General Relativity and Gravitation

, Volume 38, Issue 3, pp 425–443 | Cite as

Vaidya Space-Time in Black-Hole Evaporation

  • A. N. St. J. Farley
  • P. D. D'Eath
Research Article


We take a boundary-value approach to quantum amplitudes arising in gravitational collapse to a black hole. Pose boundary data on initial and final space-like hypersurfaces Σ F,I , separated at spatial infinity by a Lorentzian proper-time interval T. Quantum amplitudes are calculated following Feynman's approach; rotate: T→|T|exp (−iθ) into the complex, where 0< θ≤π/2, and solve the corresponding well-posed complex classical boundary-value problem. We compute the classical Lorentzian action S class and corresponding semi-classical quantum amplitude, proportional to exp (iS class). To recover the Lorentzian amplitude, take the limit θ→ 0+ of the semi-classical amplitude. For the classical boundary-value problem with given perturbative boundary data, we compute an effective spherically-symmetric energy-momentum tensor 〉 T μν EFF , averaged over several wavelengths of the radiation, describing the averaged extra energy-momentum contribution in the Einstein field equations, due to the perturbations. This takes the form of a null fluid, describing the radiation (of quantum origin) streaming radially outwards. The classical space-time metric, in this region of the space time, is of Vaidya form, justifying the adiabatic radial mode equations, for spins s = 0 and s = 2.


Black Holes Quantum evaporation Boundary-value formulation Vaidya geometry 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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