Problem 2.42. A black hole is a region of space where gravity is so strong that nothing, not even light, can escape. Throwing something into a black hole is therefore an irreversible process, at least in the everyday sense of the word. In fact. it is irreversible in the thermodynamic sense as well: Adding mass to a black hole increases the black hole's entropy. It turns out that there's no way to tell (at least from outside) what kind of matter has gone into making a black hole.' Therefore, the entropy of a black hole must be greater than the entropy of any conceivable type of matter that could have been used to create it. Knowing this, it's not hard to estimate the entropy of a black hole. (a) Use dimensional analysis to show that a black hole of mass M should have a radius of order GM/c-, where G is Newton's gravitational constant and c is the speed of light. Calculate the approximate radius of a one-solar-mass black hole (M = 2 x 10 " kg). (b) In the spirit of Problem 2.36, explain why the entropy of a black hole, in fundamental units, should be of the order of the maximum number of particles that could have been used to make it.(c) To make a black hole out of the maximum possible number of particles. you should use particles with the lowest possible energy: long-wavelength photons (or other massless particles). But the wavelength can't be any longer than the size of the black hole. By setting the total energy of the photons equal to Me', estimate the maximum number of photons that could be used to make a black hole of mass M. Aside from a factor of 8x-, your result should agree with the exact formula for the entropy of a black hole, obtained * through a much more difficult calculation: 872GM- Sh.h. = he K. (d) Calculate the entropy of a one-solar-mass black hole, and comment on the result.Problem 3.7. Use the result of Problem 2.42 to calculate the temperature of a black hole, in terms of its mass M. (The energy is Mo-.) Evaluate the resulting expression for a one-solar-mass black hole. Also sketch the entropy as a function of energy, and discuss the implications of the shape of the graph.[This problem requires some estimation and physical argumentation, rather than mathematical derivation. You may find it useful to know that the energy ofa photon is hcl, where h in Planck's constant, c is the speed of light, and FL is the photon wavelength]