MEXICOPAS2023.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f9ecc62c",
   "metadata": {},
   "source": [
    "# Sombra de Agujeros Negros"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba26128e",
   "metadata": {},
   "source": [
    "El presente cuadernillo permite determinar el contorno de la sombra de agujeros negros estáticos, simétricamente esféricos, es decir, soluciones de agujeros negros que pueden ser escritos con la siguiente estructura métrica, \n",
    "\n",
    "\\begin{equation}\n",
    "ds^2 = -\\frac{K(r)}{r^2}dt^2+\\frac{r^2}{K(r)}dr^2+r^2d\\theta^2+r^2\\sin{(\\theta)}^2d\\phi^2.\n",
    "\\end{equation}\n",
    "\n",
    "En partícular vamos a estudiar el caso de los agujeros negros de Schwarzschild y Reissner-Nordström, estas dos soluciones tienen la siguiente función métrica $K(r)$\n",
    "\n",
    "\\begin{eqnarray}\n",
    "&& K(r) = r^2 - 2Mr,\\quad\\textit{Schwarzschild}.\\\\\n",
    "&& K(r) = r^2 - 2Mr + q^2,\\quad\\textit{Reissner-Nordström}.\n",
    "\\end{eqnarray}\n",
    "\n",
    "donde $M$ y $q$ son la masa y la carga eléctrica, respectivamente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "af210acb",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Librerías\n",
    "import numpy as np\n",
    "from scipy import optimize\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "457af64a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Función Métrica\n",
    "def K(r, M, q=0):\n",
    "    if( q==0 ):\n",
    "        # Schwarzschild\n",
    "        return r**2 - 2*M*r\n",
    "    else: \n",
    "        # Reissner-Nordström\n",
    "        return r**2 - 2*M*r + q**2\n",
    "\n",
    "# M es la masa\n",
    "# q es la carga eléctrica.\n",
    "\n",
    "# Derivada de la función Métrica\n",
    "def dK(r, M):\n",
    "    F = 2*r - 2*M\n",
    "    return F\n",
    "\n",
    "# Potencial Efectivo\n",
    "def Veff(r, M, l, q=0):\n",
    "    F = K(r, M, q ) * ( l**2 / r**4 )\n",
    "    return F\n",
    "\n",
    "# l Momento angular órbital\n",
    "\n",
    "# Parámetro de Impacto \n",
    "def eta(r, M, q):\n",
    "    F = r**2/np.sqrt(K(r, M, q))\n",
    "    return F\n",
    "\n",
    "# Contorno de la sombra\n",
    "def Y(X, rc, M, q=0):\n",
    "    if (rc == rcsch and q==0):\n",
    "        return eta(rcsch, M, 0)**2 - X**2\n",
    "    else:\n",
    "        return eta(rcrn, M, q)**2 - X**2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d767b854",
   "metadata": {},
   "source": [
    "El potencial potencial efectivo para el movimiento de los fotones en el plano ecuatorial de agujeros negros estáticos con simetría esférica está dado por\n",
    "\n",
    "\\begin{equation}\n",
    "V_{eff}^{2}(r)=\\frac{l^2}{r^4}K(r),\n",
    "\\end{equation}\n",
    "\n",
    "donde $l$ es el momento angular orbital del fotón. La forma de este potencial se muestra en la siguiente gráfica:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "16dda642",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Parámetros físicos\n",
    "M = 1.0\n",
    "l = 4.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7ef6175c",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Gráfica del Potencial\n",
    "r = np.linspace( 0.0, 25, 100000 )\n",
    "fig, ax = plt.subplots(figsize=(8, 6))\n",
    "plt.plot( r, Veff(r, M, l, 0) , color = 'blue', label = 'Schwarzschild')\n",
    "#plt.plot( r, Veff(r, M, l, q), color = 'red', label = 'Reissner-Nordström' )\n",
    "ax.set_xlim( 0.0, 15.0 )\n",
    "ax.set_ylim( 0.0, 0.7 )\n",
    "ax.set_xlabel('$r$')\n",
    "ax.set_ylabel('$V^{2}_{eff}$', rotation=0)\n",
    "ax.set_title(\"Potencial Efectivo\", va='bottom')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acd5859a",
   "metadata": {},
   "source": [
    "De la gráfica se observa la existencia de un punto máximo, estos puntos están relacionados con la existencia de órbitas circulares inestables, para fotones en este punto se localiza la foton-esfera. Para que un fotón se mantenga en estas órbitas se debe de cumplir las siguientes condiciones:\n",
    "\n",
    "\\begin{eqnarray}\n",
    "&& \\dot{r}=0\\Rightarrow \\mathcal{E}^{2}-V_{eff}^2(r_{c}) = 0,\\quad\\rightarrow\\quad\\eta^{2}_{c}=\\frac{l^2}{\\mathcal{E}^2}=\\frac{r^4_{c}}{K(r_{c})},\\quad\\textit{Parámetro de Impacto.}\\\\\n",
    "&& \\ddot{r}=0\\Rightarrow \\frac{dV_{eff}^{2}(r_{c})}{dr}=0,\\quad\\textit{Posición del Máximo del Potencial.}\n",
    "\\end{eqnarray}\n",
    "\n",
    "De la primera condición se obtiene el parámetro de impacto para fotones, mientras que de la segunda condición se determina la localización de la órbita circular inestable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "9815d8df",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Punto máximo del potencial \n",
    "def dV(r, M, l, q=0 ):\n",
    "    if ( q==0 ):\n",
    "        return -(4*l**2/r**5)*K(r, M, 0) + (l**2/r**4)*dK(r, M)\n",
    "    else: \n",
    "        return -(4*l**2/r**5)*K(r, M, 0.45 ) + (l**2/r**4)*dK(r, M)\n",
    "\n",
    "r0 = 2.0\n",
    "rcsch = optimize.newton(dV, r0, args=(M, l, 0))\n",
    "#rcrn = optimize.newton(dV, r0, args=(M, l, q))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b26c5ea",
   "metadata": {},
   "source": [
    "Es necesario determinar las coordenadas celestiales, las cuales nos dará la proyección de la sombra del agujero negro en le cielo vista por un observador en el infinito, \n",
    "\n",
    "\\begin{equation}\n",
    "X=\\lim_{r_{0}\\rightarrow\\infty}\\left(-r_{0}^{2}\\sin{(\\theta_{0})}\\left.\\frac{d\\phi}{dr}\\right|_{\\theta=\\theta_{0}}\\right),\\qquad Y=\\lim_{r_{0}\\rightarrow\\infty}\\left(r_{0}^2\\left.\\frac{d\\theta}{dr}\\right|_{\\theta=\\theta_{0}}\\right).\n",
    "\\end{equation}\n",
    "\n",
    "Contorno de la sombra está dado por, \n",
    "\n",
    "\\begin{equation}\n",
    "X^2+Y^2=R_{s}^2=\\eta_{c}^2.\n",
    "\\end{equation}\n",
    "\n",
    "Observamos que para el caso de agujeros negros estáticos, simétricamente esféricos el radio de la sombra es igual al valor del parámetro de impacto en el radio de la foto-esfera. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "87046ee0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = np.linspace( -2*np.pi, 2*np.pi, 1000000 )\n",
    "fig, ax = plt.subplots(figsize=(8, 8))\n",
    "plt.plot( X , np.sqrt(Y(X, rcsch, M, 0)) , color = 'blue', label = 'Schwarzschild' )\n",
    "plt.plot( X , -np.sqrt(Y(X, rcsch, M, 0)) , color = 'blue' )\n",
    "#plt.plot( X , np.sqrt(Y(X, rcrn, M, q)) , color = 'red', label = 'Reissner-Nordström' )\n",
    "#plt.plot( X , -np.sqrt(Y(X, rcrn, M, q)) , color = 'red' )\n",
    "ax.set_xlabel('$X$')\n",
    "ax.set_ylabel('$Y$', rotation=0)\n",
    "ax.set_title(\"Sombra de Agujero Negro\", va='bottom')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6e293a0b",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8acb8270",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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