{ "cells": [ { "cell_type": "markdown", "id": "c34f9e80-a668-46a3-87fa-2eca9143fcfc", "metadata": {}, "source": [ "# Killing-Yano form and Killing tensor in Kerr off shell" ] }, { "cell_type": "markdown", "id": "e2aabbfe-6563-4653-b244-65e8d532beac", "metadata": {}, "source": [ "This [SageMath](https://www.sagemath.org/) notebook accompanies the article *Black hole photon ring beyond General Relativity: an integrable parametrization* by J. Ben Achour, E. Gourgoulhon and H. Roussille, [arXiv:2506.09882](https://arxiv.org/abs/2506.09882). The notebook makes use of tools developed \n", "developed through the [SageManifolds project](https://sagemanifolds.obspm.fr)." ] }, { "cell_type": "code", "execution_count": 1, "id": "3b150d45-6fb6-460a-97ed-a9f7b5f81437", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 10.6, Release Date: 2025-03-31'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sage.version.banner" ] }, { "cell_type": "code", "execution_count": 2, "id": "b440f8dc-6354-4b74-ba68-f4af6bc02e41", "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "id": "82179bf1-6111-4bec-b0e5-aeede8386e64", "metadata": {}, "source": [ "## Spacetime manifold" ] }, { "cell_type": "code", "execution_count": 3, "id": "29b04f2c-357a-44fb-be30-0c1767fc05a5", "metadata": {}, "outputs": [], "source": [ "M = Manifold(4, 'M', structure='Lorentzian')" ] }, { "cell_type": "code", "execution_count": 4, "id": "9bb848ed-2b1b-4713-8aac-b6a7cfc42827", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left(M,({\\tau}, r, y, {\\varphi})\\right)\\)" ], "text/latex": [ "$\\displaystyle \\left(M,({\\tau}, r, y, {\\varphi})\\right)$" ], "text/plain": [ "Chart (M, (tau, r, y, phi))" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X. = M.chart(r'tau:\\tau r y phi:(0,2*pi):\\varphi')\n", "X" ] }, { "cell_type": "code", "execution_count": 5, "id": "b24614fb-f40d-4f1a-8484-14ef1744922c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left(\\mathrm{d} {\\tau}, \\mathrm{d} r, \\mathrm{d} y, \\mathrm{d} {\\varphi}\\right)\\)" ], "text/latex": [ "$\\displaystyle \\left(\\mathrm{d} {\\tau}, \\mathrm{d} r, \\mathrm{d} y, \\mathrm{d} {\\varphi}\\right)$" ], "text/plain": [ "(1-form dtau on the 4-dimensional Lorentzian manifold M,\n", " 1-form dr on the 4-dimensional Lorentzian manifold M,\n", " 1-form dy on the 4-dimensional Lorentzian manifold M,\n", " 1-form dphi on the 4-dimensional Lorentzian manifold M)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dtau, dr, dy, dphi = X.coframe()[:]\n", "dtau, dr, dy, dphi" ] }, { "cell_type": "markdown", "id": "e0d25925-40a2-451e-8af7-42c32ac2eb0c", "metadata": {}, "source": [ "Let us introduce two basic 1-forms:" ] }, { "cell_type": "code", "execution_count": 6, "id": "4f49e827-253c-4005-824b-509ebb507308", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\omega_1 = \\mathrm{d} {\\tau} + y^{2} \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle \\omega_1 = \\mathrm{d} {\\tau} + y^{2} \\mathrm{d} {\\varphi}$" ], "text/plain": [ "om1 = dtau + y^2 dphi" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "om1 = dtau + y^2*dphi\n", "om1.set_name('om1', latex_name=r'\\omega_1')\n", "om1.display()" ] }, { "cell_type": "code", "execution_count": 7, "id": "14aff732-9687-4fe8-a1ac-39def5f222a8", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\omega_2 = \\mathrm{d} {\\tau} -r^{2} \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle \\omega_2 = \\mathrm{d} {\\tau} -r^{2} \\mathrm{d} {\\varphi}$" ], "text/plain": [ "om2 = dtau - r^2 dphi" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "om2 = dtau - r^2*dphi\n", "om2.set_name('om2', latex_name=r'\\omega_2')\n", "om2.display()" ] }, { "cell_type": "markdown", "id": "784c8c5f-8c6d-4adc-bdeb-4393a87d941c", "metadata": {}, "source": [ "## Metric tensor" ] }, { "cell_type": "code", "execution_count": 8, "id": "33535f43-0fb0-4650-8c85-d83cc934f3f2", "metadata": {}, "outputs": [], "source": [ "Sigma = r^2 + y^2\n", "Dr = function('Dr', latex_name=r'\\Delta_r')\n", "Dy = function('Dy', latex_name=r'\\Delta_y')" ] }, { "cell_type": "code", "execution_count": 9, "id": "f0955cda-5807-4044-a18f-4a9a28eea0e2", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle g = \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi} + \\left( \\frac{r^{2} + y^{2}}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{r^{2} + y^{2}}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\tau} + \\left( -\\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle g = \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi} + \\left( \\frac{r^{2} + y^{2}}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{r^{2} + y^{2}}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\tau} + \\left( -\\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$" ], "text/plain": [ "g = -(Dr(r) - Dy(y))/(r^2 + y^2) dtau⊗dtau - (y^2*Dr(r) + r^2*Dy(y))/(r^2 + y^2) dtau⊗dphi + (r^2 + y^2)/Dr(r) dr⊗dr + (r^2 + y^2)/Dy(y) dy⊗dy - (y^2*Dr(r) + r^2*Dy(y))/(r^2 + y^2) dphi⊗dtau - (y^4*Dr(r) - r^4*Dy(y))/(r^2 + y^2) dphi⊗dphi" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = M.metric()\n", "g.set( - Dr(r)/Sigma*(om1*om1)\n", " + Dy(y)/Sigma*(om2*om2) \n", " + Sigma/Dr(r)*(dr*dr)\n", " + Sigma/Dy(y)*(dy*dy) )\n", "g.display()" ] }, { "cell_type": "code", "execution_count": 10, "id": "8626e991-f2ed-41ad-9066-399549488373", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle g^{-1} = \\left( \\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\tau} }\\otimes \\frac{\\partial}{\\partial {\\tau} } + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\tau} }\\otimes \\frac{\\partial}{\\partial {\\varphi} } + \\left( \\frac{\\Delta_r\\left(r\\right)}{r^{2} + y^{2}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\frac{\\partial}{\\partial r } + \\left( \\frac{\\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\frac{\\partial}{\\partial y }\\otimes \\frac{\\partial}{\\partial y } + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\frac{\\partial}{\\partial {\\tau} } + \\left( \\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\frac{\\partial}{\\partial {\\varphi} }\\)" ], "text/latex": [ "$\\displaystyle g^{-1} = \\left( \\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\tau} }\\otimes \\frac{\\partial}{\\partial {\\tau} } + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\tau} }\\otimes \\frac{\\partial}{\\partial {\\varphi} } + \\left( \\frac{\\Delta_r\\left(r\\right)}{r^{2} + y^{2}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\frac{\\partial}{\\partial r } + \\left( \\frac{\\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\frac{\\partial}{\\partial y }\\otimes \\frac{\\partial}{\\partial y } + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\frac{\\partial}{\\partial {\\tau} } + \\left( \\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{{\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\frac{\\partial}{\\partial {\\varphi} }$" ], "text/plain": [ "inv_g = (y^4*Dr(r) - r^4*Dy(y))/((r^2*Dr(r) + y^2*Dr(r))*Dy(y)) ∂/∂tau⊗∂/∂tau - (y^2*Dr(r) + r^2*Dy(y))/((r^2*Dr(r) + y^2*Dr(r))*Dy(y)) ∂/∂tau⊗∂/∂phi + Dr(r)/(r^2 + y^2) ∂/∂r⊗∂/∂r + Dy(y)/(r^2 + y^2) ∂/∂y⊗∂/∂y - (y^2*Dr(r) + r^2*Dy(y))/((r^2*Dr(r) + y^2*Dr(r))*Dy(y)) ∂/∂phi⊗∂/∂tau + (Dr(r) - Dy(y))/((r^2*Dr(r) + y^2*Dr(r))*Dy(y)) ∂/∂phi⊗∂/∂phi" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.inverse().display()" ] }, { "cell_type": "markdown", "id": "b233d418-728a-424f-baf8-c80c563b0648", "metadata": {}, "source": [ "## Levi-Civita connection" ] }, { "cell_type": "code", "execution_count": 11, "id": "e4dbc21a-e5cb-4d1e-bf0a-03a2c0eed42b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, {\\tau}} \\, {\\tau} \\, r }^{ \\, {\\tau} \\phantom{\\, {\\tau}} \\phantom{\\, r} } & = & \\frac{r^{4} \\frac{\\partial\\,\\Delta_r}{\\partial r} + r^{2} y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r^{3} \\Delta_r\\left(r\\right) + 2 \\, r^{3} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\tau}} \\, {\\tau} \\, y }^{ \\, {\\tau} \\phantom{\\, {\\tau}} \\phantom{\\, y} } & = & \\frac{2 \\, y^{3} \\Delta_r\\left(r\\right) - 2 \\, y^{3} \\Delta_y\\left(y\\right) + {\\left(r^{2} y^{2} + y^{4}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\tau}} \\, r \\, {\\varphi} }^{ \\, {\\tau} \\phantom{\\, r} \\phantom{\\, {\\varphi}} } & = & -\\frac{2 \\, r^{5} \\Delta_y\\left(y\\right) - {\\left(r^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)\\right)} y^{4} - {\\left(r^{4} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 4 \\, r^{3} \\Delta_r\\left(r\\right)\\right)} y^{2}}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\tau}} \\, y \\, {\\varphi} }^{ \\, {\\tau} \\phantom{\\, y} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, y^{5} \\Delta_r\\left(r\\right) + 2 \\, {\\left(r^{4} y + 2 \\, r^{2} y^{3}\\right)} \\Delta_y\\left(y\\right) - {\\left(r^{4} y^{2} + r^{2} y^{4}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\tau} \\, {\\tau} }^{ \\, r \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} } & = & \\frac{r^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)^{2} + 2 \\, r \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\tau} \\, {\\varphi} }^{ \\, r \\phantom{\\, {\\tau}} \\phantom{\\, {\\varphi}} } & = & \\frac{y^{4} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} + 2 \\, r y^{2} \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) + {\\left(r^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)^{2}\\right)} y^{2}}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & -\\frac{r^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)}{2 \\, {\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, y }^{ \\, r \\phantom{\\, r} \\phantom{\\, y} } & = & \\frac{y}{r^{2} + y^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, y \\, y }^{ \\, r \\phantom{\\, y} \\phantom{\\, y} } & = & -\\frac{r \\Delta_r\\left(r\\right)}{{\\left(r^{2} + y^{2}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\varphi} \\, {\\varphi} }^{ \\, r \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & \\frac{y^{6} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} + {\\left(r^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)^{2}\\right)} y^{4} - 2 \\, {\\left(r^{5} \\Delta_r\\left(r\\right) + 2 \\, r^{3} y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, {\\tau} \\, {\\tau} }^{ \\, y \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} } & = & -\\frac{2 \\, y \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) - 2 \\, y \\Delta_y\\left(y\\right)^{2} + {\\left(r^{2} + y^{2}\\right)} \\Delta_y\\left(y\\right) \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, {\\tau} \\, {\\varphi} }^{ \\, y \\phantom{\\, {\\tau}} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, r^{2} y \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) - 2 \\, r^{2} y \\Delta_y\\left(y\\right)^{2} + {\\left(r^{4} + r^{2} y^{2}\\right)} \\Delta_y\\left(y\\right) \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, r \\, r }^{ \\, y \\phantom{\\, r} \\phantom{\\, r} } & = & -\\frac{y \\Delta_y\\left(y\\right)}{r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)} \\\\ \\Gamma_{ \\phantom{\\, y} \\, r \\, y }^{ \\, y \\phantom{\\, r} \\phantom{\\, y} } & = & \\frac{r}{r^{2} + y^{2}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, y }^{ \\, y \\phantom{\\, y} \\phantom{\\, y} } & = & \\frac{2 \\, y \\Delta_y\\left(y\\right) - {\\left(r^{2} + y^{2}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{2} + y^{2}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, y} \\, {\\varphi} \\, {\\varphi} }^{ \\, y \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, r^{4} y \\Delta_y\\left(y\\right)^{2} - {\\left(r^{6} + r^{4} y^{2}\\right)} \\Delta_y\\left(y\\right) \\frac{\\partial\\,\\Delta_y}{\\partial y} + 2 \\, {\\left(2 \\, r^{2} y^{3} \\Delta_r\\left(r\\right) + y^{5} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, {\\tau} \\, r }^{ \\, {\\varphi} \\phantom{\\, {\\tau}} \\phantom{\\, r} } & = & \\frac{r^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right) + 2 \\, r \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, {\\tau} \\, y }^{ \\, {\\varphi} \\phantom{\\, {\\tau}} \\phantom{\\, y} } & = & -\\frac{2 \\, y \\Delta_r\\left(r\\right) - 2 \\, y \\Delta_y\\left(y\\right) + {\\left(r^{2} + y^{2}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, r \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, r} \\phantom{\\, {\\varphi}} } & = & \\frac{r^{2} y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{4} \\frac{\\partial\\,\\Delta_r}{\\partial r} + 2 \\, r^{3} \\Delta_r\\left(r\\right) - 2 \\, r^{3} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, y \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, y} \\phantom{\\, {\\varphi}} } & = & -\\frac{2 \\, y^{3} \\Delta_r\\left(r\\right) - 2 \\, y^{3} \\Delta_y\\left(y\\right) - {\\left(r^{4} + r^{2} y^{2}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\end{array}\\)" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, {\\tau}} \\, {\\tau} \\, r }^{ \\, {\\tau} \\phantom{\\, {\\tau}} \\phantom{\\, r} } & = & \\frac{r^{4} \\frac{\\partial\\,\\Delta_r}{\\partial r} + r^{2} y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r^{3} \\Delta_r\\left(r\\right) + 2 \\, r^{3} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\tau}} \\, {\\tau} \\, y }^{ \\, {\\tau} \\phantom{\\, {\\tau}} \\phantom{\\, y} } & = & \\frac{2 \\, y^{3} \\Delta_r\\left(r\\right) - 2 \\, y^{3} \\Delta_y\\left(y\\right) + {\\left(r^{2} y^{2} + y^{4}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\tau}} \\, r \\, {\\varphi} }^{ \\, {\\tau} \\phantom{\\, r} \\phantom{\\, {\\varphi}} } & = & -\\frac{2 \\, r^{5} \\Delta_y\\left(y\\right) - {\\left(r^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)\\right)} y^{4} - {\\left(r^{4} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 4 \\, r^{3} \\Delta_r\\left(r\\right)\\right)} y^{2}}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\tau}} \\, y \\, {\\varphi} }^{ \\, {\\tau} \\phantom{\\, y} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, y^{5} \\Delta_r\\left(r\\right) + 2 \\, {\\left(r^{4} y + 2 \\, r^{2} y^{3}\\right)} \\Delta_y\\left(y\\right) - {\\left(r^{4} y^{2} + r^{2} y^{4}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\tau} \\, {\\tau} }^{ \\, r \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} } & = & \\frac{r^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)^{2} + 2 \\, r \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\tau} \\, {\\varphi} }^{ \\, r \\phantom{\\, {\\tau}} \\phantom{\\, {\\varphi}} } & = & \\frac{y^{4} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} + 2 \\, r y^{2} \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) + {\\left(r^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)^{2}\\right)} y^{2}}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & -\\frac{r^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)}{2 \\, {\\left(r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, y }^{ \\, r \\phantom{\\, r} \\phantom{\\, y} } & = & \\frac{y}{r^{2} + y^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, y \\, y }^{ \\, r \\phantom{\\, y} \\phantom{\\, y} } & = & -\\frac{r \\Delta_r\\left(r\\right)}{{\\left(r^{2} + y^{2}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\varphi} \\, {\\varphi} }^{ \\, r \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & \\frac{y^{6} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} + {\\left(r^{2} \\Delta_r\\left(r\\right) \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right)^{2}\\right)} y^{4} - 2 \\, {\\left(r^{5} \\Delta_r\\left(r\\right) + 2 \\, r^{3} y^{2} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, {\\tau} \\, {\\tau} }^{ \\, y \\phantom{\\, {\\tau}} \\phantom{\\, {\\tau}} } & = & -\\frac{2 \\, y \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) - 2 \\, y \\Delta_y\\left(y\\right)^{2} + {\\left(r^{2} + y^{2}\\right)} \\Delta_y\\left(y\\right) \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, {\\tau} \\, {\\varphi} }^{ \\, y \\phantom{\\, {\\tau}} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, r^{2} y \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) - 2 \\, r^{2} y \\Delta_y\\left(y\\right)^{2} + {\\left(r^{4} + r^{2} y^{2}\\right)} \\Delta_y\\left(y\\right) \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, r \\, r }^{ \\, y \\phantom{\\, r} \\phantom{\\, r} } & = & -\\frac{y \\Delta_y\\left(y\\right)}{r^{2} \\Delta_r\\left(r\\right) + y^{2} \\Delta_r\\left(r\\right)} \\\\ \\Gamma_{ \\phantom{\\, y} \\, r \\, y }^{ \\, y \\phantom{\\, r} \\phantom{\\, y} } & = & \\frac{r}{r^{2} + y^{2}} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, y }^{ \\, y \\phantom{\\, y} \\phantom{\\, y} } & = & \\frac{2 \\, y \\Delta_y\\left(y\\right) - {\\left(r^{2} + y^{2}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{2} + y^{2}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, y} \\, {\\varphi} \\, {\\varphi} }^{ \\, y \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & \\frac{2 \\, r^{4} y \\Delta_y\\left(y\\right)^{2} - {\\left(r^{6} + r^{4} y^{2}\\right)} \\Delta_y\\left(y\\right) \\frac{\\partial\\,\\Delta_y}{\\partial y} + 2 \\, {\\left(2 \\, r^{2} y^{3} \\Delta_r\\left(r\\right) + y^{5} \\Delta_r\\left(r\\right)\\right)} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{6} + 3 \\, r^{4} y^{2} + 3 \\, r^{2} y^{4} + y^{6}\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, {\\tau} \\, r }^{ \\, {\\varphi} \\phantom{\\, {\\tau}} \\phantom{\\, r} } & = & \\frac{r^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} - 2 \\, r \\Delta_r\\left(r\\right) + 2 \\, r \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, {\\tau} \\, y }^{ \\, {\\varphi} \\phantom{\\, {\\tau}} \\phantom{\\, y} } & = & -\\frac{2 \\, y \\Delta_r\\left(r\\right) - 2 \\, y \\Delta_y\\left(y\\right) + {\\left(r^{2} + y^{2}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, r \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, r} \\phantom{\\, {\\varphi}} } & = & \\frac{r^{2} y^{2} \\frac{\\partial\\,\\Delta_r}{\\partial r} + y^{4} \\frac{\\partial\\,\\Delta_r}{\\partial r} + 2 \\, r^{3} \\Delta_r\\left(r\\right) - 2 \\, r^{3} \\Delta_y\\left(y\\right)}{2 \\, {\\left(r^{4} \\Delta_r\\left(r\\right) + 2 \\, r^{2} y^{2} \\Delta_r\\left(r\\right) + y^{4} \\Delta_r\\left(r\\right)\\right)}} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, y \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, y} \\phantom{\\, {\\varphi}} } & = & -\\frac{2 \\, y^{3} \\Delta_r\\left(r\\right) - 2 \\, y^{3} \\Delta_y\\left(y\\right) - {\\left(r^{4} + r^{2} y^{2}\\right)} \\frac{\\partial\\,\\Delta_y}{\\partial y}}{2 \\, {\\left(r^{4} + 2 \\, r^{2} y^{2} + y^{4}\\right)} \\Delta_y\\left(y\\right)} \\end{array}$" ], "text/plain": [ "Gam^tau_tau,r = 1/2*(r^4*d(Dr)/dr + r^2*y^2*d(Dr)/dr - 2*r^3*Dr(r) + 2*r^3*Dy(y))/(r^4*Dr(r) + 2*r^2*y^2*Dr(r) + y^4*Dr(r)) \n", "Gam^tau_tau,y = 1/2*(2*y^3*Dr(r) - 2*y^3*Dy(y) + (r^2*y^2 + y^4)*d(Dy)/dy)/((r^4 + 2*r^2*y^2 + y^4)*Dy(y)) \n", "Gam^tau_r,phi = -1/2*(2*r^5*Dy(y) - (r^2*d(Dr)/dr - 2*r*Dr(r))*y^4 - (r^4*d(Dr)/dr - 4*r^3*Dr(r))*y^2)/(r^4*Dr(r) + 2*r^2*y^2*Dr(r) + y^4*Dr(r)) \n", "Gam^tau_y,phi = 1/2*(2*y^5*Dr(r) + 2*(r^4*y + 2*r^2*y^3)*Dy(y) - (r^4*y^2 + r^2*y^4)*d(Dy)/dy)/((r^4 + 2*r^2*y^2 + y^4)*Dy(y)) \n", "Gam^r_tau,tau = 1/2*(r^2*Dr(r)*d(Dr)/dr + y^2*Dr(r)*d(Dr)/dr - 2*r*Dr(r)^2 + 2*r*Dr(r)*Dy(y))/(r^6 + 3*r^4*y^2 + 3*r^2*y^4 + y^6) \n", "Gam^r_tau,phi = 1/2*(y^4*Dr(r)*d(Dr)/dr + 2*r*y^2*Dr(r)*Dy(y) + (r^2*Dr(r)*d(Dr)/dr - 2*r*Dr(r)^2)*y^2)/(r^6 + 3*r^4*y^2 + 3*r^2*y^4 + y^6) \n", "Gam^r_r,r = -1/2*(r^2*d(Dr)/dr + y^2*d(Dr)/dr - 2*r*Dr(r))/(r^2*Dr(r) + y^2*Dr(r)) \n", "Gam^r_r,y = y/(r^2 + y^2) \n", "Gam^r_y,y = -r*Dr(r)/((r^2 + y^2)*Dy(y)) \n", "Gam^r_phi,phi = 1/2*(y^6*Dr(r)*d(Dr)/dr + (r^2*Dr(r)*d(Dr)/dr - 2*r*Dr(r)^2)*y^4 - 2*(r^5*Dr(r) + 2*r^3*y^2*Dr(r))*Dy(y))/(r^6 + 3*r^4*y^2 + 3*r^2*y^4 + y^6) \n", "Gam^y_tau,tau = -1/2*(2*y*Dr(r)*Dy(y) - 2*y*Dy(y)^2 + (r^2 + y^2)*Dy(y)*d(Dy)/dy)/(r^6 + 3*r^4*y^2 + 3*r^2*y^4 + y^6) \n", "Gam^y_tau,phi = 1/2*(2*r^2*y*Dr(r)*Dy(y) - 2*r^2*y*Dy(y)^2 + (r^4 + r^2*y^2)*Dy(y)*d(Dy)/dy)/(r^6 + 3*r^4*y^2 + 3*r^2*y^4 + y^6) \n", "Gam^y_r,r = -y*Dy(y)/(r^2*Dr(r) + y^2*Dr(r)) \n", "Gam^y_r,y = r/(r^2 + y^2) \n", "Gam^y_y,y = 1/2*(2*y*Dy(y) - (r^2 + y^2)*d(Dy)/dy)/((r^2 + y^2)*Dy(y)) \n", "Gam^y_phi,phi = 1/2*(2*r^4*y*Dy(y)^2 - (r^6 + r^4*y^2)*Dy(y)*d(Dy)/dy + 2*(2*r^2*y^3*Dr(r) + y^5*Dr(r))*Dy(y))/(r^6 + 3*r^4*y^2 + 3*r^2*y^4 + y^6) \n", "Gam^phi_tau,r = 1/2*(r^2*d(Dr)/dr + y^2*d(Dr)/dr - 2*r*Dr(r) + 2*r*Dy(y))/(r^4*Dr(r) + 2*r^2*y^2*Dr(r) + y^4*Dr(r)) \n", "Gam^phi_tau,y = -1/2*(2*y*Dr(r) - 2*y*Dy(y) + (r^2 + y^2)*d(Dy)/dy)/((r^4 + 2*r^2*y^2 + y^4)*Dy(y)) \n", "Gam^phi_r,phi = 1/2*(r^2*y^2*d(Dr)/dr + y^4*d(Dr)/dr + 2*r^3*Dr(r) - 2*r^3*Dy(y))/(r^4*Dr(r) + 2*r^2*y^2*Dr(r) + y^4*Dr(r)) \n", "Gam^phi_y,phi = -1/2*(2*y^3*Dr(r) - 2*y^3*Dy(y) - (r^4 + r^2*y^2)*d(Dy)/dy)/((r^4 + 2*r^2*y^2 + y^4)*Dy(y)) " ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla = g.connection()\n", "nabla.display(only_nonredundant=True)" ] }, { "cell_type": "markdown", "id": "cb8d5ec7-cfae-42c4-af50-620006ca46f0", "metadata": {}, "source": [ "## Conformal Killing-Yano 2-form $p$" ] }, { "cell_type": "code", "execution_count": 12, "id": "d99edfe6-051e-4c54-9ec6-848abccd007c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle p = r \\mathrm{d} {\\tau}\\wedge \\mathrm{d} r -y \\mathrm{d} {\\tau}\\wedge \\mathrm{d} y -r y^{2} \\mathrm{d} r\\wedge \\mathrm{d} {\\varphi} -r^{2} y \\mathrm{d} y\\wedge \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle p = r \\mathrm{d} {\\tau}\\wedge \\mathrm{d} r -y \\mathrm{d} {\\tau}\\wedge \\mathrm{d} y -r y^{2} \\mathrm{d} r\\wedge \\mathrm{d} {\\varphi} -r^{2} y \\mathrm{d} y\\wedge \\mathrm{d} {\\varphi}$" ], "text/plain": [ "p = r dtau∧dr - y dtau∧dy - r*y^2 dr∧dphi - r^2*y dy∧dphi" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = y*dy.wedge(om2) - r*dr.wedge(om1)\n", "p.set_name('p')\n", "p.display()" ] }, { "cell_type": "markdown", "id": "afdc7126-c618-43bb-9b3b-88bea25f13f9", "metadata": {}, "source": [ "$p$ is a closed 2-form:" ] }, { "cell_type": "code", "execution_count": 13, "id": "bf1c18e7-66a7-437e-a42a-5f9f584f82c6", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\mathrm{d}p = 0\\)" ], "text/latex": [ "$\\displaystyle \\mathrm{d}p = 0$" ], "text/plain": [ "dp = 0" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(p).display()" ] }, { "cell_type": "markdown", "id": "c9f75520-cda4-40b9-a484-46cc15f89342", "metadata": {}, "source": [ "The covariant derivative of $p$:" ] }, { "cell_type": "code", "execution_count": 14, "id": "e98080b7-0277-4184-8056-c8d553e8b52f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\nabla_{g} p = \\left( \\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} y\\otimes \\mathrm{d} y + \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi} + \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} r + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r + \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} y -\\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi} + \\left( \\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} y\\otimes \\mathrm{d} y\\)" ], "text/latex": [ "$\\displaystyle \\nabla_{g} p = \\left( \\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} y\\otimes \\mathrm{d} y + \\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi} + \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} r + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r + \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} y + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} y -\\Delta_r\\left(r\\right) \\Delta_y\\left(y\\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi} + \\left( \\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} y\\otimes \\mathrm{d} y$" ], "text/plain": [ "nabla_g(p) = (Dr(r) - Dy(y))/Dr(r) dtau⊗dr⊗dr + (Dr(r) - Dy(y))/Dy(y) dtau⊗dy⊗dy + Dr(r)*Dy(y) dtau⊗dphi⊗dphi - (Dr(r) - Dy(y))/Dr(r) dr⊗dtau⊗dr - (y^2*Dr(r) + r^2*Dy(y))/Dr(r) dr⊗dphi⊗dr - (Dr(r) - Dy(y))/Dy(y) dy⊗dtau⊗dy - (y^2*Dr(r) + r^2*Dy(y))/Dy(y) dy⊗dphi⊗dy - Dr(r)*Dy(y) dphi⊗dtau⊗dphi + (y^2*Dr(r) + r^2*Dy(y))/Dr(r) dphi⊗dr⊗dr + (y^2*Dr(r) + r^2*Dy(y))/Dy(y) dphi⊗dy⊗dy" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab_p = nabla(p)\n", "nab_p.display()" ] }, { "cell_type": "markdown", "id": "39ef6500-9ec3-4d22-a094-2e3ffa432bda", "metadata": {}, "source": [ "The 1-form $h_a = \\frac{1}{3} \\nabla_b p^b_{\\ \\, a}$ (note that we taking the trace on the indices 0 and 2, i.e. first and last, because\n", "$\\nabla_c p^b_{\\ \\, a} = (\\nabla p^\\sharp)^b_{\\ \\, ac}$):" ] }, { "cell_type": "code", "execution_count": 15, "id": "0413a216-110c-43af-8082-32c1ae9f63ae", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle h = \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau} + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle h = \\left( -\\frac{\\Delta_r\\left(r\\right) - \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau} + \\left( -\\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}$" ], "text/plain": [ "h = -(Dr(r) - Dy(y))/(r^2 + y^2) dtau - (y^2*Dr(r) + r^2*Dy(y))/(r^2 + y^2) dphi" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h = 1/3*nab_p.up(g,0).trace(0,2)\n", "h.set_name('h')\n", "h.display()" ] }, { "cell_type": "markdown", "id": "459cad0b-9984-4532-b3ee-102209cdc4e7", "metadata": {}, "source": [ "Let us check that $p$ satisfies the **conformal Killing-Yano equation**, taking into account that $\\nabla_c p_{ab} =(\\nabla p)_{abc}$," ] }, { "cell_type": "code", "execution_count": 16, "id": "3f94c04f-bc09-4873-b712-5d6297e1f078", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\mathrm{True}\\)" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab_p == -2*(h*g).antisymmetrize(0,1)" ] }, { "cell_type": "markdown", "id": "5693cc17-3891-4047-b409-4d818fa6e4f5", "metadata": {}, "source": [ "## Killing-Yano 2-form $f$" ] }, { "cell_type": "markdown", "id": "73aa0188-a47d-4b66-bf11-3083ca21006c", "metadata": {}, "source": [ "$f$ is obtained as the Hodge dual of $p$:" ] }, { "cell_type": "code", "execution_count": 17, "id": "4d62c47e-a7cc-4110-b0e3-fd758a4eeed7", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle f = -y \\mathrm{d} {\\tau}\\wedge \\mathrm{d} r -r \\mathrm{d} {\\tau}\\wedge \\mathrm{d} y + y^{3} \\mathrm{d} r\\wedge \\mathrm{d} {\\varphi} -r^{3} \\mathrm{d} y\\wedge \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle f = -y \\mathrm{d} {\\tau}\\wedge \\mathrm{d} r -r \\mathrm{d} {\\tau}\\wedge \\mathrm{d} y + y^{3} \\mathrm{d} r\\wedge \\mathrm{d} {\\varphi} -r^{3} \\mathrm{d} y\\wedge \\mathrm{d} {\\varphi}$" ], "text/plain": [ "f = -y dtau∧dr - r dtau∧dy + y^3 dr∧dphi - r^3 dy∧dphi" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = p.hodge_dual(g)\n", "f.set_name('f')\n", "f.display()" ] }, { "cell_type": "markdown", "id": "30a2ea7f-9484-43f1-b557-bd07519d0730", "metadata": {}, "source": [ "Check that $f$ obeys the **Killing-Yano equation**, i.e. $\\nabla_{(a} f_{b)c} = 0$" ] }, { "cell_type": "code", "execution_count": 18, "id": "285efb39-5dd3-4a74-9c3f-6ecc97bc8f87", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle 0\\)" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab_f = nabla(f)\n", "nab_f.symmetrize(0,2).display()" ] }, { "cell_type": "markdown", "id": "0c64ecca-6e0d-4eca-92cf-ac878967d04f", "metadata": {}, "source": [ "Equivalently, we have $\\nabla_{[a} f_{bc]} = \\nabla_{a} f_{bc}$:" ] }, { "cell_type": "code", "execution_count": 19, "id": "9f0a7395-f3e2-48fa-bf1d-ef2ccfe508cd", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\mathrm{True}\\)" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nab_f.antisymmetrize() == nab_f" ] }, { "cell_type": "markdown", "id": "1ea2cb73-7800-4357-89da-3008603d8f42", "metadata": {}, "source": [ "Note that, contrary to $p$, $f$ is not a closed 2-form:" ] }, { "cell_type": "code", "execution_count": 20, "id": "6a761578-4bfb-4e61-896d-bfb8a8e555ca", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\mathrm{d}f = \\left( -3 \\, r^{2} - 3 \\, y^{2} \\right) \\mathrm{d} r\\wedge \\mathrm{d} y\\wedge \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle \\mathrm{d}f = \\left( -3 \\, r^{2} - 3 \\, y^{2} \\right) \\mathrm{d} r\\wedge \\mathrm{d} y\\wedge \\mathrm{d} {\\varphi}$" ], "text/plain": [ "df = (-3*r^2 - 3*y^2) dr∧dy∧dphi" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(f).display()" ] }, { "cell_type": "markdown", "id": "630721c9-4469-4a50-919c-7105957a8112", "metadata": {}, "source": [ "## Killing tensor $K$" ] }, { "cell_type": "markdown", "id": "d6e4dc5d-8804-4015-8b39-9a191eb254b7", "metadata": {}, "source": [ "We define $K$ as the square of $f$, i.e. $K_{ab} = f_{ac} f_b^{\\ \\, c}$:" ] }, { "cell_type": "code", "execution_count": 21, "id": "2b3192d2-1038-4fff-b811-a48a460b5de2", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle K = \\left( \\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\left( \\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi} + \\left( -\\frac{r^{2} y^{2} + y^{4}}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{r^{4} + r^{2} y^{2}}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( \\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\tau} + \\left( \\frac{y^{6} \\Delta_r\\left(r\\right) + r^{6} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\)" ], "text/latex": [ "$\\displaystyle K = \\left( \\frac{y^{2} \\Delta_r\\left(r\\right) + r^{2} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\tau} + \\left( \\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\tau}\\otimes \\mathrm{d} {\\varphi} + \\left( -\\frac{r^{2} y^{2} + y^{4}}{\\Delta_r\\left(r\\right)} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( \\frac{r^{4} + r^{2} y^{2}}{\\Delta_y\\left(y\\right)} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( \\frac{y^{4} \\Delta_r\\left(r\\right) - r^{4} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\tau} + \\left( \\frac{y^{6} \\Delta_r\\left(r\\right) + r^{6} \\Delta_y\\left(y\\right)}{r^{2} + y^{2}} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$" ], "text/plain": [ "K = (y^2*Dr(r) + r^2*Dy(y))/(r^2 + y^2) dtau⊗dtau + (y^4*Dr(r) - r^4*Dy(y))/(r^2 + y^2) dtau⊗dphi - (r^2*y^2 + y^4)/Dr(r) dr⊗dr + (r^4 + r^2*y^2)/Dy(y) dy⊗dy + (y^4*Dr(r) - r^4*Dy(y))/(r^2 + y^2) dphi⊗dtau + (y^6*Dr(r) + r^6*Dy(y))/(r^2 + y^2) dphi⊗dphi" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K = f['_ac']*f.up(g,1)['_b^c']\n", "K.set_name('K')\n", "K.display()" ] }, { "cell_type": "markdown", "id": "3c70d1d5-c177-42fd-8693-d0b694d1dd15", "metadata": {}, "source": [ "Check of Eq. (1.9) of the article:" ] }, { "cell_type": "code", "execution_count": 22, "id": "d97ed1dd-2227-4c34-ac8f-1cad33fda658", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\mathrm{True}\\)" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K == (y^2/Sigma*Dr(r)*om1*om1 + r^2/Sigma*Dy(y)*om2*om2 + Sigma*r^2/Dy(y)*dy*dy \n", " - Sigma*y^2/Dr(r)*dr*dr) " ] }, { "cell_type": "markdown", "id": "3e6b550f-3a8d-403f-a259-ea03fd89e5cd", "metadata": {}, "source": [ "Let us check that $K$ obeys the **Killing equation**, namely \n", "$$ \\nabla_{(a} K_{bc)} = 0$$" ] }, { "cell_type": "code", "execution_count": 23, "id": "380774f2-7bff-4df9-bbda-bf361613a1fa", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle 0\\)" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla(K).symmetrize().display()" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 10.6", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.3" } }, "nbformat": 4, "nbformat_minor": 5 }