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Black Hole Evaporation in a Spherically Symmetric Non-Commutative Space-Time

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.

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Black Hole Evaporation in a Spherically Symmetric Non-Commutative Space-Time

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  1. 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

  2. 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, 084013 (2005); Ann. Phys. 321, 1334 (2006)].

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

  4. 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.

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

  6. 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.

  7. 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.

  8. 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)].

  9. 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.

  10. Conformal infinity and wave equation [arXiv:0705.0242] • 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)].

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