In general relativity, optical scalars refer to a set of three scalar functions (expansion), (shear) and (twist/rotation/vorticity) describing the propagation of a geodesic null congruence.
In fact, these three scalars 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. Also, it is their tensorial predecessors that are adopted in tensorial equations, while the scalars mainly show up in equations written in the language of Newman–Penrose formalism.
Definitions: expansion, shear and twist
For geodesic timelike congruences
Denote the tangent vector field of an observer's worldline (in a timelike congruence) as , and then one could construct induced "spatial metrics" that
where works as a spatially projecting operator. Use to project the coordinate covariant derivative and one obtains the "spatial" auxiliary tensor ,
where represents the four-acceleration, and is purely spatial in the sense that . Specifically for an observer with a "geodesic" timelike worldline, we have
Now decompose into the symmetric part and ,
is trace-free () while is of nonzero trace, . Thus, the symmetric part can be further rewritten into its trace and trace-free part,
Hence, all in all we have
For geodesic null congruences
Now, consider a geodesic null congruence with tangent vector field . Similar to the timelike situation, we also define
which can be decomposed into
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.
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.
Applications in decomposing the propagation equations
For a geodesic timelike congruence
The propagation (or evolution) of for a geodesic timelike congruence along respects the following equation,
Take the trace of Eq(13) by contracting it with , and Eq(13) becomes
in terms of the quantities in Eq(6). Moreover, the trace-free, symmetric part of Eq(13) is
Finally, the antisymmetric component of Eq(13) yields
For a geodesic null congruence
A (generic) geodesic null congruence obeys the following propagation equation,
With the definitions summarized in Eq(9), Eq(14) could be rewritten into the following componential equations,
For a restricted geodesic null congruence
For a geodesic null congruence restricted on a null hypersurface, we have
Spin coefficients, Raychaudhuri's equation and optical scalars
For a better understanding of the previous section, we will briefly review the meanings of relevant NP spin coefficients in depicting null congruences. The tensor form of Raychaudhuri's equation governing null flows reads
where is defined such that . The quantities in Raychaudhuri's equation are related with the spin coefficients via
where Eq(24) follows directly from and
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