Variational equations on mixed Riemannian-Lorentzian metrics

Details Variational equations on mixed Riemannian-Lorentzian metrics Variational equations on mixed Riemannian-Lorentzian metrics

2007-02-01

arxiv.org/abs/math-ph/0702007v3

Journal of Geometry and Physics 58(2008) 1043-1061

Physics

Mathematical Physics

28 pages

Scientific paper

10.1016/j.geomphys.2008.03.003

A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3-space. This suggests an approach to generalized Plateau problems in 3-dimensional space-time via Hodge theory on the extended projective disc. Analogous variational problems arise on Riemannian-Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain singular Riemannian-Lorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven.

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