Feynman Path Integral Formulation

156 5 Semiclassical Gravityconservation: the mass of the black hole M needs to be decreased suitably when thevirtual particle is emitted, thus leading to a non-zero real tunneling amplitude, whichcan then be shown to agree with the original Hawking calculation.The computation is most easily carried out in Painlevé coordinates for a static,non-rotating black hole. The Painlevé line element (Painlevé, 1921; Gullstrand,1922) reads(ds 2 = − 1 − 2MG ) √2MGdt 2 + 2 dt dr + dr 2 + r 2 dΩ2 2 . (5.66)rrThe corresponding metric describes the same physics, but has several attractive featureswhen compared to the Schwarzschild metric: none of the metric (or inversemetric) components diverge on the horizon r = 2MG; furthermore it still covers theinside and outside of the black hole, and constant time slices simply correspond toflat Euclidean space. One can show that the Painlevé time is related to the originalSchwarzschild time t s byt = t s + 2 √ √ √r − 2MG2MGr + 2MGln √ √ . (5.67)r + 2MGFrom dτ/dt = 1 it follows that in these coordinates the time t is linearly related toproper time, τ = t + c, for a radially infalling observer.In this metric the radial null geodesics have a rather simple form,drdt√2MG= ±1 − , (5.68)rwhere the choice of signs depends on whether the rays go towards infinity (+),or away from it (−). One can view the above geodesic equation as arising froma classical mechanics effective potential V ef f (r) = √ 2MG/r − GM/r, showninFig. (5.1), with a total energy fixed at 1/2. Note that the maximum of this functionis precisely at r = 2GM, and that the peak is at the total energy value 1/2, whichseems to make the two classical turning points coincide with the peak.The fact that the coordinate system is stationary and non-singular allows one todefine what is meant by a vacuum: a state whose quantum fields will annihilatemodes which carry negative frequency with respect to the Painlevé time t. Butitis important to note that modifications arise when the particle’s self-gravitation istaken into account, and which are crucial in obtaining the correct result. For a nonrotatingself-gravitating shell of energy E (visualized as an s-wave state) one canshow (Kraus and Wilczek, 1995) that the shell moves on a geodesic still describedby the line element in Eq. (5.66), but with mass M → M + E, whereas if the totalmass is fixed and the black hole mass is allowed to vary, then the shell moves on ageodesic with mass M → M − E.In order to compute a tunneling amplitude one would like to use the semiclassicalor WKB approximation, which assumes point particles. One might worrythat a point particle description might not be adequate since the wavelengths