Keita Sakai’s research while affiliated with Kyoto Sangyo University and other places

One of the striking concepts of black hole is that the entropy of the black hole is proportional to the surface. The key concept to understand this statement is the uncertainty principle. As the black hole has the gravitational radius, there is the corresponding momentum. It can be speculated that the temperature of the black hole is proportional to the energy of the corresponding particle (photon). If the black hole is formed by such photons or particles, it could be explained that the temperature of the black hole is proportional to the surface gravity and the entropy of the black hole is proportional to the surface of the black hole. Recently it is confirmed the existence of cosmological constant. In the space with the cosmological constant (de Sitter space), each observer is surrounded by the event horizon of the cosmological scale. There is the characteristic length derived from the cosmological constant, so it could be speculated the temperature of the horizon from the uncertainty principle. If we assume that the universe is composed of particles (photons) with such de Brogile length, it could be derived that the entropy of the universe is related to the surface of the horizon. The similar consideration is applied to the uniformly accelerated coordinate (Rindler coordinate), where it has the characteristic length due to its acceleration. If we assume that the acceleration is due to the gravity of the gravitational mass of particles, it is derived that the entropy per surface is constant.

By a simple physical consideration and uncertain principle, we derive that temperature is proportional to the surface gravity and entropy is proportional to the surface area of the black hole. We apply the same consideration to de Sitter space and estimate the temperature and entropy of the space, then we deduce that the entropy is proportional to the boundary surface area. By the same consideration, we estimate the temperature and entropy in the uniformly accelerated system (Rindler coordinate). The cases in higher dimensions are considered.

The entropy increases enormously when a star collapses into a black hole. For the radiation dominated star, the temperature has decreased from the almost gravitational equilibrium one to the so-called black hole temperature. If we interpret this temperature decrease as the volume increase of the black hole, the entropy increase could be understood as the free expansion of radiation throughthe increased volume. The expected volume size is tentatively estimated. Under this interpretation, it is derived that the entropy is proportional to the horizon surface area and the microscopic states of the black hole entropy could be understood as the statistical states of the enlarged phase space.