The command CongruenceProperties returns a table of properties associated to a line congruence defined by a unit (time-like or space-like) vector field $U$ or a null vector field $K$.

Let $\mathrm{\ϵ} \= g\left(U\, U\right)$$\= \pm 1$, set$$$h_{\mathrm{ab}} \= g_{\mathrm{ab}}-\mathrm{\ϵ} U_a{U}_b$ . The following scalar and tensor fields are calculated by the first calling sequence.

The left-hand side of the Raychaudhuri equation $U^a{\nabla }_a$${\mathrm{\Θ}}_{} \+ R_{\mathrm{ab}} U^a{U}^b- {\mathrm{\ω}}_{\mathrm{ab}}{\mathrm{\ω}}^{\mathrm{ab}} \+ {\mathrm{\σ}}_{\mathrm{ab}}{\mathrm{\σ}}^{\mathrm{ab}} \+N {\mathrm{\Θ}}_{}^2 \= 0\,$valid when the congruence is geodesic ($A_a\=$0), where $R_{\mathrm{ab}}$is the Ricci tensor and $N \= 1\/\left(n-1\right)\,$is also calculated.

The first calling sequence returns a table with indices "Acceleration", "Expansion", "RotationTensor", "ShearTensor", "Raychaudhuri".

The remaining three calling sequences apply only to an affinely parameterized, geodesic null congruence , that is, $K^a K_a \=0$ and $K^b{\nabla }_b{K}_{a }\= 0.$

The second calling sequence requires $g\left(K\, K\right)\=0\=g\left(L\, L\right)\,$$g\left(K\, L\right) \= \mathrm{\α}\,$where ${\mathrm{\α}}_{} \= \pm 1.$Set$$$h_{\mathrm{ab}} \= g_{\mathrm{ab}}-{\mathrm{\α}}_{}\left( K_a{L}_b \+ K_b L_a\right)$and $v_{\mathrm{ab}} \= h_a^c h_b^d$${\nabla }_c K_d\. \mathrm{Define}$

The second calling sequence returns a table with 8 indices "Expansion", "RotationNormSquared" "ShearNormSquared", "RotationTensor", "RotationScalar", "ShearTensor" , "ComplexExpansion" and "Raychaudhuri".

The third calling sequence calculates: Expansion: Θ = ${\nabla }_a{K}^a\;$Rotation norm squared = ${\mathrm{\omega }}_{\mathrm{ab} }{\mathrm{\ω}}^{\mathrm{ab}} \;$and Shear norm squared = ${\mathrm{\σ}}_{\mathrm{ab}}{\mathrm{\σ}}^{\mathrm{ab}} \.$The definitions are as in the second calling sequence but, as these scalars do not in fact depend upon the choice of L, only the vector K is needed as input. The third calling sequence returns a table with indices "Expansion", "RotationNormSquared", "ShearNormSquared" and "Raychaudhuri".

Finally, from the 4th calling sequence we set $K \= \mathrm{NT}\left[1\right]\, L \=\mathrm{NT}\left[2\right]\, M \= \mathrm{NT}\left[3\right]$and$\overline{M } \= \mathrm{NT}\left[4\right]$ and calculate, in addition to the 8 quantities calculated for the second calling sequence , ${\mathrm{\σ}}_{} \= - M^a{M}^b {\nabla }_a K_b \, \mathrm{referenced} by \mathrm{the} \mathrm{index} ''sigma''\. \mathrm{In} \mathrm{this} \mathrm{case}\, \mathrm{the} \mathrm{quantities} \mathrm{\ρ} and \mathrm{\σ} \mathrm{are}$Newman-Penrose Spin Coefficients.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[\mathrm{\theta }\,\mathrm{\phi }\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(R^2\left(\mathrm{dtheta}\&t\mathrm{dtheta}+{\sin \left(\mathrm{\theta }\right)}^2\mathrm{dphi}\&t\mathrm{dphi}\right)\right)$

$g:=R^2\mathrm{dtheta}\mathrm{dtheta}\+R^2{\sin \left(\mathrm{\θ}\right)}^2\mathrm{dphi}\mathrm{dphi}$

$U≔\mathrm{evalDG}\left(\frac{1}{R\sin \left(\mathrm{\theta }\right)}\mathrm{D\_phi}\right)$

$U:=\frac{\mathrm{D\_phi}}{R\sin \left(\mathrm{\θ}\right)}$

$\mathrm{CongruenceProperties}\left(g\,U\right)$

$\mathrm{table}\left(\left[''Raychaudhuri''\=\frac{1}{R^2}\,''Acceleration''\=-\frac{\cos \left(\mathrm{\θ}\right)\mathrm{D\_theta}}{R^2\sin \left(\mathrm{\θ}\right)}\,''ShearTensor''\=0\mathrm{dtheta}\mathrm{dtheta}\,''Expansion''\=0\,''RotationTensor''\=0\mathrm{dtheta}\mathrm{dtheta}\right]\right)$

$\mathrm{DGsetup}\left(\left[u\,r\,\mathrm{\zeta }\,\mathrm{zetab}\right]\,\mathrm{RT}\right)$

$\mathrm{frame\; name:\; RT}$

$g≔\mathrm{evalDG}\left(2r^2{P\left(\mathrm{\zeta }\,\mathrm{zetab}\,u\right)}^{-2}\mathrm{dzeta}\&s\mathrm{dzetab}-2\mathrm{du}\&s\mathrm{dr}-2H\left(\mathrm{\zeta }\,\mathrm{zetab}\,r\,u\right)\mathrm{du}\&t\mathrm{du}\right)$

$g:=-2H\left(\mathrm{\ζ}\,\mathrm{zetab}\,r\,u\right)\mathrm{du}\mathrm{du}-\mathrm{du}\mathrm{dr}-\mathrm{dr}\mathrm{du}\+\frac{r^2\mathrm{dzeta}\mathrm{dzetab}}{{P\left(\mathrm{\ζ}\,\mathrm{zetab}\,u\right)}^2}\+\frac{r^2\mathrm{dzetab}\mathrm{dzeta}}{{P\left(\mathrm{\ζ}\,\mathrm{zetab}\,u\right)}^2}$

$\mathrm{NT}≔\mathrm{evalDG}\left(\left[\mathrm{D\_r}\,\mathrm{D\_u}-H\left(\mathrm{\zeta }\,\mathrm{zetab}\,r\,u\right)\mathrm{D\_r}\,\frac{P\left(\mathrm{\zeta }\,\mathrm{zetab}\,u\right)}{r}\mathrm{D\_zeta}\,\frac{P\left(\mathrm{\zeta }\,\mathrm{zetab}\,u\right)}{r}\mathrm{D\_zetab}\right]\right)$

$\mathrm{NT}:=\left[\mathrm{D\_r}\,\mathrm{D\_u}-H\left(\mathrm{\ζ}\,\mathrm{zetab}\,r\,u\right)\mathrm{D\_r}\,\frac{P\left(\mathrm{\ζ}\,\mathrm{zetab}\,u\right)\mathrm{D\_zeta}}{r}\,\frac{P\left(\mathrm{\ζ}\,\mathrm{zetab}\,u\right)\mathrm{D\_zetab}}{r}\right]$

$U≔\mathrm{NT}\left[1\right]$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{RT}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)$

$\mathrm{CongruenceProperties}\left(g\,\mathrm{D\_r}\right)$

$\mathrm{table}\left(\left[''ShearNormSquared''\=0\,''RotationNormSquared''\=0\,''Raychaudhuri''\=0\,''Expansion''\=\frac{2}{r}\right]\right)$

$\mathrm{CongruenceProperties}\left(g\,\mathrm{NT}\left[1\right]\,\mathrm{NT}\left[2\right]\right)$

$\mathrm{table}\left(\left[''ShearNormSquared''\=0\,''RotationNormSquared''\=0\,''Raychaudhuri''\=0\,''RotationScalar''\=0\,''ShearTensor''\=0\mathrm{du}\mathrm{du}\,''Expansion''\=\frac{2}{r}\,''RotationTensor''\=0\mathrm{du}\mathrm{du}\right]\right)$

$\mathrm{CongruenceProperties}\left(g\,\mathrm{NT}\right)$

$\mathrm{table}\left(\left[''ShearNormSquared''\=0\,''RotationNormSquared''\=0\,''sigma''\=0\,''Raychaudhuri''\=0\,''RotationScalar''\=0\,''ShearTensor''\=0\mathrm{du}\mathrm{du}\,''rho''\=-\frac{1}{r}\,''Expansion''\=\frac{2}{r}\,''RotationTensor''\=0\mathrm{du}\mathrm{du}\right]\right)$

$\mathrm{DGsetup}\left(\left[u\,r\,x\,y\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(r^2\mathrm{dx}\&t\mathrm{dx}+x^2\mathrm{dy}\&t\mathrm{dy}-\frac{2r}{x}\mathrm{du}\&s\mathrm{dx}-2\mathrm{du}\&s\mathrm{dr}+\frac{1}{x^2}\left(c+\ln \left(r^2{x}^4\right)\right)\mathrm{du}\&t\mathrm{du}\right)$

$g:=\frac{\left(c\+\ln \left(r^2{x}^4\right)\right)\mathrm{du}\mathrm{du}}{x^2}-\mathrm{du}\mathrm{dr}-\frac{r\mathrm{du}\mathrm{dx}}{x}-\mathrm{dr}\mathrm{du}-\frac{r\mathrm{dx}\mathrm{du}}{x}\+r^2\mathrm{dx}\mathrm{dx}\+x^2\mathrm{dy}\mathrm{dy}$

$\mathrm{NT}≔\left[\mathrm{D\_r}\,\mathrm{D\_u}+\frac{\left(c+\ln \left(r^2{x}^4\right)\right)\mathrm{D\_r}}{2x^2}\,-\frac{\mathrm{sqrt}\left(2\right)\mathrm{D\_r}}{x}+\frac{\mathrm{sqrt}\left(2\right)\mathrm{D\_x}}{2r}+\frac{\mathrm{I}\left(\frac{1}{2}\right)\mathrm{sqrt}\left(2\right)\mathrm{D\_y}}{x}\,-\frac{\mathrm{sqrt}\left(2\right)\mathrm{D\_r}}{x}+\frac{\mathrm{sqrt}\left(2\right)\mathrm{D\_x}}{2r}-\frac{\mathrm{I}\left(\frac{1}{2}\right)\mathrm{sqrt}\left(2\right)\mathrm{D\_y}}{x}\right]$

$\mathrm{NT}:=\left[\mathrm{D\_r}\,\mathrm{D\_u}\+\frac{1}{2}\frac{\left(c\+\ln \left(r^2{x}^4\right)\right)\mathrm{D\_r}}{x^2}\,-\frac{\sqrt{2}\mathrm{D\_r}}{x}\+\frac{1}{2}\frac{\sqrt{2}\mathrm{D\_x}}{r}\+\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{D\_y}}{x}\,-\frac{\sqrt{2}\mathrm{D\_r}}{x}\+\frac{1}{2}\frac{\sqrt{2}\mathrm{D\_x}}{r}-\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{D\_y}}{x}\right]$

$U≔\mathrm{D\_r}$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)$

$\mathrm{CongruenceProperties}\left(g\,U\right)$

$\mathrm{table}\left(\left[''ShearNormSquared''\=\frac{1}{2r^2}\,''RotationNormSquared''\=0\,''Raychaudhuri''\=0\,''Expansion''\=\frac{1}{r}\right]\right)$

$\mathrm{CongruenceProperties}\left(g\,\mathrm{NT}\left[1\right]\,\mathrm{NT}\left[2\right]\right)$

$\mathrm{table}\left(\left[''ShearNormSquared''\=\frac{1}{2r^2}\,''RotationNormSquared''\=0\,''Raychaudhuri''\=0\,''RotationScalar''\=0\,''ShearTensor''\=\frac{1}{2}r\mathrm{dx}\mathrm{dx}-\frac{1}{2}\frac{x^2\mathrm{dy}\mathrm{dy}}{r}\,''Expansion''\=\frac{1}{r}\,''RotationTensor''\=0\mathrm{du}\mathrm{du}\right]\right)$

$\mathrm{CongruenceProperties}\left(g\,\mathrm{NT}\right)$

$\mathrm{table}\left(\left[''ShearNormSquared''\=\frac{1}{2r^2}\,''RotationNormSquared''\=0\,''sigma''\=-\frac{1}{2r}\,''Raychaudhuri''\=0\,''RotationScalar''\=0\,''ShearTensor''\=\frac{1}{2}r\mathrm{dx}\mathrm{dx}-\frac{1}{2}\frac{x^2\mathrm{dy}\mathrm{dy}}{r}\,''rho''\=-\frac{1}{2r}\,''Expansion''\=\frac{1}{r}\,''RotationTensor''\=0\mathrm{du}\mathrm{du}\right]\right)$