In general relativity, optical scalars refer to a set of three scalar functions { θ ^ {\displaystyle \{{\hat {\theta }}} (expansion), σ ^ {\displaystyle {\hat {\sigma }}} (shear) and ω ^ {\displaystyle {\hat {\omega }}} (twist/rotation/vorticity) } {\displaystyle \}} describing the propagation of a geodesic null congruence.
In fact, these three scalars { θ ^ , σ ^ , ω ^ } {\displaystyle \{{\hat {\theta }}\,,{\hat {\sigma }}\,,{\hat {\omega }}\}} 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 { θ ^ h ^ a b , σ ^ a b , ω ^ a b } {\displaystyle \{{\hat {\theta }}{\hat {h}}_{ab}\,,{\hat {\sigma }}_{ab}\,,{\hat {\omega }}_{ab}\}} that are adopted in tensorial equations, while the scalars { θ ^ , σ ^ , ω ^ } {\displaystyle \{{\hat {\theta }}\,,{\hat {\sigma }}\,,{\hat {\omega }}\}} 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 Z a {\displaystyle Z^{a}} , and then one could construct induced "spatial metrics" that
( 1 ) h a b = g a b + Z a Z b , h a b = g a b + Z a Z b , h b a = δ b a + Z a Z b , {\displaystyle (1)\quad h^{ab}=g^{ab}+Z^{a}Z^{b}\;,\quad h_{ab}=g_{ab}+Z_{a}Z_{b}\;,\quad h_{\;\;b}^{a}=\delta _{\;\;b}^{a}+Z^{a}Z_{b}\;,}
where h b a {\displaystyle h_{\;\;b}^{a}} works as a spatially projecting operator. Use h b a {\displaystyle h_{\;\;b}^{a}} to project the coordinate covariant derivative ∇ b Z a {\displaystyle \nabla _{b}Z_{a}} and one obtains the "spatial" auxiliary tensor B a b {\displaystyle B_{ab}} ,
( 2 ) B a b = h a c h b d ∇ d Z c = ∇ b Z a + A a Z b , {\displaystyle (2)\quad B_{ab}=h_{\;\;a}^{c}\,h_{\;\;b}^{d}\,\nabla _{d}Z_{c}=\nabla _{b}Z_{a}+A_{a}Z_{b}\;,}
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