Plenty of Nothing: Black Hole Entropy in Induced Gravity

We demonstrate how Sakharov’s idea of induced gravity allows one to explain the statistical-mechanical origin of the entropy of a black hole. According to this idea, gravity becomes dynamical as the result of quantum effects in the system of heavy constituents of the underlying theory. The black hole entropy is related to the properties of the vacuum in the induced gravity in the presence of the horizon. We obtain the Bekenstein-Hawking entropy by direct counting the states of the constituents. There are physical phenomena that allow a simple description but require tremendous efforts to explain them. Without doubts the origin of black hole entropy is such a problem. A black hole of mass M radiates as a heated body [1] with temperature TH = (8πGM) −1 and has entropy [2] S = A 4G , (1) where A is the surface area of the black hole (A = 16πGM), G is Newton’s constant, and c = h̄ = 1. Statistical mechanics relates entropy to the measure of disorder and qualitatively it is the logarithm of the number of microscopically different states available for given values of the macroscopical parameters. Are there internal degrees of freedom that are responsible for the Bekenstein-Hawking entropy S? This is the question that physicists were trying to answer for almost 25 years. What makes the problem of black hole entropy so intriguing? Before discussing more technical aspects let us make simple estimations. Consider for example a supermassive black hole of 10 solar mass. According to (1), its entropy is about 10. This is seven orders of magnitude larger than the entropy of the other matter in the visible part of the Universe. What makes things even more complicated, a black hole is simply an empty