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   "source": [
    "# Riemann Curvature of the Schwarzschild Metric\n",
    "\n",
    "Outside a static spherical mass $M$, let\n",
    "\n",
    "$$\n",
    "A(r)=1-\\frac{2GM}{c^2r}.\n",
    "$$\n",
    "\n",
    "With signature $(+---)$ and coordinates $(t,r,\\theta,\\phi)$, the line\n",
    "element is\n",
    "\n",
    "$$\n",
    "ds^2=A\\,dt^2-A^{-1}dr^2-r^2d\\theta^2-r^2\\sin^2\\theta\\,d\\phi^2.\n",
    "$$\n",
    "\n",
    "The metric is Ricci-flat but not Riemann-flat.  Selected components make\n",
    "that distinction visible without requesting a costly expansion of every\n",
    "curvature component.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "459657af",
   "metadata": {},
   "source": [
    "## Local frame, metric, and inverse\n",
    "\n",
    "The chart covers $r>0$ away from $A=0$.  The surface\n",
    "$r=2GM/c^2$ is a coordinate horizon in this chart, whereas $r=0$ will be\n",
    "detected by a curvature invariant.  In a local Lorentz frame with\n",
    "$\\eta=\\operatorname{diag}(1,-1,-1,-1)$, the coordinate tangent vectors\n",
    "have components\n",
    "\n",
    "$$\n",
    "e_t=(\\sqrt A,0,0,0),\\quad\n",
    "e_r=(0,A^{-1/2},0,0),\\quad\n",
    "e_\\theta=(0,0,r,0),\\quad\n",
    "e_\\phi=(0,0,0,r\\sin\\theta).\n",
    "$$\n",
    "\n",
    "Egison obtains the metric as $g_{ij}=\\eta(e_i,e_j)$ and computes its\n",
    "inverse, instead of entering either matrix component by component.\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "20aa7dd0",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol G, M, c, t, r, θ, φ: MathValue\n",
    "\n",
    "def x : Vector MathValue := [| t, r, θ, φ |]\n",
    "def A : MathValue := `(c^2 * r - 2 * G * M) / (c^2 * r)\n",
    "\n",
    "def e_i_j : Matrix MathValue :=\n",
    "  [| [| sqrt A, 0, 0, 0 |]\n",
    "   , [| 0, 1 / sqrt A, 0, 0 |]\n",
    "   , [| 0, 0, r, 0 |]\n",
    "   , [| 0, 0, 0, r * sin θ |]\n",
    "   |]_i_j\n",
    "\n",
    "def minkowskiDot (u : Vector MathValue) (v : Vector MathValue) : MathValue :=\n",
    "  u_1 * v_1 - u_2 * v_2 - u_3 * v_3 - u_4 * v_4\n",
    "\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  generateTensor (\\[i, j] -> minkowskiDot e_i_# e_j_#) [4, 4]\n",
    "\n",
    "def g~i~j : Matrix MathValue := M.inverse g_#_#\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b9c12b39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(c^{2} r - 2 G M) c^{-2} r^{-1}$"
      ]
     },
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    }
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   "source": [
    "A\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f690f1c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} (c^{2} r - 2 G M) c^{-2} r^{-1} & 0 & 0 & 0 \\\\ 0 & -(c^{2} r - 2 G M)^{-1} c^{2} r & 0 & 0 \\\\ 0 & 0 & -r^{2} & 0 \\\\ 0 & 0 & 0 & -\\sin(θ)^{2} r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
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     "output_type": "display_data"
    }
   ],
   "source": [
    "g_#_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ad569a81",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} (c^{2} r - 2 G M)^{-1} c^{2} r & 0 & 0 & 0 \\\\ 0 & -(c^{2} r - 2 G M) c^{-2} r^{-1} & 0 & 0 \\\\ 0 & 0 & -r^{-2} & 0 \\\\ 0 & 0 & 0 & -\\sin(θ)^{-2} r^{-2} \\\\ \\end{pmatrix}_{\\;\\;}^{\\#\\#}$"
      ]
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     "output_type": "display_data"
    }
   ],
   "source": [
    "g~#~#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "014e4807",
   "metadata": {},
   "source": [
    "## Levi-Civita connection\n",
    "\n",
    "The connection is computed from the metric, rather than entered as a table.\n",
    "Angular components such as $\\Gamma^\\theta{}_{r\\theta}=1/r$ and\n",
    "$\\Gamma^\\phi{}_{\\theta\\phi}=\\cot\\theta$ are especially compact checks.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7e02c921",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Γ_i_j_k : Tensor MathValue :=\n",
    "  (1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)\n",
    "\n",
    "def Γ~i_j_k : Tensor MathValue := withSymbols [m]\n",
    "  g~i~m . Γ_m_j_k\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "cc1a96d8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$r^{-1}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Γ~3_2_3\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3c1e2019",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\cos(θ) \\sin(θ)^{-1}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Γ~4_3_4\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aeae1010",
   "metadata": {},
   "source": [
    "## Riemann and Ricci tensors\n",
    "\n",
    "We use the same sign convention as the round-sphere notebooks.  The angular\n",
    "component $R^\\theta{}_{\\phi\\theta\\phi}$ is nonzero when $M\\ne0$; a flat\n",
    "spherical-coordinate metric would make its radial and angular terms cancel.\n",
    "Ricci curvature is the contraction $R^m{}_{imj}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "94869a2a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def R~i_j_k_l : Tensor MathValue := withSymbols [m]\n",
    "  expandAll\n",
    "    (∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l\n",
    "      + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l)\n",
    "\n",
    "def Ric_i_j : Matrix MathValue := withSymbols [m]\n",
    "  sum (contract R~m_i_m_j)\n",
    "\n",
    "def scalarCurvature : MathValue := withSymbols [i, j]\n",
    "  g~i~j . Ric_i_j\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c225200e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$2 \\sin(θ)^{2} G M c^{-2} r^{-1}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expandAll (R~3_4_3_4)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "1f66f650",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$0$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expandAll (Ric_1_1)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "121ead88",
   "metadata": {},
   "source": [
    "## Interpretation\n",
    "\n",
    "The Riemann component records tidal curvature, while the Ricci component\n",
    "simplifies to zero, as required by the vacuum Einstein equations.  The\n",
    "coordinate-independent Kretschmann invariant is\n",
    "\n",
    "$$\n",
    "R_{abcd}R^{abcd}=\\frac{48G^2M^2}{c^4r^6}.\n",
    "$$\n",
    "\n",
    "Thus curvature is finite at the Schwarzschild horizon but diverges at\n",
    "$r=0$.  The full invariant contraction is stated rather than made an output\n",
    "cell because expanding all four-index combinations is needlessly expensive\n",
    "for an interactive demonstration.\n"
   ]
  }
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