calculate properties of a congruence of curves - Maple Programming Help

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Tensor[CongruenceProperties] - calculate properties of a congruence of curves

Calling Sequences

CongruenceProperties(${\mathbit{g}}$, U)

CongruenceProperties(${\mathbit{g}}$, K, L)

CongruenceProperties(${\mathbit{g}}$, K)

CongruenceProperties(${\mathbit{g}}$, NT)

Parameters

g     - a metric tensor

U     - a unit vector

K,L   - normalized null vectors, the vector defines an affinely parameterized, geodesic null congruence.

NT    - a list of 4 vectors, defining a null tetrad, the first vector in the tetrad defines the geodesic null congruence.

Description

 • 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 , set . The following scalar and tensor fields are calculated by the first calling sequence.

- Acceleration:

- Expansion: Θ = ${\nabla }_{a}{U}^{a}$ .

- Rotation Tensor : 1/2 (

- Shear Tensor: 1/2 (

 • The left-hand side of the Raychaudhuri equation ${U}^{a}{\nabla }_{a}$valid when the congruence is geodesic (${A}_{a}=$0), where is the Ricci tensor and 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,  and
 • The second calling sequence requires where Setand

- Expansion: Θ = ${\nabla }_{a}{K}^{a}$ .

- Rotation Tensor:

- Rotation Scalar:

- Complex expansion: .

- Shear Tensor:

The Raychaudhuri equation is as above but using these definitions ofand and with

 • 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 = Rotation norm squared = and Shear norm squared = 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 and and calculate, in addition to the 8 quantities calculated for the second calling sequence , Newman-Penrose Spin Coefficients.

Examples

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

Example 1.

For our first example we use the standard metric on the sphere.

 > $\mathrm{DGsetup}\left(\left[\mathrm{θ},\mathrm{φ}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 > $g≔\mathrm{evalDG}\left({R}^{2}\left(\mathrm{dtheta}&t\mathrm{dtheta}+{\mathrm{sin}\left(\mathrm{θ}\right)}^{2}\mathrm{dphi}&t\mathrm{dphi}\right)\right)$
 ${g}{:=}{{R}}^{{2}}{}{\mathrm{dtheta}}{}{\mathrm{dtheta}}{+}{{R}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}$ (2.2)

Define a unit vector field $U$.

 M > $U≔\mathrm{evalDG}\left(\frac{1\mathrm{D_phi}}{R\mathrm{sin}\left(\mathrm{θ}\right)}\right)$
 ${U}{:=}\frac{{\mathrm{D_phi}}}{{R}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}$ (2.3)

We see that the congruence is geodesic on the equator () but is accelerating elsewhere. It is shearing, rotating and non-expanding.

 M > $\mathrm{CongruenceProperties}\left(g,U\right)$
 ${\mathrm{table}}\left(\left[{"Raychaudhuri"}{=}\frac{{1}}{{{R}}^{{2}}}{,}{"Acceleration"}{=}{-}\frac{{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{\mathrm{D_theta}}}{{{R}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}{,}{"ShearTensor"}{=}{0}{}{\mathrm{dtheta}}{}{\mathrm{dtheta}}{,}{"Expansion"}{=}{0}{,}{"RotationTensor"}{=}{0}{}{\mathrm{dtheta}}{}{\mathrm{dtheta}}\right]\right)$ (2.4)

Example 2.

For the next example we consider a class of Robinson-Trautman metrics. These are of Petrov type II and admit a null congruence which is shear-free.

 M > $\mathrm{DGsetup}\left(\left[u,r,\mathrm{ζ},\mathrm{zetab}\right],\mathrm{RT}\right)$
 ${\mathrm{frame name: RT}}$ (2.5)
 RT > $g≔\mathrm{evalDG}\left(2{r}^{2}{P\left(\mathrm{ζ},\mathrm{zetab},u\right)}^{-2}\mathrm{dzeta}&s\mathrm{dzetab}-2\mathrm{du}&s\mathrm{dr}-2H\left(\mathrm{ζ},\mathrm{zetab},r,u\right)\mathrm{du}&t\mathrm{du}\right)$
 ${g}{:=}{-}{2}{}{H}{}\left({\mathrm{ζ}}{,}{\mathrm{zetab}}{,}{r}{,}{u}\right){}{\mathrm{du}}{}{\mathrm{du}}{-}{\mathrm{du}}{}{\mathrm{dr}}{-}{\mathrm{dr}}{}{\mathrm{du}}{+}\frac{{{r}}^{{2}}{}{\mathrm{dzeta}}{}{\mathrm{dzetab}}}{{{P}{}\left({\mathrm{ζ}}{,}{\mathrm{zetab}}{,}{u}\right)}^{{2}}}{+}\frac{{{r}}^{{2}}{}{\mathrm{dzetab}}{}{\mathrm{dzeta}}}{{{P}{}\left({\mathrm{ζ}}{,}{\mathrm{zetab}}{,}{u}\right)}^{{2}}}$ (2.6)

Here is a null tetrad for this metric.

 RT > $\mathrm{NT}≔\mathrm{evalDG}\left(\left[\mathrm{D_r},\mathrm{D_u}-H\left(\mathrm{ζ},\mathrm{zetab},r,u\right)\mathrm{D_r},\frac{P\left(\mathrm{ζ},\mathrm{zetab},u\right)\mathrm{D_zeta}}{r},\frac{P\left(\mathrm{ζ},\mathrm{zetab},u\right)\mathrm{D_zetab}}{r}\right]\right)$
 ${\mathrm{NT}}{:=}\left[{\mathrm{D_r}}{,}{\mathrm{D_u}}{-}{H}{}\left({\mathrm{ζ}}{,}{\mathrm{zetab}}{,}{r}{,}{u}\right){}{\mathrm{D_r}}{,}\frac{{P}{}\left({\mathrm{ζ}}{,}{\mathrm{zetab}}{,}{u}\right){}{\mathrm{D_zeta}}}{{r}}{,}\frac{{P}{}\left({\mathrm{ζ}}{,}{\mathrm{zetab}}{,}{u}\right){}{\mathrm{D_zetab}}}{{r}}\right]$ (2.7)

The null congruence is very simple:

 RT > $U≔{\mathrm{NT}}_{1}$
 ${U}{≔}{\mathrm{D_r}}$ (2.8)

First calling sequence:

 RT > $\mathrm{CongruenceProperties}\left(g,\mathrm{D_r}\right)$
 ${\mathrm{table}}\left(\left[{"ShearNormSquared"}{=}{0}{,}{"RotationNormSquared"}{=}{0}{,}{"Raychaudhuri"}{=}{0}{,}{"Expansion"}{=}\frac{{2}}{{r}}\right]\right)$ (2.9)

Third calling sequence:

 RT > $\mathrm{CongruenceProperties}\left(g,{\mathrm{NT}}_{1},{\mathrm{NT}}_{2}\right)$
 ${\mathrm{table}}\left(\left[{"ShearNormSquared"}{=}{0}{,}{"RotationNormSquared"}{=}{0}{,}{"Raychaudhuri"}{=}{0}{,}{"RotationScalar"}{=}{0}{,}{"ShearTensor"}{=}{0}{}{\mathrm{du}}{}{\mathrm{du}}{,}{"Expansion"}{=}\frac{{2}}{{r}}{,}{"RotationTensor"}{=}{0}{}{\mathrm{du}}{}{\mathrm{du}}\right]\right)$ (2.10)

Fourth calling sequence

 RT > $\mathrm{CongruenceProperties}\left(g,\mathrm{NT}\right)$
 ${\mathrm{table}}\left(\left[{"ShearNormSquared"}{=}{0}{,}{"RotationNormSquared"}{=}{0}{,}{"sigma"}{=}{0}{,}{"Raychaudhuri"}{=}{0}{,}{"RotationScalar"}{=}{0}{,}{"ShearTensor"}{=}{0}{}{\mathrm{du}}{}{\mathrm{du}}{,}{"rho"}{=}{-}\frac{{1}}{{r}}{,}{"Expansion"}{=}\frac{{2}}{{r}}{,}{"RotationTensor"}{=}{0}{}{\mathrm{du}}{}{\mathrm{du}}\right]\right)$ (2.11)

Example 3.

Here is an example of a Newman-Tamburino metric of Petrov type I and which admits a null geodesic congruence with non-vanishing shear.

 RT > $\mathrm{DGsetup}\left(\left[u,r,x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.12)
 M > $g≔\mathrm{evalDG}\left({r}^{2}\mathrm{dx}&t\mathrm{dx}+{x}^{2}\mathrm{dy}&t\mathrm{dy}-\frac{2r\mathrm{du}&s\mathrm{dx}}{x}-2\mathrm{du}&s\mathrm{dr}+\frac{1\left(c+\mathrm{ln}\left({r}^{2}{x}^{4}\right)\right)\mathrm{du}&t\mathrm{du}}{{x}^{2}}\right)$
 ${g}{:=}\frac{\left({c}{+}{\mathrm{ln}}{}\left({{r}}^{{2}}{}{{x}}^{{4}}\right)\right){}{\mathrm{du}}{}{\mathrm{du}}}{{{x}}^{{2}}}{-}{\mathrm{du}}{}{\mathrm{dr}}{-}\frac{{r}{}{\mathrm{du}}{}{\mathrm{dx}}}{{x}}{-}{\mathrm{dr}}{}{\mathrm{du}}{-}\frac{{r}{}{\mathrm{dx}}{}{\mathrm{du}}}{{x}}{+}{{r}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{x}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.13)

Here is a null tetrad for this metric.

 M > $\mathrm{NT}≔\left[\mathrm{D_r},\mathrm{D_u}+\frac{\left(c+\mathrm{ln}\left({r}^{2}{x}^{4}\right)\right)\mathrm{D_r}}{2{x}^{2}},-\frac{\sqrt{2}\mathrm{D_r}}{x}+\frac{\sqrt{2}\mathrm{D_x}}{2r}+\frac{I\frac{1}{2}\sqrt{2}\mathrm{D_y}}{x},-\frac{\sqrt{2}\mathrm{D_r}}{x}+\frac{\sqrt{2}\mathrm{D_x}}{2r}-\frac{I\frac{1}{2}\sqrt{2}\mathrm{D_y}}{x}\right]$
 ${\mathrm{NT}}{:=}\left[{\mathrm{D_r}}{,}{\mathrm{D_u}}{+}\frac{{1}}{{2}}{}\frac{\left({c}{+}{\mathrm{ln}}{}\left({{r}}^{{2}}{}{{x}}^{{4}}\right)\right){}{\mathrm{D_r}}}{{{x}}^{{2}}}{,}{-}\frac{\sqrt{{2}}{}{\mathrm{D_r}}}{{x}}{+}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{\mathrm{D_x}}}{{r}}{+}\frac{\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{D_y}}}{{x}}{,}{-}\frac{\sqrt{{2}}{}{\mathrm{D_r}}}{{x}}{+}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{\mathrm{D_x}}}{{r}}{-}\frac{\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{D_y}}}{{x}}\right]$ (2.14)

Again we consider the first leg of this tetrad.

 M > $U≔\mathrm{D_r}$
 ${U}{≔}{\mathrm{D_r}}$ (2.15)

First calling sequence:

 RT > $\mathrm{CongruenceProperties}\left(g,U\right)$
 ${\mathrm{table}}\left(\left[{"ShearNormSquared"}{=}\frac{{1}}{{2}{}{{r}}^{{2}}}{,}{"RotationNormSquared"}{=}{0}{,}{"Raychaudhuri"}{=}{0}{,}{"Expansion"}{=}\frac{{1}}{{r}}\right]\right)$ (2.16)

Third calling sequence:

 RT > $\mathrm{CongruenceProperties}\left(g,{\mathrm{NT}}_{1},{\mathrm{NT}}_{2}\right)$
 ${\mathrm{table}}\left(\left[{"ShearNormSquared"}{=}\frac{{1}}{{2}{}{{r}}^{{2}}}{,}{"RotationNormSquared"}{=}{0}{,}{"Raychaudhuri"}{=}{0}{,}{"RotationScalar"}{=}{0}{,}{"ShearTensor"}{=}\frac{{1}}{{2}}{}{r}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{1}}{{2}}{}\frac{{{x}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{r}}{,}{"Expansion"}{=}\frac{{1}}{{r}}{,}{"RotationTensor"}{=}{0}{}{\mathrm{du}}{}{\mathrm{du}}\right]\right)$ (2.17)

Fourth calling sequence:

 RT > $\mathrm{CongruenceProperties}\left(g,\mathrm{NT}\right)$
 ${\mathrm{table}}\left(\left[{"ShearNormSquared"}{=}\frac{{1}}{{2}{}{{r}}^{{2}}}{,}{"RotationNormSquared"}{=}{0}{,}{"sigma"}{=}{-}\frac{{1}}{{2}{}{r}}{,}{"Raychaudhuri"}{=}{0}{,}{"RotationScalar"}{=}{0}{,}{"ShearTensor"}{=}\frac{{1}}{{2}}{}{r}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{1}}{{2}}{}\frac{{{x}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{r}}{,}{"rho"}{=}{-}\frac{{1}}{{2}{}{r}}{,}{"Expansion"}{=}\frac{{1}}{{r}}{,}{"RotationTensor"}{=}{0}{}{\mathrm{du}}{}{\mathrm{du}}\right]\right)$ (2.18)
 M >