By Maulik Kirit Parikh

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**Example text**

Since such a particle propagates backwards in time, we have to reverse time in the equations of motion. From the line element, Eq. 13), we see that time-reversal corresponds to 2M r →− 2M . r Also, since the anti-particle sees a geometry of fixed black hole mass, the upshot of self-gravitation is to replace M by M + ω, rather than M − ω. 20) where to obtain the last equation we have used Feynman’s “hole theory” deformation of the contour: ω ′ → ω ′ + iǫ. Both channels – particle or anti-particle tunneling – contribute to the rate for the Hawking process so, in a more detailed calculation, one would have to add their amplitudes before squaring in order to obtain the semi-classical tunneling rate.

For any given value of α = (1 − 2M/r)1/2 , there is a maximum wavelength, λmax , for outgoing modes that are invisible to the observer: λmax = r − 2M (1 − 2M/r)1/2 → 2Mα . 5. Hamiltonian Formulation to exist. The regularity condition amounts to a coarse-graining over these modes. We conclude this section with an illustration of the intuitive advantage of the membrane paradigm. It is a famous result that the external state of a stationary black hole, quite unlike that of other macroscopic bodies, can be completely characterized by only four quantities: the mass, the angular momentum, and the electric and magnetic charges.

5 Tunneling To describe across-horizon phenomena, it is necessary to choose coordinates which, unlike Schwarzschild coordinates, are not singular at the horizon. 12) where ts is Schwarzschild time. With this choice, the line element reads ds2 = − 1 − 2M dt2 + 2 r 2M dt dr + dr 2 + r 2 dΩ2 . 13) There is now no singularity at r = 2M, and the true character of the spacetime, as being stationary but not static, is manifest. These coordinates were first introduced by the French mathematician Paul Painlev´e [41] and the Swedish opthalmologist and Nobel laureate Allvar Gullstrand [42], who used them to criticize general relativity for allowing singularities to come and go!