By setting the total energy of the photons equal to Mc 2, estimate the maximum number of photons that could be used to make a black hole of mass M. But the wavelength can’t be any longer than the size of the black hole. 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 mass less particles). b) In the spirit of below Problem, 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. Calculate the approximate radius of a one-solar-mass black hole ( M = 2 × 1030 kg). a) Use dimensional analysis to show that a black hole of mass M should have a radius of order GM/c 2, where G is Newton’s gravitational constant and c is the speed of light. Knowing this, it’s not hard to estimate the entropy of 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. 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. In fact, it is irreversible in the thermodynamic sense as well: Adding mass to a black hole increases the black hole’s entropy. Throwing something into a black hole is therefore an irreversible process, at least in the everyday sense of the word. Problem 42P A black hole is a region of space where gravity is so strong that nothing, not even light, can escape.
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