SM_Tomimatsu-Sato_3p1.sagews

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3+1 Einstein equations in the $\delta=2$ Tomimatsu-Sato spacetime

This worksheet is based on SageManifolds (version 0.7) and regards the 3+1 slicing of the $\delta=2$ Tomimatsu-Sato spacetime.

It 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.

Tomimatsu-Sato spacetime

The Tomimatsu-Sato solution is an exact stationary and axisymmetric  solution of the vaccum Einstein equation, which is asymptotically flat and has a non-zero angular momentum. It has been found in 1972 by A. Tomimatsu and H. Sato [Phys. Rev. Lett. 29, 1344 (1972)], as a solution of the Ernst equation. It is actually the member $\delta=2$ of a larger family of solutions parametrized by a positive integer $\delta$ and exhibited by Tomimatsu and Sato in 1973 [Prog. Theor. Phys. 50, 95 (1973)], the member $\delta=1$ being nothing but the Kerr metric. We refer to [Manko, Prog. Theor. Phys. 127, 1057 (2012)] for a discussion of the properties of this solution. 

Spacelike hypersurface

We consider some hypersurface $\Sigma$ of a spacelike foliation $(\Sigma_t)_{t\in\mathbb{R}}$ of $\delta=2$ Tomimatsu-Sato spacetime; we declare $\Sigma_t$ as a 3-dimensional manifold:

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

On $\Sigma$, we consider the prolate spheroidal coordinates $(x,y,\phi)$, with $x\in(1,+\infty)$, $y\in(-1,1)$ and $\phi\in(0,2\pi)$ :

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

chart (Sigma, (x, y, ph))

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

Riemannian metric on $\Sigma$

The Tomimatsu-Sato metric depens on three parameters: the integer $\delta$, the real number $p\in[0,1]$, and the total mass $m$:

var('d, p, m')
assume(m>0)
assumptions()

($d$, $p$, $m$)

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

We set $\delta=2$ and choose a specific value for $p=1/5$:

d = 2
p = 1/5

Furthermore, without any loss of generality, we may set $m=1$ (this simply fixes some length scale):

m = 1

The parameter $q$ is related to $p$ by $p^2+q^2=1$:

q = sqrt(1-p^2)

Some shortcut notations:

AA2 = (p^2*(x^2-1)^2+q^2*(1-y^2)^2)^2-4*p^2*q^2*(x^2-1)*(1-y^2)*(x^2-y^2)^2
BB2 = (p^2*x^4+2*p*x^3-2*p*x+q^2*y^4-1)^2+4*q^2*y^2*(p*x^3-p*x*y^2-y^2+1)^2
CC2 = p^3*x*(1-x^2)*(2*(x^4-1)+(x^2+3)*(1-y^2))+p^2*(x^2-1)*((x^2-1)*(1-y^2)-4*x^2*(x^2-y^2))+q^2*(1-y^2)^3*(p*x+1)

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

gam = Sig.riemann_metric('gam', latex_name=r'\gamma') 
gam[1,1] = m^2*BB2/(p^2*d^2*(x^2-1)*(x^2-y^2)^3)
gam[2,2] = m^2*BB2/(p^2*d^2*(y^2-1)*(-x^2+y^2)^3)
gam[3,3] = - m^2*(y^2-1)*(p^2*BB2^2*(x^2-1)+4*q^2*d^2*CC2^2*(y^2-1))/(AA2*BB2*d^2)
gam.display()

$\gamma = \left( \frac{x^{8} + 576 \, y^{8} + 20 \, x^{7} + 96 \, {\left(x^{2} + 10 \, x + 25\right)} y^{6} + 100 \, x^{6} - 20 \, x^{5} - 48 \, {\left(3 \, x^{4} + 10 \, x^{3} + 30 \, x + 125\right)} y^{4} - 250 \, x^{4} - 500 \, x^{3} + 96 \, {\left(x^{6} + 10 \, x^{3} + 25\right)} y^{2} + 100 \, x^{2} + 500 \, x + 625}{100 \, {\left(x^{8} - {\left(x^{2} - 1\right)} y^{6} - x^{6} + 3 \, {\left(x^{4} - x^{2}\right)} y^{4} - 3 \, {\left(x^{6} - x^{4}\right)} y^{2}\right)}} \right) \mathrm{d} x\otimes \mathrm{d} x + \left( \frac{x^{8} + 576 \, y^{8} + 20 \, x^{7} + 96 \, {\left(x^{2} + 10 \, x + 25\right)} y^{6} + 100 \, x^{6} - 20 \, x^{5} - 48 \, {\left(3 \, x^{4} + 10 \, x^{3} + 30 \, x + 125\right)} y^{4} - 250 \, x^{4} - 500 \, x^{3} + 96 \, {\left(x^{6} + 10 \, x^{3} + 25\right)} y^{2} + 100 \, x^{2} + 500 \, x + 625}{100 \, {\left(y^{8} - {\left(3 \, x^{2} + 1\right)} y^{6} + x^{6} + 3 \, {\left(x^{4} + x^{2}\right)} y^{4} - {\left(x^{6} + 3 \, x^{4}\right)} y^{2}\right)}} \right) \mathrm{d} y\otimes \mathrm{d} y + \left( -\frac{576 \, {\left(x^{2} - 1\right)} y^{10} - x^{10} - 40 \, x^{9} + 96 \, {\left(x^{4} + 20 \, x^{3} + 168 \, x^{2} + 980 \, x + 2431\right)} y^{8} - 699 \, x^{8} - 7920 \, x^{7} - 48 \, {\left(3 \, x^{6} + 20 \, x^{5} - x^{4} + 80 \, x^{3} + 1273 \, x^{2} + 7900 \, x + 19525\right)} y^{6} - 39450 \, x^{6} + 960 \, x^{5} + 48 \, {\left(2 \, x^{8} + x^{6} + 60 \, x^{5} - 3 \, x^{4} + 1675 \, x^{2} + 11940 \, x + 29525\right)} y^{4} + 39450 \, x^{4} + 6000 \, x^{3} + {\left(x^{10} + 40 \, x^{9} + 603 \, x^{8} + 7920 \, x^{7} + 39546 \, x^{6} - 2880 \, x^{5} - 39450 \, x^{4} - 4080 \, x^{3} - 45675 \, x^{2} - 385000 \, x - 953425\right)} y^{2} + 9675 \, x^{2} + 97000 \, x + 240625}{100 \, {\left(x^{8} + 576 \, y^{8} + 20 \, x^{7} + 96 \, {\left(x^{2} + 10 \, x + 25\right)} y^{6} + 100 \, x^{6} - 20 \, x^{5} - 48 \, {\left(3 \, x^{4} + 10 \, x^{3} + 30 \, x + 125\right)} y^{4} - 250 \, x^{4} - 500 \, x^{3} + 96 \, {\left(x^{6} + 10 \, x^{3} + 25\right)} y^{2} + 100 \, x^{2} + 500 \, x + 625\right)}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

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

gam[:]

$\left(\begin{array}{rrr} \frac{x^{8} + 576 \, y^{8} + 20 \, x^{7} + 96 \, {\left(x^{2} + 10 \, x + 25\right)} y^{6} + 100 \, x^{6} - 20 \, x^{5} - 48 \, {\left(3 \, x^{4} + 10 \, x^{3} + 30 \, x + 125\right)} y^{4} - 250 \, x^{4} - 500 \, x^{3} + 96 \, {\left(x^{6} + 10 \, x^{3} + 25\right)} y^{2} + 100 \, x^{2} + 500 \, x + 625}{100 \, {\left(x^{8} - {\left(x^{2} - 1\right)} y^{6} - x^{6} + 3 \, {\left(x^{4} - x^{2}\right)} y^{4} - 3 \, {\left(x^{6} - x^{4}\right)} y^{2}\right)}} & 0 & 0 \\ 0 & \frac{x^{8} + 576 \, y^{8} + 20 \, x^{7} + 96 \, {\left(x^{2} + 10 \, x + 25\right)} y^{6} + 100 \, x^{6} - 20 \, x^{5} - 48 \, {\left(3 \, x^{4} + 10 \, x^{3} + 30 \, x + 125\right)} y^{4} - 250 \, x^{4} - 500 \, x^{3} + 96 \, {\left(x^{6} + 10 \, x^{3} + 25\right)} y^{2} + 100 \, x^{2} + 500 \, x + 625}{100 \, {\left(y^{8} - {\left(3 \, x^{2} + 1\right)} y^{6} + x^{6} + 3 \, {\left(x^{4} + x^{2}\right)} y^{4} - {\left(x^{6} + 3 \, x^{4}\right)} y^{2}\right)}} & 0 \\ 0 & 0 & -\frac{576 \, {\left(x^{2} - 1\right)} y^{10} - x^{10} - 40 \, x^{9} + 96 \, {\left(x^{4} + 20 \, x^{3} + 168 \, x^{2} + 980 \, x + 2431\right)} y^{8} - 699 \, x^{8} - 7920 \, x^{7} - 48 \, {\left(3 \, x^{6} + 20 \, x^{5} - x^{4} + 80 \, x^{3} + 1273 \, x^{2} + 7900 \, x + 19525\right)} y^{6} - 39450 \, x^{6} + 960 \, x^{5} + 48 \, {\left(2 \, x^{8} + x^{6} + 60 \, x^{5} - 3 \, x^{4} + 1675 \, x^{2} + 11940 \, x + 29525\right)} y^{4} + 39450 \, x^{4} + 6000 \, x^{3} + {\left(x^{10} + 40 \, x^{9} + 603 \, x^{8} + 7920 \, x^{7} + 39546 \, x^{6} - 2880 \, x^{5} - 39450 \, x^{4} - 4080 \, x^{3} - 45675 \, x^{2} - 385000 \, x - 953425\right)} y^{2} + 9675 \, x^{2} + 97000 \, x + 240625}{100 \, {\left(x^{8} + 576 \, y^{8} + 20 \, x^{7} + 96 \, {\left(x^{2} + 10 \, x + 25\right)} y^{6} + 100 \, x^{6} - 20 \, x^{5} - 48 \, {\left(3 \, x^{4} + 10 \, x^{3} + 30 \, x + 125\right)} y^{4} - 250 \, x^{4} - 500 \, x^{3} + 96 \, {\left(x^{6} + 10 \, x^{3} + 25\right)} y^{2} + 100 \, x^{2} + 500 \, x + 625\right)}} \end{array}\right)$

Lapse function and shift vector

N2 = AA2/BB2 - 2*m*q*CC2*(y^2-1)/BB2*(2*m*q*CC2*(y^2-1)/(BB2*(m^2*(y^2-1)*(p^2*BB2^2*(x^2-1)+4*q^2*d^2*CC2^2*(y^2-1))/(AA2*BB2*d^2)))) 
N2.simplify_full()

$\frac{x^{10} + 20 \, x^{9} + 576 \, {\left(x^{2} - 1\right)} y^{8} + 99 \, x^{8} - 40 \, x^{7} + 96 \, {\left(x^{4} + 10 \, x^{3} + 24 \, x^{2} - 10 \, x - 25\right)} y^{6} - 350 \, x^{6} - 480 \, x^{5} - 48 \, {\left(3 \, x^{6} + 10 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 125 \, x^{2} - 30 \, x - 125\right)} y^{4} + 350 \, x^{4} + 1000 \, x^{3} + 96 \, {\left(x^{8} - x^{6} + 10 \, x^{5} - 10 \, x^{3} + 25 \, x^{2} - 25\right)} y^{2} + 525 \, x^{2} - 500 \, x - 625}{x^{10} + 40 \, x^{9} + 576 \, {\left(x^{2} - 1\right)} y^{8} + 699 \, x^{8} + 7920 \, x^{7} + 96 \, {\left(x^{4} + 20 \, x^{3} + 174 \, x^{2} + 980 \, x + 2425\right)} y^{6} + 39450 \, x^{6} - 960 \, x^{5} - 48 \, {\left(3 \, x^{6} + 20 \, x^{5} - 3 \, x^{4} + 40 \, x^{3} + 925 \, x^{2} + 5940 \, x + 14675\right)} y^{4} - 39450 \, x^{4} - 6000 \, x^{3} + 96 \, {\left(x^{8} - x^{6} + 20 \, x^{5} - 20 \, x^{3} + 375 \, x^{2} + 3000 \, x + 7425\right)} y^{2} - 9675 \, x^{2} - 97000 \, x - 240625}$

N = Sig.scalar_field(sqrt(N2.simplify_full()), name='N')
print N
N.display()

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

$\begin{array}{llcl} N:& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(x, y, {\phi}\right) & \longmapsto & \sqrt{\frac{x^{10} + 20 \, x^{9} + 576 \, {\left(x^{2} - 1\right)} y^{8} + 99 \, x^{8} - 40 \, x^{7} + 96 \, {\left(x^{4} + 10 \, x^{3} + 24 \, x^{2} - 10 \, x - 25\right)} y^{6} - 350 \, x^{6} - 480 \, x^{5} - 48 \, {\left(3 \, x^{6} + 10 \, x^{5} - 3 \, x^{4} + 20 \, x^{3} + 125 \, x^{2} - 30 \, x - 125\right)} y^{4} + 350 \, x^{4} + 1000 \, x^{3} + 96 \, {\left(x^{8} - x^{6} + 10 \, x^{5} - 10 \, x^{3} + 25 \, x^{2} - 25\right)} y^{2} + 525 \, x^{2} - 500 \, x - 625}{x^{10} + 40 \, x^{9} + 576 \, {\left(x^{2} - 1\right)} y^{8} + 699 \, x^{8} + 7920 \, x^{7} + 96 \, {\left(x^{4} + 20 \, x^{3} + 174 \, x^{2} + 980 \, x + 2425\right)} y^{6} + 39450 \, x^{6} - 960 \, x^{5} - 48 \, {\left(3 \, x^{6} + 20 \, x^{5} - 3 \, x^{4} + 40 \, x^{3} + 925 \, x^{2} + 5940 \, x + 14675\right)} y^{4} - 39450 \, x^{4} - 6000 \, x^{3} + 96 \, {\left(x^{8} - x^{6} + 20 \, x^{5} - 20 \, x^{3} + 375 \, x^{2} + 3000 \, x + 7425\right)} y^{2} - 9675 \, x^{2} - 97000 \, x - 240625}} \end{array}$

b3 = 2*m*q*CC2*(y^2-1)/(BB2*(m^2*(y^2-1)*(p^2*BB2^2*(x^2-1)+4*q^2*d^2*CC2^2*(y^2-1))/(AA2*BB2*d^2)))
b = Sig.vector_field('beta', latex_name=r'\beta') 
b[3] = b3.simplify_full()
# unset components are zero 
b.display()

$\beta = \left( -\frac{400 \, {\left(2 \, \sqrt{3} \sqrt{2} x^{7} + 20 \, \sqrt{3} \sqrt{2} x^{6} + 24 \, {\left(\sqrt{3} \sqrt{2} x + 5 \, \sqrt{3} \sqrt{2}\right)} y^{6} - \sqrt{3} \sqrt{2} x^{5} - 25 \, \sqrt{3} \sqrt{2} x^{4} - 72 \, {\left(\sqrt{3} \sqrt{2} x + 5 \, \sqrt{3} \sqrt{2}\right)} y^{4} + 10 \, \sqrt{3} \sqrt{2} x^{2} - {\left(\sqrt{3} \sqrt{2} x^{5} + 15 \, \sqrt{3} \sqrt{2} x^{4} + 2 \, \sqrt{3} \sqrt{2} x^{3} - 10 \, \sqrt{3} \sqrt{2} x^{2} - 75 \, \sqrt{3} \sqrt{2} x - 365 \, \sqrt{3} \sqrt{2}\right)} y^{2} - 25 \, \sqrt{3} \sqrt{2} x - 125 \, \sqrt{3} \sqrt{2}\right)}}{x^{10} + 40 \, x^{9} + 576 \, {\left(x^{2} - 1\right)} y^{8} + 699 \, x^{8} + 7920 \, x^{7} + 96 \, {\left(x^{4} + 20 \, x^{3} + 174 \, x^{2} + 980 \, x + 2425\right)} y^{6} + 39450 \, x^{6} - 960 \, x^{5} - 48 \, {\left(3 \, x^{6} + 20 \, x^{5} - 3 \, x^{4} + 40 \, x^{3} + 925 \, x^{2} + 5940 \, x + 14675\right)} y^{4} - 39450 \, x^{4} - 6000 \, x^{3} + 96 \, {\left(x^{8} - x^{6} + 20 \, x^{5} - 20 \, x^{3} + 375 \, x^{2} + 3000 \, x + 7425\right)} y^{2} - 9675 \, x^{2} - 97000 \, x - 240625} \right) \frac{\partial}{\partial {\phi} }$

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

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

The component $K_{13} = K_{x\phi}$:

K[1,3]

$\frac{2 \, {\left(6 \, \sqrt{3} \sqrt{2} x^{16} - 13824 \, {\left(\sqrt{3} \sqrt{2} x^{2} + 10 \, \sqrt{3} \sqrt{2} x + \sqrt{3} \sqrt{2}\right)} y^{16} + 240 \, \sqrt{3} \sqrt{2} x^{15} + 3793 \, \sqrt{3} \sqrt{2} x^{14} - 6912 \, {\left(\sqrt{3} \sqrt{2} x^{4} + 20 \, \sqrt{3} \sqrt{2} x^{3} + 150 \, \sqrt{3} \sqrt{2} x^{2} + 500 \, \sqrt{3} \sqrt{2} x + 817 \, \sqrt{3} \sqrt{2}\right)} y^{14} + 27650 \, \sqrt{3} \sqrt{2} x^{13} + 72403 \, \sqrt{3} \sqrt{2} x^{12} + 576 \, {\left(27 \, \sqrt{3} \sqrt{2} x^{6} + 310 \, \sqrt{3} \sqrt{2} x^{5} + 1033 \, \sqrt{3} \sqrt{2} x^{4} + 1060 \, \sqrt{3} \sqrt{2} x^{3} + 10493 \, \sqrt{3} \sqrt{2} x^{2} + 44870 \, \sqrt{3} \sqrt{2} x + 69503 \, \sqrt{3} \sqrt{2}\right)} y^{12} - 81820 \, \sqrt{3} \sqrt{2} x^{11} - 374975 \, \sqrt{3} \sqrt{2} x^{10} - 96 \, {\left(109 \, \sqrt{3} \sqrt{2} x^{8} + 520 \, \sqrt{3} \sqrt{2} x^{7} + 1504 \, \sqrt{3} \sqrt{2} x^{6} + 19360 \, \sqrt{3} \sqrt{2} x^{5} + 92770 \, \sqrt{3} \sqrt{2} x^{4} + 157960 \, \sqrt{3} \sqrt{2} x^{3} + 148264 \, \sqrt{3} \sqrt{2} x^{2} + 731920 \, \sqrt{3} \sqrt{2} x + 1256425 \, \sqrt{3} \sqrt{2}\right)} y^{10} - 313810 \, \sqrt{3} \sqrt{2} x^{9} + 669975 \, \sqrt{3} \sqrt{2} x^{8} + 24 \, {\left(9 \, \sqrt{3} \sqrt{2} x^{10} + 250 \, \sqrt{3} \sqrt{2} x^{9} + 6873 \, \sqrt{3} \sqrt{2} x^{8} + 40920 \, \sqrt{3} \sqrt{2} x^{7} + 63402 \, \sqrt{3} \sqrt{2} x^{6} + 146220 \, \sqrt{3} \sqrt{2} x^{5} + 1047426 \, \sqrt{3} \sqrt{2} x^{4} + 2249400 \, \sqrt{3} \sqrt{2} x^{3} + 876525 \, \sqrt{3} \sqrt{2} x^{2} + 4308810 \, \sqrt{3} \sqrt{2} x + 8401925 \, \sqrt{3} \sqrt{2}\right)} y^{8} + 1617000 \, \sqrt{3} \sqrt{2} x^{7} + 999675 \, \sqrt{3} \sqrt{2} x^{6} + 96 \, {\left(20 \, \sqrt{3} \sqrt{2} x^{11} - 179 \, \sqrt{3} \sqrt{2} x^{10} - 50 \, \sqrt{3} \sqrt{2} x^{9} - 2897 \, \sqrt{3} \sqrt{2} x^{8} - 28400 \, \sqrt{3} \sqrt{2} x^{7} - 57446 \, \sqrt{3} \sqrt{2} x^{6} - 9020 \, \sqrt{3} \sqrt{2} x^{5} - 237650 \, \sqrt{3} \sqrt{2} x^{4} - 731060 \, \sqrt{3} \sqrt{2} x^{3} - 267175 \, \sqrt{3} \sqrt{2} x^{2} - 1037250 \, \sqrt{3} \sqrt{2} x - 2111325 \, \sqrt{3} \sqrt{2}\right)} y^{6} - 2277250 \, \sqrt{3} \sqrt{2} x^{5} - 4979375 \, \sqrt{3} \sqrt{2} x^{4} - {\left(187 \, \sqrt{3} \sqrt{2} x^{14} + 3590 \, \sqrt{3} \sqrt{2} x^{13} - 5207 \, \sqrt{3} \sqrt{2} x^{12} - 73540 \, \sqrt{3} \sqrt{2} x^{11} - 454637 \, \sqrt{3} \sqrt{2} x^{10} - 1150150 \, \sqrt{3} \sqrt{2} x^{9} + 199401 \, \sqrt{3} \sqrt{2} x^{8} - 1059000 \, \sqrt{3} \sqrt{2} x^{7} - 7811175 \, \sqrt{3} \sqrt{2} x^{6} + 2899610 \, \sqrt{3} \sqrt{2} x^{5} + 1675075 \, \sqrt{3} \sqrt{2} x^{4} - 32834500 \, \sqrt{3} \sqrt{2} x^{3} - 24681575 \, \sqrt{3} \sqrt{2} x^{2} - 69684250 \, \sqrt{3} \sqrt{2} x - 122823125 \, \sqrt{3} \sqrt{2}\right)} y^{4} - 4037500 \, \sqrt{3} \sqrt{2} x^{3} + 3461875 \, \sqrt{3} \sqrt{2} x^{2} - 6 \, {\left(\sqrt{3} \sqrt{2} x^{16} + 40 \, \sqrt{3} \sqrt{2} x^{15} + 601 \, \sqrt{3} \sqrt{2} x^{14} + 4010 \, \sqrt{3} \sqrt{2} x^{13} + 12935 \, \sqrt{3} \sqrt{2} x^{12} - 1060 \, \sqrt{3} \sqrt{2} x^{11} + 10449 \, \sqrt{3} \sqrt{2} x^{10} + 139590 \, \sqrt{3} \sqrt{2} x^{9} + 57825 \, \sqrt{3} \sqrt{2} x^{8} + 146960 \, \sqrt{3} \sqrt{2} x^{7} + 781475 \, \sqrt{3} \sqrt{2} x^{6} - 702250 \, \sqrt{3} \sqrt{2} x^{5} - 2108075 \, \sqrt{3} \sqrt{2} x^{4} - 348500 \, \sqrt{3} \sqrt{2} x^{3} + 2381875 \, \sqrt{3} \sqrt{2} x^{2} + 5456250 \, \sqrt{3} \sqrt{2} x + 6941250 \, \sqrt{3} \sqrt{2}\right)} y^{2} + 7231250 \, \sqrt{3} \sqrt{2} x + 6109375 \, \sqrt{3} \sqrt{2}\right)} \sqrt{x^{10} + 40 \, x^{9} + 576 \, {\left(x^{2} - 1\right)} y^{8} + 699 \, x^{8} + 7920 \, x^{7} + 96 \, {\left(x^{4} + 20 \, x^{3} + 174 \, x^{2} + 980 \, x + 2425\right)} y^{6} + 39450 \, x^{6} - 960 \, x^{5} - 48 \, {\left(3 \, x^{6} + 20 \, x^{5} - 3 \, x^{4} + 40 \, x^{3} + 925 \, x^{2} + 5940 \, x + 14675\right)} y^{4} - 39450 \, x^{4} - 6000 \, x^{3} + 96 \, {\left(x^{8} - x^{6} + 20 \, x^{5} - 20 \, x^{3} + 375 \, x^{2} + 3000 \, x + 7425\right)} y^{2} - 9675 \, x^{2} - 97000 \, x - 240625}}{{\left(x^{18} + 60 \, x^{17} + 331776 \, {\left(x^{2} - 1\right)} y^{16} + 1599 \, x^{16} + 25880 \, x^{15} + 110592 \, {\left(x^{4} + 15 \, x^{3} + 99 \, x^{2} + 485 \, x + 1200\right)} y^{14} + 266700 \, x^{14} + 1555560 \, x^{13} - 9216 \, {\left(17 \, x^{6} + 60 \, x^{5} - 417 \, x^{4} - 3040 \, x^{3} - 13425 \, x^{2} - 31020 \, x - 16975\right)} y^{12} + 3533300 \, x^{12} - 4005000 \, x^{11} + 9216 \, {\left(9 \, x^{8} - 60 \, x^{7} - 509 \, x^{6} - 2430 \, x^{5} - 9525 \, x^{4} - 24260 \, x^{3} - 71775 \, x^{2} - 227250 \, x - 290600\right)} y^{10} - 17787450 \, x^{10} - 18420000 \, x^{9} + 5760 \, {\left(7 \, x^{10} + 90 \, x^{9} + 473 \, x^{8} + 2460 \, x^{7} + 10050 \, x^{6} + 15200 \, x^{5} + 53790 \, x^{4} + 120900 \, x^{3} + 198455 \, x^{2} + 741350 \, x + 1103625\right)} y^{8} + 15656250 \, x^{8} + 31485000 \, x^{7} - 192 \, {\left(143 \, x^{12} + 675 \, x^{11} - 1043 \, x^{10} - 7575 \, x^{9} - 52650 \, x^{8} - 224850 \, x^{7} - 156150 \, x^{6} + 1001250 \, x^{5} + 3726075 \, x^{4} + 6217375 \, x^{3} + 4145625 \, x^{2} + 19413125 \, x + 33330000\right)} y^{6} + 3527500 \, x^{6} + 12975000 \, x^{5} + 96 \, {\left(93 \, x^{14} - 105 \, x^{13} - 1693 \, x^{12} - 13470 \, x^{11} - 99575 \, x^{10} - 222675 \, x^{9} - 149025 \, x^{8} - 1024500 \, x^{7} - 2270025 \, x^{6} + 2366625 \, x^{5} + 9545625 \, x^{4} + 11931250 \, x^{3} + 451875 \, x^{2} + 11346875 \, x + 28273125\right)} y^{4} + 80032500 \, x^{4} + 102025000 \, x^{3} + 192 \, {\left(x^{16} + 30 \, x^{15} + 399 \, x^{14} + 3955 \, x^{13} + 19950 \, x^{12} + 3765 \, x^{11} + 19850 \, x^{10} + 197000 \, x^{9} + 47025 \, x^{8} + 77000 \, x^{7} + 646875 \, x^{6} - 598125 \, x^{5} - 2642500 \, x^{4} - 2896875 \, x^{3} + 1117500 \, x^{2} + 1581250 \, x - 687500\right)} y^{2} - 78609375 \, x^{2} - 180937500 \, x - 150390625\right)} \sqrt{x^{8} + 576 \, y^{8} + 20 \, x^{7} + 96 \, {\left(x^{2} + 10 \, x + 25\right)} y^{6} + 100 \, x^{6} - 20 \, x^{5} - 48 \, {\left(3 \, x^{4} + 10 \, x^{3} + 30 \, x + 125\right)} y^{4} - 250 \, x^{4} - 500 \, x^{3} + 96 \, {\left(x^{6} + 10 \, x^{3} + 25\right)} y^{2} + 100 \, x^{2} + 500 \, x + 625} \sqrt{x + 1} \sqrt{x - 1}}$

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

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

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

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

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

The Ricci tensor associated with $\gamma$:

Ric = gam.ricci()
print Ric

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

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

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

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

3+1 Einstein equations

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

We start by the constraint equations:

Hamiltonian constraint

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

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

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

a052a2f-06d2-45a2-89c6-bf636d976684i︠
%html
<p>Hence the Hamiltonian constraint is satisfied.</p>

<h3>Momentum constraint</h3>
<p>In vaccum, the momentum constraint is \[ D_j K^j_{\ \, i} - D_i K = 0 \]</p>

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

1166208-11ab-4fcd-950d-093b3c394055i︠
%html
<p>Hence the momentum constraint is satisfied.</p>

<h3>Dynamical Einstein equations</h3>
<p>Let us first evaluate the symmetric bilinear form $k_{ij} := K_{ik} K^k_{\ \, j}$:</p>

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'

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

Hence the dynamical Einstein equations are satisfied.

Finally we have checked that all the 3+1 Einstein equations are satisfied by the $\delta=2$ Tomimatsu-Sato solution.