Optical scalars

In general relativity, optical scalars refer to a set of three scalar functions ${\displaystyle \{{\hat {\theta }}}$ (expansion), ${\displaystyle {\hat {\sigma }}}$ (shear) and ${\displaystyle {\hat {\omega }}}$ (twist/rotation/vorticity)${\displaystyle \}}$ describing the propagation of a geodesic null congruence.^{[1][2][3][4][5]}

In fact, these three scalars ${\displaystyle \{{\hat {\theta }}\,,{\hat {\sigma }}\,,{\hat {\omega }}\}}$ 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 ${\displaystyle \{{\hat {\theta }}{\hat {h}}_{ab}\,,{\hat {\sigma }}_{ab}\,,{\hat {\omega }}_{ab}\}}$ that are adopted in tensorial equations, while the scalars ${\displaystyle \{{\hat {\theta }}\,,{\hat {\sigma }}\,,{\hat {\omega }}\}}$ mainly show up in equations written in the language of Newman–Penrose formalism.

Denote the tangent vector field of an observer's worldline (in a timelike congruence) as ${\displaystyle Z^{a}}$, and then one could construct induced "spatial metrics" that

${\displaystyle (1)\quad h^{ab}=g^{ab}+Z^{a}Z^{b}\;,\quad h_{ab}=g_{ab}+Z_{a}Z_{b}\;,\quad h_{\;\;b}^{a}=\delta _{\;\;b}^{a}+Z^{a}Z_{b}\;,}$

where ${\displaystyle h_{\;\;b}^{a}}$ works as a spatially projecting operator. Use ${\displaystyle h_{\;\;b}^{a}}$ to project the coordinate covariant derivative ${\displaystyle \nabla _{b}Z_{a}}$ and one obtains the "spatial" auxiliary tensor ${\displaystyle B_{ab}}$,

${\displaystyle (2)\quad B_{ab}=h_{\;\;a}^{c}\,h_{\;\;b}^{d}\,\nabla _{d}Z_{c}=\nabla _{b}Z_{a}+A_{a}Z_{b}\;,}$

where ${\displaystyle A_{a}}$ represents the four-acceleration, and ${\displaystyle B_{ab}}$ is purely spatial in the sense that ${\displaystyle B_{ab}Z^{a}=B_{ab}Z^{b}=0}$. Specifically for an observer with a geodesic timelike worldline, we have

${\displaystyle (3)\quad A_{a}=0\;,\quad \Rightarrow \quad B_{ab}=\nabla _{b}Z_{a}\;.}$

Now decompose ${\displaystyle B_{ab}}$ into its symmetric and antisymmetric parts ${\displaystyle \theta _{ab}}$ and ${\displaystyle \omega _{ab}}$,

${\displaystyle (4)\quad \theta _{ab}=B_{(ab)}\;,\quad \omega _{ab}=B_{[ab]}\;.}$

${\displaystyle \omega _{ab}=B_{[ab]}}$ is trace-free (${\displaystyle g^{ab}\omega _{ab}=0}$) while ${\displaystyle \theta _{ab}}$ has nonzero trace, ${\displaystyle g^{ab}\theta _{ab}=\theta }$. Thus, the symmetric part ${\displaystyle \theta _{ab}}$ can be further rewritten into its trace and trace-free part,

${\displaystyle (5)\quad \theta _{ab}={\frac {1}{3}}\theta h_{ab}+\sigma _{ab}\;.}$

Hence, all in all we have

${\displaystyle (6)\quad B_{ab}={\frac {1}{3}}\theta h_{ab}+\sigma _{ab}+\omega _{ab}\;,\quad \theta =g^{ab}\theta _{ab}=g^{ab}B_{(ab)}\;,\quad \sigma _{ab}=\theta _{ab}-{\frac {1}{3}}\theta h_{ab}\;,\quad \omega _{ab}=B_{[ab]}\;.}$

Now, consider a geodesic null congruence with tangent vector field ${\displaystyle k^{a}}$. Similar to the timelike situation, we also define

${\displaystyle (7)\quad {\hat {B}}_{ab}:=\nabla _{b}k_{a}\;,}$

which can be decomposed into

${\displaystyle (8)\quad {\hat {B}}_{ab}={\hat {\theta }}_{ab}+{\hat {\omega }}_{ab}={\frac {1}{2}}{\hat {\theta }}{\hat {h}}_{ab}+{\hat {\sigma }}_{ab}+{\hat {\omega }}_{ab}\;,}$

where

${\displaystyle (9)\quad {\hat {\theta }}_{ab}={\hat {B}}_{(ab)}\;,\quad {\hat {\theta }}={\hat {h}}^{ab}{\hat {B}}_{ab}\;,\quad {\hat {\sigma }}_{ab}={\hat {B}}_{(ab)}-{\frac {1}{2}}{\hat {\theta }}{\hat {h}}_{ab}\;,\quad {\hat {\omega }}_{ab}={\hat {B}}_{[ab]}\;.}$

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.

The optical scalars ${\displaystyle \{{\hat {\theta }}\,,{\hat {\sigma }}\,,{\hat {\omega }}\}}$^{[1][2][3][4][5]} come straightforwardly from "scalarization" of the tensors ${\displaystyle \{{\hat {\theta }}\,,{\hat {\sigma }}_{ab}\,,{\hat {\omega }}_{ab}\}}$ in Eq(9).

The expansion of a geodesic null congruence is defined by (where for clearance we will adopt another standard symbol "${\displaystyle ;}$" to denote the covariant derivative ${\displaystyle \nabla _{a}}$)

${\displaystyle (10)\quad {\hat {\theta }}={\frac {1}{2}}\,k^{a}{}_{;\,a}\;.}$

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 ${\displaystyle \theta _{(\ell )}}$ and ${\displaystyle \theta _{(n)}}$ respectively, are defined by

${\displaystyle (A.1)\quad \theta _{(\ell )}:=h^{ab}\nabla _{a}l_{b}\;,}$

${\displaystyle (A.2)\quad \theta _{(n)}:=h^{ab}\nabla _{a}n_{b}\;,}$

where ${\displaystyle h^{ab}=g^{ab}+l^{a}n^{b}+n^{a}l^{b}}$ represents the induced metric. Also, ${\displaystyle \theta _{(\ell )}}$ and ${\displaystyle \theta _{(n)}}$ can be calculated via

${\displaystyle (A.3)\quad \theta _{(\ell )}=g^{ab}\nabla _{a}l_{b}-\kappa _{(\ell )}\;,}$

${\displaystyle (A.4)\quad \theta _{(n)}=g^{ab}\nabla _{a}n_{b}-\kappa _{(n)}\;,}$

where ${\displaystyle \kappa _{(\ell )}}$ and ${\displaystyle \kappa _{(n)}}$ are respectively the outgoing and ingoing non-affinity coefficients defined by

${\displaystyle (A.5)\quad l^{a}\nabla _{a}l_{b}=\kappa _{(\ell )}l_{b}\;,}$

${\displaystyle (A.6)\quad n^{a}\nabla _{a}n_{b}=\kappa _{(n)}n_{b}\;.}$

Moreover, in the language of Newman–Penrose formalism with the convention ${\displaystyle \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}{\bar {m}}_{a}=1\}}$, we have

${\displaystyle (A.7)\quad \theta _{(l)}=-(\rho +{\bar {\rho }})=-2{\text{Re}}(\rho )\,,\quad \theta _{(n)}=\mu +{\bar {\mu }}=2{\text{Re}}(\mu )\,,}$

As we can see, for a geodesic null congruence, the optical scalar ${\displaystyle \theta }$ plays the same role with the expansion rates ${\displaystyle \theta _{(\ell )}}$ and ${\displaystyle \theta _{(n)}}$. Hence, for a geodesic null congruence, ${\displaystyle \theta }$ will be equal to either ${\displaystyle \theta _{(\ell )}}$ or ${\displaystyle \theta _{(n)}}$.

The shear of a geodesic null congruence is defined by

${\displaystyle (11)\quad {\hat {\sigma }}^{2}={\hat {\sigma }}_{ab}{\hat {\bar {\sigma }}}^{ab}={\frac {1}{2}}\,g^{ca}\,g^{db}\,k_{(a\,;\,b)}\,k_{c\,;\,d}-{\Big (}{\frac {1}{2}}\,k^{a}{}_{;\,a}{\Big )}^{2}=\,g^{ca}\,g^{db}{\frac {1}{2}}\,k_{(a\,;\,b)}\,k_{c\,;\,d}-{\hat {\theta }}^{2}\;.}$

The twist of a geodesic null congruence is defined by

${\displaystyle (12)\quad {\hat {\omega }}^{2}={\frac {1}{2}}\,k_{[a\,;\,b]}\,k^{a\,;\,b}=g^{ca}\,g^{db}\,k_{[a\,;\,b]}\,k_{c\,;\,d}\;.}$

In practice, a geodesic null congruence is usually defined by either its outgoing (${\displaystyle k^{a}=l^{a}}$) or ingoing (${\displaystyle k^{a}=n^{a}}$) tangent vector field (which are also its null normals). Thus, we obtain two sets of optical scalars ${\displaystyle \{{\hat {\theta }}_{(\ell )}\,,{\hat {\sigma }}_{(\ell )}\,,{\hat {\omega }}_{(\ell )}\}}$ and ${\displaystyle \{{\hat {\theta }}_{(n)}\,,{\hat {\sigma }}_{(n)}\,,{\hat {\omega }}_{(n)}\}}$, which are defined with respect to ${\displaystyle l^{a}}$ and ${\displaystyle n^{a}}$, respectively.

The propagation (or evolution) of ${\displaystyle B_{ab}}$ for a geodesic timelike congruence along ${\displaystyle Z^{c}}$ respects the following equation,

${\displaystyle (13)\quad Z^{c}\nabla _{c}B_{ab}=-B_{\;\;b}^{c}B_{ac}+R_{cbad}Z^{c}Z^{d}\;.}$

Take the trace of Eq(13) by contracting it with ${\displaystyle g^{ab}}$, and Eq(13) becomes

${\displaystyle (14)\quad Z^{c}\nabla _{c}\theta =\theta _{,\,\tau }=-{\frac {1}{3}}\theta ^{2}-\sigma _{ab}\sigma ^{ab}+\omega _{ab}\omega ^{ab}-R_{ab}Z^{a}Z^{b}}$

in terms of the quantities in Eq(6). Moreover, the trace-free, symmetric part of Eq(13) is

${\displaystyle (15)\quad Z^{c}\nabla _{c}\sigma _{ab}=-{\frac {2}{3}}\theta \sigma _{ab}-\sigma _{ac}\sigma _{\;b}^{c}-\omega _{ac}\omega _{\;b}^{c}+{\frac {1}{3}}h_{ab}\,(\sigma _{cd}\sigma ^{cd}-\omega _{cd}\omega ^{cd})+C_{cbad}Z^{c}Z^{d}+{\frac {1}{2}}{\tilde {R}}_{ab}\,.}$

Finally, the antisymmetric component of Eq(13) yields

${\displaystyle (16)\quad Z^{c}\nabla _{c}\omega _{ab}=-{\frac {2}{3}}\theta \omega _{ab}-2\sigma _{\;[b}^{c}\omega _{a]c}\;.}$

A (generic) geodesic null congruence obeys the following propagation equation,

${\displaystyle (16)\quad k^{c}\nabla _{c}{\hat {B}}_{ab}=-{\hat {B}}_{\;\;b}^{c}{\hat {B}}_{ac}+{\widehat {R_{cbad}k^{c}k^{d}}}\;.}$

With the definitions summarized in Eq(9), Eq(14) could be rewritten into the following componential equations,

${\displaystyle (17)\quad k^{c}\nabla _{c}{\hat {\theta }}={\hat {\theta }}_{,\,\lambda }=-{\frac {1}{2}}{\hat {\theta }}^{2}-{\hat {\sigma }}_{ab}{\hat {\sigma }}^{ab}+{\hat {\omega }}_{ab}{\hat {\omega }}^{ab}-{\widehat {R_{cd}k^{c}k^{d}}}\;,}$

${\displaystyle (18)\quad k^{c}\nabla _{c}{\hat {\sigma }}_{ab}=-{\hat {\theta }}{\hat {\sigma }}_{ab}+{\widehat {C_{cbad}k^{c}k^{d}}}\;,}$

${\displaystyle (19)\quad k^{c}\nabla _{c}{\hat {\omega }}_{ab}=-{\hat {\theta }}{\hat {\omega }}_{ab}\;.}$

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

${\displaystyle (20)\quad k^{c}\nabla _{c}\theta ={\hat {\theta }}_{,\,\lambda }=-{\frac {1}{2}}{\hat {\theta }}^{2}-{\hat {\sigma }}_{ab}{\hat {\sigma }}^{ab}-{\widehat {R_{cd}k^{c}k^{d}}}+\kappa _{(\ell )}{\hat {\theta }}\;,}$

${\displaystyle (21)\quad k^{c}\nabla _{c}{\hat {\sigma }}_{ab}=-{\hat {\theta }}{\hat {\sigma }}_{ab}+{\widehat {C_{cbad}k^{c}k^{d}}}+\kappa _{(\ell )}{\hat {\sigma }}_{ab}\;,}$

${\displaystyle (22)\quad k^{c}\nabla _{c}{\hat {\omega }}_{ab}=0\;.}$

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

${\displaystyle (23)\quad {\mathcal {L}}_{\ell }\theta _{(\ell )}=-{\frac {1}{2}}\theta _{(\ell )}^{2}+{\tilde {\kappa }}_{(\ell )}\theta _{(\ell )}-\sigma _{ab}\sigma ^{ab}+{\tilde {\omega }}_{ab}{\tilde {\omega }}^{ab}-R_{ab}l^{a}l^{b}\,,}$

where ${\displaystyle {\tilde {\kappa }}_{(\ell )}}$ is defined such that ${\displaystyle {\tilde {\kappa }}_{(\ell )}l^{b}:=l^{a}\nabla _{a}l^{b}}$. The quantities in Raychaudhuri's equation are related with the spin coefficients via

${\displaystyle (24)\quad \theta _{(\ell )}=-(\rho +{\bar {\rho }})=-2{\text{Re}}(\rho )\,,\quad \theta _{(n)}=\mu +{\bar {\mu }}=2{\text{Re}}(\mu )\,,}$

${\displaystyle (25)\quad \sigma _{ab}=-\sigma {\bar {m}}_{a}{\bar {m}}_{b}-{\bar {\sigma }}m_{a}m_{b}\,,}$

${\displaystyle (26)\quad {\tilde {\omega }}_{ab}={\frac {1}{2}}\,{\Big (}\rho -{\bar {\rho }}{\Big )}\,{\Big (}m_{a}{\bar {m}}_{b}-{\bar {m}}_{a}m_{b}{\Big )}={\text{Im}}(\rho )\cdot {\Big (}m_{a}{\bar {m}}_{b}-{\bar {m}}_{a}m_{b}{\Big )}\,,}$

where Eq(24) follows directly from ${\displaystyle {\hat {h}}^{ab}={\hat {h}}^{ba}=m^{b}{\bar {m}}^{a}+{\bar {m}}^{b}m^{a}}$ and

${\displaystyle (27)\quad \theta _{(\ell )}={\hat {h}}^{ba}\nabla _{a}l_{b}=m^{b}{\bar {m}}^{a}\nabla _{a}l_{b}+{\bar {m}}^{b}m^{a}\nabla _{a}l_{b}=m^{b}{\bar {\delta }}l_{b}+{\bar {m}}^{b}\delta l_{b}=-(\rho +{\bar {\rho }})\,,}$

${\displaystyle (28)\quad \theta _{(n)}={\hat {h}}^{ba}\nabla _{a}n_{b}={\bar {m}}^{b}m^{a}\nabla _{a}n_{b}+m^{b}{\bar {m}}^{a}\nabla _{a}n_{b}={\bar {m}}^{b}\delta n_{b}+m^{b}{\bar {\delta }}n_{b}=\mu +{\bar {\mu }}\,.}$

• Raychaudhari equation
• Congruence (general relativity)