Spacetime Symmetries - Mathematical Definition

Mathematical Definition

A rigorous definition of symmetries in general relativity has been given by Hall (2004). In this approach, the idea is to use (smooth) vector fields whose local flow diffeomorphisms preserve some property of the spacetime. This preserving property of the diffeomorphisms is made precise as follows. A smooth vector field on a spacetime is said to preserve a smooth tensor on (or is invariant under ) if, for each smooth local flow diffeomorphism associated with, the tensors and are equal on the domain of . This statement is equivalent to the more usable condition that the Lie derivative of the tensor under the vector field vanishes:

on . This has the consequence that, given any two points and on, the coordinates of in a coordinate system around are equal to the coordinates of in a coordinate system around . A symmetry on the spacetime is a smooth vector field whose local flow diffeomorphisms preserve some (usually geometrical) feature of the spacetime. The (geometrical) feature may refer to specific tensors (such as the metric, or the energy-momentum tensor) or to other aspects of the spacetime such as its geodesic structure. The vector fields are sometimes referred to as collineations, symmetry vector fields or just symmetries. The set of all symmetry vector fields on forms a Lie algebra under the Lie bracket operation as can be seen from the identity:

the term on the right usually being written, with an abuse of notation, as .