Optical scalars

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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.[1][2][3][4][5]

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[edit]

For geodesic timelike congruences[edit]

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[edit]

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[edit]

The optical scalars [1][2][3][4][5] 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 )

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[edit]

For a geodesic timelike congruence[edit]

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[edit]

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[edit]

For a geodesic null congruence restricted on a null hypersurface, we have

Spin coefficients, Raychaudhuri's equation and optical scalars[edit]

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

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

See also[edit]


  1. ^ a b c Eric Poisson. A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge: Cambridge University Press, 2004. Chapter 2.
  2. ^ a b Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt. Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press, 2003. Chapter 6.
  3. ^ a b Subrahmanyan Chandrasekhar. The Mathematical Theory of Black Holes. Oxford: Oxford University Press, 1998. Section 9.(a).
  4. ^ a b Jeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity. Cambridge: Cambridge University Press, 2009. Section 2.1.3.
  5. ^ a b P Schneider, J Ehlers, E E Falco. Gravitational Lenses. Berlin: Springer, 1999. Section 3.4.2.
  6. ^ Sayan Kar, Soumitra SenGupta. The Raychaudhuri equations: a brief review. Pramana, 2007, 69(1): 49-76. [arxiv.org/abs/gr-qc/0611123v1 gr-qc/0611123]