Feynman Path Integral Formulation

5.4 Black Hole Particle Emission 157involved could be comparable to the black hole size, but this is not so due to theblueshift of frequencies in the vicinity of the horizon. In the following it will thereforebe assumed that such particles can be described by point-like objects.In the WKB approximation, the imaginary part of the amplitude for an s-waveoutgoing positive energy particle which crosses the horizon outward from r in to r outis given by∫ rout∫ rout∫ prImS = Im p r dr = Im dp ′ r dr , (5.69)r inr in 0where the actual emission rate is the square of the amplitude, Γ ∼ exp(−2ImS).Using Hamilton’s equation for the classical trajectory ˙q = ∂H/∂ p, here in the formdp r =( ) dr −1dH , (5.70)dtwith H = M −E and thus dH = −dE, and inserting the radial geodesic dr/dt givenby Eq. (5.68), one obtains∫ rout∫ EImS = −Imr in 01 −drdE ′√2G(M −E ′ )r. (5.71)The r-integral can now be done by residues, first by transforming to the variablez = √ r, and then by adding a Feynman iε to the energy, which slightly displaces thepole to the upper half-plane,∫ E ∫ zoutImS = −Im dE ′0 z in2z 2 dzz − √ 2G(M − E ′ + iε) . (5.72)After closing the contour in the upper half plane and keeping only the imaginarypart of the amplitude (the real part contributes an irrelevant phase) one hasFig. 5.1 The effective potentialV ef f (r), obtained from thegeodesic equation in Painlevécoordinates (here shown forG = M = 1). The maximumoccurs on the horizonr = 2MG.