{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"# 5D Kerr-AdS spacetime with a Nambu-Goto string\n",
"\n",
"## Case a = b with global AdS coordinates"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"This [SageMath](https://www.sagemath.org/) notebook is relative to the article *Holographic drag force in 5d Kerr-AdS black hole* by Irina Ya. Aref'eva, Anastasia A. Golubtsova and Eric Gourgoulhon, [arXiv:2004.12984](https://arxiv.org/abs/2004.12984).\n",
"\n",
"The involved differential geometry computations are based on tools developed through the [SageManifolds](https://sagemanifolds.obspm.fr) project."
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"*NB:* a version of SageMath at least equal to 8.2 is required to run this notebook:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/plain": [
"'SageMath version 9.3, Release Date: 2021-05-09'"
]
},
"execution_count": 1,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"version()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"First we set up the notebook to display mathematical objects using LaTeX rendering:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"%display latex"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"Since some computations are quite long, we ask for running them in parallel on 8 cores:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"Parallelism().set(nproc=1) # only nproc=1 works on CoCalc"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"## Spacetime manifold\n",
"\n",
"We declare the Kerr-AdS spacetime as a 5-dimensional Lorentzian manifold:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"5-dimensional Lorentzian manifold M\n"
]
}
],
"source": [
"M = Manifold(5, 'M', r'\\mathcal{M}', structure='Lorentzian', metric_name='G')\n",
"print(M)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"Let us define **Boyer-Lindquist-type coordinates (rational polynomial version)** on $\\mathcal{M}$, via the method `chart()`, the argument of which is a string expressing the coordinates names, their ranges (the default is $(-\\infty,+\\infty)$) and their LaTeX symbols:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\mu}, {\\phi}, {\\psi})\\right)$$"
],
"text/plain": [
"Chart (M, (t, r, mu, ph, ps))"
]
},
"execution_count": 5,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"BL. = M.chart(r't r:(0,+oo) mu:(0,1):\\mu ph:(0,2*pi):\\phi ps:(0,2*pi):\\psi')\n",
"BL"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"The coordinate $\\mu$ is related to the standard Boyer-Lindquist coordinate $\\theta$ by\n",
"$$ \\mu = \\cos\\theta$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The coordinate ranges are"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$"
],
"text/plain": [
"t: (-oo, +oo); r: (0, +oo); mu: (0, 1); ph: (0, 2*pi); ps: (0, 2*pi)"
]
},
"execution_count": 6,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"BL.coord_range()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Note that contrary to the 4-dimensional case, the range of $\\mu$ is $(0,1)$ only (cf. Sec. 1.2 of [R.C. Myers, arXiv:1111.1903](https://arxiv.org/abs/1111.1903) or Sec. 2 of [G.W. Gibbons, H. Lüb, Don N. Page, C.N. Pope, J. Geom. Phys. **53**, 49 (2005)](https://doi.org/10.1016/j.geomphys.2004.05.001)). In other words, the range of $\\theta$ is $\\left(0, \\frac{\\pi}{2}\\right)$ only. "
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"## Metric tensor\n",
"\n",
"The 4 parameters $m$, $a$, $b$ and $\\ell$ of the Kerr-AdS spacetime are declared as symbolic variables, $a$ and $b$ being the two angular momentum parameters and $\\ell$ being related to the cosmological constant by $\\Lambda = - 6 \\ell^2$:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a, b\\right)$$"
],
"text/plain": [
"(m, a, b)"
]
},
"execution_count": 7,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"var('m a b', domain='real')"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ell}$$"
],
"text/plain": [
"l"
]
},
"execution_count": 8,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"var('l', domain='real', latex_name=r'\\ell')"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"# Particular cases\n",
"# m = 0\n",
"# a = 0\n",
"# b = 0\n",
"b = a"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"assume(a > 0)\n",
"assume(1 - a^2*l^2 > 0)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"Some auxiliary functions:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"keep_Delta = False # change to False to provide explicit expression for Delta_r, Xi_a, etc..."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"sig = (1 + r^2*l^2)/r^2\n",
"costh2 = mu^2\n",
"sinth2 = 1 - mu^2\n",
"rho2 = r^2 + a^2*mu^2 + b^2*sinth2\n",
"if keep_Delta:\n",
" Delta_r = var('Delta_r', latex_name=r'\\Delta_r', domain='real')\n",
" Delta_th = var('Delta_th', latex_name=r'\\Delta_\\theta', domain='real')\n",
" if a == b:\n",
" Xi_a = var('Xi', latex_name=r'\\Xi', domain='real')\n",
" Xi_b = Xi_a\n",
" else:\n",
" Xi_a = var('Xi_a', latex_name=r'\\Xi_a', domain='real')\n",
" Xi_b = var('Xi_b', latex_name=r'\\Xi_b', domain='real')\n",
" #Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2\n",
" Xi_a = 1 - a^2*l^2\n",
" Xi_b = 1 - b^2*l^2\n",
"else:\n",
" Delta_r = (r^2+a^2)*(r^2+b^2)*sig - 2*m\n",
" Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2\n",
" Xi_a = 1 - a^2*l^2\n",
" Xi_b = 1 - b^2*l^2"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"The metric is set by its components in the coordinate frame associated with the Boyer-Lindquist-type coordinates, which is the current manifold's default frame. These components can be deduced from\n",
"Eq. (5.22) of the article [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005) (the check of agreement with this equation is performed below):"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\psi} + \\left( \\frac{a^{2} r^{2} + r^{4}}{{\\ell}^{2} r^{6} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{4} + a^{4} + {\\left(a^{4} {\\ell}^{2} + 2 \\, a^{2} - 2 \\, m\\right)} r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\right) \\mathrm{d} {\\mu}\\otimes \\mathrm{d} {\\mu} + \\left( -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\psi} + \\left( -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\psi}$$"
],
"text/plain": [
"G = -(a^4*l^2 + l^2*r^4 + (2*a^2*l^2 + 1)*r^2 + a^2 - 2*m)/(a^2 + r^2) dt*dt - (a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dt*dph - (2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dt*dps + (a^2*r^2 + r^4)/(l^2*r^6 + (2*a^2*l^2 + 1)*r^4 + a^4 + (a^4*l^2 + 2*a^2 - 2*m)*r^2) dr*dr - (a^2 + r^2)/(a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1) dmu*dmu - (a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dph*dt - (a^6*l^2 - 2*a^2*m*mu^4 + (a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1)*r^4 - a^4 - 2*a^2*m - (a^6*l^2 - a^4 - 4*a^2*m)*mu^2 + 2*(a^4*l^2 - (a^4*l^2 - a^2)*mu^2 - a^2)*r^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dph*dph - 2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dph*dps - (2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dps*dt - 2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dps*dph + (2*a^2*m*mu^4 - (a^2*l^2 - 1)*mu^2*r^4 - 2*(a^4*l^2 - a^2)*mu^2*r^2 - (a^6*l^2 - a^4)*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dps*dps"
]
},
"execution_count": 13,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G = M.metric()\n",
"tmp = 1/rho2*( -Delta_r + Delta_th*(a^2*sinth2 + b^2*mu^2) + a^2*b^2*sig )\n",
"G[0,0] = tmp.simplify_full()\n",
"tmp = a*sinth2/(rho2*Xi_a)*( Delta_r - (r^2+a^2)*(Delta_th + b^2*sig) )\n",
"G[0,3] = tmp.simplify_full()\n",
"tmp = b*mu^2/(rho2*Xi_b)*( Delta_r - (r^2+b^2)*(Delta_th + a^2*sig) )\n",
"G[0,4] = tmp.simplify_full()\n",
"G[1,1] = (rho2/Delta_r).simplify_full()\n",
"G[2,2] = (rho2/Delta_th/(1-mu^2)).simplify_full()\n",
"tmp = sinth2/(rho2*Xi_a^2)*( - Delta_r*a^2*sinth2 + (r^2+a^2)^2*(Delta_th + sig*b^2*sinth2) ) \n",
"G[3,3] = tmp.simplify_full()\n",
"tmp = a*b*sinth2*mu^2/(rho2*Xi_a*Xi_b)*( - Delta_r + sig*(r^2+a^2)*(r^2+b^2) )\n",
"G[3,4] = tmp.simplify_full()\n",
"tmp = mu^2/(rho2*Xi_b^2)*( - Delta_r*b^2*mu^2 + (r^2+b^2)^2*(Delta_th + sig*a^2*mu^2) )\n",
"G[4,4] = tmp.simplify_full()\n",
"G.display()"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\\\ G_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ G_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ G_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{a^{2} r^{2} + r^{4}}{{\\ell}^{2} r^{6} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{4} + a^{4} + {\\left(a^{4} {\\ell}^{2} + 2 \\, a^{2} - 2 \\, m\\right)} r^{2}} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\\\ G_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ G_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ G_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\end{array}$$"
],
"text/plain": [
"G_t,t = -(a^4*l^2 + l^2*r^4 + (2*a^2*l^2 + 1)*r^2 + a^2 - 2*m)/(a^2 + r^2) \n",
"G_t,ph = -(a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n",
"G_t,ps = -(2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n",
"G_r,r = (a^2*r^2 + r^4)/(l^2*r^6 + (2*a^2*l^2 + 1)*r^4 + a^4 + (a^4*l^2 + 2*a^2 - 2*m)*r^2) \n",
"G_mu,mu = -(a^2 + r^2)/(a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1) \n",
"G_ph,ph = -(a^6*l^2 - 2*a^2*m*mu^4 + (a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1)*r^4 - a^4 - 2*a^2*m - (a^6*l^2 - a^4 - 4*a^2*m)*mu^2 + 2*(a^4*l^2 - (a^4*l^2 - a^2)*mu^2 - a^2)*r^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n",
"G_ph,ps = -2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n",
"G_ps,ps = (2*a^2*m*mu^4 - (a^2*l^2 - 1)*mu^2*r^4 - 2*(a^4*l^2 - a^2)*mu^2*r^2 - (a^6*l^2 - a^4)*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) "
]
},
"execution_count": 14,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G.display_comp(only_nonredundant=True)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Check of agreement with Eq. (5.22) of Hawking et al or Eq. (2.3) of o\n",
"\n",
"We need the 1-forms $\\mathrm{d}t$, $\\mathrm{d}r$, $\\mathrm{d}\\mu$, $\\mathrm{d}\\phi$ and $\\mathrm{d}\\psi$:"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} r, \\mathrm{d} {\\mu}, \\mathrm{d} {\\phi}, \\mathrm{d} {\\psi}\\right)$$"
],
"text/plain": [
"(1-form dt on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dr on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dmu on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dph on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dps on the 5-dimensional Lorentzian manifold M)"
]
},
"execution_count": 15,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"dt, dr, dmu, dph, dps = (BL.coframe()[i] for i in M.irange())\n",
"dt, dr, dmu, dph, dps"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1-form dt on the 5-dimensional Lorentzian manifold M\n"
]
}
],
"source": [
"print(dt)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"In agreement with $\\mu = \\cos\\theta$, we introduce the 1-form $\\mathrm{d}\\theta = - \\mathrm{d}\\mu /\\sin\\theta $, with \n",
"$\\sin\\theta = \\sqrt{1-\\mu^2}$ since $\\theta\\in\\left(0, \\frac{\\pi}{2}\\right)$:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"dth = - 1/sqrt(1 - mu^2)*dmu"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( -\\frac{a {\\mu}^{2} - a}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a {\\mu}^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$$"
],
"text/plain": [
"dt - (a*mu^2 - a)/(a^2*l^2 - 1) dph + a*mu^2/(a^2*l^2 - 1) dps"
]
},
"execution_count": 18,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = dt - a*sinth2/Xi_a*dph - b*costh2/Xi_b*dps\n",
"s1.display()"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi}$$"
],
"text/plain": [
"a dt + (a^2 + r^2)/(a^2*l^2 - 1) dph"
]
},
"execution_count": 19,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = a*dt - (r^2 + a^2)/Xi_a*dph\n",
"s2.display()"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$$"
],
"text/plain": [
"a dt + (a^2 + r^2)/(a^2*l^2 - 1) dps"
]
},
"execution_count": 20,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s3 = b*dt - (r^2 + b^2)/Xi_b*dps\n",
"s3.display()"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a^{2} \\mathrm{d} t + \\left( -\\frac{a^{3} {\\mu}^{2} - a^{3} + {\\left(a {\\mu}^{2} - a\\right)} r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a^{3} {\\mu}^{2} + a {\\mu}^{2} r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$$"
],
"text/plain": [
"a^2 dt - (a^3*mu^2 - a^3 + (a*mu^2 - a)*r^2)/(a^2*l^2 - 1) dph + (a^3*mu^2 + a*mu^2*r^2)/(a^2*l^2 - 1) dps"
]
},
"execution_count": 21,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s4 = a*b*dt - b*(r^2 + a^2)*sinth2/Xi_a * dph - a*(r^2 + b^2)*costh2/Xi_b * dps\n",
"s4.display()"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} X_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\\\ X_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{a^{2} {\\mu}^{2} - {\\left({\\mu}^{2} - 1\\right)} a^{2} + r^{2}}{2 \\, m - \\frac{{\\left({\\ell}^{2} r^{2} + 1\\right)} {\\left(a^{2} + r^{2}\\right)}^{2}}{r^{2}}} \\\\ X_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\\\ X_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\psi} \\, t }^{ \\phantom{\\, {\\psi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, {\\psi} \\, {\\phi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\end{array}$$"
],
"text/plain": [
"X_t,t = -(a^4*l^2 + l^2*r^4 + (2*a^2*l^2 + 1)*r^2 + a^2 - 2*m)/(a^2 + r^2) \n",
"X_t,ph = -(a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n",
"X_t,ps = -(2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n",
"X_r,r = -(a^2*mu^2 - (mu^2 - 1)*a^2 + r^2)/(2*m - (l^2*r^2 + 1)*(a^2 + r^2)^2/r^2) \n",
"X_mu,mu = -(a^2 + r^2)/(a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1) \n",
"X_ph,t = -(a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n",
"X_ph,ph = -(a^6*l^2 - 2*a^2*m*mu^4 + (a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1)*r^4 - a^4 - 2*a^2*m - (a^6*l^2 - a^4 - 4*a^2*m)*mu^2 + 2*(a^4*l^2 - (a^4*l^2 - a^2)*mu^2 - a^2)*r^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n",
"X_ph,ps = -2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n",
"X_ps,t = -(2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n",
"X_ps,ph = -2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n",
"X_ps,ps = (2*a^2*m*mu^4 - (a^2*l^2 - 1)*mu^2*r^4 - 2*(a^4*l^2 - a^2)*mu^2*r^2 - (a^6*l^2 - a^4)*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) "
]
},
"execution_count": 22,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G0 = - Delta_r/rho2 * s1*s1 + Delta_th*sinth2/rho2 * s2*s2 + Delta_th*costh2/rho2 * s3*s3 \\\n",
" + rho2/Delta_r * dr*dr + rho2/Delta_th * dth*dth + sig/rho2 * s4*s4\n",
"G0.display_comp(only_nonredundant=True)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$"
],
"text/plain": [
"True"
]
},
"execution_count": 23,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G0 == G"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"## Einstein equation\n",
"\n",
"The Ricci tensor of $g$ is"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"if not keep_Delta:\n",
" # Ric = G.ricci()\n",
" # print(Ric)\n",
" pass"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"if not keep_Delta:\n",
" # show(Ric.display_comp(only_nonredundant=True))\n",
" pass"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"Let us check that $g$ is a solution of the vacuum Einstein equation with the cosmological constant $\\Lambda = - 6 \\ell^2$:"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"Lambda = -6*l^2\n",
"if not keep_Delta:\n",
" # print(Ric == 2/3*Lambda*G)\n",
" pass"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Check of Eq. (2.10)\n",
"\n",
"One must have $a=b$ and `keep_Delta == False` for the test to pass:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"True\n"
]
}
],
"source": [
"if a == b and not keep_Delta:\n",
" G1 = - (1 + rho2*l^2 - 2*m/rho2) * dt*dt + rho2/Delta_r * dr*dr \\\n",
" + rho2/Delta_th * dth*dth \\\n",
" + sinth2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*sinth2) * dph * dph \\\n",
" + costh2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*costh2) * dps * dps \\\n",
" + a*sinth2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dph + dph*dt) \\\n",
" + a*costh2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dps + dps*dt) \\\n",
" + 2*m*a^2*sinth2*costh2/Xi_a^2/rho2 * (dph*dps + dps*dph)\n",
" print(G1 == G)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Global AdS coordinates"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, y, {\\mu}, {\\Phi}, {\\Psi})\\right)$$"
],
"text/plain": [
"Chart (M, (t, y, mu, Ph, Ps))"
]
},
"execution_count": 28,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"ADS. = M.chart(r't y:(a/sqrt(1-a^2*l^2),+oo) mu:(0,1):\\mu Ph:(0,2*pi):\\Phi Ps:(0,2*pi):\\Psi')\n",
"ADS"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( \\frac{a}{\\sqrt{-a^{2} {\\ell}^{2} + 1}} , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\Phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\Psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$"
],
"text/plain": [
"t: (-oo, +oo); y: (a/sqrt(-a^2*l^2 + 1), +oo); mu: (0, 1); Ph: (0, 2*pi); Ps: (0, 2*pi)"
]
},
"execution_count": 29,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"ADS.coord_range()"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\verb|t|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|r|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, r > 0, \\verb|mu|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\mu} > 0, {\\mu} < 1, \\verb|ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\phi} > 0, {\\phi} < 2 \\, \\pi, \\verb|ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\psi} > 0, {\\psi} < 2 \\, \\pi, \\verb|m|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|a|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|b|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|l|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, a > 0, -a^{2} {\\ell}^{2} + 1 > 0, \\verb|y|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, y > \\frac{a}{\\sqrt{-a^{2} {\\ell}^{2} + 1}}, \\verb|Ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Phi} > 0, {\\Phi} < 2 \\, \\pi, \\verb|Ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Psi} > 0, {\\Psi} < 2 \\, \\pi\\right]$$"
],
"text/plain": [
"[t is real,\n",
" r is real,\n",
" r > 0,\n",
" mu is real,\n",
" mu > 0,\n",
" mu < 1,\n",
" ph is real,\n",
" ph > 0,\n",
" ph < 2*pi,\n",
" ps is real,\n",
" ps > 0,\n",
" ps < 2*pi,\n",
" m is real,\n",
" a is real,\n",
" b is real,\n",
" l is real,\n",
" a > 0,\n",
" -a^2*l^2 + 1 > 0,\n",
" y is real,\n",
" y > a/sqrt(-a^2*l^2 + 1),\n",
" Ph is real,\n",
" Ph > 0,\n",
" Ph < 2*pi,\n",
" Ps is real,\n",
" Ps > 0,\n",
" Ps < 2*pi]"
]
},
"execution_count": 30,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"assumptions()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Transition from the Boyer-Lindquist coordinates to the AdS global coordinates, according to Eq. (5.24) of [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005):"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ y & = & \\frac{\\sqrt{a^{2} + r^{2}}}{\\sqrt{-a^{2} {\\ell}^{2} + 1}} \\\\ {\\mu} & = & {\\mu} \\\\ {\\Phi} & = & a {\\ell}^{2} t + {\\phi} \\\\ {\\Psi} & = & a {\\ell}^{2} t + {\\psi} \\end{array}\\right.$$"
],
"text/plain": [
"t = t\n",
"y = sqrt(a^2 + r^2)/sqrt(-a^2*l^2 + 1)\n",
"mu = mu\n",
"Ph = a*l^2*t + ph\n",
"Ps = a*l^2*t + ps"
]
},
"execution_count": 31,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"BL_to_ADS = BL.transition_map(ADS, [t, sqrt(r^2 + a^2)/sqrt(Xi_a), mu, \n",
" ph + a*l^2*t, ps + a*l^2*t])\n",
"BL_to_ADS.display()"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Check of the inverse coordinate transformation:\n",
" t == t *passed*\n",
" r == r *passed*\n",
" mu == mu *passed*\n",
" ph == ph *passed*\n",
" ps == ps *passed*\n",
" t == t *passed*\n",
" y == abs(y) **failed**\n",
" mu == mu *passed*\n",
" Ph == Ph *passed*\n",
" Ps == Ps *passed*\n",
"NB: a failed report can reflect a mere lack of simplification.\n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ r & = & \\sqrt{-{\\left(a^{2} {\\ell}^{2} - 1\\right)} y^{2} - a^{2}} \\\\ {\\mu} & = & {\\mu} \\\\ {\\phi} & = & -a {\\ell}^{2} t + {\\Phi} \\\\ {\\psi} & = & -a {\\ell}^{2} t + {\\Psi} \\end{array}\\right.$$"
],
"text/plain": [
"t = t\n",
"r = sqrt(-(a^2*l^2 - 1)*y^2 - a^2)\n",
"mu = mu\n",
"ph = -a*l^2*t + Ph\n",
"ps = -a*l^2*t + Ps"
]
},
"execution_count": 32,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"BL_to_ADS.set_inverse(t, sqrt(Xi_a*y^2 - a^2), mu, Ph - a*l^2*t, Ps - a*l^2*t, \n",
" verbose=True)\n",
"BL_to_ADS.inverse().display()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Metric tensor is global AdS coordinates"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + 2 \\, m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left(a m {\\mu}^{2} - a m\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, a m {\\mu}^{2}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4}}{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4} - 2 \\, {\\left(a^{2} {\\ell}^{2} - 1\\right)} m y^{2} - 2 \\, a^{2} m} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{y^{2}}{{\\mu}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - 4 \\, a^{2} m {\\mu}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} y^{4} + 2 \\, a^{2} m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} y^{4}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\end{array}$$"
],
"text/plain": [
"G_t,t = -((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^4 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2 + 2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_t,Ph = -2*(a*m*mu^2 - a*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_t,Ps = 2*a*m*mu^2/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_y,y = (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4/((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^6 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4 - 2*(a^2*l^2 - 1)*m*y^2 - 2*a^2*m) \n",
"G_mu,mu = -y^2/(mu^2 - 1) \n",
"G_Ph,Ph = -(2*a^2*m*mu^4 - 4*a^2*m*mu^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2 - 1)*y^4 + 2*a^2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_Ph,Ps = 2*(a^2*m*mu^4 - a^2*m*mu^2)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_Ps,Ps = -(2*a^2*m*mu^4 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2*y^4)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) "
]
},
"execution_count": 33,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G.display_comp(chart=ADS, only_nonredundant=True)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"From now on, we set the AdS coordinates as the default chart on $\\mathcal{M}$: "
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"M.set_default_chart(ADS)\n",
"M.set_default_frame(ADS.frame())"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Then"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + 2 \\, m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left(a m {\\mu}^{2} - a m\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, a m {\\mu}^{2}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4}}{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4} - 2 \\, {\\left(a^{2} {\\ell}^{2} - 1\\right)} m y^{2} - 2 \\, a^{2} m} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{y^{2}}{{\\mu}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - 4 \\, a^{2} m {\\mu}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} y^{4} + 2 \\, a^{2} m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} y^{4}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\end{array}$$"
],
"text/plain": [
"G_t,t = -((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^4 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2 + 2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_t,Ph = -2*(a*m*mu^2 - a*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_t,Ps = 2*a*m*mu^2/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_y,y = (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4/((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^6 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4 - 2*(a^2*l^2 - 1)*m*y^2 - 2*a^2*m) \n",
"G_mu,mu = -y^2/(mu^2 - 1) \n",
"G_Ph,Ph = -(2*a^2*m*mu^4 - 4*a^2*m*mu^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2 - 1)*y^4 + 2*a^2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_Ph,Ps = 2*(a^2*m*mu^4 - a^2*m*mu^2)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n",
"G_Ps,Ps = -(2*a^2*m*mu^4 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2*y^4)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) "
]
},
"execution_count": 35,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G.display_comp(only_nonredundant=True)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Comparison with Eq. (5.32) of [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005) (or Eq. (2.18) of our paper):"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} y, \\mathrm{d} {\\mu}, \\mathrm{d} {\\Phi}, \\mathrm{d} {\\Psi}\\right)$$"
],
"text/plain": [
"(1-form dt on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dy on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dmu on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dPh on the 5-dimensional Lorentzian manifold M,\n",
" 1-form dPs on the 5-dimensional Lorentzian manifold M)"
]
},
"execution_count": 36,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"dt, dy, dmu, dPh, dPs = (ADS.coframe()[i] for i in M.irange())\n",
"dt, dy, dmu, dPh, dPs"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( a {\\mu}^{2} - a \\right) \\mathrm{d} {\\Phi} -a {\\mu}^{2} \\mathrm{d} {\\Psi}$$"
],
"text/plain": [
"dt + (a*mu^2 - a) dPh - a*mu^2 dPs"
]
},
"execution_count": 37,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s = dt - a*sinth2*dPh - a*costh2*dPs\n",
"s.display()"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{1}{\\sqrt{{\\mu} + 1} \\sqrt{-{\\mu} + 1}} \\right) \\mathrm{d} {\\mu}$$"
],
"text/plain": [
"-1/(sqrt(mu + 1)*sqrt(-mu + 1)) dmu"
]
},
"execution_count": 38,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"dth.display()"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$"
],
"text/plain": [
"True"
]
},
"execution_count": 39,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G1 = - (1 + y^2*l^2)* dt*dt \\\n",
" + y^2*(dth*dth + sinth2* dPh*dPh + costh2* dPs*dPs) \\\n",
" + 2*m/(y^2*Xi_a^3)* s*s \\\n",
" + y^4/(y^4*(1 + y^2*l^2) - 2*m*y^2/Xi_a^2 + 2*m*a^2/Xi_a^3)* dy*dy\n",
"# NB: note the Xi_a^3 term in the factor of s*s differs from Eq. (5.32) of Hawking et al (1999)\n",
"G == G1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"## String worldsheet"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"The string worldsheet as a 2-dimensional pseudo-Riemannian manifold (we don't assume Lorentzian signature here):"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2-dimensional Riemannian manifold W\n"
]
}
],
"source": [
"W = Manifold(2, 'W', structure='pseudo-Riemannian')\n",
"print(W)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"Let us assume that the string worldsheet is parametrized by $(t,y)$:"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(W,(t, y)\\right)$$"
],
"text/plain": [
"Chart (W, (t, y))"
]
},
"execution_count": 41,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"XW. = W.chart(r't y:(a/sqrt(1-a^2*l^2),+oo)')\n",
"XW"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"The string embedding in Kerr-AdS spacetime, as an expansion about a straight string solution in AdS (Eqs. (4.30)-(4.32) of the paper)"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& W & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, y\\right) & \\longmapsto & \\left(t, r, {\\mu}, {\\phi}, {\\psi}\\right) = \\left(t, \\sqrt{-{\\left(a^{2} {\\ell}^{2} - 1\\right)} y^{2} - a^{2}}, a^{2} \\mu_{1}\\left(y\\right) + {\\mu_0}, {\\left(a {\\beta_1} - a\\right)} {\\ell}^{2} t + a {\\beta_1} \\Phi_{1}\\left(y\\right) + {\\Phi_0}, {\\left(a {\\beta_2} - a\\right)} {\\ell}^{2} t + a {\\beta_2} \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\\\ & \\left(t, y\\right) & \\longmapsto & \\left(t, y, {\\mu}, {\\Phi}, {\\Psi}\\right) = \\left(t, y, a^{2} \\mu_{1}\\left(y\\right) + {\\mu_0}, a {\\beta_1} {\\ell}^{2} t + a {\\beta_1} \\Phi_{1}\\left(y\\right) + {\\Phi_0}, a {\\beta_2} {\\ell}^{2} t + a {\\beta_2} \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\end{array}$$"
],
"text/plain": [
"F: W --> M\n",
" (t, y) |--> (t, r, mu, ph, ps) = (t, sqrt(-(a^2*l^2 - 1)*y^2 - a^2), a^2*mu_1(y) + Mu0, (a*beta1 - a)*l^2*t + a*beta1*Phi_1(y) + Phi0, (a*beta2 - a)*l^2*t + a*beta2*Psi_1(y) + Psi0)\n",
" (t, y) |--> (t, y, mu, Ph, Ps) = (t, y, a^2*mu_1(y) + Mu0, a*beta1*l^2*t + a*beta1*Phi_1(y) + Phi0, a*beta2*l^2*t + a*beta2*Psi_1(y) + Psi0)"
]
},
"execution_count": 42,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Mu0 = var('Mu0', latex_name=r'\\mu_0', domain='real')\n",
"Phi0 = var('Phi0', latex_name=r'\\Phi_0', domain='real')\n",
"Psi0 = var('Psi0', latex_name=r'\\Psi_0', domain='real')\n",
"beta1 = var('beta1', latex_name=r'\\beta_1', domain='real')\n",
"beta2 = var('beta2', latex_name=r'\\beta_2', domain='real')\n",
"\n",
"cosTh0 = Mu0\n",
"sinTh0 = sqrt(1 - Mu0^2)\n",
"\n",
"mu_s = Mu0 + a^2*function('mu_1')(y)\n",
"Ph_s = Phi0 + beta1*a*l^2*t + beta1*a*function('Phi_1')(y)\n",
"Ps_s = Psi0 + beta2*a*l^2*t + beta2*a*function('Psi_1')(y)\n",
"\n",
"F = W.diff_map(M, {(XW, ADS): [t, y, mu_s, Ph_s, Ps_s]}, name='F') \n",
"F.display()"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
"1 & 0 \\\\\n",
"0 & 1 \\\\\n",
"0 & a^{2} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) \\\\\n",
"a {\\beta_1} {\\ell}^{2} & a {\\beta_1} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) \\\\\n",
"a {\\beta_2} {\\ell}^{2} & a {\\beta_2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[ 1 0]\n",
"[ 0 1]\n",
"[ 0 a^2*diff(mu_1(y), y)]\n",
"[ a*beta1*l^2 a*beta1*diff(Phi_1(y), y)]\n",
"[ a*beta2*l^2 a*beta2*diff(Psi_1(y), y)]"
]
},
"execution_count": 43,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"F.jacobian_matrix()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false,
"size": 4
},
"source": [
"### Induced metric on the string worldsheet\n",
"\n",
"The string worldsheet metric is the metric $g$ induced by the spacetime metric $G$, i.e. the pullback of $G$ by the embedding $F$:"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
],
"source": [
"g = W.metric()\n",
"g.set(F.pullback(G))"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": false,
"size": 4
},
"outputs": [
{
"data": {
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""
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\left(a^{12} {\\beta_1}^{2} - 2 \\, a^{12} {\\beta_1} {\\beta_2} + a^{12} {\\beta_2}^{2}\\right)} {\\ell}^{4} m \\mu_{1}\\left(y\\right)^{4} + 8 \\, {\\left({\\mu_0} a^{10} {\\beta_1}^{2} - 2 \\, {\\mu_0} a^{10} {\\beta_1} {\\beta_2} + {\\mu_0} a^{10} {\\beta_2}^{2}\\right)} {\\ell}^{4} m \\mu_{1}\\left(y\\right)^{3} - {\\left({\\left({\\mu_0}^{2} a^{8} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1}^{2}\\right)} {\\ell}^{10} - {\\left(3 \\, {\\mu_0}^{2} a^{6} {\\beta_2}^{2} - 3 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{6} {\\beta_1}^{2} + a^{6}\\right)} {\\ell}^{8} + 3 \\, {\\left({\\mu_0}^{2} a^{4} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{4} {\\beta_1}^{2} + a^{4}\\right)} {\\ell}^{6} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} + 3 \\, a^{2}\\right)} {\\ell}^{4} + {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + {\\left({\\left({\\left(a^{12} {\\beta_1}^{2} - a^{12} {\\beta_2}^{2}\\right)} {\\ell}^{10} - 3 \\, {\\left(a^{10} {\\beta_1}^{2} - a^{10} {\\beta_2}^{2}\\right)} {\\ell}^{8} + 3 \\, {\\left(a^{8} {\\beta_1}^{2} - a^{8} {\\beta_2}^{2}\\right)} {\\ell}^{6} - {\\left(a^{6} {\\beta_1}^{2} - a^{6} {\\beta_2}^{2}\\right)} {\\ell}^{4}\\right)} y^{4} + 4 \\, {\\left({\\left(3 \\, {\\mu_0}^{2} a^{8} {\\beta_2}^{2} + {\\left(3 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1}^{2} - {\\left(6 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{4} + {\\left(a^{6} {\\beta_1} - a^{6} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)^{2} + 2 \\, {\\left({\\left({\\mu_0}^{4} a^{4} {\\beta_2}^{2} + {\\left({\\mu_0}^{4} - 2 \\, {\\mu_0}^{2} + 1\\right)} a^{4} {\\beta_1}^{2} - 2 \\, {\\left({\\mu_0}^{4} - {\\mu_0}^{2}\\right)} a^{4} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{4} - 2 \\, {\\left({\\mu_0}^{2} a^{2} {\\beta_2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}\\right)} {\\ell}^{2} + 1\\right)} m + 2 \\, {\\left({\\left({\\left({\\mu_0} a^{10} {\\beta_1}^{2} - {\\mu_0} a^{10} {\\beta_2}^{2}\\right)} {\\ell}^{10} - 3 \\, {\\left({\\mu_0} a^{8} {\\beta_1}^{2} - {\\mu_0} a^{8} {\\beta_2}^{2}\\right)} {\\ell}^{8} + 3 \\, {\\left({\\mu_0} a^{6} {\\beta_1}^{2} - {\\mu_0} a^{6} {\\beta_2}^{2}\\right)} {\\ell}^{6} - {\\left({\\mu_0} a^{4} {\\beta_1}^{2} - {\\mu_0} a^{4} {\\beta_2}^{2}\\right)} {\\ell}^{4}\\right)} y^{4} + 4 \\, {\\left({\\left({\\mu_0}^{3} a^{6} {\\beta_2}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1}^{2} - {\\left(2 \\, {\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{4} + {\\left({\\mu_0} a^{4} {\\beta_1} - {\\mu_0} a^{4} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}}$$"
],
"text/plain": [
"-(2*(a^12*beta1^2 - 2*a^12*beta1*beta2 + a^12*beta2^2)*l^4*m*mu_1(y)^4 + 8*(Mu0*a^10*beta1^2 - 2*Mu0*a^10*beta1*beta2 + Mu0*a^10*beta2^2)*l^4*m*mu_1(y)^3 - ((Mu0^2*a^8*beta2^2 - (Mu0^2 - 1)*a^8*beta1^2)*l^10 - (3*Mu0^2*a^6*beta2^2 - 3*(Mu0^2 - 1)*a^6*beta1^2 + a^6)*l^8 + 3*(Mu0^2*a^4*beta2^2 - (Mu0^2 - 1)*a^4*beta1^2 + a^4)*l^6 - (Mu0^2*a^2*beta2^2 - (Mu0^2 - 1)*a^2*beta1^2 + 3*a^2)*l^4 + l^2)*y^4 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2 + (((a^12*beta1^2 - a^12*beta2^2)*l^10 - 3*(a^10*beta1^2 - a^10*beta2^2)*l^8 + 3*(a^8*beta1^2 - a^8*beta2^2)*l^6 - (a^6*beta1^2 - a^6*beta2^2)*l^4)*y^4 + 4*((3*Mu0^2*a^8*beta2^2 + (3*Mu0^2 - 1)*a^8*beta1^2 - (6*Mu0^2 - 1)*a^8*beta1*beta2)*l^4 + (a^6*beta1 - a^6*beta2)*l^2)*m)*mu_1(y)^2 + 2*((Mu0^4*a^4*beta2^2 + (Mu0^4 - 2*Mu0^2 + 1)*a^4*beta1^2 - 2*(Mu0^4 - Mu0^2)*a^4*beta1*beta2)*l^4 - 2*(Mu0^2*a^2*beta2 - (Mu0^2 - 1)*a^2*beta1)*l^2 + 1)*m + 2*(((Mu0*a^10*beta1^2 - Mu0*a^10*beta2^2)*l^10 - 3*(Mu0*a^8*beta1^2 - Mu0*a^8*beta2^2)*l^8 + 3*(Mu0*a^6*beta1^2 - Mu0*a^6*beta2^2)*l^6 - (Mu0*a^4*beta1^2 - Mu0*a^4*beta2^2)*l^4)*y^4 + 4*((Mu0^3*a^6*beta2^2 + (Mu0^3 - Mu0)*a^6*beta1^2 - (2*Mu0^3 - Mu0)*a^6*beta1*beta2)*l^4 + (Mu0*a^4*beta1 - Mu0*a^4*beta2)*l^2)*m)*mu_1(y))/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2)"
]
},
"execution_count": 45,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"g[0,0]"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(a {\\ell} + 1\\right)}^{3} {\\left(a {\\ell} - 1\\right)}^{3} y^{2}$$"
],
"text/plain": [
"(a*l + 1)^3*(a*l - 1)^3*y^2"
]
},
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"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"g[0,0].expr().denominator().factor()"
]
},
{
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"execution_count": 47,
"metadata": {
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"outputs": [
{
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(2 \\, {\\left(a^{12} {\\beta_1}^{2} - a^{12} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2} m \\mu_{1}\\left(y\\right)^{4} + 8 \\, {\\left({\\mu_0} a^{10} {\\beta_1}^{2} - {\\mu_0} a^{10} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2} m \\mu_{1}\\left(y\\right)^{3} + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1}^{2} {\\ell}^{8} - 3 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{6} {\\beta_1}^{2} {\\ell}^{6} + 3 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{4} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2}\\right)} y^{4} + {\\left({\\left(a^{12} {\\beta_1}^{2} {\\ell}^{8} - 3 \\, a^{10} {\\beta_1}^{2} {\\ell}^{6} + 3 \\, a^{8} {\\beta_1}^{2} {\\ell}^{4} - a^{6} {\\beta_1}^{2} {\\ell}^{2}\\right)} y^{4} + 2 \\, {\\left(a^{6} {\\beta_1} + {\\left(2 \\, {\\left(3 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1}^{2} - {\\left(6 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)^{2} + 2 \\, {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1} + {\\left({\\left({\\mu_0}^{4} - 2 \\, {\\mu_0}^{2} + 1\\right)} a^{4} {\\beta_1}^{2} - {\\left({\\mu_0}^{4} - {\\mu_0}^{2}\\right)} a^{4} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m + 2 \\, {\\left({\\left({\\mu_0} a^{10} {\\beta_1}^{2} {\\ell}^{8} - 3 \\, {\\mu_0} a^{8} {\\beta_1}^{2} {\\ell}^{6} + 3 \\, {\\mu_0} a^{6} {\\beta_1}^{2} {\\ell}^{4} - {\\mu_0} a^{4} {\\beta_1}^{2} {\\ell}^{2}\\right)} y^{4} + 2 \\, {\\left({\\mu_0} a^{4} {\\beta_1} + {\\left(2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1}^{2} - {\\left(2 \\, {\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)\\right)} \\frac{\\partial\\,\\Phi_{1}}{\\partial y} - {\\left(2 \\, {\\left(a^{12} {\\beta_1} {\\beta_2} - a^{12} {\\beta_2}^{2}\\right)} {\\ell}^{2} m \\mu_{1}\\left(y\\right)^{4} + 8 \\, {\\left({\\mu_0} a^{10} {\\beta_1} {\\beta_2} - {\\mu_0} a^{10} {\\beta_2}^{2}\\right)} {\\ell}^{2} m \\mu_{1}\\left(y\\right)^{3} + {\\left({\\mu_0}^{2} a^{8} {\\beta_2}^{2} {\\ell}^{8} - 3 \\, {\\mu_0}^{2} a^{6} {\\beta_2}^{2} {\\ell}^{6} + 3 \\, {\\mu_0}^{2} a^{4} {\\beta_2}^{2} {\\ell}^{4} - {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2}\\right)} y^{4} + {\\left({\\left(a^{12} {\\beta_2}^{2} {\\ell}^{8} - 3 \\, a^{10} {\\beta_2}^{2} {\\ell}^{6} + 3 \\, a^{8} {\\beta_2}^{2} {\\ell}^{4} - a^{6} {\\beta_2}^{2} {\\ell}^{2}\\right)} y^{4} + 2 \\, {\\left(a^{6} {\\beta_2} - {\\left(6 \\, {\\mu_0}^{2} a^{8} {\\beta_2}^{2} - {\\left(6 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)^{2} + 2 \\, {\\left({\\mu_0}^{2} a^{2} {\\beta_2} - {\\left({\\mu_0}^{4} a^{4} {\\beta_2}^{2} - {\\left({\\mu_0}^{4} - {\\mu_0}^{2}\\right)} a^{4} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m + 2 \\, {\\left({\\left({\\mu_0} a^{10} {\\beta_2}^{2} {\\ell}^{8} - 3 \\, {\\mu_0} a^{8} {\\beta_2}^{2} {\\ell}^{6} + 3 \\, {\\mu_0} a^{6} {\\beta_2}^{2} {\\ell}^{4} - {\\mu_0} a^{4} {\\beta_2}^{2} {\\ell}^{2}\\right)} y^{4} + 2 \\, {\\left({\\mu_0} a^{4} {\\beta_2} - {\\left(2 \\, {\\mu_0}^{3} a^{6} {\\beta_2}^{2} - {\\left(2 \\, {\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)\\right)} \\frac{\\partial\\,\\Psi_{1}}{\\partial y}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}}$$"
],
"text/plain": [
"-((2*(a^12*beta1^2 - a^12*beta1*beta2)*l^2*m*mu_1(y)^4 + 8*(Mu0*a^10*beta1^2 - Mu0*a^10*beta1*beta2)*l^2*m*mu_1(y)^3 + ((Mu0^2 - 1)*a^8*beta1^2*l^8 - 3*(Mu0^2 - 1)*a^6*beta1^2*l^6 + 3*(Mu0^2 - 1)*a^4*beta1^2*l^4 - (Mu0^2 - 1)*a^2*beta1^2*l^2)*y^4 + ((a^12*beta1^2*l^8 - 3*a^10*beta1^2*l^6 + 3*a^8*beta1^2*l^4 - a^6*beta1^2*l^2)*y^4 + 2*(a^6*beta1 + (2*(3*Mu0^2 - 1)*a^8*beta1^2 - (6*Mu0^2 - 1)*a^8*beta1*beta2)*l^2)*m)*mu_1(y)^2 + 2*((Mu0^2 - 1)*a^2*beta1 + ((Mu0^4 - 2*Mu0^2 + 1)*a^4*beta1^2 - (Mu0^4 - Mu0^2)*a^4*beta1*beta2)*l^2)*m + 2*((Mu0*a^10*beta1^2*l^8 - 3*Mu0*a^8*beta1^2*l^6 + 3*Mu0*a^6*beta1^2*l^4 - Mu0*a^4*beta1^2*l^2)*y^4 + 2*(Mu0*a^4*beta1 + (2*(Mu0^3 - Mu0)*a^6*beta1^2 - (2*Mu0^3 - Mu0)*a^6*beta1*beta2)*l^2)*m)*mu_1(y))*d(Phi_1)/dy - (2*(a^12*beta1*beta2 - a^12*beta2^2)*l^2*m*mu_1(y)^4 + 8*(Mu0*a^10*beta1*beta2 - Mu0*a^10*beta2^2)*l^2*m*mu_1(y)^3 + (Mu0^2*a^8*beta2^2*l^8 - 3*Mu0^2*a^6*beta2^2*l^6 + 3*Mu0^2*a^4*beta2^2*l^4 - Mu0^2*a^2*beta2^2*l^2)*y^4 + ((a^12*beta2^2*l^8 - 3*a^10*beta2^2*l^6 + 3*a^8*beta2^2*l^4 - a^6*beta2^2*l^2)*y^4 + 2*(a^6*beta2 - (6*Mu0^2*a^8*beta2^2 - (6*Mu0^2 - 1)*a^8*beta1*beta2)*l^2)*m)*mu_1(y)^2 + 2*(Mu0^2*a^2*beta2 - (Mu0^4*a^4*beta2^2 - (Mu0^4 - Mu0^2)*a^4*beta1*beta2)*l^2)*m + 2*((Mu0*a^10*beta2^2*l^8 - 3*Mu0*a^8*beta2^2*l^6 + 3*Mu0*a^6*beta2^2*l^4 - Mu0*a^4*beta2^2*l^2)*y^4 + 2*(Mu0*a^4*beta2 - (2*Mu0^3*a^6*beta2^2 - (2*Mu0^3 - Mu0)*a^6*beta1*beta2)*l^2)*m)*mu_1(y))*d(Psi_1)/dy)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2)"
]
},
"execution_count": 47,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"g[0,1]"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"## Nambu-Goto action"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"detg = g.determinant().expr()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Expanding at second order in $a$:"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} - 2 \\, {\\left({\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1} - a^{2}\\right)} {\\ell}^{2} - 1\\right)} m y^{2} - y^{4} + 2 \\, a^{2} m + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m y^{4} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m^{2} y^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2} {\\beta_2}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2}}{{\\ell}^{2} y^{6} + y^{4} - 2 \\, m y^{2}}$$"
],
"text/plain": [
"(((Mu0^2*a^2*beta2^2 - (Mu0^2 - 1)*a^2*beta1^2)*l^4 - l^2)*y^6 - 2*((2*Mu0^2*a^2*beta2 - 2*(Mu0^2 - 1)*a^2*beta1 - a^2)*l^2 - 1)*m*y^2 - y^4 + 2*a^2*m + ((Mu0^2 - 1)*a^2*beta1^2*l^4*y^10 + 2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^8 - 4*(Mu0^2 - 1)*a^2*beta1^2*m*y^4 + 4*(Mu0^2 - 1)*a^2*beta1^2*m^2*y^2 - (4*(Mu0^2 - 1)*a^2*beta1^2*l^2*m - (Mu0^2 - 1)*a^2*beta1^2)*y^6)*diff(Phi_1(y), y)^2 - (Mu0^2*a^2*beta2^2*l^4*y^10 + 2*Mu0^2*a^2*beta2^2*l^2*y^8 - 4*Mu0^2*a^2*beta2^2*m*y^4 + 4*Mu0^2*a^2*beta2^2*m^2*y^2 - (4*Mu0^2*a^2*beta2^2*l^2*m - Mu0^2*a^2*beta2^2)*y^6)*diff(Psi_1(y), y)^2)/(l^2*y^6 + y^4 - 2*m*y^2)"
]
},
"execution_count": 49,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"detg_a2 = detg.series(a, 3).truncate().simplify_full()\n",
"detg_a2 "
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The Nambu-Goto Lagrangian at second order in $a$:"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left({\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} {\\ell}^{4} - 2 \\, {\\ell}^{2}\\right)} y^{6} - 2 \\, {\\left({\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1} - a^{2}\\right)} {\\ell}^{2} - 2\\right)} m y^{2} - 2 \\, y^{4} + 2 \\, a^{2} m + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m y^{4} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m^{2} y^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2} {\\beta_2}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2}}{2 \\, {\\left({\\ell}^{2} y^{6} + y^{4} - 2 \\, m y^{2}\\right)}}$$"
],
"text/plain": [
"-1/2*(((Mu0^2*a^2*beta2^2 - (Mu0^2 - 1)*a^2*beta1^2)*l^4 - 2*l^2)*y^6 - 2*((2*Mu0^2*a^2*beta2 - 2*(Mu0^2 - 1)*a^2*beta1 - a^2)*l^2 - 2)*m*y^2 - 2*y^4 + 2*a^2*m + ((Mu0^2 - 1)*a^2*beta1^2*l^4*y^10 + 2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^8 - 4*(Mu0^2 - 1)*a^2*beta1^2*m*y^4 + 4*(Mu0^2 - 1)*a^2*beta1^2*m^2*y^2 - (4*(Mu0^2 - 1)*a^2*beta1^2*l^2*m - (Mu0^2 - 1)*a^2*beta1^2)*y^6)*diff(Phi_1(y), y)^2 - (Mu0^2*a^2*beta2^2*l^4*y^10 + 2*Mu0^2*a^2*beta2^2*l^2*y^8 - 4*Mu0^2*a^2*beta2^2*m*y^4 + 4*Mu0^2*a^2*beta2^2*m^2*y^2 - (4*Mu0^2*a^2*beta2^2*l^2*m - Mu0^2*a^2*beta2^2)*y^6)*diff(Psi_1(y), y)^2)/(l^2*y^6 + y^4 - 2*m*y^2)"
]
},
"execution_count": 50,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"L_a2 = (sqrt(-detg_a2)).series(a, 3).truncate().simplify_full()\n",
"L_a2"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{6} - {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{6} + 2 \\, a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\beta_1}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{6} - {\\mu_0}^{2} a^{2} {\\beta_1}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + a^{2} {\\beta_1}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\beta_1}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1} {\\ell}^{2} m y^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2} {\\ell}^{2} m y^{2} + 4 \\, a^{2} {\\beta_1}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, a^{2} {\\beta_1} {\\ell}^{2} m y^{2} + 2 \\, {\\ell}^{2} y^{6} - 2 \\, a^{2} {\\ell}^{2} m y^{2} + 2 \\, y^{4} - 2 \\, a^{2} m - 4 \\, m y^{2}$$"
],
"text/plain": [
"-Mu0^2*a^2*beta1^2*l^4*y^10*diff(Phi_1(y), y)^2 + Mu0^2*a^2*beta2^2*l^4*y^10*diff(Psi_1(y), y)^2 + a^2*beta1^2*l^4*y^10*diff(Phi_1(y), y)^2 - 2*Mu0^2*a^2*beta1^2*l^2*y^8*diff(Phi_1(y), y)^2 + 2*Mu0^2*a^2*beta2^2*l^2*y^8*diff(Psi_1(y), y)^2 + 4*Mu0^2*a^2*beta1^2*l^2*m*y^6*diff(Phi_1(y), y)^2 - 4*Mu0^2*a^2*beta2^2*l^2*m*y^6*diff(Psi_1(y), y)^2 + Mu0^2*a^2*beta1^2*l^4*y^6 - Mu0^2*a^2*beta2^2*l^4*y^6 + 2*a^2*beta1^2*l^2*y^8*diff(Phi_1(y), y)^2 - 4*a^2*beta1^2*l^2*m*y^6*diff(Phi_1(y), y)^2 - a^2*beta1^2*l^4*y^6 - Mu0^2*a^2*beta1^2*y^6*diff(Phi_1(y), y)^2 + Mu0^2*a^2*beta2^2*y^6*diff(Psi_1(y), y)^2 + 4*Mu0^2*a^2*beta1^2*m*y^4*diff(Phi_1(y), y)^2 - 4*Mu0^2*a^2*beta2^2*m*y^4*diff(Psi_1(y), y)^2 - 4*Mu0^2*a^2*beta1^2*m^2*y^2*diff(Phi_1(y), y)^2 + a^2*beta1^2*y^6*diff(Phi_1(y), y)^2 + 4*Mu0^2*a^2*beta2^2*m^2*y^2*diff(Psi_1(y), y)^2 - 4*a^2*beta1^2*m*y^4*diff(Phi_1(y), y)^2 - 4*Mu0^2*a^2*beta1*l^2*m*y^2 + 4*Mu0^2*a^2*beta2*l^2*m*y^2 + 4*a^2*beta1^2*m^2*y^2*diff(Phi_1(y), y)^2 + 4*a^2*beta1*l^2*m*y^2 + 2*l^2*y^6 - 2*a^2*l^2*m*y^2 + 2*y^4 - 2*a^2*m - 4*m*y^2"
]
},
"execution_count": 51,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"L_a2.numerator()"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\ell}^{2} y^{6} + 2 \\, y^{4} - 4 \\, m y^{2}$$"
],
"text/plain": [
"2*l^2*y^6 + 2*y^4 - 4*m*y^2"
]
},
"execution_count": 52,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"L_a2.denominator()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Euler-Lagrange equations"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"def euler_lagrange(lagr, qs, var):\n",
" r\"\"\"\n",
" Derive the Euler-Lagrange equations from a given Lagrangian.\n",
"\n",
" INPUT:\n",
"\n",
" - ``lagr`` -- symbolic expression representing the Lagrangian density\n",
" - ``qs`` -- either a single symbolic function or a list/tuple of\n",
" symbolic functions, representing the `q`'s; these functions must\n",
" appear in ``lagr`` up to at most their first derivatives\n",
" - ``var`` -- either a single variable, typically `t` (1-dimensional\n",
" problem) or a list/tuple of symbolic variables\n",
"\n",
" OUTPUT:\n",
"\n",
" - list of Euler-Lagrange equations; if only one function is involved, the\n",
" single Euler-Lagrannge equation is returned instead.\n",
"\n",
" \"\"\"\n",
" if not isinstance(qs, (list, tuple)):\n",
" qs = [qs]\n",
" if not isinstance(var, (list, tuple)):\n",
" var = [var]\n",
" n = len(qs)\n",
" d = len(var)\n",
" qv = [SR.var('qxxxx{}'.format(q)) for q in qs]\n",
" dqv = [[SR.var('qxxxx{}_{}'.format(q, v)) for v in var] for q in qs]\n",
" subs = {qs[i](*var): qv[i] for i in range(n)}\n",
" subs_inv = {qv[i]: qs[i](*var) for i in range(n)}\n",
" for i in range(n):\n",
" subs.update({diff(qs[i](*var), var[j]): dqv[i][j]\n",
" for j in range(d)})\n",
" subs_inv.update({dqv[i][j]: diff(qs[i](*var), var[j])\n",
" for j in range(d)})\n",
" lg = lagr.substitute(subs)\n",
" eqs = []\n",
" for i in range(n):\n",
" dLdq = diff(lg, qv[i]).simplify_full()\n",
" dLdq = dLdq.substitute(subs_inv)\n",
" ddL = 0\n",
" for j in range(d):\n",
" h = diff(lg, dqv[i][j]).simplify_full()\n",
" h = h.substitute(subs_inv)\n",
" ddL += diff(h, var[j])\n",
" eqs.append((dLdq - ddL).simplify_full() == 0)\n",
" if n == 1:\n",
" return eqs[0]\n",
" return eqs"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"We compute the Euler-Lagrange equations at order $a^2$ for $\\phi_1$ and $\\psi_1$:"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0, -2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{3} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0\\right]$$"
],
"text/plain": [
"[2*(2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^3 + (Mu0^2 - 1)*a^2*beta1^2*y)*diff(Phi_1(y), y) + ((Mu0^2 - 1)*a^2*beta1^2*l^2*y^4 + (Mu0^2 - 1)*a^2*beta1^2*y^2 - 2*(Mu0^2 - 1)*a^2*beta1^2*m)*diff(Phi_1(y), y, y) == 0,\n",
" -2*(2*Mu0^2*a^2*beta2^2*l^2*y^3 + Mu0^2*a^2*beta2^2*y)*diff(Psi_1(y), y) - (Mu0^2*a^2*beta2^2*l^2*y^4 + Mu0^2*a^2*beta2^2*y^2 - 2*Mu0^2*a^2*beta2^2*m)*diff(Psi_1(y), y, y) == 0]"
]
},
"execution_count": 54,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eqs = euler_lagrange(L_a2, [Phi_1, Psi_1], y)\n",
"eqs"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"#### Solving the equation for $\\phi_1$ (check of Eq. (4.34))"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$$"
],
"text/plain": [
"2*(2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^3 + (Mu0^2 - 1)*a^2*beta1^2*y)*diff(Phi_1(y), y) + ((Mu0^2 - 1)*a^2*beta1^2*l^2*y^4 + (Mu0^2 - 1)*a^2*beta1^2*y^2 - 2*(Mu0^2 - 1)*a^2*beta1^2*m)*diff(Phi_1(y), y, y) == 0"
]
},
"execution_count": 55,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_phi1 = eqs[0]\n",
"eq_phi1"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$$"
],
"text/plain": [
"2*(2*l^2*y^3 + y)*diff(Phi_1(y), y) + (l^2*y^4 + y^2 - 2*m)*diff(Phi_1(y), y, y) == 0"
]
},
"execution_count": 56,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_phi1 = (eq_phi1/(a^2*(Mu0^2-1)*beta1^2)).simplify_full()\n",
"eq_phi1"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$$"
],
"text/plain": [
"_K1*integrate(1/(l^2*y^4 + y^2 - 2*m), y) + _K2"
]
},
"execution_count": 57,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Phi1_sol(y) = desolve(eq_phi1, Phi_1(y), ivar=y)\n",
"Phi1_sol(y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The symbolic constants $K_1$ and $K_2$ are actually denoted `_K1` and `_K2` by SageMath, as the `print` reveals:"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"_K1*integrate(1/(l^2*y^4 + y^2 - 2*m), y) + _K2\n"
]
}
],
"source": [
"print(Phi1_sol(y))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Hence we perform the substitutions with `SR.var('_K1')` and `SR.var('_K2')`:"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"P*integrate(1/(l^2*y^4 + y^2 - 2*m), y)\n"
]
}
],
"source": [
"P = var(\"P\", latex_name=r\"\\mathcal{P}'\")\n",
"Phi1_sol(y) = Phi1_sol(y).subs({SR.var('_K1'): P, SR.var('_K2'): 0})\n",
"print(Phi1_sol(y))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"#### Solving the equation for $\\psi_1$ (check of Eq. (4.34))"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{3} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$$"
],
"text/plain": [
"-2*(2*Mu0^2*a^2*beta2^2*l^2*y^3 + Mu0^2*a^2*beta2^2*y)*diff(Psi_1(y), y) - (Mu0^2*a^2*beta2^2*l^2*y^4 + Mu0^2*a^2*beta2^2*y^2 - 2*Mu0^2*a^2*beta2^2*m)*diff(Psi_1(y), y, y) == 0"
]
},
"execution_count": 60,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_psi1 = eqs[1]\n",
"eq_psi1"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\ell}^{2} y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$$"
],
"text/plain": [
"-2*(2*l^2*y^3 + y)*diff(Psi_1(y), y) - (l^2*y^4 + y^2 - 2*m)*diff(Psi_1(y), y, y) == 0"
]
},
"execution_count": 61,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_psi1 = (eq_psi1/(a^2*Mu0^2*beta2^2)).simplify_full()\n",
"eq_psi1"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$$"
],
"text/plain": [
"_K1*integrate(1/(l^2*y^4 + y^2 - 2*m), y) + _K2"
]
},
"execution_count": 62,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Psi1_sol(y) = desolve(eq_psi1, Psi_1(y), ivar=y)\n",
"Psi1_sol(y)"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Q*integrate(1/(l^2*y^4 + y^2 - 2*m), y)\n"
]
}
],
"source": [
"Q = var('Q', latex_name=r\"\\mathcal{Q}'\")\n",
"Psi1_sol(y) = Psi1_sol(y).subs({SR.var('_K1'): Q, SR.var('_K2'): 0})\n",
"print(Psi1_sol(y))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Nambu-Goto Lagrangian at fourth order in $a$"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"detg_a4 = detg.series(a, 5).truncate().simplify_full()"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"L_a4 = (sqrt(-detg_a4)).series(a, 5).truncate().simplify_full()"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"eqs = euler_lagrange(L_a4, [Phi_1, Psi_1, mu_1], y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### The equation for $\\mu_1$ (check of Eq. (4.35))"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}\\right)} {\\ell}^{2} m - {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 2 \\, {\\left(2 \\, a^{4} {\\ell}^{4} y^{7} + 3 \\, a^{4} {\\ell}^{2} y^{5} - 2 \\, a^{4} m y - {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} y^{3}\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left(a^{4} {\\ell}^{4} y^{8} + 2 \\, a^{4} {\\ell}^{2} y^{6} - 4 \\, a^{4} m y^{2} + 4 \\, a^{4} m^{2} - {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} y^{4}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right)}{{\\left({\\mu_0}^{2} - 1\\right)} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} m} = 0$$"
],
"text/plain": [
"(((Mu0^3 - Mu0)*a^4*beta1^2 - (Mu0^3 - Mu0)*a^4*beta2^2)*l^4*y^4 - 4*((Mu0^3 - Mu0)*a^4*beta1 - (Mu0^3 - Mu0)*a^4*beta2)*l^2*m - ((Mu0^3 - Mu0)*a^4*beta1^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*a^4*beta1^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*a^4*beta1^2*m*y^2 + 4*(Mu0^3 - Mu0)*a^4*beta1^2*m^2 - (4*(Mu0^3 - Mu0)*a^4*beta1^2*l^2*m - (Mu0^3 - Mu0)*a^4*beta1^2)*y^4)*diff(Phi_1(y), y)^2 + ((Mu0^3 - Mu0)*a^4*beta2^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*a^4*beta2^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*a^4*beta2^2*m*y^2 + 4*(Mu0^3 - Mu0)*a^4*beta2^2*m^2 - (4*(Mu0^3 - Mu0)*a^4*beta2^2*l^2*m - (Mu0^3 - Mu0)*a^4*beta2^2)*y^4)*diff(Psi_1(y), y)^2 + 2*(2*a^4*l^4*y^7 + 3*a^4*l^2*y^5 - 2*a^4*m*y - (4*a^4*l^2*m - a^4)*y^3)*diff(mu_1(y), y) + (a^4*l^4*y^8 + 2*a^4*l^2*y^6 - 4*a^4*m*y^2 + 4*a^4*m^2 - (4*a^4*l^2*m - a^4)*y^4)*diff(mu_1(y), y, y))/((Mu0^2 - 1)*l^2*y^4 + (Mu0^2 - 1)*y^2 - 2*(Mu0^2 - 1)*m) == 0"
]
},
"execution_count": 67,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_mu1 = eqs[2]\n",
"eq_mu1"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# eq_mu1.lhs().numerator().simplify_full()"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# eq_mu1.lhs().denominator().simplify_full()"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"eq_mu1 = eq_mu1.lhs().numerator().simplify_full() == 0"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m - {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$$"
],
"text/plain": [
"((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m - ((Mu0^3 - Mu0)*beta1^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*beta1^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*beta1^2*m*y^2 + 4*(Mu0^3 - Mu0)*beta1^2*m^2 - (4*(Mu0^3 - Mu0)*beta1^2*l^2*m - (Mu0^3 - Mu0)*beta1^2)*y^4)*diff(Phi_1(y), y)^2 + ((Mu0^3 - Mu0)*beta2^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*beta2^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*beta2^2*m*y^2 + 4*(Mu0^3 - Mu0)*beta2^2*m^2 - (4*(Mu0^3 - Mu0)*beta2^2*l^2*m - (Mu0^3 - Mu0)*beta2^2)*y^4)*diff(Psi_1(y), y)^2 + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y) + (l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)*diff(mu_1(y), y, y) == 0"
]
},
"execution_count": 71,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_mu1 = (eq_mu1/a^4).simplify_full()\n",
"eq_mu1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"We plug the solutions obtained previously for $\\phi_1(r)$ and $\\psi_1(r)$ in this equation:"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$$"
],
"text/plain": [
"((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y) + (l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)*diff(mu_1(y), y, y) == 0"
]
},
"execution_count": 72,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_mu1 = eq_mu1.substitute_function(Phi_1, Phi1_sol).substitute_function(Psi_1, Psi1_sol)\n",
"eq_mu1 = eq_mu1.simplify_full()\n",
"eq_mu1"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right)$$"
],
"text/plain": [
"((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y) + (l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)*diff(mu_1(y), y, y)"
]
},
"execution_count": 73,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"lhs = eq_mu1.lhs()\n",
"lhs = lhs.simplify_full()\n",
"lhs"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}$$"
],
"text/plain": [
"(l^2*y^4 + y^2 - 2*m)^2"
]
},
"execution_count": 74,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s = lhs.coefficient(diff(mu_1(y), y, 2)) # coefficient of mu_1''\n",
"s.factor()"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}}$$"
],
"text/plain": [
"(((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y))/(l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)"
]
},
"execution_count": 75,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = (lhs/s - diff(mu_1(y), y, 2)).simplify_full()\n",
"s1"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} y^{2} + 1\\right)} y}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$"
],
"text/plain": [
"2*(2*l^2*y^2 + 1)*y/(l^2*y^4 + y^2 - 2*m)"
]
},
"execution_count": 76,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"b1 = s1.coefficient(diff(mu_1(y), y)).factor()\n",
"b1"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}}$$"
],
"text/plain": [
"(((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m)/(l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)"
]
},
"execution_count": 77,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = (s1 - b1*diff(mu_1(y), y)).simplify_full()\n",
"s2"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\beta_1}^{2} {\\ell}^{4} y^{4} - {\\beta_2}^{2} {\\ell}^{4} y^{4} - {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\beta_1} {\\ell}^{2} m + 4 \\, {\\beta_2} {\\ell}^{2} m\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}}$$"
],
"text/plain": [
"(beta1^2*l^4*y^4 - beta2^2*l^4*y^4 - P^2*beta1^2 + Q^2*beta2^2 - 4*beta1*l^2*m + 4*beta2*l^2*m)*(Mu0 + 1)*(Mu0 - 1)*Mu0/(l^2*y^4 + y^2 - 2*m)^2"
]
},
"execution_count": 78,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2.factor()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The equation for $\\mu_1$ is thus:"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\beta_1}^{2} {\\ell}^{4} y^{4} - {\\beta_2}^{2} {\\ell}^{4} y^{4} - {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\beta_1} {\\ell}^{2} m + 4 \\, {\\beta_2} {\\ell}^{2} m\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}} + \\frac{2 \\, {\\left(2 \\, {\\ell}^{2} y^{2} + 1\\right)} y \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m} + \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$$"
],
"text/plain": [
"(beta1^2*l^4*y^4 - beta2^2*l^4*y^4 - P^2*beta1^2 + Q^2*beta2^2 - 4*beta1*l^2*m + 4*beta2*l^2*m)*(Mu0 + 1)*(Mu0 - 1)*Mu0/(l^2*y^4 + y^2 - 2*m)^2 + 2*(2*l^2*y^2 + 1)*y*diff(mu_1(y), y)/(l^2*y^4 + y^2 - 2*m) + diff(mu_1(y), y, y) == 0"
]
},
"execution_count": 79,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_mu1 = diff(mu_1(y), y, 2) + b1*diff(mu_1(y), y) + s2.factor() == 0\n",
"eq_mu1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Given that \n",
"$$ \\mu_1(y) = - \\sin\\Theta_0 \\; \\theta_1(y) = - \\sqrt{1-\\mu_0^2} \\; \\theta_1(y), \\qquad \\sin2\\Theta_0 = 2\\mu_0\\sqrt{1-\\mu_0^2}$$\n",
"and\n",
"$$\\mathcal{P}' = \\mathcal{P}/\\beta_1^2 \\qquad\\mbox{and}\\qquad \\mathcal{Q}' = \\mathcal{Q}/\\beta_1^2,$$ \n",
"we get for the equation for $\\theta_1$:\n",
"$$ \\theta_1'' + \\frac{2y(2\\ell^2 y^2 + 1)}{\\ell^2 y^4 + y^2 - 2m} \\, \\theta_1' + \\frac{\\beta_2^{-2}\\mathcal{Q}^2 - \\beta_1^{-2}\\mathcal{P}^2 - 4 (\\beta_1 - \\beta_2) \\ell^2 m + (\\beta_1^2 - \\beta_2^2) \\ell^4 y^4}{2(\\ell^2 y^4 + y^2 - 2m)^2}\\sin(2\\Theta_0) = 0 $$ \n",
"This agrees with Eq. (4.35) of the paper."
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Solving the equation for $\\mu_1$"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$$"
],
"text/plain": [
"_K2 - integrate((((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^2*y + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2)*l^2*y^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)/(l^2*y^4 + y^2 - 2*m), y) - _K1)/(l^2*y^4 + y^2 - 2*m), y)"
]
},
"execution_count": 80,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"mu1_sol(y) = desolve(eq_mu1, mu_1(y), ivar=y)\n",
"mu1_sol(y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Let us check that `mu1_sol` is indeed a solution of the equation for $\\mu_1$:"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$$"
],
"text/plain": [
"0 == 0"
]
},
"execution_count": 81,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_mu1.substitute_function(mu_1, mu1_sol).simplify_full()"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$$"
],
"text/plain": [
"_K2 - integrate((((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^2*y + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2)*l^2*y^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)/(l^2*y^4 + y^2 - 2*m), y) - _K1)/(l^2*y^4 + y^2 - 2*m), y)"
]
},
"execution_count": 82,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"mu1_sol(y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The innermost integral can be written\n",
"$$ (\\beta_1^2 - \\beta_2^2) \\ell^2 \\; s_1(y) + \\left({\\mathcal{P}'}^2 \\beta_1^2 - {\\mathcal{Q}'}\\beta_2^2 - 2 (\\beta_1^2-\\beta_2^2 - 2 (\\beta_1-\\beta_2))\\ell^2 m \\right) \\; s_2(y)$$\n",
"with \n",
"$$ s_1(y) := \\int^y \\frac{\\bar{y}^2}{\\ell^2 \\bar{y}^4 + \\bar{y}^2 - 2m} \\, \\mathrm{d}\\bar{y} \\qquad \\mbox{and}\\qquad s_2(y) := \\int^y \\frac{\\mathrm{d}\\bar{y}}{\\ell^2 \\bar{y}^4 + \\bar{y}^2 - 2m} .$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Let us evaluate $s_1$ by means of FriCAS:"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) + \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right)$$"
],
"text/plain": [
"1/2*sqrt(1/2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))*log(sqrt(1/2)*(8*l^4*m + l^2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y) - 1/2*sqrt(1/2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))*log(-sqrt(1/2)*(8*l^4*m + l^2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y) - 1/2*sqrt(1/2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))*log(sqrt(1/2)*(8*l^4*m + l^2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y) + 1/2*sqrt(1/2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))*log(-sqrt(1/2)*(8*l^4*m + l^2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y)"
]
},
"execution_count": 83,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = integrate(y^2/(l^2*y^4 + y^2 - 2*m), y, algorithm='fricas')\n",
"s1"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y - \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y + \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + \\sqrt{2} \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y + \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y - \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} {\\ell}}$$"
],
"text/plain": [
"1/4*(sqrt(2)*sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1)*log((sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y - sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1))/(sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y + sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1))) + sqrt(2)*sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1)*log((sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y + sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1))/(sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y - sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1))))/((8*l^2*m + 1)^(3/4)*l)"
]
},
"execution_count": 84,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = s1.canonicalize_radical().simplify_log()\n",
"s1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Check:"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$"
],
"text/plain": [
"y^2/(l^2*y^4 + y^2 - 2*m)"
]
},
"execution_count": 85,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"diff(s1, y).simplify_full()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Similarly, we evaluate $s_2$ by means of FriCAS:"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) - \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right)$$"
],
"text/plain": [
"-1/4*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))*log(2*l^2*y + 1/2*(8*l^2*m - (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))) + 1/4*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))*log(2*l^2*y - 1/2*(8*l^2*m - (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))) - 1/4*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))*log(2*l^2*y + 1/2*(8*l^2*m + (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))) + 1/4*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))*log(2*l^2*y - 1/2*(8*l^2*m + (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m)))"
]
},
"execution_count": 86,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = integrate(1/(l^2*y^4 + y^2 - 2*m), y, algorithm='fricas')\n",
"s2"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y - \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y + \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}\\right) + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y - \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} \\sqrt{m}}$$"
],
"text/plain": [
"1/4*(sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*log((4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y - sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*(sqrt(8*l^2*m + 1) + 1))/(4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y + sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*(sqrt(8*l^2*m + 1) + 1))) + sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*log((4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y - sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*(sqrt(8*l^2*m + 1) - 1))/(4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y + sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*(sqrt(8*l^2*m + 1) - 1))))/((8*l^2*m + 1)^(3/4)*sqrt(m))"
]
},
"execution_count": 87,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = s2.canonicalize_radical().simplify_log()\n",
"s2"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Check:"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$"
],
"text/plain": [
"1/(l^2*y^4 + y^2 - 2*m)"
]
},
"execution_count": 88,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"diff(s2, y).simplify_full()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"In the above expressions for $s_1(y)$ and $s_2(y)$ there appears the factor \n",
"$$\\mathfrak{P} = \\sqrt{1 + 8\\ell^2 m},$$\n",
"which we represent by the symbolic variable `B`"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"B = var('B')\n",
"assume(B > 1)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Let us make $B$ appear in $s_1$:"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{B^{2} - B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{B^{2} - B}}{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{B^{2} - B}}\\right) + \\sqrt{2} \\sqrt{-B^{2} - B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{-B^{2} - B}}{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{-B^{2} - B}}\\right)}{4 \\, B^{\\frac{3}{2}} {\\ell}}$$"
],
"text/plain": [
"1/4*(sqrt(2)*sqrt(B^2 - B)*log((sqrt(2)*sqrt(B)*l*y - sqrt(B^2 - B))/(sqrt(2)*sqrt(B)*l*y + sqrt(B^2 - B))) + sqrt(2)*sqrt(-B^2 - B)*log((sqrt(2)*sqrt(B)*l*y + sqrt(-B^2 - B))/(sqrt(2)*sqrt(B)*l*y - sqrt(-B^2 - B))))/(B^(3/2)*l)"
]
},
"execution_count": 90,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = s1.subs({l^2: (B^2 - 1)/(8*m)}).simplify_full()\n",
"s1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"In this expression, there appears the term $\\sqrt{-B^2-B}$ which is imaginary since $B>1$. \n",
"We there rewrite it as $i\\sqrt{B}\\sqrt{B+1}$:"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, \\sqrt{2} \\sqrt{B + 1} \\sqrt{B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y + i \\, \\sqrt{B + 1} \\sqrt{B}}{\\sqrt{2} \\sqrt{B} {\\ell} y - i \\, \\sqrt{B + 1} \\sqrt{B}}\\right) + \\sqrt{2} \\sqrt{B - 1} \\sqrt{B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{B - 1} \\sqrt{B}}{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{B - 1} \\sqrt{B}}\\right)}{4 \\, B^{\\frac{3}{2}} {\\ell}}$$"
],
"text/plain": [
"1/4*(I*sqrt(2)*sqrt(B + 1)*sqrt(B)*log((sqrt(2)*sqrt(B)*l*y + I*sqrt(B + 1)*sqrt(B))/(sqrt(2)*sqrt(B)*l*y - I*sqrt(B + 1)*sqrt(B))) + sqrt(2)*sqrt(B - 1)*sqrt(B)*log((sqrt(2)*sqrt(B)*l*y - sqrt(B - 1)*sqrt(B))/(sqrt(2)*sqrt(B)*l*y + sqrt(B - 1)*sqrt(B))))/(B^(3/2)*l)"
]
},
"execution_count": 91,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = s1.subs({sqrt(-B^2 - B): I*sqrt(B)*sqrt(B + 1), \n",
" sqrt(B^2 - B): sqrt(B)*sqrt(B - 1)})\n",
"s1"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, \\sqrt{2} \\sqrt{B + 1} \\log\\left(\\frac{\\sqrt{2} {\\ell} y + i \\, \\sqrt{B + 1}}{\\sqrt{2} {\\ell} y - i \\, \\sqrt{B + 1}}\\right) + \\sqrt{2} \\sqrt{B - 1} \\log\\left(\\frac{\\sqrt{2} {\\ell} y - \\sqrt{B - 1}}{\\sqrt{2} {\\ell} y + \\sqrt{B - 1}}\\right)}{4 \\, B {\\ell}}$$"
],
"text/plain": [
"1/4*(I*sqrt(2)*sqrt(B + 1)*log((sqrt(2)*l*y + I*sqrt(B + 1))/(sqrt(2)*l*y - I*sqrt(B + 1))) + sqrt(2)*sqrt(B - 1)*log((sqrt(2)*l*y - sqrt(B - 1))/(sqrt(2)*l*y + sqrt(B - 1))))/(B*l)"
]
},
"execution_count": 92,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = s1.simplify_log()\n",
"s1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"In the first $\\log$, we recognize the $\\mathrm{arctan}$ function, via the identity\n",
"$$\n",
" \\mathrm{arctan}\\, x = \\frac{i}{2} \\ln\\left( \\frac{i + x}{i - x} \\right), \n",
"$$\n",
"which we use in the form\n",
"$$\n",
"i \\ln\\left( \\frac{x + i}{x - i} \\right) = 2 \\mathrm{arctan}(x) - \\pi\n",
"$$\n",
"as we can check:"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$$"
],
"text/plain": [
"0"
]
},
"execution_count": 93,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"taylor(I*ln((x+I)/(x-I)) - 2*atan(x) + pi, x, 0, 10)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Thus, we set, disregarding the additive constant $-\\pi$,"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)}}{4 \\, B {\\ell}}$$"
],
"text/plain": [
"1/4*sqrt(2)*(2*sqrt(B + 1)*arctan(sqrt(2)*l*y/sqrt(B + 1)) + sqrt(B - 1)*log((sqrt(2)*l*y/sqrt(B - 1) - 1)/(sqrt(2)*l*y/sqrt(B - 1) + 1)))/(B*l)"
]
},
"execution_count": 94,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s1 = sqrt(2)/(4*B*l)*(2*sqrt(B+1)*atan(sqrt(2)*l/sqrt(B+1)*y)\n",
" + sqrt(B-1)*ln((sqrt(2)*l/sqrt(B-1)*y - 1)/(sqrt(2)*l/sqrt(B-1)*y + 1)))\n",
"s1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Let us check that we have indeed a primitive of $y\\mapsto \\frac{y^2}{\\ell^2 y^4 + y^2 - 2m}$:"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, {\\ell}^{2} y^{2}}{4 \\, {\\ell}^{4} y^{4} + 4 \\, {\\ell}^{2} y^{2} - B^{2} + 1}$$"
],
"text/plain": [
"4*l^2*y^2/(4*l^4*y^4 + 4*l^2*y^2 - B^2 + 1)"
]
},
"execution_count": 95,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Ds1 = diff(s1, y).simplify_full()\n",
"Ds1"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$"
],
"text/plain": [
"y^2/(l^2*y^4 + y^2 - 2*m)"
]
},
"execution_count": 96,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Ds1.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Similarly, we can express $s_2$ in terms of $B$:"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B^{2} + B} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{B^{2} + B} m}{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{B^{2} + B} m}\\right) + \\sqrt{-B^{2} + B} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{-B^{2} + B} m}{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{-B^{2} + B} m}\\right)}{4 \\, B^{\\frac{3}{2}} \\sqrt{m}}$$"
],
"text/plain": [
"1/4*(sqrt(B^2 + B)*log(((B + 1)*sqrt(B)*sqrt(m)*y - 2*sqrt(B^2 + B)*m)/((B + 1)*sqrt(B)*sqrt(m)*y + 2*sqrt(B^2 + B)*m)) + sqrt(-B^2 + B)*log(((B - 1)*sqrt(B)*sqrt(m)*y - 2*sqrt(-B^2 + B)*m)/((B - 1)*sqrt(B)*sqrt(m)*y + 2*sqrt(-B^2 + B)*m)))/(B^(3/2)*sqrt(m))"
]
},
"execution_count": 97,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = s2.subs({l^2: (B^2 - 1)/(8*m)}).simplify_full()\n",
"s2"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Since $B>1$, we replace $\\sqrt{-B^2 + B}$ by $i\\sqrt{B}\\sqrt{B-1}$:"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B + 1} \\sqrt{B} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{B + 1} \\sqrt{B} m}{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{B + 1} \\sqrt{B} m}\\right) + i \\, \\sqrt{B - 1} \\sqrt{B} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 i \\, \\sqrt{B - 1} \\sqrt{B} m}{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 i \\, \\sqrt{B - 1} \\sqrt{B} m}\\right)}{4 \\, B^{\\frac{3}{2}} \\sqrt{m}}$$"
],
"text/plain": [
"1/4*(sqrt(B + 1)*sqrt(B)*log(((B + 1)*sqrt(B)*sqrt(m)*y - 2*sqrt(B + 1)*sqrt(B)*m)/((B + 1)*sqrt(B)*sqrt(m)*y + 2*sqrt(B + 1)*sqrt(B)*m)) + I*sqrt(B - 1)*sqrt(B)*log(((B - 1)*sqrt(B)*sqrt(m)*y - 2*I*sqrt(B - 1)*sqrt(B)*m)/((B - 1)*sqrt(B)*sqrt(m)*y + 2*I*sqrt(B - 1)*sqrt(B)*m)))/(B^(3/2)*sqrt(m))"
]
},
"execution_count": 98,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = s2.subs({sqrt(-B^2 + B): I*sqrt(B)*sqrt(B - 1), \n",
" sqrt(B^2 + B): sqrt(B)*sqrt(B + 1)})\n",
"s2"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B + 1} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{m} y - 2 \\, \\sqrt{B + 1} m}{{\\left(B + 1\\right)} \\sqrt{m} y + 2 \\, \\sqrt{B + 1} m}\\right) + i \\, \\sqrt{B - 1} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{m} y - 2 i \\, \\sqrt{B - 1} m}{{\\left(B - 1\\right)} \\sqrt{m} y + 2 i \\, \\sqrt{B - 1} m}\\right)}{4 \\, B \\sqrt{m}}$$"
],
"text/plain": [
"1/4*(sqrt(B + 1)*log(((B + 1)*sqrt(m)*y - 2*sqrt(B + 1)*m)/((B + 1)*sqrt(m)*y + 2*sqrt(B + 1)*m)) + I*sqrt(B - 1)*log(((B - 1)*sqrt(m)*y - 2*I*sqrt(B - 1)*m)/((B - 1)*sqrt(m)*y + 2*I*sqrt(B - 1)*m)))/(B*sqrt(m))"
]
},
"execution_count": 99,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = s2.simplify_log()\n",
"s2"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Again, we use the identity\n",
"$$\n",
"i \\ln\\left( \\frac{x + i}{x - i} \\right) = 2 \\mathrm{arctan}(x) - \\pi\n",
"$$\n",
"to rewrite $s_2$ as"
]
},
{
"cell_type": "code",
"execution_count": 100,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)}{4 \\, B \\sqrt{m}}$$"
],
"text/plain": [
"-1/4*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))/(B*sqrt(m))"
]
},
"execution_count": 100,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s2 = 1/(4*B*sqrt(m))*(sqrt(B+1)*ln( (sqrt(B+1)/(2*sqrt(m))*y - 1)\n",
" /(sqrt(B+1)/(2*sqrt(m))*y + 1) )\n",
" - 2*sqrt(B-1)*atan(sqrt(B-1)/(2*sqrt(m))*y))\n",
"s2"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Let us check that we have indeed a primitive of $y\\mapsto \\frac{1}{\\ell^2 y^4 + y^2 - 2m}$:"
]
},
{
"cell_type": "code",
"execution_count": 101,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{8 \\, m}{{\\left(B^{2} - 1\\right)} y^{4} + 8 \\, m y^{2} - 16 \\, m^{2}}$$"
],
"text/plain": [
"8*m/((B^2 - 1)*y^4 + 8*m*y^2 - 16*m^2)"
]
},
"execution_count": 101,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Ds2 = diff(s2, y).simplify_full()\n",
"Ds2"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$"
],
"text/plain": [
"1/(l^2*y^4 + y^2 - 2*m)"
]
},
"execution_count": 102,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Ds2.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Given the above expressions for $s_1(y)$ and $s_2(y)$ we rewrite the solution"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$$"
],
"text/plain": [
"_K2 - integrate((((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^2*y + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2)*l^2*y^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)/(l^2*y^4 + y^2 - 2*m), y) - _K1)/(l^2*y^4 + y^2 - 2*m), y)"
]
},
"execution_count": 103,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"mu1_sol(y)"
]
},
{
"cell_type": "code",
"execution_count": 104,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{C_{2}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}} + \\frac{{\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)} C_{1}}{4 \\, \\sqrt{-{\\mu_0}^{2} + 1} B {\\mu_0} \\sqrt{m}} + \\int \\frac{4 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y - \\frac{\\sqrt{2} {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)} {\\ell}}{B} + \\frac{{\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} {\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)}}{B \\sqrt{m}}}{4 \\, {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}}\\,{d y}$$"
],
"text/plain": [
"-C_2/(sqrt(-Mu0^2 + 1)*Mu0) + 1/4*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))*C_1/(sqrt(-Mu0^2 + 1)*B*Mu0*sqrt(m)) + integrate(1/4*(4*(beta1^2 - beta2^2)*l^2*y - sqrt(2)*(beta1^2 - beta2^2)*(2*sqrt(B + 1)*arctan(sqrt(2)*l*y/sqrt(B + 1)) + sqrt(B - 1)*log((sqrt(2)*l*y/sqrt(B - 1) - 1)/(sqrt(2)*l*y/sqrt(B - 1) + 1)))*l/B + (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))/(B*sqrt(m)))/(l^2*y^4 + y^2 - 2*m), y)"
]
},
"execution_count": 104,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"C1, C2 = var('C_1', 'C_2') \n",
"# mu1 / mu0(1-mu0^2) : \n",
"mu1s0 = - C2/(Mu0*sqrt(1-Mu0^2)) - C1/(Mu0*sqrt(1-Mu0^2))*s2 \\\n",
" + integrate(((beta1^2 - beta2^2)*l^2*y \n",
" - (beta1^2 - beta2^2)*l^2 * s1\n",
" - (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*(beta1-beta2))*l^2*m) * s2)\n",
" / (l^2*y^4 + y^2 - 2*m), \n",
" y, hold=True)\n",
"mu1s0"
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{4} \\, {\\left({\\mu_0}^{2} - 1\\right)} {\\mu_0} {\\left(\\frac{4 \\, C_{2}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}} - \\frac{{\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)} C_{1}}{\\sqrt{-{\\mu_0}^{2} + 1} B {\\mu_0} \\sqrt{m}} - 4 \\, \\int \\frac{4 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y - \\frac{\\sqrt{2} {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)} {\\ell}}{B} + \\frac{{\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} {\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)}}{B \\sqrt{m}}}{4 \\, {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}}\\,{d y}\\right)}$$"
],
"text/plain": [
"1/4*(Mu0^2 - 1)*Mu0*(4*C_2/(sqrt(-Mu0^2 + 1)*Mu0) - (2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))*C_1/(sqrt(-Mu0^2 + 1)*B*Mu0*sqrt(m)) - 4*integrate(1/4*(4*(beta1^2 - beta2^2)*l^2*y - sqrt(2)*(beta1^2 - beta2^2)*(2*sqrt(B + 1)*arctan(sqrt(2)*l*y/sqrt(B + 1)) + sqrt(B - 1)*log((sqrt(2)*l*y/sqrt(B - 1) - 1)/(sqrt(2)*l*y/sqrt(B - 1) + 1)))*l/B + (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))/(B*sqrt(m)))/(l^2*y^4 + y^2 - 2*m), y))"
]
},
"execution_count": 105,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"mu1_sol(y) = mu1s0 * Mu0*(1-Mu0^2)\n",
"mu1_sol(y)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Let us check that we do have a solution of the equation for $\\mu_1$:"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$$"
],
"text/plain": [
"0 == 0"
]
},
"execution_count": 106,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"eq_mu1.substitute_function(mu_1, mu1_sol).simplify_full().subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"### Conjugate momenta"
]
},
{
"cell_type": "code",
"execution_count": 107,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"def conjugate_momenta(lagr, qs, var):\n",
" r\"\"\"\n",
" Compute the conjugate momenta from a given Lagrangian.\n",
"\n",
" INPUT:\n",
"\n",
" - ``lagr`` -- symbolic expression representing the Lagrangian density\n",
" - ``qs`` -- either a single symbolic function or a list/tuple of\n",
" symbolic functions, representing the `q`'s; these functions must\n",
" appear in ``lagr`` up to at most their first derivatives\n",
" - ``var`` -- either a single variable, typically `t` (1-dimensional\n",
" problem) or a list/tuple of symbolic variables; in the latter case the\n",
" time coordinate must the first one\n",
"\n",
" OUTPUT:\n",
"\n",
" - list of conjugate momenta; if only one function is involved, the\n",
" single conjugate momentum is returned instead.\n",
"\n",
" \"\"\"\n",
" if not isinstance(qs, (list, tuple)):\n",
" qs = [qs]\n",
" if not isinstance(var, (list, tuple)):\n",
" var = [var]\n",
" n = len(qs)\n",
" d = len(var)\n",
" dqvt = [SR.var('qxxxx{}_t'.format(q)) for q in qs]\n",
" subs = {diff(qs[i](*var), var[0]): dqvt[i] for i in range(n)}\n",
" subs_inv = {dqvt[i]: diff(qs[i](*var), var[0]) for i in range(n)}\n",
" lg = lagr.substitute(subs)\n",
" ps = [diff(lg, dotq).simplify_full().substitute(subs_inv) for dotq in dqvt]\n",
" if n == 1:\n",
" return ps[0]\n",
" return ps"
]
},
{
"cell_type": "code",
"execution_count": 108,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-{\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right), {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\\right]$$"
],
"text/plain": [
"[-(Mu0^2 - 1)*a^2*beta1^2*l^2*y^4*diff(Phi_1(y), y) - (Mu0^2 - 1)*a^2*beta1^2*y^2*diff(Phi_1(y), y) + 2*(Mu0^2 - 1)*a^2*beta1^2*m*diff(Phi_1(y), y),\n",
" Mu0^2*a^2*beta2^2*l^2*y^4*diff(Psi_1(y), y) + Mu0^2*a^2*beta2^2*y^2*diff(Psi_1(y), y) - 2*Mu0^2*a^2*beta2^2*m*diff(Psi_1(y), y)]"
]
},
"execution_count": 108,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"pis = conjugate_momenta(L_a2, [Phi_1, Psi_1], y)\n",
"pis"
]
},
{
"cell_type": "code",
"execution_count": 109,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\mu_0}^{2} - 1\\right)} {\\mathcal{P}'} a {\\beta_1}$$"
],
"text/plain": [
"-(Mu0^2 - 1)*P*a*beta1"
]
},
"execution_count": 109,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"pi_phi_y = (pis[0]/(a*beta1)).substitute_function(Phi_1, Phi1_sol).simplify_full()\n",
"pi_phi_y"
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu_0}^{2} {\\mathcal{Q}'} a {\\beta_2}$$"
],
"text/plain": [
"Mu0^2*Q*a*beta2"
]
},
"execution_count": 110,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"pi_psi_y = (pis[1]/(a*beta2)).substitute_function(Psi_1, Psi1_sol).simplify_full()\n",
"pi_psi_y"
]
},
{
"cell_type": "code",
"execution_count": 111,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"pis4 = conjugate_momenta(L_a4, [Phi_1, Psi_1, mu_1], y)"
]
},
{
"cell_type": "code",
"execution_count": 112,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{4} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + a^{4} y^{2} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) - 2 \\, a^{4} m \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\mu_0}^{2} - 1}$$"
],
"text/plain": [
"-(a^4*l^2*y^4*diff(mu_1(y), y) + a^4*y^2*diff(mu_1(y), y) - 2*a^4*m*diff(mu_1(y), y))/(Mu0^2 - 1)"
]
},
"execution_count": 112,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"pis4[2]"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The quantity $\\pi_\\theta^y / (a^2 \\sin\\Theta_0\\cos\\Theta_0)$:"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(2 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + {\\left({\\mu_0} {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mu_0} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2} - 2 \\, {\\mu_0} {\\beta_1} + 2 \\, {\\mu_0} {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\log\\left(\\frac{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - 2 \\, m}{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + 2 \\, m}\\right)\\right)} \\sqrt{-{\\mu_0}^{2} + 1} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + {\\left({\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\log\\left(\\frac{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}\\right) - 2 \\, {\\left({\\mu_0} {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mu_0} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2} - 2 \\, {\\mu_0} {\\beta_1} + 2 \\, {\\mu_0} {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\arctan\\left(\\frac{y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{2 \\, \\sqrt{m}}\\right)\\right)} \\sqrt{-{\\mu_0}^{2} + 1} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - 4 \\, {\\left({\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{-{\\mu_0}^{2} + 1} {\\ell}^{2} \\sqrt{m} y - C_{1} \\sqrt{m}\\right)} \\sqrt{8 \\, {\\ell}^{2} m + 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0} \\sqrt{m}}$$"
],
"text/plain": [
"1/4*((2*(sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*l*sqrt(m)*arctan(sqrt(2)*l*y/sqrt(sqrt(8*l^2*m + 1) + 1)) + (Mu0*P^2*beta1^2 - Mu0*Q^2*beta2^2 - 2*(Mu0*beta1^2 - Mu0*beta2^2 - 2*Mu0*beta1 + 2*Mu0*beta2)*l^2*m)*log((sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) - 2*m)/(sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) + 2*m)))*sqrt(-Mu0^2 + 1)*sqrt(sqrt(8*l^2*m + 1) + 1) + ((sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*l*sqrt(m)*log((sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) - sqrt(8*l^2*m + 1) + 1)/(sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) + sqrt(8*l^2*m + 1) - 1)) - 2*(Mu0*P^2*beta1^2 - Mu0*Q^2*beta2^2 - 2*(Mu0*beta1^2 - Mu0*beta2^2 - 2*Mu0*beta1 + 2*Mu0*beta2)*l^2*m)*arctan(1/2*y*sqrt(sqrt(8*l^2*m + 1) - 1)/sqrt(m)))*sqrt(-Mu0^2 + 1)*sqrt(sqrt(8*l^2*m + 1) - 1) - 4*((Mu0*beta1^2 - Mu0*beta2^2)*sqrt(-Mu0^2 + 1)*l^2*sqrt(m)*y - C_1*sqrt(m))*sqrt(8*l^2*m + 1))/(sqrt(8*l^2*m + 1)*sqrt(-Mu0^2 + 1)*Mu0*sqrt(m))"
]
},
"execution_count": 113,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"pi_theta_y_a2sT0 = (- pis4[2] / (a^4*Mu0)).substitute_function(mu_1, mu1_sol).simplify_full()\n",
"pi_theta_y_a2sT0 = pi_theta_y_a2sT0.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()\n",
"pi_theta_y_a2sT0"
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{16 \\, B C_{1} m^{\\frac{3}{2}} - {\\left(2 \\, {\\left(2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_1} + {\\left(4 \\, {\\mu_0} {\\mathcal{P}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_1}^{2} - 2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_2} - {\\left(4 \\, {\\mu_0} {\\mathcal{Q}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} \\sqrt{B - 1} m \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - {\\left(2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_1} + {\\left(4 \\, {\\mu_0} {\\mathcal{P}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_1}^{2} - 2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_2} - {\\left(4 \\, {\\mu_0} {\\mathcal{Q}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} \\sqrt{B + 1} m \\log\\left(\\frac{\\sqrt{B + 1} \\sqrt{m} y - 2 \\, m}{\\sqrt{B + 1} \\sqrt{m} y + 2 \\, m}\\right) - 2 \\, {\\left(4 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{B + 1} {\\ell} m \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + 2 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{B - 1} {\\ell} m \\log\\left(\\frac{\\sqrt{2} \\sqrt{B - 1} {\\ell} y - B + 1}{\\sqrt{2} \\sqrt{B - 1} {\\ell} y + B - 1}\\right) - {\\left({\\left(B^{3} - B\\right)} {\\mu_0} {\\beta_1}^{2} - {\\left(B^{3} - B\\right)} {\\mu_0} {\\beta_2}^{2}\\right)} y\\right)} \\sqrt{m}\\right)} \\sqrt{-{\\mu_0}^{2} + 1}}{4 \\, m}$$"
],
"text/plain": [
"1/4*(16*B*C_1*m^(3/2) - (2*(2*(B^2 - 1)*Mu0*beta1 + (4*Mu0*P^2 - (B^2 - 1)*Mu0)*beta1^2 - 2*(B^2 - 1)*Mu0*beta2 - (4*Mu0*Q^2 - (B^2 - 1)*Mu0)*beta2^2)*sqrt(B - 1)*m*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - (2*(B^2 - 1)*Mu0*beta1 + (4*Mu0*P^2 - (B^2 - 1)*Mu0)*beta1^2 - 2*(B^2 - 1)*Mu0*beta2 - (4*Mu0*Q^2 - (B^2 - 1)*Mu0)*beta2^2)*sqrt(B + 1)*m*log((sqrt(B + 1)*sqrt(m)*y - 2*m)/(sqrt(B + 1)*sqrt(m)*y + 2*m)) - 2*(4*(sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*sqrt(B + 1)*l*m*arctan(sqrt(2)*l*y/sqrt(B + 1)) + 2*(sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*sqrt(B - 1)*l*m*log((sqrt(2)*sqrt(B - 1)*l*y - B + 1)/(sqrt(2)*sqrt(B - 1)*l*y + B - 1)) - ((B^3 - B)*Mu0*beta1^2 - (B^3 - B)*Mu0*beta2^2)*y)*sqrt(m))*sqrt(-Mu0^2 + 1))/m"
]
},
"execution_count": 114,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"pi_theta_y_a2sT0.numerator().subs({l^2: (B^2 - 1)/(8*m)}).simplify_full()"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0} \\sqrt{m}$$"
],
"text/plain": [
"4*sqrt(8*l^2*m + 1)*sqrt(-Mu0^2 + 1)*Mu0*sqrt(m)"
]
},
"execution_count": 115,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"pi_theta_y_a2sT0.denominator()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The quantity \n",
"$$\\frac{\\pi_\\theta^y}{(a^2/2) \\sin 2\\Theta_0} + (\\beta_1^2 - \\beta_2^2)\\ell^2 y - \\frac{C_1}{\\sin\\Theta_0\\cos\\Theta_0}$$"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(2 \\, {\\left(\\sqrt{2} {\\beta_1}^{2} - \\sqrt{2} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + {\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\log\\left(\\frac{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - 2 \\, m}{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + 2 \\, m}\\right)\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + {\\left({\\left(\\sqrt{2} {\\beta_1}^{2} - \\sqrt{2} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\log\\left(\\frac{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}\\right) - 2 \\, {\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\arctan\\left(\\frac{y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{2 \\, \\sqrt{m}}\\right)\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}}$$"
],
"text/plain": [
"1/4*((2*(sqrt(2)*beta1^2 - sqrt(2)*beta2^2)*l*sqrt(m)*arctan(sqrt(2)*l*y/sqrt(sqrt(8*l^2*m + 1) + 1)) + (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*log((sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) - 2*m)/(sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) + 2*m)))*sqrt(sqrt(8*l^2*m + 1) + 1) + ((sqrt(2)*beta1^2 - sqrt(2)*beta2^2)*l*sqrt(m)*log((sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) - sqrt(8*l^2*m + 1) + 1)/(sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) + sqrt(8*l^2*m + 1) - 1)) - 2*(P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*arctan(1/2*y*sqrt(sqrt(8*l^2*m + 1) - 1)/sqrt(m)))*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m))"
]
},
"execution_count": 116,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"part1 = - (beta1^2 - beta2^2)*l^2*y + C1/(Mu0*sqrt(1-Mu0^2))\n",
"s = (pi_theta_y_a2sT0 - part1).simplify_full()\n",
"s"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Let us perform an expansion in $1/y$ for $y\\rightarrow +\\infty$:"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y}$$"
],
"text/plain": [
"1/4*((sqrt(2)*pi*beta1^2 - sqrt(2)*pi*beta2^2)*l*sqrt(m)*sqrt(sqrt(8*l^2*m + 1) + 1) - (pi*P^2*beta1^2 - pi*Q^2*beta2^2 - 2*(pi*beta1^2 - pi*beta2^2 - 2*pi*beta1 + 2*pi*beta2)*l^2*m)*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m)) - (beta1^2 - beta2^2)/y"
]
},
"execution_count": 117,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"u = var('u')\n",
"assume(u > 0)\n",
"s = s.subs({y: 1/u}).simplify_log()\n",
"assume(l>0)\n",
"s = s.taylor(u, 0, 2)\n",
"s = s.subs({u: 1/y})\n",
"s"
]
},
{
"cell_type": "code",
"execution_count": 118,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y}$$"
],
"text/plain": [
"1/4*((sqrt(2)*pi*beta1^2 - sqrt(2)*pi*beta2^2)*l*sqrt(m)*sqrt(sqrt(8*l^2*m + 1) + 1) - (pi*P^2*beta1^2 - pi*Q^2*beta2^2 - 2*(pi*beta1^2 - pi*beta2^2 - 2*pi*beta1 + 2*pi*beta2)*l^2*m)*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m)) - (beta1^2 - beta2^2)/y"
]
},
"execution_count": 118,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"s"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Final result for $\\frac{\\pi_\\theta^y}{(a^2/2) \\sin 2\\Theta_0}$:"
]
},
{
"cell_type": "code",
"execution_count": 119,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y + \\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y} + \\frac{C_{1}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}}$$"
],
"text/plain": [
"-(beta1^2 - beta2^2)*l^2*y + 1/4*((sqrt(2)*pi*beta1^2 - sqrt(2)*pi*beta2^2)*l*sqrt(m)*sqrt(sqrt(8*l^2*m + 1) + 1) - (pi*P^2*beta1^2 - pi*Q^2*beta2^2 - 2*(pi*beta1^2 - pi*beta2^2 - 2*pi*beta1 + 2*pi*beta2)*l^2*m)*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m)) - (beta1^2 - beta2^2)/y + C_1/(sqrt(-Mu0^2 + 1)*Mu0)"
]
},
"execution_count": 119,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"part1 + s"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"The terms in $C_1$, $y$ and $y^{-1}$ agree with Eq. (4.37) of the paper."
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
}
],
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