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

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For geodesic timelike congruences

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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 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

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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

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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  )


 

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

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For a geodesic timelike congruence

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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

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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

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


 


 


 

Spin coefficients, Raychaudhuri's equation and optical scalars

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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

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References

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  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]