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Kerr-Schild symmetries. (English) Zbl 0982.83015
Summary: We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any $$n$$-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixed. The properties of these Lie algebras are briefly analyzed and we show that they are generically finite-dimensional but that they may have infinite dimension in some relevant situations. The most general vector fields of the above type are explicitly constructed for the following cases: any two-dimensional metric, the general spherically symmetric metric and deformation direction, and the flat metric with parallel or cylindrical deformation directions.

##### MSC:
 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory 53C80 Applications of global differential geometry to the sciences 17B81 Applications of Lie (super)algebras to physics, etc.
##### Keywords:
Kerr-Schild symmetries; space-time transformation
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##### References:
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