Curvature decomposition in circular space-times

Advisor: Oldřich Semerák (ITP MFF CUNI)

Funding: Basic scholarship


When computing Kretschmann or other curvature scalar for a not-totally-simple metric, one often gets stuck in a cumbersome expression whose individual terms have no clear meaning. Mainly problematic is to evaluate such scalars on a black-hole horizon: they are usually finite there, yet their coordinate expressions contain many divergent terms (which cancel against each other, but in a complicated way). At the same time, it is known that in simple cases (static and axisymmetric electro-vacuum, in particular) the Kretschmann and possibly other scalars are very simply related to the clear lower-dimensional geometrical quantities like Gauss' curvature of the horizon surface.

I thus propose to derive, for circular (i.e., stationary, axisymmetric and orthogonally transitive) space-times, the decomposition of curvature in terms of quantities describing suitably chosen submanifolds and/or suitably chosen "observer" congruence. An ultimate goal is to obtain such a decomposition of curvature which would enable one to express the curvature scalars in a simple and -mainly- "geometrical" manner. (For those who know of a usual splitting with respect to meridional planes: this one is different.)

Actually, I have already done quite some work in this direction and even wrote it up in a paper style (unpublished). However, the formulas derived up to now should first be checked and then the whole idea should be finished (which need not be easy). The necessary basis are Gauss-Codazzi-Ricci equations, and, for checking the formulas, the knowledge of Maple or Mathematica (or similar) is required. If reaching such a goal, we could use it in many ways (to analyse the invariants in specific space-times).