In this problem we will derive the expression for the lifetime of a nonrotating, evaporating black hole (Eq. 17.30).
(a) Consider a black hole to be a perfectly radiating blackbody of temperature T, given by Eq. (17.34) in Problem 17.22. Assuming that the surface area of the black hole is given by 4πRS2, where RS is the Schwarzschild radius, show that the luminosity of the black hole due to Hawking radiation is
L=15360πG2M2ℏc6=3840πRS2ℏc2.(b) The luminosity of the black hole must originate from a loss in the black hole's internal energy. Assuming that the energy of the black hole is given by E=Mc2 and that L=dE/dt, show that the time required for the black hole to lose all of its mass to Hawking radiation is given by Eq. (17.30).