In general relativity, optical scalars refer to a set of three scalar describing the propagation of a geodesic null congruence.

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, and then one could construct induced "spatial metrics" that

,

represents the four-acceleration. Specifically for an observer with a geodesic timelike worldline, we have

,

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. Similar to the timelike situation, we also define

which can be decomposed into

where

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

in Eq(9).

The expansion)

Comparison with the "expansion rates of a null congruence": As shown in the article "Expansion rate of a null congruence respectively, are defined by

represents the induced metric can be calculated via

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

Moreover, in the language of Newman–Penrose formalism, we have

.

The shear of a geodesic null congruence is defined by

The twist of a geodesic null congruence is defined by

, respectively.

Applications in decomposing the propagation equations

For a geodesic timelike congruence

respects the following equation,

, 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

. The quantities in Raychaudhuri's equation are related with the spin coefficients via

and

See also

• Raychaudhari equation

• Congruence (general relativity)

References