding Hawking temperature is investigated. The results are shown in the form of graphs. Further, we examine the tunneling process of the charged massive particles through the quantum horizon. It is found that the tunneling amplitude is modified due to noncommutativity. Also, it turns out that black hole evaporates completely in the limits of large time and horizon radius. The effect of charge is to reduce the temperature from maximum value to zero. It is mentioned here that the final stage of black hole evaporation turns out to be a naked singularity.Comment: 25 pages, 36 figures, accepted for publication in J. Exp. Theor. Phy