SM_Simon-Mars_3p1_Kerr.sagews

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3+1 Simon-Mars tensor in Kerr spacetime

This worksheet demonstrates a few capabilities of SageManifolds (version 0.7) in computations regarding 3+1 slicing of Kerr spacetime. In particular, it implements the computation of the 3+1 decomposition of the Simon-Mars tensor as given in the article arXiv:1412.6542.

The worksheet is released under the GNU General Public License version 2.

(c) Claire Somé, Eric Gourgoulhon (2015)

The worksheet file in Sage notebook format is here.

Spacelike hypersurface

We consider some hypersurface $\Sigma$ of a spacelike foliation $(\Sigma_t)_{t\in\mathbb{R}}$ of Kerr spacetime; we declare $\Sigma_t$ as a 3-dimensional manifold:

Sig = Manifold(3, 'Sigma', r'\Sigma', start_index=1)

The two Kerr parameters:

var('m, a')
assume(m>0)
assume(a>0)

($m$, $a$)

Riemannian metric on $\Sigma$

The variables introduced so far satisfy the following assumptions:

Without any loss of generality (for $m\not =0$), we may set $m=1$:

m=1
assume(a<1)

#a=1 # extreme Kerr

On the hypersurface $\Sigma$, we are using coordinates $(r,y,\phi)$ that are related to the standard Boyer-Lindquist coordinates $(r,\theta,\phi)$ by $y=\cos\theta$:

X.<r,y,ph> = Sig.chart(r'r:(1+sqrt(1-a^2),+oo) y:(-1,1) ph:(0,2*pi):\phi')
print X ; X

chart (Sigma, (r, y, ph))

$\left(\Sigma,(r, y, {\phi})\right)$

Riemannian metric on $\Sigma$

The variables introduced so far obey the following assumptions:

assumptions()

[$m > 0$, $a > 0$, $a < 1$, $\text{\texttt{r{ }is{ }real}}$, $\text{\texttt{y{ }is{ }real}}$, $y > \left(-1\right)$, $y < 1$, $\text{\texttt{ph{ }is{ }real}}$, ${\phi} > 0$, ${\phi} < 2 \, \pi$]

Some shortcut notations:

rho2 = r^2 + a^2*y^2 
Del = r^2 -2*m*r + a^2
AA2 = rho2*(r^2 + a^2) + 2*a^2*m*r*(1-y^2)
BB2 = r^2 + a^2 + 2*a^2*m*r*(1-y^2)/rho2

The metric $h$ induced by the spacetime metric $g$ on $\Sigma$:

gam = Sig.riemann_metric('gam', latex_name=r'\gamma') 
gam[1,1] = rho2/Del
gam[2,2] = rho2/(1-y^2)
gam[3,3] = BB2*(1-y^2)
gam.display()

$\gamma = \left( \frac{a^{2} y^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -\frac{a^{2} y^{2} + r^{2}}{y^{2} - 1} \right) \mathrm{d} y\otimes \mathrm{d} y + \left( -\frac{{\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \, a^{2} r - {\left(a^{4} - r^{4} - 4 \, a^{2} r\right)} y^{2}}{a^{2} y^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

A matrix view of the components w.r.t. coordinates $(r,y,\phi)$:

gam[:]

$\left(\begin{array}{rrr} \frac{a^{2} y^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r} & 0 & 0 \\ 0 & -\frac{a^{2} y^{2} + r^{2}}{y^{2} - 1} & 0 \\ 0 & 0 & -\frac{{\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \, a^{2} r - {\left(a^{4} - r^{4} - 4 \, a^{2} r\right)} y^{2}}{a^{2} y^{2} + r^{2}} \end{array}\right)$

Lapse function and shift vector

N = Sig.scalar_field(sqrt(Del / BB2), name='N')
print N
N.display()

scalar field 'N' on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} N:& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & \sqrt{-\frac{a^{2} + r^{2} - 2 \, r}{\frac{2 \, {\left(y^{2} - 1\right)} a^{2} r}{a^{2} y^{2} + r^{2}} - a^{2} - r^{2}}} \end{array}$

b = Sig.vector_field('beta', latex_name=r'\beta') 
b[3] = -2*m*r*a/AA2   
# unset components are zero 
b.display()

$\beta = \left( -\frac{2 \, a r}{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \right) \frac{\partial}{\partial {\phi} }$

Extrinsic curvature of $\Sigma$

We use the formula $K_{ij} = \frac{1}{2N} \mathcal{L}_{\beta} \gamma_{ij}$ which is valid for any stationary spacetime:

K = gam.lie_der(b) / (2*N)
K.set_name('K')
print K ; K.display()

field of symmetric bilinear forms 'K' on the 3-dimensional manifold 'Sigma'

$K = \left( \frac{{\left(a^{3} r^{2} + 3 \, a r^{4} + {\left(a^{5} - a^{3} r^{2}\right)} y^{4} - {\left(a^{5} + 3 \, a r^{4}\right)} y^{2}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}}}{{\left(a^{2} r^{4} + r^{6} + 2 \, a^{2} r^{3} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} y^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( -\frac{2 \, \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left({\left(a^{5} r + a^{3} r^{3} - 2 \, a^{3} r^{2}\right)} y^{3} - {\left(a^{5} r + a^{3} r^{3} - 2 \, a^{3} r^{2}\right)} y\right)}}{{\left(a^{2} r^{4} + r^{6} + 2 \, a^{2} r^{3} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} y^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} y\otimes \mathrm{d} {\phi} + \left( \frac{{\left(a^{3} r^{2} + 3 \, a r^{4} + {\left(a^{5} - a^{3} r^{2}\right)} y^{4} - {\left(a^{5} + 3 \, a r^{4}\right)} y^{2}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}}}{{\left(a^{2} r^{4} + r^{6} + 2 \, a^{2} r^{3} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} y^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( -\frac{2 \, \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left({\left(a^{5} r + a^{3} r^{3} - 2 \, a^{3} r^{2}\right)} y^{3} - {\left(a^{5} r + a^{3} r^{3} - 2 \, a^{3} r^{2}\right)} y\right)}}{{\left(a^{2} r^{4} + r^{6} + 2 \, a^{2} r^{3} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} y^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} y$

Check (comparison with known formulas):

Krp = a*m*(1-y^2)*(3*r^4+a^2*r^2+a^2*(r^2-a^2)*y^2) / rho2^2/sqrt(Del*BB2)
Krp

$\frac{{\left({\left(a^{2} - r^{2}\right)} a^{2} y^{2} - a^{2} r^{2} - 3 \, r^{4}\right)} {\left(y^{2} - 1\right)} a}{{\left(a^{2} y^{2} + r^{2}\right)}^{2} \sqrt{-{\left(\frac{2 \, {\left(y^{2} - 1\right)} a^{2} r}{a^{2} y^{2} + r^{2}} - a^{2} - r^{2}\right)} {\left(a^{2} + r^{2} - 2 \, r\right)}}}$

K[1,3] - Krp

$0$

Kyp = 2*m*r*a^3*(1-y^2)*y*sqrt(Del)/rho2^2/sqrt(BB2)
Kyp

$-\frac{2 \, \sqrt{a^{2} + r^{2} - 2 \, r} {\left(y^{2} - 1\right)} a^{3} r y}{{\left(a^{2} y^{2} + r^{2}\right)}^{2} \sqrt{-\frac{2 \, {\left(y^{2} - 1\right)} a^{2} r}{a^{2} y^{2} + r^{2}} + a^{2} + r^{2}}}$

K[2,3] - Kyp

$0$

For now on, we use the expressions Krp and Kyp above for $K_{r\phi}$ and $K_{ry}$, respectively:

K1 = Sig.sym_bilin_form_field('K')
K1[1,3] = Krp
K1[2,3] = Kyp
K = K1
K.display()

$K = \left( \frac{{\left(a^{3} r^{2} + 3 \, a r^{4} + {\left(a^{5} - a^{3} r^{2}\right)} y^{4} - {\left(a^{5} + 3 \, a r^{4}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}}}{{\left(a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( -\frac{2 \, {\left(a^{3} r y^{3} - a^{3} r y\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}}} \right) \mathrm{d} y\otimes \mathrm{d} {\phi} + \left( \frac{{\left(a^{3} r^{2} + 3 \, a r^{4} + {\left(a^{5} - a^{3} r^{2}\right)} y^{4} - {\left(a^{5} + 3 \, a r^{4}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}}}{{\left(a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( -\frac{2 \, {\left(a^{3} r y^{3} - a^{3} r y\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} y$

The type-(1,1) tensor $K^\sharp$ of components $K^i_{\ \, j} = \gamma^{ik} K_{kj}$:

Ku = K.up(gam, 0)
print Ku ; Ku.display()

tensor field of type (1,1) on the 3-dimensional manifold 'Sigma'

$\left( \frac{{\left(a^{3} r^{2} + 3 \, a r^{4} + {\left(a^{5} - a^{3} r^{2}\right)} y^{4} - {\left(a^{5} + 3 \, a r^{4}\right)} y^{2}\right)} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} y^{2} + r^{2}}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} {\phi} + \left( \frac{2 \, {\left(a^{3} r y^{5} - 2 \, a^{3} r y^{3} + a^{3} r y\right)} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} y^{2} + r^{2}}} \right) \frac{\partial}{\partial y }\otimes \mathrm{d} {\phi} + \left( \frac{{\left(a^{3} r^{2} + 3 \, a r^{4} - {\left(a^{5} - a^{3} r^{2}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}}}{{\left(a^{2} r^{4} + r^{6} + 2 \, a^{2} r^{3} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} y^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\right)} y^{2}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} r + \left( \frac{2 \, \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r} a^{3} r y}{{\left(a^{2} r^{4} + r^{6} + 2 \, a^{2} r^{3} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} y^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\right)} y^{2}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} y$

We may check that the hypersurface $\Sigma$ is maximal, i.e. that $K^k_{\ \, k} = 0$:

trK = Ku.trace()
print trK

scalar field on the 3-dimensional manifold 'Sigma'

Connection and curvature

Let us call $D$ the Levi-Civita connection associated with $\gamma$:

D = gam.connection(name='D')
print D ; D

Levi-Civita connection 'D' associated with the Riemannian metric 'gam' on the 3-dimensional manifold 'Sigma'

$D$

The Ricci tensor associated with $\gamma$:

Ric = gam.ricci()
print Ric ; Ric

field of symmetric bilinear forms 'Ric(gam)' on the 3-dimensional manifold 'Sigma'

$\mathrm{Ric}\left(\gamma\right)$

Ric[1,1]

$-\frac{8 \, a^{4} r^{7} + 7 \, a^{2} r^{9} + 2 \, r^{11} + 5 \, a^{6} r^{4} + 2 \, a^{4} r^{6} - 7 \, a^{2} r^{8} + {\left(3 \, a^{10} r + 3 \, a^{6} r^{5} + a^{10} - 14 \, a^{8} r^{2} - 11 \, a^{6} r^{4} + 6 \, {\left(a^{8} + 2 \, a^{6}\right)} r^{3}\right)} y^{6} + {\left(3 \, a^{6} - 4 \, a^{4}\right)} r^{5} - {\left(9 \, a^{10} r + 4 \, a^{4} r^{7} + a^{10} - 30 \, a^{8} r^{2} - 35 \, a^{6} r^{4} - 16 \, a^{4} r^{6} + {\left(17 \, a^{6} + 4 \, a^{4}\right)} r^{5} + 2 \, {\left(11 \, a^{8} + 12 \, a^{6}\right)} r^{3}\right)} y^{4} - {\left(16 \, a^{4} r^{7} + 5 \, a^{2} r^{9} + 16 \, a^{8} r^{2} + 29 \, a^{6} r^{4} + 18 \, a^{4} r^{6} - 7 \, a^{2} r^{8} + {\left(17 \, a^{6} - 8 \, a^{4}\right)} r^{5} + 6 \, {\left(a^{8} - 2 \, a^{6}\right)} r^{3}\right)} y^{2}}{3 \, a^{2} r^{12} + r^{14} + 6 \, a^{4} r^{9} - 2 \, r^{13} + 4 \, a^{6} r^{6} + {\left(3 \, a^{4} - 8 \, a^{2}\right)} r^{10} + {\left(a^{6} - 4 \, a^{4}\right)} r^{8} + {\left(a^{14} + a^{8} r^{6} - 6 \, a^{12} r - 6 \, a^{8} r^{5} + 3 \, {\left(a^{10} + 4 \, a^{8}\right)} r^{4} - 4 \, {\left(3 \, a^{10} + 2 \, a^{8}\right)} r^{3} + 3 \, {\left(a^{12} + 4 \, a^{10}\right)} r^{2}\right)} y^{8} + 4 \, {\left(a^{6} - 2 \, a^{4}\right)} r^{7} + 4 \, {\left(a^{6} r^{8} + a^{12} r - 5 \, a^{6} r^{7} + {\left(3 \, a^{8} + 8 \, a^{6}\right)} r^{6} - {\left(9 \, a^{8} + 4 \, a^{6}\right)} r^{5} + {\left(3 \, a^{10} + 4 \, a^{8}\right)} r^{4} - {\left(3 \, a^{10} - 4 \, a^{8}\right)} r^{3} + {\left(a^{12} - 4 \, a^{10}\right)} r^{2}\right)} y^{6} + 2 \, {\left(3 \, a^{4} r^{10} - 12 \, a^{4} r^{9} + 2 \, a^{10} r^{2} + 16 \, a^{6} r^{5} + {\left(9 \, a^{6} + 14 \, a^{4}\right)} r^{8} - 2 \, {\left(9 \, a^{6} + 2 \, a^{4}\right)} r^{7} + 3 \, {\left(3 \, a^{8} - 2 \, a^{6}\right)} r^{6} + 3 \, {\left(a^{10} - 6 \, a^{8}\right)} r^{4} + 2 \, {\left(3 \, a^{10} - 2 \, a^{8}\right)} r^{3}\right)} y^{4} + 4 \, {\left(a^{2} r^{12} - 3 \, a^{4} r^{9} - 3 \, a^{2} r^{11} + 2 \, a^{8} r^{4} + {\left(3 \, a^{4} + 2 \, a^{2}\right)} r^{10} + 3 \, {\left(a^{6} - 2 \, a^{4}\right)} r^{8} + {\left(3 \, a^{6} + 4 \, a^{4}\right)} r^{7} + {\left(a^{8} - 6 \, a^{6}\right)} r^{6} + {\left(3 \, a^{8} - 4 \, a^{6}\right)} r^{5}\right)} y^{2}}$

Ric[1,2]

$\frac{{\left(3 \, a^{10} + 6 \, a^{8} r^{2} + 3 \, a^{6} r^{4} - 4 \, a^{8} r - 8 \, a^{6} r^{3}\right)} y^{5} - 2 \, {\left(3 \, a^{8} r^{2} + 6 \, a^{6} r^{4} + 3 \, a^{4} r^{6} - 2 \, a^{8} r - 12 \, a^{6} r^{3} - 6 \, a^{4} r^{5}\right)} y^{3} - {\left(9 \, a^{6} r^{4} + 18 \, a^{4} r^{6} + 9 \, a^{2} r^{8} + 16 \, a^{6} r^{3} + 12 \, a^{4} r^{5}\right)} y}{a^{4} r^{8} + 2 \, a^{2} r^{10} + r^{12} + 4 \, a^{4} r^{7} + 4 \, a^{2} r^{9} + 4 \, a^{4} r^{6} + {\left(a^{12} + a^{8} r^{4} - 4 \, a^{10} r - 4 \, a^{8} r^{3} + 2 \, {\left(a^{10} + 2 \, a^{8}\right)} r^{2}\right)} y^{8} + 4 \, {\left(a^{6} r^{6} + a^{10} r - 2 \, a^{8} r^{3} - 3 \, a^{6} r^{5} + 2 \, {\left(a^{8} + a^{6}\right)} r^{4} + {\left(a^{10} - 2 \, a^{8}\right)} r^{2}\right)} y^{6} + 2 \, {\left(3 \, a^{4} r^{8} + 6 \, a^{8} r^{3} - 6 \, a^{4} r^{7} + 2 \, a^{8} r^{2} + 2 \, {\left(3 \, a^{6} + a^{4}\right)} r^{6} + {\left(3 \, a^{8} - 8 \, a^{6}\right)} r^{4}\right)} y^{4} + 4 \, {\left(2 \, a^{4} r^{8} + a^{2} r^{10} + 3 \, a^{6} r^{5} + 2 \, a^{4} r^{7} - a^{2} r^{9} + 2 \, a^{6} r^{4} + {\left(a^{6} - 2 \, a^{4}\right)} r^{6}\right)} y^{2}}$

Ric[1,3]

$0$

Ric[2,2]

$\frac{7 \, a^{4} r^{7} + 5 \, a^{2} r^{9} + r^{11} + 6 \, a^{6} r^{4} + 4 \, a^{4} r^{6} - 2 \, a^{2} r^{8} + 2 \, {\left(3 \, a^{10} r + 3 \, a^{6} r^{5} - 10 \, a^{8} r^{2} - 10 \, a^{6} r^{4} + 2 \, {\left(3 \, a^{8} + 4 \, a^{6}\right)} r^{3}\right)} y^{6} + {\left(3 \, a^{6} - 8 \, a^{4}\right)} r^{5} - {\left(9 \, a^{10} r - a^{4} r^{7} - 34 \, a^{8} r^{2} - 36 \, a^{6} r^{4} - 2 \, a^{4} r^{6} + {\left(7 \, a^{6} + 8 \, a^{4}\right)} r^{5} + {\left(17 \, a^{8} + 32 \, a^{6}\right)} r^{3}\right)} y^{4} - 2 \, {\left(7 \, a^{4} r^{7} + 2 \, a^{2} r^{9} + 7 \, a^{8} r^{2} + 11 \, a^{6} r^{4} + 3 \, a^{4} r^{6} - a^{2} r^{8} + 8 \, {\left(a^{6} - a^{4}\right)} r^{5} + {\left(3 \, a^{8} - 8 \, a^{6}\right)} r^{3}\right)} y^{2}}{a^{4} r^{8} + 2 \, a^{2} r^{10} + r^{12} + 4 \, a^{4} r^{7} + 4 \, a^{2} r^{9} - {\left(a^{12} + a^{8} r^{4} - 4 \, a^{10} r - 4 \, a^{8} r^{3} + 2 \, {\left(a^{10} + 2 \, a^{8}\right)} r^{2}\right)} y^{10} + 4 \, a^{4} r^{6} + {\left(a^{12} - 4 \, a^{6} r^{6} - 8 \, a^{10} r + 4 \, a^{8} r^{3} + 12 \, a^{6} r^{5} - {\left(7 \, a^{8} + 8 \, a^{6}\right)} r^{4} - 2 \, {\left(a^{10} - 6 \, a^{8}\right)} r^{2}\right)} y^{8} - 2 \, {\left(3 \, a^{4} r^{8} - 2 \, a^{10} r + 10 \, a^{8} r^{3} + 6 \, a^{6} r^{5} - 6 \, a^{4} r^{7} + 2 \, {\left(2 \, a^{6} + a^{4}\right)} r^{6} - {\left(a^{8} + 12 \, a^{6}\right)} r^{4} - 2 \, {\left(a^{10} - 3 \, a^{8}\right)} r^{2}\right)} y^{6} - 2 \, {\left(a^{4} r^{8} + 2 \, a^{2} r^{10} - 6 \, a^{8} r^{3} + 6 \, a^{6} r^{5} + 10 \, a^{4} r^{7} - 2 \, a^{2} r^{9} - 2 \, a^{8} r^{2} - 2 \, {\left(2 \, a^{6} + 3 \, a^{4}\right)} r^{6} - 3 \, {\left(a^{8} - 4 \, a^{6}\right)} r^{4}\right)} y^{4} + {\left(7 \, a^{4} r^{8} + 2 \, a^{2} r^{10} - r^{12} + 12 \, a^{6} r^{5} + 4 \, a^{4} r^{7} - 8 \, a^{2} r^{9} + 8 \, a^{6} r^{4} + 4 \, {\left(a^{6} - 3 \, a^{4}\right)} r^{6}\right)} y^{2}}$

Ric[2,3]

$0$

Ric[3,3]

$\frac{a^{4} r^{7} + 2 \, a^{2} r^{9} + r^{11} + a^{6} r^{4} + 10 \, a^{4} r^{6} + 13 \, a^{2} r^{8} + 4 \, a^{4} r^{5} + {\left(3 \, a^{10} r + 3 \, a^{6} r^{5} + a^{10} - 18 \, a^{8} r^{2} - 15 \, a^{6} r^{4} + 2 \, {\left(3 \, a^{8} + 10 \, a^{6}\right)} r^{3}\right)} y^{8} - {\left(3 \, a^{10} r - 5 \, a^{4} r^{7} + 2 \, a^{10} - 38 \, a^{8} r^{2} - 22 \, a^{6} r^{4} + 2 \, a^{4} r^{6} - {\left(7 \, a^{6} - 4 \, a^{4}\right)} r^{5} + {\left(a^{8} + 60 \, a^{6}\right)} r^{3}\right)} y^{6} - {\left(3 \, a^{4} r^{7} - a^{2} r^{9} - a^{10} + 22 \, a^{8} r^{2} - 2 \, a^{6} r^{4} - 14 \, a^{4} r^{6} - 13 \, a^{2} r^{8} + 3 \, {\left(3 \, a^{6} - 4 \, a^{4}\right)} r^{5} + 5 \, {\left(a^{8} - 12 \, a^{6}\right)} r^{3}\right)} y^{4} - {\left(3 \, a^{4} r^{7} + 3 \, a^{2} r^{9} + r^{11} - 2 \, a^{8} r^{2} + 10 \, a^{6} r^{4} + 22 \, a^{4} r^{6} + 26 \, a^{2} r^{8} + 20 \, a^{6} r^{3} + {\left(a^{6} + 12 \, a^{4}\right)} r^{5}\right)} y^{2}}{a^{2} r^{10} + r^{12} + 2 \, a^{2} r^{9} + {\left(a^{12} + a^{10} r^{2} - 2 \, a^{10} r\right)} y^{10} + {\left(5 \, a^{10} r^{2} + 5 \, a^{8} r^{4} + 2 \, a^{10} r - 8 \, a^{8} r^{3}\right)} y^{8} + 2 \, {\left(5 \, a^{8} r^{4} + 5 \, a^{6} r^{6} + 4 \, a^{8} r^{3} - 6 \, a^{6} r^{5}\right)} y^{6} + 2 \, {\left(5 \, a^{6} r^{6} + 5 \, a^{4} r^{8} + 6 \, a^{6} r^{5} - 4 \, a^{4} r^{7}\right)} y^{4} + {\left(5 \, a^{4} r^{8} + 5 \, a^{2} r^{10} + 8 \, a^{4} r^{7} - 2 \, a^{2} r^{9}\right)} y^{2}}$

The scalar curvature $R = \gamma^{ij} R_{ij}$:

R = gam.ricci_scalar(name='R')
print R
R.display()

scalar field 'R' on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} \mathrm{r}\left(\gamma\right):& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & \frac{2 \, {\left(a^{6} r^{4} + 6 \, a^{4} r^{6} + 9 \, a^{2} r^{8} - {\left(a^{10} - 6 \, a^{8} r^{2} - 3 \, a^{6} r^{4} + 8 \, a^{6} r^{3}\right)} y^{6} + {\left(a^{10} - 8 \, a^{8} r^{2} - 3 \, a^{6} r^{4} - 6 \, a^{4} r^{6} + 16 \, a^{6} r^{3}\right)} y^{4} + {\left(2 \, a^{8} r^{2} - a^{6} r^{4} - 9 \, a^{2} r^{8} - 8 \, a^{6} r^{3}\right)} y^{2}\right)}}{a^{4} r^{10} + 2 \, a^{2} r^{12} + r^{14} + 4 \, a^{4} r^{9} + 4 \, a^{2} r^{11} + 4 \, a^{4} r^{8} + {\left(a^{14} + a^{10} r^{4} - 4 \, a^{12} r - 4 \, a^{10} r^{3} + 2 \, {\left(a^{12} + 2 \, a^{10}\right)} r^{2}\right)} y^{10} + {\left(5 \, a^{8} r^{6} + 4 \, a^{12} r - 12 \, a^{10} r^{3} - 16 \, a^{8} r^{5} + 2 \, {\left(5 \, a^{10} + 6 \, a^{8}\right)} r^{4} + {\left(5 \, a^{12} - 8 \, a^{10}\right)} r^{2}\right)} y^{8} + 2 \, {\left(5 \, a^{6} r^{8} + 8 \, a^{10} r^{3} - 4 \, a^{8} r^{5} - 12 \, a^{6} r^{7} + 2 \, a^{10} r^{2} + 2 \, {\left(5 \, a^{8} + 3 \, a^{6}\right)} r^{6} + {\left(5 \, a^{10} - 12 \, a^{8}\right)} r^{4}\right)} y^{6} + 2 \, {\left(5 \, a^{4} r^{10} + 12 \, a^{8} r^{5} + 4 \, a^{6} r^{7} - 8 \, a^{4} r^{9} + 6 \, a^{8} r^{4} + 2 \, {\left(5 \, a^{6} + a^{4}\right)} r^{8} + {\left(5 \, a^{8} - 12 \, a^{6}\right)} r^{6}\right)} y^{4} + {\left(10 \, a^{4} r^{10} + 5 \, a^{2} r^{12} + 16 \, a^{6} r^{7} + 12 \, a^{4} r^{9} - 4 \, a^{2} r^{11} + 12 \, a^{6} r^{6} + {\left(5 \, a^{6} - 8 \, a^{4}\right)} r^{8}\right)} y^{2}} \end{array}$

Test: 3+1 Einstein equations

Let us check that the vacuum 3+1 Einstein equations are satisfied.

We start by the contraint equations:

Hamiltonian constraint

Let us first evaluate the term $K_{ij} K^{ij}$:

Kuu = Ku.up(gam, 1)
trKK = K['_ij']*Kuu['^ij']
print trKK ; trKK.display()

scalar field on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & \frac{2 \, {\left(a^{6} r^{4} + 6 \, a^{4} r^{6} + 9 \, a^{2} r^{8} - {\left(a^{10} - 6 \, a^{8} r^{2} - 3 \, a^{6} r^{4} + 8 \, a^{6} r^{3}\right)} y^{6} + {\left(a^{10} - 8 \, a^{8} r^{2} - 3 \, a^{6} r^{4} - 6 \, a^{4} r^{6} + 16 \, a^{6} r^{3}\right)} y^{4} + {\left(2 \, a^{8} r^{2} - a^{6} r^{4} - 9 \, a^{2} r^{8} - 8 \, a^{6} r^{3}\right)} y^{2}\right)}}{a^{4} r^{10} + 2 \, a^{2} r^{12} + r^{14} + 4 \, a^{4} r^{9} + 4 \, a^{2} r^{11} + 4 \, a^{4} r^{8} + {\left(a^{14} + a^{10} r^{4} - 4 \, a^{12} r - 4 \, a^{10} r^{3} + 2 \, {\left(a^{12} + 2 \, a^{10}\right)} r^{2}\right)} y^{10} + {\left(5 \, a^{8} r^{6} + 4 \, a^{12} r - 12 \, a^{10} r^{3} - 16 \, a^{8} r^{5} + 2 \, {\left(5 \, a^{10} + 6 \, a^{8}\right)} r^{4} + {\left(5 \, a^{12} - 8 \, a^{10}\right)} r^{2}\right)} y^{8} + 2 \, {\left(5 \, a^{6} r^{8} + 8 \, a^{10} r^{3} - 4 \, a^{8} r^{5} - 12 \, a^{6} r^{7} + 2 \, a^{10} r^{2} + 2 \, {\left(5 \, a^{8} + 3 \, a^{6}\right)} r^{6} + {\left(5 \, a^{10} - 12 \, a^{8}\right)} r^{4}\right)} y^{6} + 2 \, {\left(5 \, a^{4} r^{10} + 12 \, a^{8} r^{5} + 4 \, a^{6} r^{7} - 8 \, a^{4} r^{9} + 6 \, a^{8} r^{4} + 2 \, {\left(5 \, a^{6} + a^{4}\right)} r^{8} + {\left(5 \, a^{8} - 12 \, a^{6}\right)} r^{6}\right)} y^{4} + {\left(10 \, a^{4} r^{10} + 5 \, a^{2} r^{12} + 16 \, a^{6} r^{7} + 12 \, a^{4} r^{9} - 4 \, a^{2} r^{11} + 12 \, a^{6} r^{6} + {\left(5 \, a^{6} - 8 \, a^{4}\right)} r^{8}\right)} y^{2}} \end{array}$

The vacuum Hamiltonian constraint equation is $R + K^2 -K_{ij} K^{ij} = 0$

Ham = R + trK^2 - trKK
print Ham ; Ham.display()

scalar field 'zero' on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} 0:& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & 0 \end{array}$

Momentum constraint

In vaccum, the momentum constraint is $D_j K^j_{\ \, i} - D_i K = 0$

mom = D(Ku).trace(0,2) - D(trK)
print mom
mom.display()

1-form on the 3-dimensional manifold 'Sigma'

$0$

Dynamical Einstein equations

Let us first evaluate the symmetric bilinear form $k_{ij} := K_{ik} K^k_{\ \, j}$:

KK = K['_ik']*Ku['^k_j']
print KK

tensor field of type (0,2) on the 3-dimensional manifold 'Sigma'

KK1 = KK.symmetrize()
KK == KK1

$\mathrm{True}$

KK = KK1
print KK

field of symmetric bilinear forms on the 3-dimensional manifold 'Sigma'

KK[1,1]

$\frac{a^{6} r^{4} + 6 \, a^{4} r^{6} + 9 \, a^{2} r^{8} - {\left(a^{10} - 2 \, a^{8} r^{2} + a^{6} r^{4}\right)} y^{6} + {\left(a^{10} + 5 \, a^{6} r^{4} - 6 \, a^{4} r^{6}\right)} y^{4} - {\left(2 \, a^{8} r^{2} + 5 \, a^{6} r^{4} + 9 \, a^{2} r^{8}\right)} y^{2}}{3 \, a^{2} r^{12} + r^{14} + 6 \, a^{4} r^{9} - 2 \, r^{13} + 4 \, a^{6} r^{6} + {\left(3 \, a^{4} - 8 \, a^{2}\right)} r^{10} + {\left(a^{6} - 4 \, a^{4}\right)} r^{8} + {\left(a^{14} + a^{8} r^{6} - 6 \, a^{12} r - 6 \, a^{8} r^{5} + 3 \, {\left(a^{10} + 4 \, a^{8}\right)} r^{4} - 4 \, {\left(3 \, a^{10} + 2 \, a^{8}\right)} r^{3} + 3 \, {\left(a^{12} + 4 \, a^{10}\right)} r^{2}\right)} y^{8} + 4 \, {\left(a^{6} - 2 \, a^{4}\right)} r^{7} + 4 \, {\left(a^{6} r^{8} + a^{12} r - 5 \, a^{6} r^{7} + {\left(3 \, a^{8} + 8 \, a^{6}\right)} r^{6} - {\left(9 \, a^{8} + 4 \, a^{6}\right)} r^{5} + {\left(3 \, a^{10} + 4 \, a^{8}\right)} r^{4} - {\left(3 \, a^{10} - 4 \, a^{8}\right)} r^{3} + {\left(a^{12} - 4 \, a^{10}\right)} r^{2}\right)} y^{6} + 2 \, {\left(3 \, a^{4} r^{10} - 12 \, a^{4} r^{9} + 2 \, a^{10} r^{2} + 16 \, a^{6} r^{5} + {\left(9 \, a^{6} + 14 \, a^{4}\right)} r^{8} - 2 \, {\left(9 \, a^{6} + 2 \, a^{4}\right)} r^{7} + 3 \, {\left(3 \, a^{8} - 2 \, a^{6}\right)} r^{6} + 3 \, {\left(a^{10} - 6 \, a^{8}\right)} r^{4} + 2 \, {\left(3 \, a^{10} - 2 \, a^{8}\right)} r^{3}\right)} y^{4} + 4 \, {\left(a^{2} r^{12} - 3 \, a^{4} r^{9} - 3 \, a^{2} r^{11} + 2 \, a^{8} r^{4} + {\left(3 \, a^{4} + 2 \, a^{2}\right)} r^{10} + 3 \, {\left(a^{6} - 2 \, a^{4}\right)} r^{8} + {\left(3 \, a^{6} + 4 \, a^{4}\right)} r^{7} + {\left(a^{8} - 6 \, a^{6}\right)} r^{6} + {\left(3 \, a^{8} - 4 \, a^{6}\right)} r^{5}\right)} y^{2}}$

KK[1,2]

$\frac{2 \, {\left({\left(a^{8} r - a^{6} r^{3}\right)} y^{5} - {\left(a^{8} r + 3 \, a^{4} r^{5}\right)} y^{3} + {\left(a^{6} r^{3} + 3 \, a^{4} r^{5}\right)} y\right)}}{a^{4} r^{8} + 2 \, a^{2} r^{10} + r^{12} + 4 \, a^{4} r^{7} + 4 \, a^{2} r^{9} + 4 \, a^{4} r^{6} + {\left(a^{12} + a^{8} r^{4} - 4 \, a^{10} r - 4 \, a^{8} r^{3} + 2 \, {\left(a^{10} + 2 \, a^{8}\right)} r^{2}\right)} y^{8} + 4 \, {\left(a^{6} r^{6} + a^{10} r - 2 \, a^{8} r^{3} - 3 \, a^{6} r^{5} + 2 \, {\left(a^{8} + a^{6}\right)} r^{4} + {\left(a^{10} - 2 \, a^{8}\right)} r^{2}\right)} y^{6} + 2 \, {\left(3 \, a^{4} r^{8} + 6 \, a^{8} r^{3} - 6 \, a^{4} r^{7} + 2 \, a^{8} r^{2} + 2 \, {\left(3 \, a^{6} + a^{4}\right)} r^{6} + {\left(3 \, a^{8} - 8 \, a^{6}\right)} r^{4}\right)} y^{4} + 4 \, {\left(2 \, a^{4} r^{8} + a^{2} r^{10} + 3 \, a^{6} r^{5} + 2 \, a^{4} r^{7} - a^{2} r^{9} + 2 \, a^{6} r^{4} + {\left(a^{6} - 2 \, a^{4}\right)} r^{6}\right)} y^{2}}$

KK[1,3]

$0$

KK[2,2]

$-\frac{4 \, {\left({\left(a^{8} r^{2} + a^{6} r^{4} - 2 \, a^{6} r^{3}\right)} y^{4} - {\left(a^{8} r^{2} + a^{6} r^{4} - 2 \, a^{6} r^{3}\right)} y^{2}\right)}}{a^{4} r^{8} + 2 \, a^{2} r^{10} + r^{12} + 4 \, a^{4} r^{7} + 4 \, a^{2} r^{9} + 4 \, a^{4} r^{6} + {\left(a^{12} + a^{8} r^{4} - 4 \, a^{10} r - 4 \, a^{8} r^{3} + 2 \, {\left(a^{10} + 2 \, a^{8}\right)} r^{2}\right)} y^{8} + 4 \, {\left(a^{6} r^{6} + a^{10} r - 2 \, a^{8} r^{3} - 3 \, a^{6} r^{5} + 2 \, {\left(a^{8} + a^{6}\right)} r^{4} + {\left(a^{10} - 2 \, a^{8}\right)} r^{2}\right)} y^{6} + 2 \, {\left(3 \, a^{4} r^{8} + 6 \, a^{8} r^{3} - 6 \, a^{4} r^{7} + 2 \, a^{8} r^{2} + 2 \, {\left(3 \, a^{6} + a^{4}\right)} r^{6} + {\left(3 \, a^{8} - 8 \, a^{6}\right)} r^{4}\right)} y^{4} + 4 \, {\left(2 \, a^{4} r^{8} + a^{2} r^{10} + 3 \, a^{6} r^{5} + 2 \, a^{4} r^{7} - a^{2} r^{9} + 2 \, a^{6} r^{4} + {\left(a^{6} - 2 \, a^{4}\right)} r^{6}\right)} y^{2}}$

KK[2,3]

$0$

KK[3,3]

$\frac{a^{6} r^{4} + 6 \, a^{4} r^{6} + 9 \, a^{2} r^{8} + {\left(a^{10} - 6 \, a^{8} r^{2} - 3 \, a^{6} r^{4} + 8 \, a^{6} r^{3}\right)} y^{8} - 2 \, {\left(a^{10} - 7 \, a^{8} r^{2} - 3 \, a^{6} r^{4} - 3 \, a^{4} r^{6} + 12 \, a^{6} r^{3}\right)} y^{6} + {\left(a^{10} - 10 \, a^{8} r^{2} - 2 \, a^{6} r^{4} - 6 \, a^{4} r^{6} + 9 \, a^{2} r^{8} + 24 \, a^{6} r^{3}\right)} y^{4} + 2 \, {\left(a^{8} r^{2} - a^{6} r^{4} - 3 \, a^{4} r^{6} - 9 \, a^{2} r^{8} - 4 \, a^{6} r^{3}\right)} y^{2}}{a^{2} r^{10} + r^{12} + 2 \, a^{2} r^{9} + {\left(a^{12} + a^{10} r^{2} - 2 \, a^{10} r\right)} y^{10} + {\left(5 \, a^{10} r^{2} + 5 \, a^{8} r^{4} + 2 \, a^{10} r - 8 \, a^{8} r^{3}\right)} y^{8} + 2 \, {\left(5 \, a^{8} r^{4} + 5 \, a^{6} r^{6} + 4 \, a^{8} r^{3} - 6 \, a^{6} r^{5}\right)} y^{6} + 2 \, {\left(5 \, a^{6} r^{6} + 5 \, a^{4} r^{8} + 6 \, a^{6} r^{5} - 4 \, a^{4} r^{7}\right)} y^{4} + {\left(5 \, a^{4} r^{8} + 5 \, a^{2} r^{10} + 8 \, a^{4} r^{7} - 2 \, a^{2} r^{9}\right)} y^{2}}$

In vacuum and for stationary spacetimes, the dynamical Einstein equations are $\mathcal{L}_\beta K_{ij} - D_i D_j N + N \left( R_{ij} + K K_{ij} - 2 K_{ik} K^k_{\ \, j}\right) = 0$

dyn = K.lie_der(b) - D(D(N)) + N*( Ric + trK*K - 2*KK )
print dyn
dyn.display()

tensor field of type (0,2) on the 3-dimensional manifold 'Sigma'

$0$

Hence, we have checked that all the vacuum 3+1 Einstein equations are fulfilled.

Electric and magnetic parts of the Weyl tensor

The electric part is the bilinear form $E$ given by $E_{ij} = R_{ij} + K K_{ij} - K_{ik} K^k_{\ \, j}$

E = Ric + trK*K - KK
print E

field of symmetric bilinear forms on the 3-dimensional manifold 'Sigma'

E[1,1]

$-\frac{3 \, a^{4} r^{3} + 5 \, a^{2} r^{5} + 2 \, r^{7} - 2 \, a^{2} r^{4} + 3 \, {\left(a^{6} r + a^{4} r^{3} - 2 \, a^{4} r^{2}\right)} y^{4} - {\left(9 \, a^{6} r + 16 \, a^{4} r^{3} + 7 \, a^{2} r^{5} - 6 \, a^{4} r^{2} - 2 \, a^{2} r^{4}\right)} y^{2}}{2 \, a^{2} r^{8} + r^{10} + 2 \, a^{4} r^{5} - 2 \, r^{9} + {\left(a^{4} - 4 \, a^{2}\right)} r^{6} + {\left(a^{10} + a^{6} r^{4} - 4 \, a^{8} r - 4 \, a^{6} r^{3} + 2 \, {\left(a^{8} + 2 \, a^{6}\right)} r^{2}\right)} y^{6} + {\left(3 \, a^{4} r^{6} + 2 \, a^{8} r - 8 \, a^{6} r^{3} - 10 \, a^{4} r^{5} + 2 \, {\left(3 \, a^{6} + 4 \, a^{4}\right)} r^{4} + {\left(3 \, a^{8} - 4 \, a^{6}\right)} r^{2}\right)} y^{4} + {\left(3 \, a^{2} r^{8} + 4 \, a^{6} r^{3} - 4 \, a^{4} r^{5} - 8 \, a^{2} r^{7} + 2 \, {\left(3 \, a^{4} + 2 \, a^{2}\right)} r^{6} + {\left(3 \, a^{6} - 8 \, a^{4}\right)} r^{4}\right)} y^{2}}$

E[1,1].factor()

$-\frac{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} - 3 \, a^{4} - 5 \, a^{2} r^{2} - 2 \, r^{4} + 2 \, a^{2} r\right)} {\left(3 \, a^{2} y^{2} - r^{2}\right)} r}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{2} {\left(a^{2} + r^{2} - 2 \, r\right)}}$

E[1,2]

$\frac{3 \, {\left({\left(a^{6} + a^{4} r^{2}\right)} y^{3} - 3 \, {\left(a^{4} r^{2} + a^{2} r^{4}\right)} y\right)}}{a^{2} r^{6} + r^{8} + 2 \, a^{2} r^{5} + {\left(a^{8} + a^{6} r^{2} - 2 \, a^{6} r\right)} y^{6} + {\left(3 \, a^{6} r^{2} + 3 \, a^{4} r^{4} + 2 \, a^{6} r - 4 \, a^{4} r^{3}\right)} y^{4} + {\left(3 \, a^{4} r^{4} + 3 \, a^{2} r^{6} + 4 \, a^{4} r^{3} - 2 \, a^{2} r^{5}\right)} y^{2}}$

E[1,2].factor()

$\frac{3 \, {\left(a^{2} y^{2} - 3 \, r^{2}\right)} {\left(a^{2} + r^{2}\right)} a^{2} y}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{2}}$

E[1,3]

$0$

E[2,2]

$-\frac{3 \, a^{4} r^{3} + 4 \, a^{2} r^{5} + r^{7} - 4 \, a^{2} r^{4} + 6 \, {\left(a^{6} r + a^{4} r^{3} - 2 \, a^{4} r^{2}\right)} y^{4} - {\left(9 \, a^{6} r + 14 \, a^{4} r^{3} + 5 \, a^{2} r^{5} - 12 \, a^{4} r^{2} - 4 \, a^{2} r^{4}\right)} y^{2}}{{\left(a^{8} + a^{6} r^{2} - 2 \, a^{6} r\right)} y^{8} - a^{2} r^{6} - r^{8} - 2 \, a^{2} r^{5} - {\left(a^{8} - 2 \, a^{6} r^{2} - 3 \, a^{4} r^{4} - 4 \, a^{6} r + 4 \, a^{4} r^{3}\right)} y^{6} - {\left(3 \, a^{6} r^{2} - 3 \, a^{2} r^{6} + 2 \, a^{6} r - 8 \, a^{4} r^{3} + 2 \, a^{2} r^{5}\right)} y^{4} - {\left(3 \, a^{4} r^{4} + 2 \, a^{2} r^{6} - r^{8} + 4 \, a^{4} r^{3} - 4 \, a^{2} r^{5}\right)} y^{2}}$

E[2,2].factor()

$-\frac{{\left(2 \, a^{4} y^{2} + 2 \, a^{2} r^{2} y^{2} - 4 \, a^{2} r y^{2} - 3 \, a^{4} - 4 \, a^{2} r^{2} - r^{4} + 4 \, a^{2} r\right)} {\left(3 \, a^{2} y^{2} - r^{2}\right)} r}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{2} {\left(y + 1\right)} {\left(y - 1\right)}}$

E[2,3]

$0$

E[3,3]

$\frac{a^{2} r^{5} + r^{7} + 3 \, {\left(a^{6} r + a^{4} r^{3} - 2 \, a^{4} r^{2}\right)} y^{6} + 2 \, a^{2} r^{4} - {\left(3 \, a^{6} r + a^{4} r^{3} - 2 \, a^{2} r^{5} - 12 \, a^{4} r^{2} - 2 \, a^{2} r^{4}\right)} y^{4} - {\left(2 \, a^{4} r^{3} + 3 \, a^{2} r^{5} + r^{7} + 6 \, a^{4} r^{2} + 4 \, a^{2} r^{4}\right)} y^{2}}{a^{8} y^{8} + 4 \, a^{6} r^{2} y^{6} + 6 \, a^{4} r^{4} y^{4} + 4 \, a^{2} r^{6} y^{2} + r^{8}}$

E[3,3].factor()

$\frac{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(3 \, a^{2} y^{2} - r^{2}\right)} r {\left(y + 1\right)} {\left(y - 1\right)}}{{\left(a^{2} y^{2} + r^{2}\right)}^{4}}$

The magnetic part is the bilinear form $B$ defined by $B_{ij} = \epsilon^k_{\ \, l i} D_k K^l_{\ \, j},$

where $\epsilon^k_{\ \, l i}$ are the components of the type-(1,2) tensor $\epsilon^\sharp$, related to the Levi-Civita alternating tensor $\epsilon$ associated with $\gamma$ by $\epsilon^k_{\ \, l i} = \gamma^{km} \epsilon_{m l i}$. In SageManifolds, $\epsilon$ is obtained by the command volume_form() and $\epsilon^\sharp$ by the command volume_form(1) (1 = 1 index raised):

eps = gam.volume_form() 
print eps ; eps.display()

3-form 'eps_gam' on the 3-dimensional manifold 'Sigma'

$\epsilon_{\gamma} = \left( \frac{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} y^{2} + r^{2}}}{\sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} r\wedge \mathrm{d} y\wedge \mathrm{d} {\phi}$

epsu = gam.volume_form(1)
print epsu ; epsu.display()

tensor field of type (1,2) on the 3-dimensional manifold 'Sigma'

$\left( \frac{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}}{\sqrt{a^{2} y^{2} + r^{2}}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} y\otimes \mathrm{d} {\phi} + \left( -\frac{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}}{\sqrt{a^{2} y^{2} + r^{2}}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} y + \left( \frac{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left(y^{2} - 1\right)}}{\sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \frac{\partial}{\partial y }\otimes \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( -\frac{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left(y^{2} - 1\right)}}{\sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \frac{\partial}{\partial y }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( -\frac{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left(a^{2} y^{2} + r^{2}\right)}^{\frac{3}{2}}}{{\left({\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \, a^{2} r - {\left(a^{4} - r^{4} - 4 \, a^{2} r\right)} y^{2}\right)} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} r\otimes \mathrm{d} y + \left( \frac{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left(a^{2} y^{2} + r^{2}\right)}^{\frac{3}{2}}}{{\left({\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{4} - a^{2} r^{2} - r^{4} - 2 \, a^{2} r - {\left(a^{4} - r^{4} - 4 \, a^{2} r\right)} y^{2}\right)} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} y\otimes \mathrm{d} r$

DKu = D(Ku)
B = epsu['^k_li']*DKu['^l_jk'] 
print B

tensor field of type (0,2) on the 3-dimensional manifold 'Sigma'

Let us check that $B$ is symmetric:

B1 = B.symmetrize()
B == B1

$\mathrm{True}$

Accordingly, we set

B = B1
B.set_name('B')
print B

field of symmetric bilinear forms 'B' on the 3-dimensional manifold 'Sigma'

B[1,1]

$-\frac{{\left(a^{7} + a^{5} r^{2} - 2 \, a^{5} r\right)} y^{5} - {\left(3 \, a^{7} + 8 \, a^{5} r^{2} + 5 \, a^{3} r^{4} - 2 \, a^{5} r - 6 \, a^{3} r^{3}\right)} y^{3} + 3 \, {\left(3 \, a^{5} r^{2} + 5 \, a^{3} r^{4} + 2 \, a r^{6} - 2 \, a^{3} r^{3}\right)} y}{2 \, a^{2} r^{8} + r^{10} + 2 \, a^{4} r^{5} - 2 \, r^{9} + {\left(a^{4} - 4 \, a^{2}\right)} r^{6} + {\left(a^{10} + a^{6} r^{4} - 4 \, a^{8} r - 4 \, a^{6} r^{3} + 2 \, {\left(a^{8} + 2 \, a^{6}\right)} r^{2}\right)} y^{6} + {\left(3 \, a^{4} r^{6} + 2 \, a^{8} r - 8 \, a^{6} r^{3} - 10 \, a^{4} r^{5} + 2 \, {\left(3 \, a^{6} + 4 \, a^{4}\right)} r^{4} + {\left(3 \, a^{8} - 4 \, a^{6}\right)} r^{2}\right)} y^{4} + {\left(3 \, a^{2} r^{8} + 4 \, a^{6} r^{3} - 4 \, a^{4} r^{5} - 8 \, a^{2} r^{7} + 2 \, {\left(3 \, a^{4} + 2 \, a^{2}\right)} r^{6} + {\left(3 \, a^{6} - 8 \, a^{4}\right)} r^{4}\right)} y^{2}}$

B[1,1].factor()

$-\frac{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} - 3 \, a^{4} - 5 \, a^{2} r^{2} - 2 \, r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} - 3 \, r^{2}\right)} a y}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{2} {\left(a^{2} + r^{2} - 2 \, r\right)}}$

B[1,2]

$\frac{3 \, {\left(a^{3} r^{3} + a r^{5} - 3 \, {\left(a^{5} r + a^{3} r^{3}\right)} y^{2}\right)}}{a^{2} r^{6} + r^{8} + 2 \, a^{2} r^{5} + {\left(a^{8} + a^{6} r^{2} - 2 \, a^{6} r\right)} y^{6} + {\left(3 \, a^{6} r^{2} + 3 \, a^{4} r^{4} + 2 \, a^{6} r - 4 \, a^{4} r^{3}\right)} y^{4} + {\left(3 \, a^{4} r^{4} + 3 \, a^{2} r^{6} + 4 \, a^{4} r^{3} - 2 \, a^{2} r^{5}\right)} y^{2}}$

B[1,2].factor()

$-\frac{3 \, {\left(3 \, a^{2} y^{2} - r^{2}\right)} {\left(a^{2} + r^{2}\right)} a r}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{2}}$

B[1,3]

$0$

B[2,2]

$-\frac{2 \, {\left(a^{7} + a^{5} r^{2} - 2 \, a^{5} r\right)} y^{5} - {\left(3 \, a^{7} + 10 \, a^{5} r^{2} + 7 \, a^{3} r^{4} - 4 \, a^{5} r - 12 \, a^{3} r^{3}\right)} y^{3} + 3 \, {\left(3 \, a^{5} r^{2} + 4 \, a^{3} r^{4} + a r^{6} - 4 \, a^{3} r^{3}\right)} y}{{\left(a^{8} + a^{6} r^{2} - 2 \, a^{6} r\right)} y^{8} - a^{2} r^{6} - r^{8} - 2 \, a^{2} r^{5} - {\left(a^{8} - 2 \, a^{6} r^{2} - 3 \, a^{4} r^{4} - 4 \, a^{6} r + 4 \, a^{4} r^{3}\right)} y^{6} - {\left(3 \, a^{6} r^{2} - 3 \, a^{2} r^{6} + 2 \, a^{6} r - 8 \, a^{4} r^{3} + 2 \, a^{2} r^{5}\right)} y^{4} - {\left(3 \, a^{4} r^{4} + 2 \, a^{2} r^{6} - r^{8} + 4 \, a^{4} r^{3} - 4 \, a^{2} r^{5}\right)} y^{2}}$

B[2,2].factor()

$-\frac{{\left(2 \, a^{4} y^{2} + 2 \, a^{2} r^{2} y^{2} - 4 \, a^{2} r y^{2} - 3 \, a^{4} - 4 \, a^{2} r^{2} - r^{4} + 4 \, a^{2} r\right)} {\left(a^{2} y^{2} - 3 \, r^{2}\right)} a y}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{2} {\left(y + 1\right)} {\left(y - 1\right)}}$

B[2,3]

$0$

B[3,3]

$\frac{{\left(a^{7} + a^{5} r^{2} - 2 \, a^{5} r\right)} y^{7} - {\left(a^{7} + 3 \, a^{5} r^{2} + 2 \, a^{3} r^{4} - 4 \, a^{5} r - 6 \, a^{3} r^{3}\right)} y^{5} + {\left(2 \, a^{5} r^{2} - a^{3} r^{4} - 3 \, a r^{6} - 2 \, a^{5} r - 12 \, a^{3} r^{3}\right)} y^{3} + 3 \, {\left(a^{3} r^{4} + a r^{6} + 2 \, a^{3} r^{3}\right)} y}{a^{8} y^{8} + 4 \, a^{6} r^{2} y^{6} + 6 \, a^{4} r^{4} y^{4} + 4 \, a^{2} r^{6} y^{2} + r^{8}}$

B[3,3].factor()

$\frac{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} - 3 \, r^{2}\right)} a {\left(y + 1\right)} {\left(y - 1\right)} y}{{\left(a^{2} y^{2} + r^{2}\right)}^{4}}$

3+1 decomposition of the Simon-Mars tensor

We follow the computation presented in arXiv:1412.6542. We start by the tensor $E^\sharp$ of components $E^i_ {\ \, j}$:

Eu = E.up(gam, 0) 
print Eu

tensor field of type (1,1) on the 3-dimensional manifold 'Sigma'

Tensor $B^\sharp$ of components $B^i_{\ \, j}$:

Bu = B.up(gam, 0)
print Bu

tensor field of type (1,1) on the 3-dimensional manifold 'Sigma'

1-form $\beta^\flat$ of components $\beta_i$ and its exterior derivative:

bd = b.down(gam)
xdb = bd.exterior_der()
print xdb ; xdb.display()

2-form on the 3-dimensional manifold 'Sigma'

$\left( \frac{2 \, {\left(a^{3} y^{4} + a r^{2} - {\left(a^{3} + a r^{2}\right)} y^{2}\right)}}{a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}} \right) \mathrm{d} r\wedge \mathrm{d} {\phi} + \left( \frac{4 \, {\left(a^{3} r + a r^{3}\right)} y}{a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}} \right) \mathrm{d} y\wedge \mathrm{d} {\phi}$

Scalar square of shift $\beta_i \beta^i$:

b2 = bd(b)
print b2 ; b2.display()

scalar field on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & -\frac{4 \, {\left(a^{2} r^{2} y^{2} - a^{2} r^{2}\right)}}{a^{2} r^{4} + r^{6} + 2 \, a^{2} r^{3} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} y^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} + a^{4} r - a^{2} r^{3}\right)} y^{2}} \end{array}$

Scalar $Y = E(\beta,\beta) = E_{ij} \beta^i \beta^j$:

Ebb = E(b,b)
Y = Ebb
print Y ; Y.display()

scalar field on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & \frac{4 \, {\left(3 \, a^{4} r^{3} y^{4} + a^{2} r^{5} - {\left(3 \, a^{4} r^{3} + a^{2} r^{5}\right)} y^{2}\right)}}{a^{2} r^{10} + r^{12} + 2 \, a^{2} r^{9} + {\left(a^{12} + a^{10} r^{2} - 2 \, a^{10} r\right)} y^{10} + {\left(5 \, a^{10} r^{2} + 5 \, a^{8} r^{4} + 2 \, a^{10} r - 8 \, a^{8} r^{3}\right)} y^{8} + 2 \, {\left(5 \, a^{8} r^{4} + 5 \, a^{6} r^{6} + 4 \, a^{8} r^{3} - 6 \, a^{6} r^{5}\right)} y^{6} + 2 \, {\left(5 \, a^{6} r^{6} + 5 \, a^{4} r^{8} + 6 \, a^{6} r^{5} - 4 \, a^{4} r^{7}\right)} y^{4} + {\left(5 \, a^{4} r^{8} + 5 \, a^{2} r^{10} + 8 \, a^{4} r^{7} - 2 \, a^{2} r^{9}\right)} y^{2}} \end{array}$

Ebb.function_chart().factor()

$\frac{4 \, {\left(3 \, a^{2} y^{2} - r^{2}\right)} a^{2} r^{3} {\left(y + 1\right)} {\left(y - 1\right)}}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{4}}$

Ebb.display()

$\begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & \frac{4 \, {\left(3 \, a^{2} y^{2} - r^{2}\right)} a^{2} r^{3} {\left(y + 1\right)} {\left(y - 1\right)}}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{4}} \end{array}$

Scalar $\bar Y = B(\beta,\beta) = B_{ij}\beta^i \beta^j$:

Bbb = B(b,b)
Y_bar = Bbb
print Y_bar ; Y_bar.display()

scalar field 'B(beta,beta)' on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} B\left(\beta,\beta\right):& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & \frac{4 \, {\left(a^{5} r^{2} y^{5} + 3 \, a^{3} r^{4} y - {\left(a^{5} r^{2} + 3 \, a^{3} r^{4}\right)} y^{3}\right)}}{a^{2} r^{10} + r^{12} + 2 \, a^{2} r^{9} + {\left(a^{12} + a^{10} r^{2} - 2 \, a^{10} r\right)} y^{10} + {\left(5 \, a^{10} r^{2} + 5 \, a^{8} r^{4} + 2 \, a^{10} r - 8 \, a^{8} r^{3}\right)} y^{8} + 2 \, {\left(5 \, a^{8} r^{4} + 5 \, a^{6} r^{6} + 4 \, a^{8} r^{3} - 6 \, a^{6} r^{5}\right)} y^{6} + 2 \, {\left(5 \, a^{6} r^{6} + 5 \, a^{4} r^{8} + 6 \, a^{6} r^{5} - 4 \, a^{4} r^{7}\right)} y^{4} + {\left(5 \, a^{4} r^{8} + 5 \, a^{2} r^{10} + 8 \, a^{4} r^{7} - 2 \, a^{2} r^{9}\right)} y^{2}} \end{array}$

Bbb.function_chart().factor()

$\frac{4 \, {\left(a^{2} y^{2} - 3 \, r^{2}\right)} a^{3} r^{2} {\left(y + 1\right)} {\left(y - 1\right)} y}{{\left(a^{4} y^{2} + a^{2} r^{2} y^{2} - 2 \, a^{2} r y^{2} + a^{2} r^{2} + r^{4} + 2 \, a^{2} r\right)} {\left(a^{2} y^{2} + r^{2}\right)}^{4}}$

1-form of components $Eb_i = E_{ij} \beta^j$:

Eb = E.contract(b)
print Eb ; Eb.display()

1-form on the 3-dimensional manifold 'Sigma'

$\left( -\frac{2 \, {\left(3 \, a^{3} r^{2} y^{4} + a r^{4} - {\left(3 \, a^{3} r^{2} + a r^{4}\right)} y^{2}\right)}}{a^{8} y^{8} + 4 \, a^{6} r^{2} y^{6} + 6 \, a^{4} r^{4} y^{4} + 4 \, a^{2} r^{6} y^{2} + r^{8}} \right) \mathrm{d} {\phi}$

Vector field of components $Eub^i = E^i_{\ \, j} \beta^j$:

Eub = Eu.contract(b)
print Eub ; Eub.display()

vector field on the 3-dimensional manifold 'Sigma'

$\left( \frac{2 \, {\left(3 \, a^{3} r^{2} y^{2} - a r^{4}\right)}}{a^{2} r^{8} + r^{10} + 2 \, a^{2} r^{7} + {\left(a^{10} + a^{8} r^{2} - 2 \, a^{8} r\right)} y^{8} + 2 \, {\left(2 \, a^{8} r^{2} + 2 \, a^{6} r^{4} + a^{8} r - 3 \, a^{6} r^{3}\right)} y^{6} + 6 \, {\left(a^{6} r^{4} + a^{4} r^{6} + a^{6} r^{3} - a^{4} r^{5}\right)} y^{4} + 2 \, {\left(2 \, a^{4} r^{6} + 2 \, a^{2} r^{8} + 3 \, a^{4} r^{5} - a^{2} r^{7}\right)} y^{2}} \right) \frac{\partial}{\partial {\phi} }$

1-form of components $Bb_i = B_{ij} \beta^j$:

Bb = B.contract(b)
print Bb ; Bb.display()

1-form on the 3-dimensional manifold 'Sigma'

$\left( -\frac{2 \, {\left(a^{4} r y^{5} + 3 \, a^{2} r^{3} y - {\left(a^{4} r + 3 \, a^{2} r^{3}\right)} y^{3}\right)}}{a^{8} y^{8} + 4 \, a^{6} r^{2} y^{6} + 6 \, a^{4} r^{4} y^{4} + 4 \, a^{2} r^{6} y^{2} + r^{8}} \right) \mathrm{d} {\phi}$

Vector field of components $Bub^i = B^i_{\ \, j} \beta^j$:

Bub = Bu.contract(b)
print Bub ; Bub.display()

vector field on the 3-dimensional manifold 'Sigma'

$\left( \frac{2 \, {\left(a^{4} r y^{3} - 3 \, a^{2} r^{3} y\right)}}{a^{2} r^{8} + r^{10} + 2 \, a^{2} r^{7} + {\left(a^{10} + a^{8} r^{2} - 2 \, a^{8} r\right)} y^{8} + 2 \, {\left(2 \, a^{8} r^{2} + 2 \, a^{6} r^{4} + a^{8} r - 3 \, a^{6} r^{3}\right)} y^{6} + 6 \, {\left(a^{6} r^{4} + a^{4} r^{6} + a^{6} r^{3} - a^{4} r^{5}\right)} y^{4} + 2 \, {\left(2 \, a^{4} r^{6} + 2 \, a^{2} r^{8} + 3 \, a^{4} r^{5} - a^{2} r^{7}\right)} y^{2}} \right) \frac{\partial}{\partial {\phi} }$

Vector field of components $Kub^i = K^i_{\ \, j} \beta^j$:

Kub = Ku.contract(b)
print Kub ; Kub.display()

vector field on the 3-dimensional manifold 'Sigma'

$\left( -\frac{2 \, {\left(a^{4} r^{3} + 3 \, a^{2} r^{5} + {\left(a^{6} r - a^{4} r^{3}\right)} y^{4} - {\left(a^{6} r + 3 \, a^{2} r^{5}\right)} y^{2}\right)} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{2} r^{6} + r^{8} + 2 \, a^{2} r^{5} + {\left(a^{8} + a^{6} r^{2} - 2 \, a^{6} r\right)} y^{6} + {\left(3 \, a^{6} r^{2} + 3 \, a^{4} r^{4} + 2 \, a^{6} r - 4 \, a^{4} r^{3}\right)} y^{4} + {\left(3 \, a^{4} r^{4} + 3 \, a^{2} r^{6} + 4 \, a^{4} r^{3} - 2 \, a^{2} r^{5}\right)} y^{2}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} y^{2} + r^{2}}} \right) \frac{\partial}{\partial r } + \left( -\frac{4 \, {\left(a^{4} r^{2} y^{5} - 2 \, a^{4} r^{2} y^{3} + a^{4} r^{2} y\right)} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{2} r^{6} + r^{8} + 2 \, a^{2} r^{5} + {\left(a^{8} + a^{6} r^{2} - 2 \, a^{6} r\right)} y^{6} + {\left(3 \, a^{6} r^{2} + 3 \, a^{4} r^{4} + 2 \, a^{6} r - 4 \, a^{4} r^{3}\right)} y^{4} + {\left(3 \, a^{4} r^{4} + 3 \, a^{2} r^{6} + 4 \, a^{4} r^{3} - 2 \, a^{2} r^{5}\right)} y^{2}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} y^{2} + r^{2}}} \right) \frac{\partial}{\partial y }$

T = 2*b(N) - 2*K(b,b)
print T ; T.display()

scalar field 'zero' on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} 0:& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & 0 \end{array}$

Db = D(b)  #  Db^i_j = D_j b^i
Dbu = Db.up(gam, 1)  # Dbu^{ij} = D^j b^i
bDb = b*Dbu  # bDb^{ijk} = b^i D^k b^j
T_bar = eps['_ijk']*bDb['^ikj']
print T_bar ; T_bar.display()

scalar field on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & 0 \end{array}$

epsb = eps.contract(b) 
print epsb
epsb.display()

2-form on the 3-dimensional manifold 'Sigma'

$\left( -\frac{2 \, \sqrt{a^{2} y^{2} + r^{2}} a r}{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}} \right) \mathrm{d} r\wedge \mathrm{d} y$

epsB = eps['_ijl']*Bu['^l_k']
print epsB

tensor field of type (0,3) on the 3-dimensional manifold 'Sigma'

epsB.symmetries()

no symmetry; antisymmetry: (0, 1)

epsB[1,2,3]

$-\frac{{\left(a^{3} y^{3} - 3 \, a r^{2} y\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} \sqrt{a^{2} y^{2} + r^{2}}}{{\left(a^{6} y^{6} + 3 \, a^{4} r^{2} y^{4} + 3 \, a^{2} r^{4} y^{2} + r^{6}\right)} \sqrt{a^{2} + r^{2} - 2 \, r}}$

Z = 2*N*( D(N)  -K.contract(b)) + b.contract(xdb)
print Z ; Z.display()

1-form on the 3-dimensional manifold 'Sigma'

$\left( -\frac{2 \, {\left(a^{2} y^{2} - r^{2}\right)}}{a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}} \right) \mathrm{d} r + \left( \frac{4 \, a^{2} r y}{a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}} \right) \mathrm{d} y$

DNu = D(N).up(gam)
A = 2*(DNu - Ku.contract(b))*b + N*Dbu
Z_bar = eps['_ijk']*A['^kj']
print Z_bar ; Z_bar.display()

1-form on the 3-dimensional manifold 'Sigma'

$\left( \frac{4 \, a r y}{a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}} \right) \mathrm{d} r + \left( \frac{2 \, {\left(a^{3} y^{2} - a r^{2}\right)}}{a^{4} y^{4} + 2 \, a^{2} r^{2} y^{2} + r^{4}} \right) \mathrm{d} y$

# Test:
Dbdu = D(bd).up(gam,1).up(gam,1)  # (Db)^{ij} = D^i b^j
A = 2*b*(DNu - Ku.contract(b)) + N*Dbdu
Z_bar0 = eps['_ijk']*A['^jk']  # NB: '^jk' and not 'kj'
Z_bar0 == Z_bar

$\mathrm{True}$

W = N*Eb + epsb.contract(Bub)
print W ; W.display()

1-form on the 3-dimensional manifold 'Sigma'

$\left( -\frac{2 \, {\left(3 \, a^{3} r^{2} y^{4} + a r^{4} - {\left(3 \, a^{3} r^{2} + a r^{4}\right)} y^{2}\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{8} y^{8} + 4 \, a^{6} r^{2} y^{6} + 6 \, a^{4} r^{4} y^{4} + 4 \, a^{2} r^{6} y^{2} + r^{8}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}}} \right) \mathrm{d} {\phi}$

W_bar = N*Bb - epsb.contract(Eub)
print W_bar ; W_bar.display()

1-form on the 3-dimensional manifold 'Sigma'

$\left( -\frac{2 \, {\left(a^{4} r y^{5} + 3 \, a^{2} r^{3} y - {\left(a^{4} r + 3 \, a^{2} r^{3}\right)} y^{3}\right)} \sqrt{a^{2} y^{2} + r^{2}} \sqrt{a^{2} + r^{2} - 2 \, r}}{{\left(a^{8} y^{8} + 4 \, a^{6} r^{2} y^{6} + 6 \, a^{4} r^{4} y^{4} + 4 \, a^{2} r^{6} y^{2} + r^{8}\right)} \sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}}} \right) \mathrm{d} {\phi}$

W[3].factor()

$-\frac{2 \, {\left(3 \, a^{2} y^{2} - r^{2}\right)} \sqrt{a^{2} + r^{2} - 2 \, r} a r^{2} {\left(y + 1\right)} {\left(y - 1\right)}}{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left(a^{2} y^{2} + r^{2}\right)}^{\frac{7}{2}}}$

W_bar[3].factor()

$-\frac{2 \, {\left(a^{2} y^{2} - 3 \, r^{2}\right)} \sqrt{a^{2} + r^{2} - 2 \, r} a^{2} r {\left(y + 1\right)} {\left(y - 1\right)} y}{\sqrt{a^{2} r^{2} + r^{4} + 2 \, a^{2} r + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} y^{2}} {\left(a^{2} y^{2} + r^{2}\right)}^{\frac{7}{2}}}$

M = - 4*Eb(Kub - DNu) - 2*(epsB['_ij.']*Dbu['^ji'])(b)
print M ; M.display()

scalar field 'zero' on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} 0:& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & 0 \end{array}$

M_bar = 2*(eps.contract(Eub))['_ij']*Dbu['^ji'] - 4*Bb(Kub - DNu)
print M_bar ; M_bar.display()

scalar field 'zero' on the 3-dimensional manifold 'Sigma'

$\begin{array}{llcl} 0:& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, y, {\phi}\right) & \longmapsto & 0 \end{array}$

A = epsB['_ilk']*b['^l'] + epsB['_ikl']*b['^l'] + Bu['^m_i']*epsb['_mk'] - 2*N*E
xdbE = xdb['_kl']*Eu['^k_i']
L = 2*N*epsB['_kli']*Dbu['^kl'] + 2*xdb['_ij']*Eub['^j'] + 2*xdbE['_li']*b['^l'] \
    + 2*A['_ik']*(Kub - DNu)['^k']
print L

1-form on the 3-dimensional manifold 'Sigma'

L[1]

$-\frac{8 \, {\left(5 \, a^{4} r y^{4} - 10 \, a^{2} r^{3} y^{2} + r^{5}\right)}}{a^{10} y^{10} + 5 \, a^{8} r^{2} y^{8} + 10 \, a^{6} r^{4} y^{6} + 10 \, a^{4} r^{6} y^{4} + 5 \, a^{2} r^{8} y^{2} + r^{10}}$

L[1].factor()

$-\frac{8 \, {\left(5 \, a^{4} y^{4} - 10 \, a^{2} r^{2} y^{2} + r^{4}\right)} r}{{\left(a^{2} y^{2} + r^{2}\right)}^{5}}$

L[2]

$-\frac{8 \, {\left(a^{6} y^{5} - 10 \, a^{4} r^{2} y^{3} + 5 \, a^{2} r^{4} y\right)}}{a^{10} y^{10} + 5 \, a^{8} r^{2} y^{8} + 10 \, a^{6} r^{4} y^{6} + 10 \, a^{4} r^{6} y^{4} + 5 \, a^{2} r^{8} y^{2} + r^{10}}$

L[2].factor()

$-\frac{8 \, {\left(a^{4} y^{4} - 10 \, a^{2} r^{2} y^{2} + 5 \, r^{4}\right)} a^{2} y}{{\left(a^{2} y^{2} + r^{2}\right)}^{5}}$

L[3]

$0$

N2pbb = N^2 + b2
V = N2pbb*E - 2*(b.contract(E)*bd).symmetrize() + Ebb*gam + 2*N*(b.contract(epsB).symmetrize())
print V

V[1,1]

V[1,1].factor()

V[1,2]

V[1,2].factor()

V[1,3]

V[2,2]

V[2,2].factor()

V[2,3]

V[3,3]

V[3,3].factor()

beps = b.contract(eps)
V_bar = N2pbb*B - 2*(b.contract(B)*bd).symmetrize() + Bbb*gam \
        -2*N*(beps['_il']*Eu['^l_j']).symmetrize()
print V_bar

V_bar[1,1]

V_bar[1,1].factor()

V_bar[1,2]

V_bar[1,2].factor()

V_bar[1,3]

V_bar[2,2]

V_bar[2,2].factor()

V_bar[2,3]

V_bar[3,3]

V_bar[3,3].factor()

G = (N^2 - b2)*gam + bd*bd
print G

G.display()

3+1 decomposition of the real part of the Simon-Mars tensor

We follow Eqs. (77)-(80) of arXiv:1412.6542:

S1 = (4*(V*Z - V_bar*Z_bar) + G*L).antisymmetrize(1,2)
print S1

S1.display()

S2 = 4*(T*V - T_bar*V_bar - W*Z + W_bar*Z_bar) + M*G - N*bd*L
print S2

S2.display()

S3 = (4*(W*Z - W_bar*Z_bar) + N*bd*L).antisymmetrize()
print S3

S3.display()

S2[3,1] == -2*S3[3,1]

S2[3,2] == -2*S3[3,2]

S4 = 4*(T*W - T_bar*W_bar) -4*(Y*Z - Y_bar*Z_bar) + N*M*bd - b2*L
print S4

S4.display()

Hence all the tensors $S^1$, $S^2$, $S^3$ and $S^4$ involved in the 3+1 decomposition of the real part of the Simon-Mars are zero, as they should since the Simon-Mars tensor vanishes identically for the Kerr spacetime.

3+1 decomposition of the imaginary part of the Simon-Mars tensor

We follow Eqs. (82)-(85) of arXiv:1412.6542.

epsE = eps['_ijl']*Eu['^l_k']
print epsE

A = - epsE['_ilk']*b['^l'] - epsE['_ikl']*b['^l'] - Eu['^m_i']*epsb['_mk'] - 2*N*B
xdbB = xdb['_kl']*Bu['^k_i']
L_bar = - 2*N*epsE['_kli']*Dbu['^kl'] + 2*xdb['_ij']*Bub['^j'] + 2*xdbB['_li']*b['^l'] \
    + 2*A['_ik']*(Kub - DNu)['^k']
print L_bar

L_bar.display()

S1_bar = (4*(V*Z_bar + V_bar*Z) + G*L_bar).antisymmetrize(1,2)
print S1_bar

S1_bar.display()

S2_bar = 4*(T_bar*V + T*V_bar) - 4*(W*Z_bar + W_bar*Z) + M_bar*G - N*bd*L_bar
print S2_bar

S2_bar.display()

S3_bar = (4*(W*Z_bar + W_bar*Z) + N*bd*L_bar).antisymmetrize()
print S3_bar

S3_bar.display()

S4_bar = 4*(T_bar*W + T*W_bar - Y*Z_bar - Y_bar*Z) + M_bar*N*bd - b2*L_bar
print S4_bar

S4_bar.display()

Hence all the tensors ${\bar S}^1$, ${\bar S}^2$, ${\bar S}^3$ and ${\bar S}^4$ involved in the 3+1 decomposition of the imaginary part of the Simon-Mars are zero, as they should since the Simon-Mars tensor vanishes identically for the Kerr spacetime.