Feynman Path Integral Formulation

154 5 Semiclassical Gravityessentially non-perturbative, and non-analytic in the external field perturbation E.Furthermore, a comparison of the exact answer in Eq. (5.61) with the WKB resultof Eq. (5.54) shows that the latter only gives the leading term in an infinite series ofprogressively smaller contribution.The result obtained for electrons and positrons in strong uniform electric fieldsare clearly not transferable as is to the gravitational case. For once, there is no notionof oppositely charged particles in gravity. Thus the naive replacement |eE|→m/4MG for a particle-antiparticle pair, say close to the horizon of a black hole, doesnot seem to make much sense. Yet the strong electric field QED calculation showsthat quantum mechanical tunneling events can take place, and that their effects canbe computed to first order using the WKB approximation. A second lesson fromQED is that in general higher order corrections should be expected.5.4 Black Hole Particle EmissionOriginally quantum gravitational effects for black holes were ignored, since the radiusof curvature outside the black hole is much larger than the Planck length, thelength scale on which one would expect quantum fluctuations of the metric to becomeimportant. If the gravitational field is able to create locally virtual pairs, thelocal energy density associated with such a pair would be much smaller than theenergy scale associated with the local curvature. It can be shown though that in thevicinity of a black hole horizon particle production is possible, due to vacuum fluctuationsand tunneling. The resulting effects add up over time, are therefore macroscopicand could in principle be observable.Normally when describing a stationary non-rotating black hole one uses theSchwarzschild metric in standard formds 2 = −(1 − 2GMr)dt 2 +(1 − 2MGr) −1dr 2 + r 2 dΩ 2 2 , (5.62)with dΩ2 2 ≡ dθ 2 + sin 2 θdφ 2 . The metric shows a singularity at r = 2MG. Sincenone of the curvature invariants are singular on the horizon r = 2MG, one wouldexpect the singularity to be perhaps an artifact of the coordinate system, here mostsuitably describing the viewpoint of an observer stationary at infinity. On the otherhand a freely falling observer is expected to pass initially unscathed through theblack hole horizon, and indeed a singularity-free coordinate system can be found describingsuch an observer (Kruskal, 1960). In these new coordinates, defined by thetransformation from the original Schwarzschild coordinates (r,θ,φ,t) → (r ′ ,θ,φ,t ′ )(r ′ 2 − t′2 = T2 r) ( r)2GM − 1 exp2GM2r ′ t ′ ( t)r ′2 + t ′ 2 = tanh , (5.63)2GM