类光测地线的变分和动态黑洞的熵
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摘要
本文包含三部分主要内容:第一部分,论证并推广了类光测地线之加速度的定义,并指出可无限延伸的类光测地线之加速度为无穷大;第二部分,对具有共轭点的类光测地线进行变分,由此得到类时曲线在逼近此类光测地线时,其固有加速度趋于无穷大;第三部分,应用动态黑洞局部热平衡的概念,利用膜模型计算了动态Kinnersley黑洞的熵。
Rindler研究了平直时空中作匀加速直线运动的粒子,发现当此粒子的固有加速度A→∞时,该粒子的世界线趋于光子的世界线。因而Rindler认为,光子的加速度,可以看作“趋于它的匀加速直线运动的类时粒子的加速度”,即可以认为光子的加速度是无穷大[1]。
Rindler对光子加速度的这种定义,只是在平直时空中进行的;同时,光子在运动过程中受到镜面的反射作用,它的世界线本身不是一条完整的类光测地线,而是由两段类光测地线组成的,两测地线的连接点是一个拐点(即镜面反射点);再者,趋于类光线的粒子的运动状态也很特殊,是匀加速直线运动(四维语言是沿双曲类时线运动的粒子),因而这种定义有一定的局限性。
我们在第一章中,对Rindler的结果加以推广。首先,我们对于任意运动形式的趋于类光测地线(自由光子)的粒子,用类光测地线的变分对其进行数学描述,并且利用李导数概念写出粒子的世界线上的切矢量的表达式,最后证明对于弯曲时空中的任意类时线,只要其满足“一致趋于类光测地线”的条件,则该类时线的固有加速度一定趋于无穷大,因而类光测地线(自由光子)的加速度为无穷大的观点具有普适意义。
第二章前四节围绕奇点定理展开,介绍了非类空测地线族的相对运动规律(即相对转动,剪切,膨胀满足的规律),简介非类空测地线上的Jacobi场与共轭点的性质,以及同共轭点相关的许多定理。结合时空的因果结构,对广义相对论中的奇点概念及奇点定理作了介绍。
本章第五节和第七节是我们的主要工作。即对具有共轭点的类光测地线
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 x_0 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 conjugate
points 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 charged
as 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.
Rindler研究了平直时空中作匀加速直线运动的粒子,发现当此粒子的固有加速度A→∞时,该粒子的世界线趋于光子的世界线。因而Rindler认为,光子的加速度,可以看作“趋于它的匀加速直线运动的类时粒子的加速度”,即可以认为光子的加速度是无穷大[1]。
Rindler对光子加速度的这种定义,只是在平直时空中进行的;同时,光子在运动过程中受到镜面的反射作用,它的世界线本身不是一条完整的类光测地线,而是由两段类光测地线组成的,两测地线的连接点是一个拐点(即镜面反射点);再者,趋于类光线的粒子的运动状态也很特殊,是匀加速直线运动(四维语言是沿双曲类时线运动的粒子),因而这种定义有一定的局限性。
我们在第一章中,对Rindler的结果加以推广。首先,我们对于任意运动形式的趋于类光测地线(自由光子)的粒子,用类光测地线的变分对其进行数学描述,并且利用李导数概念写出粒子的世界线上的切矢量的表达式,最后证明对于弯曲时空中的任意类时线,只要其满足“一致趋于类光测地线”的条件,则该类时线的固有加速度一定趋于无穷大,因而类光测地线(自由光子)的加速度为无穷大的观点具有普适意义。
第二章前四节围绕奇点定理展开,介绍了非类空测地线族的相对运动规律(即相对转动,剪切,膨胀满足的规律),简介非类空测地线上的Jacobi场与共轭点的性质,以及同共轭点相关的许多定理。结合时空的因果结构,对广义相对论中的奇点概念及奇点定理作了介绍。
本章第五节和第七节是我们的主要工作。即对具有共轭点的类光测地线
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 x_0 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 conjugate
points 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 charged
as 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.
引文
1. W. Rindler. Essential Relativity. New York: Springer-Verlag, 1977.
2. Guihua Tian, ZhaoZheng and C Liang, Classical and Quantum Gravity, Vol19, No 11, (2002).
3.赵峥。黑洞的热性质与时空奇异性。北京:北京师范大学出版社,1999年。
4.梁灿彬。微分几何入门与广义相对论(上册),北京:北京师范大学出版社,2000年
5.梁灿彬。微分几何入门与广义相对论(下册),北京:北京师范大学出版社,2001年
6.刘辽。广义相对论。北京:高等教育出版社,1987年。
7.张元仲。狭义相对论实验基础。北京:科学出版社,1994年。
8.俞允强。广义相对论引论。北京:北京大学出版社,1997年。
9. S. W. Hawking and G. F. R. Ellis. The large scale structure of space-time. Cambridge: Cambridge University Press, 1973.
10. R. M. Wald, General Relativity. Chicago: The University of Chicago Press, 1984.
2. Guihua Tian, ZhaoZheng and C Liang, Classical and Quantum Gravity, Vol19, No 11, (2002).
3.赵峥。黑洞的热性质与时空奇异性。北京:北京师范大学出版社,1999年。
4.梁灿彬。微分几何入门与广义相对论(上册),北京:北京师范大学出版社,2000年
5.梁灿彬。微分几何入门与广义相对论(下册),北京:北京师范大学出版社,2001年
6.刘辽。广义相对论。北京:高等教育出版社,1987年。
7.张元仲。狭义相对论实验基础。北京:科学出版社,1994年。
8.俞允强。广义相对论引论。北京:北京大学出版社,1997年。
9. S. W. Hawking and G. F. R. Ellis. The large scale structure of space-time. Cambridge: Cambridge University Press, 1973.
10. R. M. Wald, General Relativity. Chicago: The University of Chicago Press, 1984.