The unification of quantum theory with general relativity must be one of the most interesting problem for theoretical physics. The answer to this problem is also expected to tell us about the structure of space and time.

There are various approaches to this problem. It is widely believed that the thermodynamical behavior of the black holes and the Hawking radiation will play a key role in this problem. The black hole is a region of space time in which the gravitational field is so strong that even light cannot escape from it to infinity. However, people noticed later that there is an analogy between black hole physics and thermodynamics. And after that, Hawking showed that due to the quantum effect, the black hole has a thermal radiation with a black body spectrum. This means that the black hole thermodynamics is not just an analogy. It should have some physical meaning. This is a very surprisingly fact since the black hole itself is just a solution of Einstein equation which is a hyperbolic second order partial differential equation. This fact gives many implications about microscopic structure of space time. For example, if the thermodynamical quantities of black hole are physical, then how do we explain them from statistical mechanics? Or more precisely, how to count the number of states and obtain the black hole entropy? The theory of quantum gravity should answer these questions. Indeed there are varieties of works to explain the black hole entropy from microscopic point of view.

On the other hand, the event horizon plays an important role in the thermodynamics of black holes. Unruh found that the Minkowski vacuum appears as a thermal state for an uniformly accelerated observer. This effect is known as the Unruh effect. And the Unruh effect is related to Hawking radiation via equivalence theorem. Just as the black hole case, the event horizon also emerges for the uniformly accelerated observer. There are many kinds of derivations of the Hawking radiation. From either of them, one can see that the existence of event horizon is very essential for the Hawking radiation. Furthermore, Ted Jacobson even found that one can derive Einstein equation by assuming the thermodynamics of horizons.

So the thermodynamics of event horizon is closely related to gravity, and must be a key to the quantum aspect of gravity. Understanding these thermodynamical properties of horizons better should be be important to understand the structure of space time. Until now, most of the discussions of the horizons were done in equilibrium region. In this thesis, we would like to investigate the non-equilibrium fluctuations related to the event horizons.

Our works are done on two cases. The first one is a stochastic approach to the Unruh effect and the second one is to show a fluctuation theorem for black holes.

When a particle uniformly accelerated in Minkowski vacuum, it will observe a thermal bath. Due to interactions with this thermal bath, the motion of the particle will become stochastic. Using the stochastic approach, we investigated the fluctuations of this particle and proved the equipartition theorem for the transverse fluctuations. We also obtained the relaxation time of the fluctuations and the extra radiation due to the fluctuations (the Unruh radiation). These results are also useful in experiments which are under planning to detect such radiation by using ultrahigh intensity lasers which are in constructing at Europe.

For black holes. We applied the recent developments in non-equilibrium statistical physics to area changing processes of the black hole interacting with external matter. We derived the non-equilibrium fluctuation theorems corresponding to Crooks and Jarzynski’s theorems. This will also give another derivation of the generalized second law of black hole thermodynamics. The second law holds only after taking a thermodynamic average, and it should be violated as an individual process in a way to satisfy the Jarzynski equality. This is a first step to understand the non-equilibrium nature of black hole horizons.

Behind the horizon thermodynamics, there should be a fundamental structure of spacetime. I hope that our results would make some help on this problem.

, application/pdf, 総研大甲第1417号}, title = {Quantum Fluctuations and Space-time Horizons}, year = {} }