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arXiv 1202.0540gr-qcmath-phmath.MP
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弯曲时空中的焦散线与波的传播

Caustics and wave propagation in curved spacetimes

Abraham I. Harte, Theodore D. Drivas

摘要:

研究了一般四维时空中光锥焦散线对线性标量场传播的影响。特别地,我们分析了相关格林函数的奇异结构。正如一般定理所预期的那样,与波动方程相关的格林函数沿着一大类零测地线是全局奇异的。尽管如此,给定测地线上奇点的"性质"并不一定保持不变。当遇到光锥的焦散线时,它可以改变特性。这些变化进行了研究,首先推导全球格林函数的标量场光滑平面波时空上传播。然后,我们使用彭罗斯极限认为,有一个意义上的“领先的阶奇异行为”的(通常未知)格林函数与一个通用的时空总是可以理解使用(已知)格林函数与一个适当的平面波时空。这种对应关系被用来推导出一个简单的规则,描述格林函数如何改变其奇异结构附近的一些参考零测地线。这样的变化只取决于沿着参考测地线遇到的共扼点的多重性。使用sigma(p,p ')表示Synge世界函数的适当推广,具有重数1的共扼点将涉及delta(sigma)的格林函数奇点转换为涉及1/pi sigma的奇点(反之亦然)。重数为2的共轭点可以被视为具有连续两次通过重数为1的共轭点的效果。

英文摘要:

We investigate the effects of light cone caustics on the propagation of linear scalar fields in generic four-dimensional spacetimes. In particular, we analyze the singular structure of relevant Green functions. As expected from general theorems, Green functions associated with wave equations are globally singular along a large class of null geodesics. Despite this, the "nature" of the singularity on a given geodesic does not necessarily remain fixed. It can change character on encountering caustics of the light cone. These changes are studied by first deriving global Green functions for scalar fields propagating on smooth plane wave spacetimes. We then use Penrose limits to argue that there is a sense in which the "leading order singular behavior" of a (typically unknown) Green function associated with a generic spacetime can always be understood using a (known) Green function associated with an appropriate plane wave spacetime. This correspondence is used to derive a simple rule describing how Green functions change their singular structure near some reference null geodesic. Such changes depend only on the multiplicities of the conjugate points encountered along the reference geodesic. Using sigma(p,p') to denote a suitable generalization of Synge's world function, conjugate points with multiplicity 1 convert Green function singularities involving delta(sigma) into singularities involving 1/pi sigma (and vice-versa). Conjugate points with multiplicity 2 may be viewed as having the effect of two successive passes through conjugate points with multiplicity 1.

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