Class Light Geodesic Variational And Dynamic Black Hole Entropy

The thesis consists three parts: the first part generalize the definition of the proper acceleration of a null geodesic to curved space-time and prove that the future-complete null geodesic's proper acceleration is infinity;we study the variation of a null geodesic with conjugate points on it, and make the conclusion that the proper acceleration of the time-like curves generally approach infinity as they approach the null geodesic in the second part of the thesis;finally, making use of the local equilibrium, we obtain the entropy of the 'arbitrarily accelerating' Kinnersley black hole.In chapter I,given a null geodesic in Minkowski space-time, there exists a one-parameter family of observers in 'hyperbolic' motion which approaches the world-line of photon as the parameter x₀ approaches zero. It is well-known that the proper acceleration of the observers in the family approaches infinity as his world line approaches the world-line of photon-this is the main reason for Rindler to suggest that "the photon' proper acceleration can be taken to be infinty". the world-line of photon is not a full null geodesic, and consists of two null geodesic segements that is connected by a point (where a reflecting mirror acts on the photon). We consider the proper acceleration of the free photon whose world-line is a full null geodesic. The main purpose of the first chapter is to generalize this result to future-complete null geodesies in curved space-times.In chapter II, we first introduce non-space-like geodesic congruence, the property of the Jacobi field on a non-space-like geodesic, and the corresponding theorems. Then the causal strcture,singularity,and the sigularity theurems are introduced.Secondly, we consider the variation of a null geodesic with conjugatepoints on it. It is well-known fact that given a null geodesic 70(A) with a point r in (p, q) conjugate to p along 70(A) , there will be a variation of 70(A) which will give a time-like curve from p to q. This is proved in the famous book[10]. In the first part of the second chapter, we prove that the time-like curves coming from the above-mentioned variation have a proper acceleration approaching infinity as the time-like curve approaches the null geodesic.Because the curve obtained from variation must be everywhere time-like, we discuss the constrain of the 'acceleration' of the variation vector field on the null geodesic 70(A) in the third part of this chapter. The 'acceleration' of the variation vector field on the null geodesic 70 (A) can not be zero, we also discuss the condition satisfied by the 'acceleration' of the deviation vector field on the null geodesies [70 (A)]The first part of the chapter III introduces the fundamental property of the black-hole thermodynamics and the calculation of the blackhole entropy in the brick-wall model and the thin film model, second , by the generalized tortoise coordinate transformation we get the Hawking radiation temperature and spectrum, which indicates the particle has chemical potential originated from the acceleration of the black-hole. Later, we disscuss the codition required by the the assumption of local equilibrium near the horizon, that is , the evaporation of the hole is negligible and the change of the acceleration is slow,-this could be met by the actual black-hole. With the thin film model and the notion of the local equilibrium, we obtain the entropy of the Kinner-sley blackhole ,which is proportional to its area with the same geometrical cutoff relationship as in the static case.Finally, we try to intrduce a new coordinate system in which the black-hole horizon is identical with the infinite red-shift surface. We then use the thin-film model to calculate the entropy of nonstationary blackhole. We use an arbitrarily accelerating black hole that is electrically and magnetically chargedas an example. We introduce the new coordinate system in which g00 is zero at the event horizon's surface r = r/j, and calculate the entropy locally via the thin film model with the locally thermal equilibrium satisfied. The results confirm that the entropy is propositional to its area both in the stationary space-time and non-stationary one.