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Spin-2 Amplitudes in Black-Hole Evaporation
Quantum amplitudes for gravitational-wave perturbations of
Einstein/scalar collapse to a black hole are treated by analogy with
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 , for natural 'coordinate' variables are given by the magnetic
part of the Weyl tensor, which can be taken as boundary
data on a final space-like hypersurface . For simplicity, we take the
data on the initial surface to be exactly spherically-symmetric. The
(large) Lorentzian proper-time interval between and ,
measured at spatial infinity, is denoted by . We follow Feynman's
prescription and rotate into the complex: , for . 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 . For
boundary data well below the Planck scale, and for a locally supersymmetric
theory, this involves only the semi-classical amplitude , where denotes the second-variation classical
action. The relations between the and natural boundary data,
involving supersymmetry, are investigated using 2-component spinor language in
terms of the Maxwell field strength and the Weyl spinor
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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
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