Optical scalars


In general relativity, optical scalars refer to a set of three scalar functions , and 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 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 its symmetric and antisymmetric parts and,




is trace-free while has 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




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

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


The expansion of a geodesic null congruence is defined by





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. 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 of for a geodesic timelike congruence along respects the following equation,




Take the trace of Eq by contracting it with, and Eq becomes




in terms of the quantities in Eq. Moreover, the trace-free, symmetric part of Eq is




Finally, the antisymmetric component of Eq yields

For a geodesic null congruence

A geodesic null congruence obeys the following propagation equation,




With the definitions summarized in Eq, Eq 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 follows directly from and