Mathematics – Mathematical Physics
Scientific paper
Sep 1983
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1983cmaph..88..339r&link_type=abstract
Communications in Mathematical Physics, Volume 88, Issue 3, pp.339-355
Mathematics
Mathematical Physics
2
Scientific paper
Caustics formed by timelike and null geodesics in a space-time M are investigated. Care is taken to distinguish the conjugate points in the tangent space ( T-conjugate points) from conjugate points in the manifold ( M-conjugate points). It is shown that most nonspacelike conjugate points are regular, i.e. with all neighbouring conjugate points having the same degree of degeneracy. The regular timelike T-conjugate locus is shown to be a smooth 3-dimensional submanifold of the tangent space. Analogously, the regular null T-conjugate locus is shown to be a smooth 2-dimensional submanifold of the light cone in the tangent space. The smoothness properties of the null caustic are used to show that if an observer sees focusing in all directions, then there will necessarily be a cusp in the caustic. If, in addition, all the null conjugate points have maximal degree of degeneracy (as in the closed Friedmann-Robertson-Walker universes), then the space-time is closed.
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