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3+1 Simon-Mars tensor in the $\delta=2$ Tomimatsu-Sato spacetime
This is a SageManifolds (version 0.7) worksheet implementing the computation of the 3+1 decomposition of the Simon-Mars tensor in the $\delta=2$ Tomimatsu-Sato spacetime. The results obtained here are used 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.
Tomimatsu-Sato spacetime
The Tomimatsu-Sato metric is an exact stationary and axisymmetric solution of the vacuum 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:
On $\Sigma$, we consider the prolate spheroidal coordinates $(x,y,\phi)$, with $x\in(1,+\infty)$, $y\in(-1,1)$ and $\phi\in(0,2\pi)$ :
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 $m$:
We set $\delta=2$ and choose a specific value for $p$, namely $p=1/5$:
Furthermore, without any loss of generality, we may set $m=1$ (this simply fixes some length scale):
The parameter $q$ is related to $p$ by $p^2+q^2=1$:
Some shortcut notations:
The Riemannian metric $\gamma$ induced by the spacetime metric $g$ on $\Sigma$:
A matrix view of the components w.r.t. coordinates $(x,y,\phi)$:
Lapse function and shift vector
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:
The component $K_{13} = K_{x\phi}$:
The type-(1,1) tensor $K^\sharp$ of components $K^i_{\ \, j} = \gamma^{ik} K_{kj}$:
We may check that the hypersurface $\Sigma$ is maximal, i.e. that $K^k_{\ \, k} = 0$:
Connection and curvature
Let us call $D$ the Levi-Civita connection associated with $\gamma$:
The Ricci tensor associated with $\gamma$:
The scalar curvature $R = \gamma^{ij} R_{ij}$:
Terms related to the extrinsic curvature
Let us first evaluate the term $K_{ij} K^{ij}$:
Then we compute the symmetric bilinear form $k_{ij} := K_{ik} K^k_{\ \, j}$:
We check that this tensor field is symmetric:
Accordingly, we work with the explicitly symmetric version:
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Electric and magnetic parts of the Weyl tensor
The electric part is the bilinear form $E$ given by \[ E_{ij} = R_{ij} + K K_{ij} - K_{ik} K^k_{\ \, j} \]
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):
Let us check that $B$ is symmetric:
Accordingly, we set
3+1 decomposition of the Simon-Mars tensor
We proceed according to the computation presented in arXiv:1412.6542.
Tensor $E^\sharp$ of components $E^i_ {\ \, j}$:
Tensor $B^\sharp$ of components $B^i_{\ \, j}$:
1-form $\beta^\flat$ of components $\beta_i$ and its exterior derivative:
Scalar square of shift $\beta_i \beta^i$:
Scalar $Y = E(\beta,\beta) = E_{ij} \beta^i \beta^j$:
Scalar $\bar Y = B(\beta,\beta) = B_{ij}\beta^i \beta^j$:
1-form of components $Eb_i = E_{ij} \beta^j$:
Vector field of components $Eub^i = E^i_{\ \, j} \beta^j$:
1-form of components $Bb_i = B_{ij} \beta^j$:
Vector field of components $Bub^i = B^i_{\ \, j} \beta^j$:
Vector field of components $Kub^i = K^i_{\ \, j} \beta^j$:
Simon-Mars scalars
Simon-Mars scalars expressed in terms of the coordinates $X=-1/x,y$:
Definition of the ergoregion:
Due to the very high degree of the polynomials involved in the expression of the Simon-Mars scalars, the floating-point precision of Sage's contour_plot function (53 bits) is not sufficient. Taking avantage that Sage is open-source, we modify the function to allow for an arbitrary precision. First, we define a sampling function with a floating-point precision specified by the user (argument precis):Â
Then we redefine contour_plot so that it uses the sampling function with a floating-point precision of 200 bits:
Then we are able to draw the contour plot of the two Simon-Mars scalars, in terms of the coordinates $(X,y)$ (Figure 11 of arXiv:1412.6542):
Let us do the same in terms of the Weyl-Lewis-Papapetrou cylindrical coordinates $(\rho,z)$, which are related to the prolate spheroidal coordinates $(x,y)$ by \[ \rho = \sqrt{(x^2-1)(1-y^2)}Â \quad\mbox{and}\quad z=xy . \]Â
For simplicity, we denote $\rho$ by $r$:
3D plots
Contour plots of the two Simon-Mars scalar fields in terms of coordinates $(\rho,z)$ (Figure 12 of arXiv:1412.6542)