A couple metrics on generalrepytivity.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Importing the library "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import generalrepytivity as gr\n",
    "import sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sympy.init_printing(use_latex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Creating tensors "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "t, x, y, z = sympy.symbols('t x y z')\n",
    "values = {\n",
    "    ((0,1), (1, 3)): x**2 + y**2,\n",
    "    ((1,2), (0, 0)): sympy.sin(t)\n",
    "}\n",
    "\n",
    "Gamma = gr.Tensor([t, x, y, z], (2,2), values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(x^{2} + y^{2})\\partial/\\partial t \\otimes \\partial/\\partial x \\otimes dx \\otimes dz + (\\sin{\\left (t \\right )})\\partial/\\partial x \\otimes \\partial/\\partial y \\otimes dt \\otimes dt$"
      ],
      "text/plain": [
       "(x**2 + y**2)t \\otimes x \\otimes x* \\otimes z* + (sin(t))x \\otimes y \\otimes t* \\otimes t*"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "Gamma[(0,1), (1,3)] == x**2 + y**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Metrics "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gödel "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(e^{x_{1}})dx_0 \\otimes dx_2 + (1)dx_0 \\otimes dx_0 + (-1)dx_1 \\otimes dx_1 + (\\frac{e^{2 x_{1}}}{2})dx_2 \\otimes dx_2 + (e^{x_{1}})dx_2 \\otimes dx_0 + (-1)dx_3 \\otimes dx_3$"
      ],
      "text/plain": [
       "(exp(x_1))x_0* \\otimes x_2* + (1)x_0* \\otimes x_0* + (-1)x_1* \\otimes x_1* + (exp(2*x_1)/2)x_2* \\otimes x_2* + (exp(x_1))x_2* \\otimes x_0* + (-1)x_3* \\otimes x_3*"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3')\n",
    "a = sympy.Symbol('a')\n",
    "A = sympy.Matrix([\n",
    "    [1, 0, sympy.exp(x1), 0],\n",
    "    [0, -1, 0, 0],\n",
    "    [sympy.exp(x1), 0, sympy.exp(2*x1)/2, 0],\n",
    "    [0,0,0,-1]])\n",
    "\n",
    "g_godel = gr.get_tensor_from_matrix(A, [x0, x1, x2, x3])\n",
    "g_godel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Schwarzschild "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(r^{2})d\\theta \\otimes d\\theta + (\\frac{1}{- \\frac{2 m}{r} + 1})dr \\otimes dr + (\\frac{2 m}{r} - 1)dt \\otimes dt + (r^{2} \\sin^{2}{\\left (\\theta \\right )})d\\phi \\otimes d\\phi$"
      ],
      "text/plain": [
       "(r**2)\\theta* \\otimes \\theta* + (1/(-2*m/r + 1))r* \\otimes r* + (2*m/r - 1)t* \\otimes t* + (r**2*sin(\\theta)**2)\\phi* \\otimes \\phi*"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t, r, theta, phi = sympy.symbols('t r \\\\theta \\\\phi')\n",
    "m = sympy.Symbol('m')\n",
    "values = {\n",
    "    ((), (0,0)): -(1-2*m/r),\n",
    "    ((), (1,1)): 1/(1-2*m/r),\n",
    "    ((), (2,2)): r**2,\n",
    "    ((), (3,3)): r**2*sympy.sin(theta)**2\n",
    "}\n",
    "g_sch = gr.Tensor([t, r, theta, phi], (0, 2), values)\n",
    "g_sch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## FRLW "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(\\epsilon^{2} x_{0}^{2 q})dx_2 \\otimes dx_2 + (\\epsilon^{2} x_{0}^{2 q})dx_1 \\otimes dx_1 + (-1)dx_0 \\otimes dx_0 + (\\epsilon^{2} x_{0}^{2 q})dx_3 \\otimes dx_3$"
      ],
      "text/plain": [
       "(\\epsilon**2*x_0**(2*q))x_2* \\otimes x_2* + (\\epsilon**2*x_0**(2*q))x_1* \\otimes x_1* + (-1)x_0* \\otimes x_0* + (\\epsilon**2*x_0**(2*q))x_3* \\otimes x_3*"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3')\n",
    "epsilon, q = sympy.symbols('\\\\epsilon q')\n",
    "A = epsilon * (x0 ** q)\n",
    "g_matrix = sympy.diag(-1, A**2, A**2, A**2)\n",
    "g_FRLW = gr.get_tensor_from_matrix(g_matrix, [x0, x1, x2, x3])\n",
    "g_FRLW"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# `Spacetime` objects "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "Godel_spacetime = gr.Spacetime(g_godel)\n",
    "Sch_spacetime = gr.Spacetime(g_sch)\n",
    "FRLW_spacetime = gr.Spacetime(g_FRLW)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sch_spacetime.Ric == 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Printing a summary in LaTeX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "Godel_spacetime.print_summary('godel.tex', _format='tex')"
   ]
  }
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