A Quick Overview of Killing Vector Fields
Hi, Ateowa and a belated welcome to PF,
The short answer is: "read in a good gtr textbook about Killing vector fields on a Lorentzian manifold".
A slightly longer answer:
The Lorentz transformations are self-isometries of Minkowski spacetime, that is, transformations which preserve "intervals" between all events, or IOW preserve the metric tensor. More precisely, the Lorentz group is a six-dimensional real Lie group which consists of all self-isometries which take the origin to itself; it is a subgroup of the ten-dimensional real Lie group of all self-isometries of Minkowski spacetime, which is called the Poincare group. The Lie algebra of a Lie group is a kind of "linearization" of the group in which elements of the group are replaced by vector fields on our manifold (which, it turns out, can be thought of as first order linear partial differential operators on the manifold). The Lie algebra of the Poincare group is generated by ten vector fields: three rotations, three boosts, three spatial translations, and one time translation 3+3+3+1=10.
This is exactly analogous to saying that the (six-dimensional) euclidean group is the group of all self-isometries of three-dimensional euclidean space, and the (three-dimensional) rotation group is the subgroup consisting of all self-isometries taking the origin to itself. The Lie algebra of the euclidean group is generated by six vector fields: three rotations and three translations.
This suggests that the analogue of the Poincare group for a general Lorentzian spacetime M should be the Lie group of all self-isometries on M. The Lie algebra of this group is generated by vector fields called Killing vector fields, and they are found by solving the Killing equations. (Killing was a student of Lie who did much to help develop Lie's ideas.)
It turns out that no spacetime has more independent Killing vectors (ten) than Minkowski spacetime, and most have far less or even none at all. The possible Lie algebras of Killing vectors have been enumerated and many theorems are known to the effect that no solution of the EFE has such and such symmetry with a finite number of known exceptional cases. I can give arbitrarly many specific examples of solutions of the Einstein field equation illustrating the various possible symmetry groups (equivalently, Lie algebras of Killing vectors), but perhaps it would be easier to refer interested PF readers to the monograph by Stephani et al, Exact Solutions of Einstein's Field Equations, Cambridge University Press, 2nd edition.
