{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 3+1 Einstein equations in the $\\delta=2$ Tomimatsu-Sato spacetime\n", "\n", "This worksheet demonstrates a few capabilities of SageMath in computations regarding the 3+1 slicing of the $\\delta=2$ Tomimatsu-Sato spacetime. The corresponding tools have been developed within the [SageManifolds](http://sagemanifolds.obspm.fr) project (version 1.2, as included in SageMath 8.2).\n", "\n", "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v1.2/SM_Tomimatsu-Sato_3p1.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, via the command `sage -n jupyter`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*NB:* a version of SageMath at least equal to 8.2 is required to run this worksheet:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 8.2, Release Date: 2018-05-05'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since some computations are quite long, we ask for running them in parallel on 8 cores:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Tomimatsu-Sato spacetime

\n", "

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.

\n", "

Spacelike hypersurface

\n", "

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 Riemannian manifold:

" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "Sig = Manifold(3, 'Sigma', latex_name=r'\\Sigma', structure='Riemannian',\n", " metric_name='gam', metric_latex_name=r'\\gamma',\n", " start_index=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (Sigma, (x, y, ph))\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Chart (Sigma, (x, y, ph))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X. = Sig.chart(r'x:(1,+oo) y:(-1,1) ph:(0,2*pi):\\phi')\n", "print(X) ; X" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Riemannian metric on $\\Sigma$

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[x is real,\n", " x > 1,\n", " y is real,\n", " y > -1,\n", " y < 1,\n", " ph is real,\n", " ph > 0,\n", " ph < 2*pi,\n", " m > 0]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var('d, p, m')\n", "assume(m>0)\n", "assumptions()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "d = 2\n", "p = 1/5" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "m = 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "q = sqrt(1-p^2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Some shortcut notations:

" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "AA2 = (p^2*(x^2-1)^2+q^2*(1-y^2)^2)^2 \\\n", " - 4*p^2*q^2*(x^2-1)*(1-y^2)*(x^2-y^2)^2\n", "BB2 = (p^2*x^4+2*p*x^3-2*p*x+q^2*y^4-1)^2 \\\n", " + 4*q^2*y^2*(p*x^3-p*x*y^2-y^2+1)^2\n", "CC2 = p^3*x*(1-x^2)*(2*(x^4-1)+(x^2+3)*(1-y^2)) \\\n", " + p^2*(x^2-1)*((x^2-1)*(1-y^2)-4*x^2*(x^2-y^2)) \\\n", " + q^2*(1-y^2)^3*(p*x+1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "gam = 1/100*(96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)/((x^2 - y^2)^3*(x^2 - 1)) dx*dx - 1/100*(96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)/((x^2 - y^2)^3*(y^2 - 1)) dy*dy - 1/100*((96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)^2*(x^2 - 1) + 9600*(24*(y^2 - 1)^3*(x + 5) + (2*x^4 - (x^2 + 3)*(y^2 - 1) - 2)*(x^2 - 1)*x + 5*(4*(x^2 - y^2)*x^2 + (x^2 - 1)*(y^2 - 1))*(x^2 - 1))^2*(y^2 - 1))*(y^2 - 1)/((96*(x^2 - y^2)^2*(x^2 - 1)*(y^2 - 1) + ((x^2 - 1)^2 + 24*(y^2 - 1)^2)^2)*(96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)) dph*dph" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "gam = Sig.metric()\n", "gam[1,1] = m^2*BB2/(p^2*d^2*(x^2-1)*(x^2-y^2)^3)\n", "gam[2,2] = m^2*BB2/(p^2*d^2*(y^2-1)*(-x^2+y^2)^3)\n", "gam[3,3] = - m^2*(y^2-1)*(p^2*BB2^2*(x^2-1)\n", " + 4*q^2*d^2*CC2^2*(y^2-1))/(AA2*BB2*d^2)\n", "gam.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A view of the non-vanishing components of $\\gamma$ w.r.t. coordinates $(x,y,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "gam_x,x = 1/100*(96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)/((x^2 - y^2)^3*(x^2 - 1)) \n", "gam_y,y = -1/100*(96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)/((x^2 - y^2)^3*(y^2 - 1)) \n", "gam_ph,ph = -1/100*((96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)^2*(x^2 - 1) + 9600*(24*(y^2 - 1)^3*(x + 5) + (2*x^4 - (x^2 + 3)*(y^2 - 1) - 2)*(x^2 - 1)*x + 5*(4*(x^2 - y^2)*x^2 + (x^2 - 1)*(y^2 - 1))*(x^2 - 1))^2*(y^2 - 1))*(y^2 - 1)/((96*(x^2 - y^2)^2*(x^2 - 1)*(y^2 - 1) + ((x^2 - 1)^2 + 24*(y^2 - 1)^2)^2)*(96*(x^3 - x*y^2 - 5*y^2 + 5)^2*y^2 + (x^4 + 24*y^4 + 10*x^3 - 10*x - 25)^2)) " ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "gam.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Lapse function and shift vector

" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(x^10 + 20*x^9 + 576*(x^2 - 1)*y^8 + 99*x^8 - 40*x^7 + 96*(x^4 + 10*x^3 + 24*x^2 - 10*x - 25)*y^6 - 350*x^6 - 480*x^5 - 48*(3*x^6 + 10*x^5 - 3*x^4 + 20*x^3 + 125*x^2 - 30*x - 125)*y^4 + 350*x^4 + 1000*x^3 + 96*(x^8 - x^6 + 10*x^5 - 10*x^3 + 25*x^2 - 25)*y^2 + 525*x^2 - 500*x - 625)/(x^10 + 40*x^9 + 576*(x^2 - 1)*y^8 + 699*x^8 + 7920*x^7 + 96*(x^4 + 20*x^3 + 174*x^2 + 980*x + 2425)*y^6 + 39450*x^6 - 960*x^5 - 48*(3*x^6 + 20*x^5 - 3*x^4 + 40*x^3 + 925*x^2 + 5940*x + 14675)*y^4 - 39450*x^4 - 6000*x^3 + 96*(x^8 - x^6 + 20*x^5 - 20*x^3 + 375*x^2 + 3000*x + 7425)*y^2 - 9675*x^2 - 97000*x - 240625)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N2 = AA2/BB2 - 2*m*q*CC2*(y^2-1)/BB2*(2*m*q*CC2*(y^2-1)\n", " /(BB2*(m^2*(y^2-1)*(p^2*BB2^2*(x^2-1)\n", " +4*q^2*d^2*CC2^2*(y^2-1))/(AA2*BB2*d^2)))) \n", "N2.simplify_full()" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field N on the 3-dimensional Riemannian manifold Sigma\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "N: Sigma --> R\n", " (x, y, ph) |--> sqrt((x^10 + 20*x^9 + 576*(x^2 - 1)*y^8 + 99*x^8 - 40*x^7 + 96*(x^4 + 10*x^3 + 24*x^2 - 10*x - 25)*y^6 - 350*x^6 - 480*x^5 - 48*(3*x^6 + 10*x^5 - 3*x^4 + 20*x^3 + 125*x^2 - 30*x - 125)*y^4 + 350*x^4 + 1000*x^3 + 96*(x^8 - x^6 + 10*x^5 - 10*x^3 + 25*x^2 - 25)*y^2 + 525*x^2 - 500*x - 625)/(x^10 + 40*x^9 + 576*(x^2 - 1)*y^8 + 699*x^8 + 7920*x^7 + 96*(x^4 + 20*x^3 + 174*x^2 + 980*x + 2425)*y^6 + 39450*x^6 - 960*x^5 - 48*(3*x^6 + 20*x^5 - 3*x^4 + 40*x^3 + 925*x^2 + 5940*x + 14675)*y^4 - 39450*x^4 - 6000*x^3 + 96*(x^8 - x^6 + 20*x^5 - 20*x^3 + 375*x^2 + 3000*x + 7425)*y^2 - 9675*x^2 - 97000*x - 240625))" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = Sig.scalar_field(sqrt(N2.simplify_full()), name='N')\n", "print(N)\n", "N.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinate expression of the scalar field $N$:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "sqrt((x^10 + 20*x^9 + 576*(x^2 - 1)*y^8 + 99*x^8 - 40*x^7 + 96*(x^4 + 10*x^3 + 24*x^2 - 10*x - 25)*y^6 - 350*x^6 - 480*x^5 - 48*(3*x^6 + 10*x^5 - 3*x^4 + 20*x^3 + 125*x^2 - 30*x - 125)*y^4 + 350*x^4 + 1000*x^3 + 96*(x^8 - x^6 + 10*x^5 - 10*x^3 + 25*x^2 - 25)*y^2 + 525*x^2 - 500*x - 625)/(x^10 + 40*x^9 + 576*(x^2 - 1)*y^8 + 699*x^8 + 7920*x^7 + 96*(x^4 + 20*x^3 + 174*x^2 + 980*x + 2425)*y^6 + 39450*x^6 - 960*x^5 - 48*(3*x^6 + 20*x^5 - 3*x^4 + 40*x^3 + 925*x^2 + 5940*x + 14675)*y^4 - 39450*x^4 - 6000*x^3 + 96*(x^8 - x^6 + 20*x^5 - 20*x^3 + 375*x^2 + 3000*x + 7425)*y^2 - 9675*x^2 - 97000*x - 240625))" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N.expr()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "beta^x = 0 \n", "beta^y = 0 \n", "beta^ph = -400*(2*sqrt(6)*x^7 + 24*(sqrt(6)*x + 5*sqrt(6))*y^6 + 20*sqrt(6)*x^6 - sqrt(6)*x^5 - 72*(sqrt(6)*x + 5*sqrt(6))*y^4 - 25*sqrt(6)*x^4 - (sqrt(6)*x^5 + 15*sqrt(6)*x^4 + 2*sqrt(6)*x^3 - 10*sqrt(6)*x^2 - 75*sqrt(6)*x - 365*sqrt(6))*y^2 + 10*sqrt(6)*x^2 - 25*sqrt(6)*x - 125*sqrt(6))/(x^10 + 40*x^9 + 576*(x^2 - 1)*y^8 + 699*x^8 + 7920*x^7 + 96*(x^4 + 20*x^3 + 174*x^2 + 980*x + 2425)*y^6 + 39450*x^6 - 960*x^5 - 48*(3*x^6 + 20*x^5 - 3*x^4 + 40*x^3 + 925*x^2 + 5940*x + 14675)*y^4 - 39450*x^4 - 6000*x^3 + 96*(x^8 - x^6 + 20*x^5 - 20*x^3 + 375*x^2 + 3000*x + 7425)*y^2 - 9675*x^2 - 97000*x - 240625) " ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b3 = 2*m*q*CC2*(y^2-1)/(BB2*(m^2*(y^2-1)*(p^2*BB2^2*(x^2-1)\n", " +4*q^2*d^2*CC2^2*(y^2-1))/(AA2*BB2*d^2)))\n", "b = Sig.vector_field('beta', latex_name=r'\\beta') \n", "b[3] = b3.simplify_full()\n", "# unset components are zero \n", "b.display_comp(only_nonzero=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Extrinsic curvature of $\\Sigma$

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms K on the 3-dimensional Riemannian manifold Sigma\n" ] } ], "source": [ "K = gam.lie_derivative(b) / (2*N)\n", "K.set_name('K')\n", "print(K)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "2*(6*sqrt(3)*sqrt(2)*x^16 - 13824*(sqrt(3)*sqrt(2)*x^2 + 10*sqrt(3)*sqrt(2)*x + sqrt(3)*sqrt(2))*y^16 + 240*sqrt(3)*sqrt(2)*x^15 + 3793*sqrt(3)*sqrt(2)*x^14 - 6912*(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))*y^14 + 27650*sqrt(3)*sqrt(2)*x^13 + 72403*sqrt(3)*sqrt(2)*x^12 + 576*(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))*y^12 - 81820*sqrt(3)*sqrt(2)*x^11 - 374975*sqrt(3)*sqrt(2)*x^10 - 96*(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))*y^10 - 313810*sqrt(3)*sqrt(2)*x^9 + 669975*sqrt(3)*sqrt(2)*x^8 + 24*(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))*y^8 + 1617000*sqrt(3)*sqrt(2)*x^7 + 999675*sqrt(3)*sqrt(2)*x^6 + 96*(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))*y^6 - 2277250*sqrt(3)*sqrt(2)*x^5 - 4979375*sqrt(3)*sqrt(2)*x^4 - (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))*y^4 - 4037500*sqrt(3)*sqrt(2)*x^3 + 3461875*sqrt(3)*sqrt(2)*x^2 - 6*(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))*y^2 + 7231250*sqrt(3)*sqrt(2)*x + 6109375*sqrt(3)*sqrt(2))*sqrt(abs(x^10 + 40*x^9 + 576*(x^2 - 1)*y^8 + 699*x^8 + 7920*x^7 + 96*(x^4 + 20*x^3 + 174*x^2 + 980*x + 2425)*y^6 + 39450*x^6 - 960*x^5 - 48*(3*x^6 + 20*x^5 - 3*x^4 + 40*x^3 + 925*x^2 + 5940*x + 14675)*y^4 - 39450*x^4 - 6000*x^3 + 96*(x^8 - x^6 + 20*x^5 - 20*x^3 + 375*x^2 + 3000*x + 7425)*y^2 - 9675*x^2 - 97000*x - 240625))/((x^18 + 60*x^17 + 331776*(x^2 - 1)*y^16 + 1599*x^16 + 25880*x^15 + 110592*(x^4 + 15*x^3 + 99*x^2 + 485*x + 1200)*y^14 + 266700*x^14 + 1555560*x^13 - 9216*(17*x^6 + 60*x^5 - 417*x^4 - 3040*x^3 - 13425*x^2 - 31020*x - 16975)*y^12 + 3533300*x^12 - 4005000*x^11 + 9216*(9*x^8 - 60*x^7 - 509*x^6 - 2430*x^5 - 9525*x^4 - 24260*x^3 - 71775*x^2 - 227250*x - 290600)*y^10 - 17787450*x^10 - 18420000*x^9 + 5760*(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)*y^8 + 15656250*x^8 + 31485000*x^7 - 192*(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)*y^6 + 3527500*x^6 + 12975000*x^5 + 96*(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)*y^4 + 80032500*x^4 + 102025000*x^3 + 192*(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)*y^2 - 78609375*x^2 - 180937500*x - 150390625)*sqrt(x^8 + 576*y^8 + 20*x^7 + 96*(x^2 + 10*x + 25)*y^6 + 100*x^6 - 20*x^5 - 48*(3*x^4 + 10*x^3 + 30*x + 125)*y^4 - 250*x^4 - 500*x^3 + 96*(x^6 + 10*x^3 + 25)*y^2 + 100*x^2 + 500*x + 625)*sqrt(x + 1)*sqrt(x - 1))" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K[1,3]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (1,1) on the 3-dimensional Riemannian manifold Sigma\n" ] } ], "source": [ "Ku = K.up(gam, 0)\n", "print(Ku)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Scalar field zero on the 3-dimensional Riemannian manifold Sigma" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "trK = Ku.trace()\n", "trK" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Connection and curvature

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection D associated with the Riemannian metric gam on the 3-dimensional Riemannian manifold Sigma\n" ] } ], "source": [ "D = gam.connection(name='D')\n", "print(D)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

The Ricci tensor associated with $\\gamma$:

" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(gam) on the 3-dimensional Riemannian manifold Sigma\n" ] } ], "source": [ "Ric = gam.ricci()\n", "print(Ric)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field R on the 3-dimensional Riemannian manifold Sigma\n" ] } ], "source": [ "R = gam.ricci_scalar(name='R')\n", "print(R)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinate expression of the Ricci scalar is huge:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "480000*(36*x^38 + 2880*x^37 - 191102976*(3*x^4 + 20*x^3 - 6*x^2 - 60*x - 101)*y^36 + 103116*x^36 + 2152440*x^35 + 382205952*(5*x^6 + 45*x^5 + 195*x^4 + 1210*x^3 + 3555*x^2 + 4065*x - 291)*y^34 + 28527685*x^34 + 243524500*x^33 - 63700992*(45*x^8 + 605*x^7 + 4434*x^6 + 21165*x^5 + 36233*x^4 - 28965*x^3 - 244428*x^2 - 457885*x - 503220)*y^32 + 1269998358*x^32 + 3199445660*x^31 + 10616832*(279*x^10 + 4375*x^9 + 30087*x^8 + 71400*x^7 - 238782*x^6 - 1698210*x^5 - 6419238*x^4 - 18093840*x^3 - 36663561*x^2 - 47421645*x - 39429937)*y^30 - 2269601041*x^30 - 34623715080*x^29 - 331776*(5729*x^12 + 49180*x^11 - 74502*x^10 - 3734980*x^9 - 26371593*x^8 - 109948680*x^7 - 472804404*x^6 - 1606840680*x^5 - 3848737185*x^4 - 6740837780*x^3 - 9099798310*x^2 - 9444811860*x - 7651340375)*y^28 - 59087224000*x^28 + 65688034640*x^27 - 110592*(2649*x^14 + 238580*x^13 + 3051495*x^12 + 19491840*x^11 + 91830265*x^10 + 456692580*x^9 + 2183925951*x^8 + 7465185120*x^7 + 19582237971*x^6 + 43189401660*x^5 + 79002052285*x^4 + 114585899040*x^3 + 122428856475*x^2 + 101684757740*x + 84896596125)*y^26 + 384825320925*x^26 + 395426661500*x^25 + 9216*(181479*x^16 + 4048360*x^15 + 34156308*x^14 + 188537160*x^13 + 1036205812*x^12 + 5987029560*x^11 + 26294717532*x^10 + 87261273880*x^9 + 261260973090*x^8 + 690772283160*x^7 + 1471695406220*x^6 + 2533618367160*x^5 + 3691666652100*x^4 + 4791547622600*x^3 + 4502416196100*x^2 + 2923338273000*x + 2562270874175)*y^24 - 689419898350*x^24 - 2430744577500*x^23 - 9216*(146361*x^18 + 2352670*x^17 + 16966899*x^16 + 120818880*x^15 + 809501212*x^14 + 4167280920*x^13 + 15760668420*x^12 + 56142463840*x^11 + 200275537254*x^10 + 588282675780*x^9 + 1405136391410*x^8 + 2938902756480*x^7 + 5350560752700*x^6 + 7979584965560*x^5 + 9571504935300*x^4 + 11338546428000*x^3 + 10122365599625*x^2 + 5038926732750*x + 4632320521875)*y^22 - 1857563145625*x^22 + 3726507914000*x^21 + 576*(872259*x^20 + 10405140*x^19 + 82612818*x^18 + 902230980*x^17 + 5601145999*x^16 + 20778680080*x^15 + 69911685912*x^14 + 343436041360*x^13 + 1497770805142*x^12 + 4970942130840*x^11 + 14590852687852*x^10 + 37040501042680*x^9 + 75185699557350*x^8 + 126956136629840*x^7 + 200548351151000*x^6 + 280689612306000*x^5 + 289604848243175*x^4 + 307553520288500*x^3 + 274894209791250*x^2 + 104726374470500*x + 99270616476875)*y^20 + 10725964057500*x^20 + 7153101875000*x^19 - 1152*(25696*x^22 - 631175*x^21 - 161828*x^20 + 43569130*x^19 - 147524592*x^18 - 2734696695*x^17 - 8429504836*x^16 + 8540896280*x^15 + 142172247840*x^14 + 800517208530*x^13 + 3285944502360*x^12 + 9449079586780*x^11 + 21612057340800*x^10 + 45038295323450*x^9 + 82051701857400*x^8 + 117838236487000*x^7 + 161281016060800*x^6 + 224621201119125*x^5 + 211864359447500*x^4 + 193342988674250*x^3 + 177597398470000*x^2 + 52756443228125*x + 49969255607500)*y^18 - 13398254925625*x^18 - 34661922712500*x^17 - 48*(1317855*x^24 + 35094860*x^23 + 301540716*x^22 + 3001862540*x^21 + 23519567706*x^20 + 106393862660*x^19 + 311028711012*x^18 + 719292194340*x^17 + 1077335948157*x^16 - 2382833534280*x^15 - 20289271710280*x^14 - 74864124423240*x^13 - 228136339665700*x^12 - 553418865892920*x^11 - 1008839074137400*x^10 - 1647179273211000*x^9 - 2848515077153175*x^8 - 3846168576082500*x^7 - 4546922858182500*x^6 - 6417482452730500*x^5 - 5915475415953750*x^4 - 4443051200687500*x^3 - 4189051440907500*x^2 - 1001111910587500*x - 909098726953125)*y^16 - 22308384343750*x^16 + 31657039062500*x^15 + 48*(826719*x^26 + 23430200*x^25 + 272358873*x^24 + 2539133640*x^23 + 14381842762*x^22 + 50227933080*x^21 + 147297993214*x^20 + 463374453880*x^19 + 1166546488285*x^18 + 1655137640160*x^17 - 793319453005*x^16 - 18257108470320*x^15 - 75659008152900*x^14 - 188155382315600*x^13 - 390580154803500*x^12 - 800760869162000*x^11 - 1281957887842775*x^10 - 1605940798661000*x^9 - 2752224360140625*x^8 - 4005975929887000*x^7 - 4199432221443750*x^6 - 5753278142925000*x^5 - 5450901493411250*x^4 - 3307170487425000*x^3 - 3112402899453125*x^2 - 615820486250000*x - 513017261046875)*y^14 + 81676588453125*x^14 + 58656861875000*x^13 - (16632123*x^28 + 499946740*x^27 + 6196259958*x^26 + 48354916500*x^25 + 210671691233*x^24 + 676264764600*x^23 + 2480527625628*x^22 + 8835658984280*x^21 + 23129361505347*x^20 + 48962891419740*x^19 + 67475269064330*x^18 - 64483145981220*x^17 - 634130008786575*x^16 - 2378356273612720*x^15 - 6843388543399800*x^14 - 13403795491878000*x^13 - 17968166150880775*x^12 - 29491543705444500*x^11 - 49841887049313750*x^10 - 51645998160896500*x^9 - 87332506037923125*x^8 - 154714164855525000*x^7 - 155450250997622500*x^6 - 188899684974225000*x^5 - 186644230989984375*x^4 - 92578967313437500*x^3 - 82813950742406250*x^2 - 13688637908437500*x - 10052326473828125)*y^12 - 38894490625000*x^12 - 127794000000000*x^11 + (5287413*x^30 + 153553860*x^29 + 1732590387*x^28 + 10983605840*x^27 + 41705752809*x^26 + 182167871700*x^25 + 742487990119*x^24 + 1812065771520*x^23 + 4180994474577*x^22 + 16676823221140*x^21 + 45941187607623*x^20 + 60767412776400*x^19 - 46828949980475*x^18 - 541654529965980*x^17 - 1641522681341925*x^16 - 3216999939440000*x^15 - 6214406047779825*x^14 - 10658201858008500*x^13 - 8192115508664375*x^12 - 7387864891722000*x^11 - 24095715523603125*x^10 - 28509120447162500*x^9 - 43068217775386875*x^8 - 93542930451600000*x^7 - 99595604072703125*x^6 - 100441157954062500*x^5 - 100220408582671875*x^4 - 41492192986250000*x^3 - 33314791784765625*x^2 - 4477008820312500*x - 2805643465234375)*y^10 - 107249503515625*x^10 + 734164062500*x^9 - (1235529*x^32 + 28614420*x^31 + 242301933*x^30 + 1525335720*x^29 + 8566281302*x^28 + 47434915320*x^27 + 106842591351*x^26 - 41497935380*x^25 + 156438129780*x^24 + 3581296064220*x^23 + 12192876368965*x^22 + 29274331168800*x^21 + 45233459539650*x^20 - 12021024328400*x^19 - 270597672490425*x^18 - 896327607811500*x^17 - 1938183310470350*x^16 - 2243010663184500*x^15 - 1658353427210625*x^14 - 2945595506411000*x^13 - 265352800083750*x^12 + 4005477027675000*x^11 - 5626322615511875*x^10 - 13997280486187500*x^9 - 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3+1 Einstein equations

\n", "

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

\n", "

We start by the constraint equations:

\n", "

Hamiltonian constraint

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field on the 3-dimensional Riemannian manifold Sigma\n" ] } ], "source": [ "Kuu = Ku.up(gam, 1)\n", "trKK = K['_ij']*Kuu['^ij']\n", "print(trKK)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field on the 3-dimensional Riemannian manifold Sigma\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Sigma --> R\n", "(x, y, ph) |--> 480000*(36*x^48 + 4320*x^47 + 243480*x^46 + 8575320*x^45 - 110075314176*(3*x^6 + 20*x^5 - 9*x^4 - 80*x^3 - 95*x^2 + 60*x + 101)*y^44 + 210933169*x^44 + 3819447620*x^43 + 18345885696*(57*x^8 + 460*x^7 + 1364*x^6 + 7740*x^5 + 15790*x^4 + 4100*x^3 + 44772*x^2 + 195700*x + 248417)*y^42 + 52062896173*x^42 + 534863205320*x^41 - 9172942848*(151*x^10 + 1720*x^9 + 9323*x^8 + 11120*x^7 - 346534*x^6 - 2578480*x^5 - 10710838*x^4 - 30392560*x^3 - 50713057*x^2 - 34977800*x + 6317755)*y^40 + 4061416393951*x^40 + 21663482892380*x^39 + 6115295232*(180*x^12 + 2105*x^11 + 1061*x^10 - 214915*x^9 - 2324973*x^8 - 12436470*x^7 - 45071934*x^6 - 102072870*x^5 - 78278550*x^4 + 241992285*x^3 + 898906729*x^2 + 1479417865*x + 1274799487)*y^38 + 71037156892391*x^38 + 70714187108320*x^37 - 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1651873588562011718750*x - 15549421310424804687500)*y^2 - 20104440701007843017578125*x^2 - 12276974916458129882812500*x - 3401435911655426025390625)" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ham = R + trK^2 - trKK\n", "print(Ham)\n", "Ham.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Hence the Hamiltonian constraint is satisfied.

\n", "\n", "

Momentum constraint

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form on the 3-dimensional Riemannian manifold Sigma\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "0" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mom = D(Ku).trace(0,2) - D(trK)\n", "print(mom)\n", "mom.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Hence the momentum constraint is satisfied.

\n", "\n", "

Dynamical Einstein equations

\n", "

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

" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field of type (0,2) on the 3-dimensional Riemannian manifold Sigma\n" ] } ], "source": [ "KK = K['_ik']*Ku['^k_j']\n", "print(KK)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "KK1 = KK.symmetrize()\n", "KK == KK1" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "KK = KK1\n", "print(KK)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

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

" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "dyn = K.lie_derivative(b) - D(D(N)) + N*( Ric + trK*K - 2*KK )\n", "print(dyn)\n", "dyn.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Hence the dynamical Einstein equations are satisfied.

\n", "\n", "

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

" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.2", "language": "", "name": "sagemath" }, "language": "python", "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 1 }