Problem 1: Black holes 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, in the everyday sense of the word. It is true 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 the 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 its entropy. a) Use dimensional analysis to show that a black hole of mass M should have a radius of order GM/c², 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 × 103ºkg). b) 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 (maximizing its entropy) you should use particles with the lowest possible energy: long-wavelength photons. 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 Mc², estimate the maximum number of photons that could be used to make a black hole of mass M. Aside from a factor of 8n², your result should agree with the exact formula obtained by Stephen Hawking (1973), through much more difficult calculations: 8n?GM² -kg hc Splack hole = d) Calculate the entropy of a one-solar mass black hole, and comment on the result.

Problem 1: Black holes 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, in the everyday sense of the word. It is true 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 the 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 its entropy. a) Use dimensional analysis to show that a black hole of mass M should have a radius of order GM/c², 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 × 103ºkg). b) 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 (maximizing its entropy) you should use particles with the lowest possible energy: long-wavelength photons. 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 Mc², estimate the maximum number of photons that could be used to make a black hole of mass M. Aside from a factor of 8n², your result should agree with the exact formula obtained by Stephen Hawking (1973), through much more difficult calculations: 8n?GM² -kg hc Splack hole = d) Calculate the entropy of a one-solar mass black hole, and comment on the result.