Optical Scalars - Definitions: Optical Scalars For Null Congruences

Definitions: Optical Scalars For Null Congruences

The optical scalars come straightforwardly from "scalarization" of the tensors in Eq(9).

The expansion of a geodesic null congruence is defined by (where for clearance we will adopt another standard symbol "" to denote the covariant derivative )

Box A: Comparison with the "expansion rates of a null congruence"

As shown in the article "Expansion rate of a null congruence", the outgoing and ingoing expansion rates, denoted by and respectively, are defined by

where represents the induced metric. Also, and can be calculated via

where and are respectively the outgoing and ingoing non-affinity coefficients defined by

Moreover, in the language of Newman-Penrose formalism with the convention, we have

As we can see, for a geodesic null congruence, the optical scalar plays the same role with the expansion rates and . Hence, for a geodesic null congruence, will be equal to either or .

The shear of a geodesic null congruence is defined by

The twist of a geodesic null congruence is defined by

In practice, a geodesic null congruence is usually defined by either its outgoing or ingoing tangent vector field (which are also its null normals). Thus, we obtain two sets of optical scalars and, which are defined with respect to and, respectively.