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キーワード:
General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
要旨:
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.
