describing the propagation of a geodesic null congruence.
can be defined for both timelike and null geodesic congruences in an identical spirit, but they are called "optical scalars" only for the null case.
that are adopted in tensorial equations, while the scalars
mainly show up in equations written in the language of Newman–Penrose formalism.
Denote the tangent vector field of an observer's worldline (in a timelike congruence) as
, and then one could construct induced "spatial metrics" that
works as a spatially projecting operator.
to project the coordinate covariant derivative
and one obtains the "spatial" auxiliary tensor
Specifically for an observer with a geodesic timelike worldline, we have
Now, consider a geodesic null congruence with tangent vector field
Similar to the timelike situation, we also define
Here, "hatted" quantities are utilized to stress that these quantities for null congruences are two-dimensional as opposed to the three-dimensional timelike case.
However, if we only discuss null congruences in a paper, the hats can be omitted for simplicity.
The expansion of a geodesic null congruence is defined by (where for clearance we will adopt another standard symbol "
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
are respectively the outgoing and ingoing non-affinity coefficients defined by
As we can see, for a geodesic null congruence, the optical scalar
plays the same role with the expansion rates
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 (
) tangent vector field (which are also its null normals).
Thus, we obtain two sets of optical scalars
Moreover, the trace-free, symmetric part of Eq(13) is
Finally, the antisymmetric component of Eq(13) yields
A (generic) geodesic null congruence obeys the following propagation equation,
For a better understanding of the previous section, we will briefly review the meanings of relevant NP spin coefficients in depicting null congruences.
[1] The tensor form of Raychaudhuri's equation[6] governing null flows reads
The quantities in Raychaudhuri's equation are related with the spin coefficients via