Bi-conformal vector fields and their applications

Details Bi-conformal vector fields and their applications Bi-conformal vector fields and their applications

2003-11-10

arxiv.org/abs/math-ph/0311014v2

Class.Quant.Grav. 21 (2004) 2153-2178

Physics

Mathematical Physics

Replaced version with some changes in the terminology and a new theorem. To appear in Classical and Quantum Gravity

Scientific paper

10.1088/0264-9381/21/8/017

We introduce the concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric $\G$ to be scaled by different conformal factors. In particular, we study their infinitesimal version, called bi-conformal vector fields. We show the differential conditions characterizing them in terms of a "square root" of the metric, or equivalently of two complementary orthogonal projectors. Keeping these fixed, the set of bi-conformal vector fields is a Lie algebra which can be finite or infinite dimensional according to the dimensionality of the projectors. We determine (i) when an infinite-dimensional case is feasible and its properties, and (ii) a normal system for the generators in the finite-dimensional case. Its integrability conditions are also analyzed, which in particular provides the maximum number of linearly independent solutions. We identify the corresponding maximal spaces, and show a necessary geometric condition for a metric tensor to be a double-twisted product. More general ``breakable'' spaces are briefly considered. Many known symmetries are included, such as conformal Killing vectors, Kerr-Schild vector fields, kinematic self-similarity, causal symmetries, and rigid motions.

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