2. Black hole heat loss, mass and temperature The properties of a black hole can, in many cases, be well-understood by simple thermody- namics that we are discussing. The temperature, T, of a black hole is related to it's mass, m, via the following expression: o T= 1 " M 1 PHYS 133b, Spring Quarter 2024 Problem Set #4, Thursday Apr 11th, 2024 and the change in heat, @, for a given change in mass of the black hole is given by: dQ = 2 am (2) where is the speed of light (a constant). Assume that the heat lost during a temperature change can also be written as i% = (), just as we did in class, where C, is the effective specific heat of the black hole at constant pressure. (a) Calculate the derivative of the temperature with respect to mass. State in words how the temperature changes when mass is gained or lost. (b) Use this to calculate the specific heat, C, of the black hole. What does this mean in terms of change in heat for a given change in temperature