Optical scalars

In general relativity, optical scalars refer to a set of three scalar functions formula_1 (expansion), formula_2 (shear) and formula_3 (twist/rotation/vorticity)formula_4 describing the propagation of a geodesic null congruence.
In fact, these three scalars formula_5 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 formula_6 that are adopted in tensorial equations, while the scalars formula_5 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 formula_8, and then one could construct induced "spatial metrics" that
formula_9
where formula_10 works as a spatially projecting operator. Use formula_10 to project the coordinate covariant derivative formula_12 and one obtains the "spatial" auxiliary tensor formula_13,
formula_14
where formula_15 represents the four-acceleration, and formula_13 is purely spatial in the sense that formula_17. Specifically for an observer with a geodesic timelike worldline, we have
formula_18
Now decompose formula_13 into its symmetric and antisymmetric parts formula_20 and formula_21,
formula_22
formula_23 is trace-free (formula_24) while formula_20 has nonzero trace, formula_26. Thus, the symmetric part formula_20 can be further rewritten into its trace and trace-free part,
formula_28
Hence, all in all we have
formula_29
For geodesic null congruences.
Now, consider a geodesic "null" congruence with tangent vector field formula_30. Similar to the timelike situation, we also define
formula_31
which can be decomposed into
formula_32
where
formula_33
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 formula_5 come straightforwardly from "scalarization" of the tensors formula_35 in Eq(9).
The expansion of a geodesic null congruence is defined by (where for clearance we will adopt another standard symbol "formula_36" to denote the covariant derivative formula_37)
formula_38
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 formula_39 and formula_40 respectively, are defined by
formula_41
formula_42
where formula_43 represents the induced metric. Also, formula_39 and formula_40 can be calculated via
formula_46
formula_47
where formula_48 and formula_49 are respectively the outgoing and ingoing non-affinity coefficients defined by
formula_50
formula_51
Moreover, in the language of Newman–Penrose formalism with the convention formula_52, we have
formula_53
As we can see, for a geodesic null congruence, the optical scalar formula_54 plays the same role with the expansion rates formula_39 and formula_40. Hence, for a geodesic null congruence, formula_54 will be equal to either formula_39 or formula_40.
The shear of a geodesic null congruence is defined by
formula_60
The twist of a geodesic null congruence is defined by
formula_61
In practice, a geodesic null congruence is usually defined by either its outgoing (formula_62) or ingoing (formula_63) tangent vector field (which are also its null normals). Thus, we obtain two sets of optical scalars formula_64 and formula_65, which are defined with respect to formula_66 and formula_67, respectively.
Applications in decomposing the propagation equations.
For a geodesic timelike congruence.
The propagation (or evolution) of formula_13 for a geodesic timelike congruence along formula_69 respects the following equation,
formula_70
Take the trace of Eq(13) by contracting it with formula_71, and Eq(13) becomes
formula_72
in terms of the quantities in Eq(6). Moreover, the trace-free, symmetric part of Eq(13) is
formula_73
Finally, the antisymmetric component of Eq(13) yields
formula_74
For a geodesic null congruence.
A (generic) geodesic null congruence obeys the following propagation equation,
formula_75
With the definitions summarized in Eq(9), Eq(14) could be rewritten into the following componential equations,
formula_76
formula_77
formula_78
For a restricted geodesic null congruence.
For a geodesic null congruence restricted on a null hypersurface, we have
formula_79
formula_80
formula_81
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
formula_82
where formula_83 is defined such that formula_84. The quantities in Raychaudhuri's equation are related with the spin coefficients via
formula_85
formula_86
formula_87
where Eq(24) follows directly from formula_88 and
formula_89
formula_90