Hawking, by combining methods from quantum mechanics and general relativity, calculated the emission of radiation from a black hole." found a wonderful result: black holes emit per- fect black-body radiation at a temperature he 8TGMKB Th (5.38) According to Einstein's theory, the energy of the black hole is E = Me. (a) Calculate the specific heat of the black hole. The specific heat of a black hole is negative. That is, it gets cooler as you add energy to it. In a bulk material, this would lead to an instability; the cold regions would suck in more heat and get colder. Indeed, a population of black holes is unstable; the larger ones will eat the smaller Ones, 45 (b) Calculate the entropy of the black hole, by us- ing the definition of temperature 1/T = as/aE and assuming the entropy is zero at mass M = 0. Express your result in terms of the surface area A = 4n R, mcasured in units of the Planck length L = VhG/E squared. As it happens, Bekenstein had deduced this for- mula for the entropy somewhat earlier, by think- ing about analogies between thermodynamics, information theory, and statistical mechanics. On the one hand, when black holes interact or change charge and angular momentum, one can prove in classical general relativity that the area can only increase. So it made sense to assume that the entropy was somehow proportional to the area. He then recognized that if you had some waste material of high entropy to dispose of, you could ship it into a black hole and never worry about it again. Indeed, given that the en- tropy represents your lack of knowledge about a system, once matter goes into a black hole one can say that our knowledge about it completely vanishes. 0 (More specifically, the entropy of a black hole represents the inaccessibility of all in- formation about what it was built out of.) By carefully dropping various physical systems into a black hole (theoretically) and measuring the area increase compared to the entropy increase,