Tutorial-Psi4.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<script async src=\"https://www.googletagmanager.com/gtag/js?id=UA-59152712-8\"></script>\n",
    "<script>\n",
    "  window.dataLayer = window.dataLayer || [];\n",
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    "\n",
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    "\n",
    "# The Outgoing Gravitational Wave Weyl scalar $\\psi_4$\n",
    "\n",
    "## Author: Zach Etienne\n",
    "\n",
    "[comment]: <> (Abstract: TODO)\n",
    "\n",
    "**Notebook Status:** <font color='green'><b> Validated </b></font>\n",
    "\n",
    "**Validation Notes:** This module has been validated as follows:\n",
    "\n",
    "* Agreement (to roundoff error) with the WeylScal4 ETK thorn in Cartesian coordinates (as it agrees to roundoff error with Patrick Nelson's [Cartesian Weyl Scalars & Invariants NRPy+ tutorial notebook](Tutorial-WeylScalarsInvariants-Cartesian.ipynb), which itself was validated against WeylScal4). \n",
    "* In SinhSpherical coordinates this module yields amplitude falloff and phase agreement with black hole perturbation theory for the $l=2,m=0$ (dominant, spin-weight -2 spherical harmonic) mode of a ringing Brill-Lindquist black hole remnant to more than 7 decades in amplitude, surpassing the agreement seen in Fig. 6 of [Ruchlin, Etienne, & Baumgarte](https://arxiv.org/pdf/1712.07658.pdf). (as shown in [corresponding start-to-finish notebook](Tutorial-Start_to_Finish-BSSNCurvilinear-Two_BHs_Collide-Psi4.ipynb); must choose EvolOption = \"high resolution\").\n",
    "* Above head-on collision calculation was performed with the [Einstein Toolkit](https://einsteintoolkit.org/), and results were found to match in the case of all spherical harmonic modes.\n",
    "* An equal-mass binary black hole calculation (QC-0) was performed using this module for $\\psi_4$ (SinhSpherical coordinates in region where gravitational waves extracted), and excellent agreement was observed for all (spin-weight -2 spherical harmonic) modes up to and including $l=4$, when compared with the same calculation performed with the [Einstein Toolkit](https://einsteintoolkit.org/).\n",
    "\n",
    "### NRPy+ Source Code for this module: [BSSN/Psi4.py](../edit/BSSN/Psi4.py)\n",
    "\n",
    "## Introduction:\n",
    "This module constructs $\\psi_4$, a quantity that is immensely useful when extracting gravitational wave content from a numerical relativity simulation. $\\psi_4$ is related to the gravitational wave strain via\n",
    "\n",
    "$$\n",
    "\\psi_4 = \\ddot{h}_+ - i \\ddot{h}_\\times.\n",
    "$$\n",
    "\n",
    "We construct $\\psi_4$ from the standard ADM spatial metric $\\gamma_{ij}$ and extrinsic curvature $K_{ij}$, and their derivatives. The full expression is given by Eq. 5.1 in [Baker, Campanelli, Lousto (2001)](https://arxiv.org/pdf/gr-qc/0104063.pdf):\n",
    "\n",
    "\\begin{align}\n",
    "\\psi_4 &= \\left[ {R}_{ijkl}+2K_{i[k}K_{l]j}\\right]\n",
    "{n}^i\\bar{m}^j{n}^k\\bar{m}^l  \\\\\n",
    "& -8\\left[ K_{j[k,l]}+{\\Gamma }_{j[k}^pK_{l]p}\\right]\n",
    "{n}^{[0}\\bar{m}^{j]}{n}^k\\bar{m}^l \\\\\n",
    "& +4\\left[ {R}_{jl}-K_{jp}K_l^p+KK_{jl}\\right]\n",
    "{n}^{[0}\\bar{m}^{j]}{n}^{[0}\\bar{m}^{l]},\n",
    "\\end{align}\n",
    "\n",
    "Note that $\\psi_4$ is complex, with the imaginary components originating from the tetrad vector $m^\\mu$. This module does not specify a tetrad; instead it only constructs the above expression leaving $m^\\mu$ and $n^\\mu$ unspecified. The [next module on tetrads defines these tetrad quantities](Tutorial-Psi4_tetrads.ipynb) (currently only a quasi-Kinnersley tetrad is supported).\n",
    "\n",
    "### A Note on Notation:\n",
    "\n",
    "As is standard in NRPy+, \n",
    "\n",
    "* Greek indices range from 0 to 3, inclusive, with the zeroth component denoting the temporal (time) component.\n",
    "* Latin indices range from 0 to 2, inclusive, with the zeroth component denoting the first spatial component.\n",
    "\n",
    "As a corollary, any expressions involving mixed Greek and Latin indices will need to offset one set of indices by one: A Latin index in a four-vector will be incremented and a Greek index in a three-vector will be decremented (however, the latter case does not occur in this tutorial notebook).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='toc'></a>\n",
    "\n",
    "# Table of Contents\n",
    "$$\\label{toc}$$\n",
    "\n",
    "This tutorial notebook is organized as follows\n",
    "\n",
    "1. [Step 1](#initializenrpy): Initialize needed NRPy+ modules\n",
    "1. [Step 2](#riemann): Constructing the 3-Riemann tensor $R_{ik\\ell m}$\n",
    "1. [Step 3](#rank4termone): Constructing the rank-4 tensor in Term 1 of $\\psi_4$: $R_{ijkl} + 2 K_{i[k} K_{l]j}$\n",
    "1. [Step 4](#rank3termtwo): Constructing the rank-3 tensor in Term 2 of $\\psi_4$: $-8 \\left(K_{j[k,l]} + \\Gamma^{p}_{j[k} K_{l]p} \\right)$\n",
    "1. [Step 5](#rank2termthree): Constructing the rank-2 tensor in term 3 of $\\psi_4$: $+4 \\left(R_{jl} - K_{jp} K^p_l + K K_{jl} \\right)$\n",
    "1. [Step 6](#psifour): Constructing $\\psi_4$ through contractions of the above terms with arbitrary tetrad vectors $n^\\mu$ and $m^\\mu$\n",
    "1. [Step 7](#code_validation): Code Validation against `BSSN.Psi4` NRPy+ module\n",
    "1. [Step 8](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='initializenrpy'></a>\n",
    "\n",
    "# Step 1: Initialize core NRPy+ modules \\[Back to [top](#toc)\\]\n",
    "$$\\label{initializenrpy}$$\n",
    "\n",
    "Let's start by importing all the needed modules from NRPy+:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 1.a: import all needed modules from NRPy+:\n",
    "import sympy as sp\n",
    "import NRPy_param_funcs as par\n",
    "import indexedexp as ixp\n",
    "import grid as gri\n",
    "import finite_difference as fin\n",
    "import reference_metric as rfm\n",
    "\n",
    "# Step 1.b: Set the coordinate system for the numerical grid\n",
    "#           Note that this parameter is assumed to be set\n",
    "#           prior to calling the Python Psi4.py module,\n",
    "#           so this Step will not appear there.\n",
    "par.set_parval_from_str(\"reference_metric::CoordSystem\",\"Spherical\")\n",
    "\n",
    "# Step 1.c: Given the chosen coordinate system, set up\n",
    "#           corresponding reference metric and needed\n",
    "#           reference metric quantities\n",
    "# The following function call sets up the reference metric\n",
    "#    and related quantities, including rescaling matrices ReDD,\n",
    "#    ReU, and hatted quantities.\n",
    "rfm.reference_metric()\n",
    "\n",
    "# Step 1.d: Set spatial dimension (must be 3 for BSSN, as BSSN is\n",
    "#           a 3+1-dimensional decomposition of the general\n",
    "#           relativistic field equations)\n",
    "DIM = 3\n",
    "\n",
    "# Step 1.e: Import all ADM quantities as written in terms of BSSN quantities\n",
    "import BSSN.ADM_in_terms_of_BSSN as AB\n",
    "AB.ADM_in_terms_of_BSSN()\n",
    "\n",
    "# Step 1.f: Initialize tetrad vectors.\n",
    "#           mre4U = $\\text{Re}(m^\\mu)$\n",
    "#           mim4U = $\\text{Im}(m^\\mu)$, and\n",
    "#           n4U   = $n^\\mu$\n",
    "#           Note that in the separate Python Psi4.py\n",
    "#           module, these will be set to the tetrad\n",
    "#           chosen within the Psi4_tetrads.py module.\n",
    "\n",
    "#           We choose the most general form for the\n",
    "#           tetrad vectors here instead, to ensure complete\n",
    "#           code validation.\n",
    "mre4U = ixp.declarerank1(\"mre4U\",DIM=4)\n",
    "mim4U = ixp.declarerank1(\"mim4U\",DIM=4)\n",
    "n4U   = ixp.declarerank1(\"n4U\"  ,DIM=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='riemann'></a>\n",
    "\n",
    "# Step 2: Constructing the 3-Riemann tensor $R_{ik\\ell m}$ \\[Back to [top](#toc)\\]\n",
    "$$\\label{riemann}$$\n",
    "\n",
    "Analogously to Christoffel symbols, the Riemann tensor is a measure of the curvature of an $N$-dimensional manifold. Thus the 3-Riemann tensor is not simply a projection of the 4-Riemann tensor (see e.g., Eq. 2.7 of [Campanelli *et al* (1998)](https://arxiv.org/pdf/gr-qc/9803058.pdf) for the relation between 4-Riemann and 3-Riemann), as $N$-dimensional Riemann tensors are meant to define a notion of curvature given only the associated $N$-dimensional metric. \n",
    "\n",
    "So, given the ADM 3-metric, the Riemann tensor in arbitrary dimension is given by the 3-dimensional version of Eq. 1.19 in Baumgarte & Shapiro's *Numerical Relativity*. I.e.,\n",
    "\n",
    "$$\n",
    "R^i_{jkl} = \\partial_k \\Gamma^{i}_{jl} - \\partial_l \\Gamma^{i}_{jk} + \\Gamma^i_{mk} \\Gamma^m_{jl} - \\Gamma^{i}_{ml} \\Gamma^{m}_{jk},\n",
    "$$\n",
    "where $\\Gamma^i_{jk}$ is the Christoffel symbol associated with the 3-metric $\\gamma_{ij}$:\n",
    "\n",
    "$$\n",
    "\\Gamma^l_{ij} = \\frac{1}{2} \\gamma^{lk} \\left(\\gamma_{ki,j} + \\gamma_{kj,i} - \\gamma_{ij,k} \\right) \n",
    "$$\n",
    "\n",
    "Notice that this equation for the Riemann tensor is equivalent to the equation given in the Wikipedia article on [Formulas in Riemannian geometry](https://en.wikipedia.org/w/index.php?title=List_of_formulas_in_Riemannian_geometry&oldid=882667524):\n",
    "\n",
    "$$\n",
    "R^\\ell{}_{ijk}=\n",
    "\\partial_j \\Gamma^\\ell{}_{ik}-\\partial_k\\Gamma^\\ell{}_{ij}\n",
    "+\\Gamma^\\ell{}_{js}\\Gamma_{ik}^s-\\Gamma^\\ell{}_{ks}\\Gamma^s{}_{ij},\n",
    "$$\n",
    "with the replacements $i\\to \\ell$, $j\\to i$, $k\\to j$, $l\\to k$, and $s\\to m$. Wikipedia also provides a simpler form in terms of second-derivatives of three-metric itself (using the definition of Christoffel symbol), so that we need not define derivatives of the Christoffel symbol:\n",
    "\n",
    "$$\n",
    "R_{ik\\ell m}=\\frac{1}{2}\\left(\n",
    "\\gamma_{im,k\\ell} \n",
    "+ \\gamma_{k\\ell,im}\n",
    "- \\gamma_{i\\ell,km}\n",
    "- \\gamma_{km,i\\ell} \\right)\n",
    "+\\gamma_{np} \\left(\n",
    "\\Gamma^n{}_{k\\ell} \\Gamma^p{}_{im} - \n",
    "\\Gamma^n{}_{km} \\Gamma^p{}_{i\\ell} \\right).\n",
    "$$\n",
    "\n",
    "First we construct the term on the left:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 2: Construct the (rank-4) Riemann curvature tensor associated with the ADM 3-metric:\n",
    "RDDDD = ixp.zerorank4()\n",
    "gammaDDdDD = AB.gammaDDdDD\n",
    "\n",
    "for i in range(DIM):\n",
    "    for k in range(DIM):\n",
    "        for l in range(DIM):\n",
    "            for m in range(DIM):\n",
    "                RDDDD[i][k][l][m] = sp.Rational(1,2) * \\\n",
    "                    (gammaDDdDD[i][m][k][l] + gammaDDdDD[k][l][i][m] - gammaDDdDD[i][l][k][m] - gammaDDdDD[k][m][i][l])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... then we add the term on the right:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# ... then we add the term on the right:\n",
    "gammaDD = AB.gammaDD\n",
    "GammaUDD = AB.GammaUDD\n",
    "\n",
    "for i in range(DIM):\n",
    "    for k in range(DIM):\n",
    "        for l in range(DIM):\n",
    "            for m in range(DIM):\n",
    "                for n in range(DIM):\n",
    "                    for p in range(DIM):\n",
    "                        RDDDD[i][k][l][m] += gammaDD[n][p] * \\\n",
    "                            (GammaUDD[n][k][l]*GammaUDD[p][i][m] - GammaUDD[n][k][m]*GammaUDD[p][i][l])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='rank4termone'></a>\n",
    "\n",
    "# Step 3: Constructing the rank-4 tensor in Term 1 of $\\psi_4$: $R_{ijkl} + 2 K_{i[k} K_{l]j}$ \\[Back to [top](#toc)\\]\n",
    "$$\\label{rank4termone}$$\n",
    "\n",
    "Following Eq. 5.1 in [Baker, Campanelli, Lousto (2001)](https://arxiv.org/pdf/gr-qc/0104063.pdf), the rank-4 tensor in the first term of $\\psi_4$ is given by\n",
    "\n",
    "$$\n",
    "R_{ijkl} + 2 K_{i[k} K_{l]j} = R_{ijkl} + K_{ik} K_{lj} - K_{il} K_{kj}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 3: Construct the (rank-4) tensor in term 1 of psi_4 (referring to Eq 5.1 in\n",
    "#   Baker, Campanelli, Lousto (2001); https://arxiv.org/pdf/gr-qc/0104063.pdf\n",
    "rank4term1DDDD = ixp.zerorank4()\n",
    "KDD = AB.KDD\n",
    "\n",
    "for i in range(DIM):\n",
    "    for j in range(DIM):\n",
    "        for k in range(DIM):\n",
    "            for l in range(DIM):\n",
    "                rank4term1DDDD[i][j][k][l] = RDDDD[i][j][k][l] + KDD[i][k]*KDD[l][j] - KDD[i][l]*KDD[k][j]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='rank3termtwo'></a>\n",
    "\n",
    "# Step 4: Constructing the rank-3 tensor in Term 2 of $\\psi_4$: $-8 \\left(K_{j[k,l]} + \\Gamma^{p}_{j[k} K_{l]p} \\right)$ \\[Back to [top](#toc)\\]\n",
    "$$\\label{rank3termtwo}$$\n",
    "\n",
    "Following Eq. 5.1 in [Baker, Campanelli, Lousto (2001)](https://arxiv.org/pdf/gr-qc/0104063.pdf), the rank-3 tensor in the second term of $\\psi_4$ is given by\n",
    "\n",
    "$$\n",
    "-8 \\left(K_{j[k,l]} + \\Gamma^{p}_{j[k} K_{l]p} \\right)\n",
    "$$\n",
    "First let's construct the first term in this sum: $K_{j[k,l]} = \\frac{1}{2} (K_{jk,l} - K_{jl,k})$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 4: Construct the (rank-3) tensor in term 2 of psi_4 (referring to Eq 5.1 in\n",
    "#   Baker, Campanelli, Lousto (2001); https://arxiv.org/pdf/gr-qc/0104063.pdf\n",
    "rank3term2DDD = ixp.zerorank3()\n",
    "KDDdD = AB.KDDdD\n",
    "\n",
    "for j in range(DIM):\n",
    "    for k in range(DIM):\n",
    "        for l in range(DIM):\n",
    "            rank3term2DDD[j][k][l] = sp.Rational(1,2)*(KDDdD[j][k][l] - KDDdD[j][l][k])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... then we construct the second term in this sum: $\\Gamma^{p}_{j[k} K_{l]p} = \\frac{1}{2} (\\Gamma^{p}_{jk} K_{lp}-\\Gamma^{p}_{jl} K_{kp})$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# ... then we construct the second term in this sum:\n",
    "#  \\Gamma^{p}_{j[k} K_{l]p} = \\frac{1}{2} (\\Gamma^{p}_{jk} K_{lp}-\\Gamma^{p}_{jl} K_{kp}):\n",
    "for j in range(DIM):\n",
    "    for k in range(DIM):\n",
    "        for l in range(DIM):\n",
    "            for p in range(DIM):\n",
    "                rank3term2DDD[j][k][l] += sp.Rational(1,2)*(GammaUDD[p][j][k]*KDD[l][p] - GammaUDD[p][j][l]*KDD[k][p])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we multiply the term by $-8$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Finally, we multiply the term by $-8$:\n",
    "for j in range(DIM):\n",
    "    for k in range(DIM):\n",
    "        for l in range(DIM):\n",
    "            rank3term2DDD[j][k][l] *= sp.sympify(-8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='rank2termthree'></a>\n",
    "\n",
    "# Step 5: Constructing the rank-2 tensor in term 3 of $\\psi_4$: $+4 \\left(R_{jl} - K_{jp} K^p_l + K K_{jl} \\right)$ \\[Back to [top](#toc)\\]\n",
    "$$\\label{rank2termthree}$$\n",
    "\n",
    "Following Eq. 5.1 in [Baker, Campanelli, Lousto (2001)](https://arxiv.org/pdf/gr-qc/0104063.pdf), the rank-2 tensor in the third term of $\\psi_4$ is given by\n",
    "\n",
    "$$\n",
    "+4 \\left(R_{jl} - K_{jp} K^p_l + K K_{jl} \\right),\n",
    "$$\n",
    "where\n",
    "\\begin{align}\n",
    "R_{jl} &= R^i_{jil} \\\\\n",
    "&= \\gamma^{im} R_{ijml} \\\\\n",
    "K &= K^i_i \\\\\n",
    "&= \\gamma^{im} K_{im}\n",
    "\\end{align}\n",
    "\n",
    "Let's build the components of this term: $R_{jl}$, $K^p_l$, and $K$, as defined above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 5: Construct the (rank-2) tensor in term 3 of psi_4 (referring to Eq 5.1 in\n",
    "#   Baker, Campanelli, Lousto (2001); https://arxiv.org/pdf/gr-qc/0104063.pdf\n",
    "\n",
    "# Step 5.1: Construct 3-Ricci tensor R_{ij} = gamma^{im} R_{ijml}\n",
    "RDD = ixp.zerorank2()\n",
    "gammaUU = AB.gammaUU\n",
    "for j in range(DIM):\n",
    "    for l in range(DIM):\n",
    "        for i in range(DIM):\n",
    "            for m in range(DIM):\n",
    "                RDD[j][l] += gammaUU[i][m]*RDDDD[i][j][m][l]\n",
    "\n",
    "# Step 5.2: Construct K^p_l = gamma^{pi} K_{il}\n",
    "KUD = ixp.zerorank2()\n",
    "for p in range(DIM):\n",
    "    for l in range(DIM):\n",
    "        for i in range(DIM):\n",
    "            KUD[p][l] += gammaUU[p][i]*KDD[i][l]\n",
    "\n",
    "# Step 5.3: Construct trK = gamma^{ij} K_{ij}\n",
    "trK = sp.sympify(0)\n",
    "for i in range(DIM):\n",
    "    for j in range(DIM):\n",
    "        trK += gammaUU[i][j]*KDD[i][j]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we put these terms together to construct the entire term:\n",
    "$$\n",
    "+4 \\left(R_{jl} - K_{jp} K^p_l + K K_{jl} \\right),\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Next we put these terms together to construct the entire term in parentheses:\n",
    "# +4 \\left(R_{jl} - K_{jp} K^p_l + K K_{jl} \\right),\n",
    "rank2term3DD = ixp.zerorank2()\n",
    "for j in range(DIM):\n",
    "    for l in range(DIM):\n",
    "        rank2term3DD[j][l] = RDD[j][l] + trK*KDD[j][l]\n",
    "        for p in range(DIM):\n",
    "            rank2term3DD[j][l] += - KDD[j][p]*KUD[p][l]\n",
    "# Finally we multiply by +4:\n",
    "for j in range(DIM):\n",
    "    for l in range(DIM):\n",
    "        rank2term3DD[j][l] *= sp.sympify(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='psifour'></a>\n",
    "\n",
    "# Step 6: Constructing $\\psi_4$ through contractions of the above terms with an arbitrary tetrad vectors $m^\\mu$ and $n^\\mu$ \\[Back to [top](#toc)\\]\n",
    "$$\\label{psifour}$$\n",
    "\n",
    "Eq. 5.1 in [Baker, Campanelli, Lousto (2001)](https://arxiv.org/pdf/gr-qc/0104063.pdf) writes $\\psi_4$ (which is complex) as the contraction of each of the above terms with products of tetrad vectors:\n",
    "\n",
    "\\begin{align}\n",
    "\\psi_4 &= \\left[ {R}_{ijkl}+2K_{i[k}K_{l]j}\\right]\n",
    "{n}^i\\bar{m}^j{n}^k\\bar{m}^l  \\\\\n",
    "& -8\\left[ K_{j[k,l]}+{\\Gamma }_{j[k}^pK_{l]p}\\right]\n",
    "{n}^{[0}\\bar{m}^{j]}{n}^k\\bar{m}^l \\\\\n",
    "& +4\\left[ {R}_{jl}-K_{jp}K_l^p+KK_{jl}\\right]\n",
    "{n}^{[0}\\bar{m}^{j]}{n}^{[0}\\bar{m}^{l]},\n",
    "\\end{align}\n",
    "where $\\bar{m}^\\mu$ is the complex conjugate of $m^\\mu$, and $n^\\mu$ is real. The third term is given by\n",
    "\\begin{align}\n",
    "{n}^{[0}\\bar{m}^{j]}{n}^{[0}\\bar{m}^{l]}\n",
    "&= \\frac{1}{2}({n}^{0}\\bar{m}^{j} - {n}^{j}\\bar{m}^{0} )\\frac{1}{2}({n}^{0}\\bar{m}^{l} - {n}^{l}\\bar{m}^{0} )\\\\\n",
    "&= \\frac{1}{4}({n}^{0}\\bar{m}^{j} - {n}^{j}\\bar{m}^{0} )({n}^{0}\\bar{m}^{l} - {n}^{l}\\bar{m}^{0} )\\\\\n",
    "&= \\frac{1}{4}({n}^{0}\\bar{m}^{j}{n}^{0}\\bar{m}^{l} - {n}^{j}\\bar{m}^{0}{n}^{0}\\bar{m}^{l} - {n}^{0}\\bar{m}^{j}{n}^{l}\\bar{m}^{0} +  {n}^{j}\\bar{m}^{0}{n}^{l}\\bar{m}^{0})\n",
    "\\end{align}\n",
    "\n",
    "Only $m^\\mu$ is complex, so we can separate the real and imaginary parts of $\\psi_4$ by hand, defining $M^\\mu$ to now be the real part of $\\bar{m}^\\mu$ and $\\mathcal{M}^\\mu$ to be the imaginary part. All of the above products are of the form ${n}^\\mu\\bar{m}^\\nu{n}^\\eta\\bar{m}^\\delta$, so let's evalute the real and imaginary parts of this product once, for all such terms:\n",
    "\n",
    "\\begin{align}\n",
    "{n}^\\mu\\bar{m}^\\nu{n}^\\eta\\bar{m}^\\delta\n",
    "&= {n}^\\mu(M^\\nu - i \\mathcal{M}^\\nu){n}^\\eta(M^\\delta - i \\mathcal{M}^\\delta) \\\\\n",
    "&= \\left({n}^\\mu M^\\nu {n}^\\eta M^\\delta -\n",
    "{n}^\\mu \\mathcal{M}^\\nu {n}^\\eta \\mathcal{M}^\\delta \\right)+\n",
    "i \\left(\n",
    "-{n}^\\mu M^\\nu {n}^\\eta \\mathcal{M}^\\delta\n",
    "-{n}^\\mu \\mathcal{M}^\\nu {n}^\\eta M^\\delta\n",
    "\\right)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 6: Construct real & imaginary parts of psi_4\n",
    "#         by contracting constituent rank 2, 3, and 4\n",
    "#         tensors with input tetrads mre4U, mim4U, & n4U.\n",
    "\n",
    "def tetrad_product__Real_psi4(n,Mre,Mim,  mu,nu,eta,delta):\n",
    "    return +n[mu]*Mre[nu]*n[eta]*Mre[delta] - n[mu]*Mim[nu]*n[eta]*Mim[delta]\n",
    "\n",
    "def tetrad_product__Imag_psi4(n,Mre,Mim,  mu,nu,eta,delta):\n",
    "    return -n[mu]*Mre[nu]*n[eta]*Mim[delta] - n[mu]*Mim[nu]*n[eta]*Mre[delta]\n",
    "\n",
    "\n",
    "# We split psi_4 into three pieces, to expedite & possibly parallelize C code generation.\n",
    "psi4_re_pt = [sp.sympify(0),sp.sympify(0),sp.sympify(0)]\n",
    "psi4_im_pt = [sp.sympify(0),sp.sympify(0),sp.sympify(0)]\n",
    "\n",
    "# First term:\n",
    "for i in range(DIM):\n",
    "    for j in range(DIM):\n",
    "        for k in range(DIM):\n",
    "            for l in range(DIM):\n",
    "                psi4_re_pt[0] += rank4term1DDDD[i][j][k][l] * \\\n",
    "                                 tetrad_product__Real_psi4(n4U,mre4U,mim4U, i+1,j+1,k+1,l+1)\n",
    "                psi4_im_pt[0] += rank4term1DDDD[i][j][k][l] * \\\n",
    "                                 tetrad_product__Imag_psi4(n4U,mre4U,mim4U, i+1,j+1,k+1,l+1)\n",
    "\n",
    "# Second term:\n",
    "for j in range(DIM):\n",
    "    for k in range(DIM):\n",
    "        for l in range(DIM):\n",
    "            psi4_re_pt[1] += rank3term2DDD[j][k][l] * \\\n",
    "                       sp.Rational(1,2)*(+tetrad_product__Real_psi4(n4U,mre4U,mim4U, 0,j+1,k+1,l+1)\n",
    "                                         -tetrad_product__Real_psi4(n4U,mre4U,mim4U, j+1,0,k+1,l+1) )\n",
    "            psi4_im_pt[1] += rank3term2DDD[j][k][l] * \\\n",
    "                       sp.Rational(1,2)*(+tetrad_product__Imag_psi4(n4U,mre4U,mim4U, 0,j+1,k+1,l+1)\n",
    "                                         -tetrad_product__Imag_psi4(n4U,mre4U,mim4U, j+1,0,k+1,l+1) )\n",
    "# Third term:\n",
    "for j in range(DIM):\n",
    "    for l in range(DIM):\n",
    "        psi4_re_pt[2] += rank2term3DD[j][l] * \\\n",
    "                       (sp.Rational(1,4)*(+tetrad_product__Real_psi4(n4U,mre4U,mim4U, 0,j+1,0,l+1)\n",
    "                                          -tetrad_product__Real_psi4(n4U,mre4U,mim4U, j+1,0,0,l+1)\n",
    "                                          -tetrad_product__Real_psi4(n4U,mre4U,mim4U, 0,j+1,l+1,0)\n",
    "                                          +tetrad_product__Real_psi4(n4U,mre4U,mim4U, j+1,0,l+1,0)))\n",
    "        psi4_im_pt[2] += rank2term3DD[j][l] * \\\n",
    "                       (sp.Rational(1,4)*(+tetrad_product__Imag_psi4(n4U,mre4U,mim4U, 0,j+1,0,l+1)\n",
    "                                          -tetrad_product__Imag_psi4(n4U,mre4U,mim4U, j+1,0,0,l+1)\n",
    "                                          -tetrad_product__Imag_psi4(n4U,mre4U,mim4U, 0,j+1,l+1,0)\n",
    "                                          +tetrad_product__Imag_psi4(n4U,mre4U,mim4U, j+1,0,l+1,0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='code_validation'></a>\n",
    "\n",
    "# Step 7: Code validation against `BSSN.Psi4` NRPy+ module \\[Back to [top](#toc)\\]\n",
    "$$\\label{code_validation}$$\n",
    "\n",
    "As a code validation check, we verify agreement in the SymPy expressions for the RHSs of the BSSN equations between\n",
    "1. this tutorial and \n",
    "2. the NRPy+ [BSSN.Psi4](../edit/BSSN/Psi4.py) module.\n",
    "\n",
    "By default, we compare all quantities in Spherical coordinates, though other coordinate systems may be chosen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Consistency check between this tutorial and BSSN.Psi4 NRPy+ module: ALL SHOULD BE ZERO.\n",
      "psi4_im_pt[0] - BP4.psi4_im_pt[0] = 0\n",
      "psi4_re_pt[0] - BP4.psi4_re_pt[0] = 0\n",
      "psi4_im_pt[1] - BP4.psi4_im_pt[1] = 0\n",
      "psi4_re_pt[1] - BP4.psi4_re_pt[1] = 0\n",
      "psi4_im_pt[2] - BP4.psi4_im_pt[2] = 0\n",
      "psi4_re_pt[2] - BP4.psi4_re_pt[2] = 0\n"
     ]
    }
   ],
   "source": [
    "# Call the BSSN_RHSs() function from within the\n",
    "#          BSSN/BSSN_RHSs.py module,\n",
    "#          which should do exactly the same as in Steps 1-16 above.\n",
    "import BSSN.Psi4 as BP4\n",
    "BP4.Psi4(specify_tetrad=False)\n",
    "\n",
    "print(\"Consistency check between this tutorial and BSSN.Psi4 NRPy+ module: ALL SHOULD BE ZERO.\")\n",
    "\n",
    "for part in range(3):\n",
    "    print(\"psi4_im_pt[\"+str(part)+\"] - BP4.psi4_im_pt[\"+str(part)+\"] = \" + str(psi4_im_pt[part] - BP4.psi4_im_pt[part]))\n",
    "    print(\"psi4_re_pt[\"+str(part)+\"] - BP4.psi4_re_pt[\"+str(part)+\"] = \" + str(psi4_re_pt[part] - BP4.psi4_re_pt[part]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='latex_pdf_output'></a>\n",
    "\n",
    "# Step 8: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n",
    "$$\\label{latex_pdf_output}$$\n",
    "\n",
    "The following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n",
    "[Tutorial-Psi4.pdf](Tutorial-Psi4.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Created Tutorial-Psi4.tex, and compiled LaTeX file to PDF file Tutorial-\n",
      "    Psi4.pdf\n"
     ]
    }
   ],
   "source": [
    "import cmdline_helper as cmd    # NRPy+: Multi-platform Python command-line interface\n",
    "cmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-Psi4\")"
   ]
  }
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