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   "source": [
    "# Hodge Star in Minkowski Spacetime\n",
    "\n",
    "Electromagnetism is naturally expressed with differential forms on\n",
    "spacetime.  With metric signature $(-,+,+,+)$, the Hodge star maps\n",
    "two-forms to two-forms, but time-like basis elements introduce signs:\n",
    "\n",
    "$$\\star(dt\\wedge dx)=-dy\\wedge dz,\\qquad\n",
    "  \\star(dy\\wedge dz)=dt\\wedge dx.$$\n"
   ]
  },
  {
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   "id": "cde9f17e",
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   "source": [
    "## Lorentzian metric\n",
    "\n",
    "The determinant has negative sign, so the volume density uses\n",
    "$\\sqrt{|\\det g|}$.  Raising a time index contributes the additional\n",
    "minus sign visible in the result.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b789a048",
   "metadata": {},
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   "source": [
    "declare symbol t, x, y, z : MathValue\n",
    "\n",
    "def N : Integer := 4\n",
    "def params : Vector MathValue := [| t, x, y, z |]\n",
    "def g : Matrix MathValue :=\n",
    "  [| [| -1, 0, 0, 0 |]\n",
    "   , [| 0, 1, 0, 0 |]\n",
    "   , [| 0, 0, 1, 0 |]\n",
    "   , [| 0, 0, 0, 1 |] |]\n",
    "\n",
    "def dt : DiffForm MathValue := [| 1, 0, 0, 0 |]\n",
    "def dx : DiffForm MathValue := [| 0, 1, 0, 0 |]\n",
    "def dy : DiffForm MathValue := [| 0, 0, 1, 0 |]\n",
    "def dz : DiffForm MathValue := [| 0, 0, 0, 1 |]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5cce8ed",
   "metadata": {},
   "source": [
    "## Metric-dependent duality\n",
    "\n",
    "The definition is the same contraction used in Euclidean space.  Only\n",
    "the dimension and metric have changed.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d05c698f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def hodge (A : DiffForm MathValue) : DiffForm MathValue :=\n",
    "  let k := dfOrder A\n",
    "   in withSymbols [i, j]\n",
    "        sqrt (abs (M.det g_#_#)) *\n",
    "        foldl\n",
    "          (.)\n",
    "          ((ε' N k)_(i_1)..._(i_N) . A..._(j_1)..._(j_k))\n",
    "          (map (\\n -> g~(i_n)~(j_n)) [1..k])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f0e75a0",
   "metadata": {},
   "source": [
    "First dualize a two-form containing the time direction.\n"
   ]
  },
  {
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   "execution_count": 3,
   "id": "5b0ba917",
   "metadata": {},
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       "$\\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & -1 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{pmatrix}$"
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   "source": [
    "hodge (wedge dt dx)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a1e435e",
   "metadata": {},
   "source": [
    "A purely spatial area form dualizes to a time-space area form.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "dfcf5583",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "$\\begin{pmatrix} 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{pmatrix}$"
      ]
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     "output_type": "display_data"
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   ],
   "source": [
    "hodge (wedge dy dz)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc98eb43",
   "metadata": {},
   "source": [
    "These signs imply $\\star^2=-1$ on two-forms for this Lorentzian\n",
    "convention.  They are also the signs that exchange electric and\n",
    "magnetic components when the electromagnetic field tensor is dualized.\n"
   ]
  }
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