Black Hole Evaporation in a Spherically Symmetric Non- Commutative Space-Time G. Esposito, INFN, Naples (QFEXT07, Leipzig, September 2007) with E. Di Grezia, G. Miele

Framework Recent work by Hawking, Farley & D’Eath, has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat spacetime and weak radiation at a very late time [Phys. Rev. D72, (2005); Ann. Phys. 321, 1334 (2006)].

Farley-D’Eath: p. 1 Their method can be applied to: (i)Quantum radiation associated with gravitational collapse to a black hole; (ii)Local collapse which is not sufficient to lead to Lorentzian curvature singularities; (iii)Quantum processes in cosmology, e.g. small fluctuations of an isotropic homogeneous universe.

Farley-D’Eath: p. 2 (i) There are quantum amplitudes for final outcomes; (ii) The end-state of black-hole evaporation is a combination of outgoing radiation states; (iii) They evaluate the quantum amplitude to go from data on an initial spacelike hypersurface to data on a final spacelike hypersurface. Space- time is taken to be asymptotically flat, with the above surfaces diffeomorphic to Euclidean space in three dimensions.

Farley-D’Eath: p. 3 (iv) Following Feynman, the proper time- interval T between such hypersurfaces is rotated into the complex:

Farley-D’Eath: p. 4 The Riemannian (Euclidean) quantum amplitude is proportional to the exponential of the imaginary unit times the part of the classical action quadratic in the non- spherical perturbative part. The Lorentzian amplitude is the limit for going to zero from the right of the above amplitude.

Farley-D’Eath: p. 5 The imaginary part of yields a Gaussian probability density. The real part of describes rapid oscillations through the phase of the quantum amplitude.

Non-commutativity effects We rely upon the recently obtained non- commutativity effect on a static, spherically symmetric metric (Nicolini, Smailagic & Spallucci) to consider from a new perspective the quantum amplitudes in black hole evaporation [Phys. Lett. B632, 547 (2006)].

The F factor [CQG,23,6425 (2006)] The general relativity analysis of spin-2 amplitudes is modified by a multiplicative factor F depending on a constant non- commutativity parameter and on the upper limit of the radial coordinate. Limiting forms of F are derived which are compatible with the adiabatic approximation here exploited.

Conformal infinity and wave equation [arXiv: ] For the scalar wave equation in a non- commutative spherically symmetric spacetime, we build the associated conformal infinity. The analysis of the wave equation is reduced to solving an inhomogeneous Euler-Poisson- Darboux equation. The scalar field has an asymptotic behaviour with a fall-off going on rather more slowly than in flat spacetime [Schmidt & Stewart in Proc. R. Soc. Lond. A367, 503 (1979)].