{ "cells": [ { "cell_type": "markdown", "id": "597d65fd", "metadata": {}, "source": [ "# An Einstein Metric on $S^2\\times S^3$\n", "\n", "The historical Egison demo bearing the name “Riemann curvature tensor of\n", "$S^2\\times S^3$” uses a non-diagonal five-dimensional metric, not the\n", "elementary block product of two round metrics. Its goal is the Einstein\n", "identity\n", "\n", "$$\n", "\\operatorname{Ric}_{ij}=4g_{ij},\\qquad \\mathcal R=20.\n", "$$\n", "\n", "This notebook preserves that metric and makes its symbolic workload\n", "explicit. Small metric and connection components are suitable interactive\n", "outputs; the full Einstein residual is defined for deliberate evaluation.\n" ] }, { "cell_type": "markdown", "id": "3b2a5c7e", "metadata": {}, "source": [ "## Coordinates and abbreviations\n", "\n", "In the coordinate order $x=(\\phi,\\theta,\\psi,y,\\alpha)$, introduce\n", "\n", "$$\n", "p=1-y,\\quad u=a-y^2,\\quad\n", "v=a-3y^2+2y^3,\\quad q=a-2y+y^2.\n", "$$\n", "\n", "These factors expose the repeated structure of the source metric and make\n", "its coordinate domain restrictions ($p u v\\ne0$) visible.\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "eb17d249", "metadata": {}, "outputs": [], "source": [ "declare symbol φ, θ, ψ, y, α, a: MathValue\n", "\n", "def x : Vector MathValue := [| φ, θ, ψ, y, α |]\n", "\n", "def p : MathValue := `(1 - y)\n", "def u : MathValue := `(a - y^2)\n", "def v : MathValue := `(a - 3 * y^2 + 2 * y^3)\n", "def q : MathValue := `(a - 2 * y + y^2)\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "a1cae8df", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$((-y + 1), (-y^{2} + a), (2 y^{3} - 3 y^{2} + a), (y^{2} - 2 y + a))$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "(p, u, v, q)\n" ] }, { "cell_type": "markdown", "id": "40c363a8", "metadata": {}, "source": [ "## Local frame and induced metric\n", "\n", "Put $f=q/(6u)$. The metric has the orthonormal-frame decomposition\n", "\n", "$$\n", "\\begin{split}\n", "ds^2={}&\\frac p6(d\\theta^2+\\sin^2\\theta\\,d\\phi^2)\n", " +\\frac{p}{2v}dy^2\n", " +\\frac{v}{9u}(d\\psi-\\cos\\theta\\,d\\phi)^2\\\\\n", " &+\\frac{2u}{p}\n", " \\bigl(d\\alpha+f(d\\psi-\\cos\\theta\\,d\\phi)\\bigr)^2.\n", "\\end{split}\n", "$$\n", "\n", "Each row of `e_i_j` below is the coordinate tangent vector $e_i$\n", "expressed in this orthonormal frame. As in the round-sphere notebooks,\n", "Egison constructs every metric component from tangent-vector inner\n", "products, $g_{ij}=e_i\\mathbin{\\cdot}e_j$. The coupled $(1,3,5)$ block is\n", "produced by the two appearances of $d\\psi-\\cos\\theta\\,d\\phi$.\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "b7f8fffe", "metadata": {}, "outputs": [], "source": [ "def f : MathValue := q / (6 * u)\n", "\n", "def e_i_j : Matrix MathValue :=\n", " [| [| sqrt (p / 6) * sin θ\n", " , 0\n", " , -(sqrt (v / u) / 3) * cos θ\n", " , 0\n", " , -sqrt (2 * u / p) * f * cos θ |]\n", " , [| 0, sqrt (p / 6), 0, 0, 0 |]\n", " , [| 0, 0, sqrt (v / u) / 3, 0, sqrt (2 * u / p) * f |]\n", " , [| 0, 0, 0, sqrt (p / (2 * v)), 0 |]\n", " , [| 0, 0, 0, 0, sqrt (2 * u / p) |]\n", " |]_i_j\n", "\n", "def g_i_j : Matrix MathValue :=\n", " generateTensor (\\[i, j] -> V.* e_i_# e_j_#) [5, 5]\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "97c43a04", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\frac{1}{6} (-y + 1)$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g_2_2\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "c3b4ad22", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\frac{1}{2} (-y + 1) (2 y^{3} - 3 y^{2} + a)^{-1}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g_4_4\n" ] }, { "cell_type": "code", "execution_count": 6, "id": "34a97d65", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$(\\frac{-1}{9} \\cos(θ) (-y^{2} + a)^{-1} (2 y^{3} - 3 y^{2} + a) + \\frac{-1}{18} \\cos(θ) (-y + 1)^{-1} (-y^{2} + a)^{-1} (y^{2} - 2 y + a)^{2}, \\frac{-1}{9} \\cos(θ) (-y^{2} + a)^{-1} (2 y^{3} - 3 y^{2} + a) + \\frac{-1}{18} \\cos(θ) (-y + 1)^{-1} (-y^{2} + a)^{-1} (y^{2} - 2 y + a)^{2}, \\frac{-1}{3} \\cos(θ) (-y + 1)^{-1} (y^{2} - 2 y + a), \\frac{-1}{3} \\cos(θ) (-y + 1)^{-1} (y^{2} - 2 y + a))$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "(g_1_3, g_3_1, g_1_5, g_5_1)\n" ] }, { "cell_type": "markdown", "id": "2a675a89", "metadata": {}, "source": [ "## Connection and curvature pipeline\n", "\n", "The first-kind symbols require only derivatives of the displayed metric.\n", "Raising an index introduces the inverse of the coupled block. The Riemann\n", "and Ricci definitions are nevertheless identical to the lower-dimensional\n", "examples:\n", "\n", "$$\n", "R^i{}_{jkl}=\\partial_k\\Gamma^i{}_{jl}-\\partial_l\\Gamma^i{}_{jk}\n", " +\\Gamma^m{}_{jl}\\Gamma^i{}_{mk}-\\Gamma^m{}_{jk}\\Gamma^i{}_{ml},\n", "\\qquad\n", "\\operatorname{Ric}_{ij}=R^m{}_{imj}.\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 7, "id": "a5dfe69c", "metadata": {}, "outputs": [], "source": [ "def g~i~j : Matrix MathValue := M.inverse g_#_#\n", "\n", "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", "\n", "def R~i_j_k_l : Tensor MathValue := withSymbols [m]\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 einsteinResidual_i_j : Matrix MathValue := withSymbols [i, j]\n", " expandAll' (Ric_i_j -' 4 *' g_i_j)\n" ] }, { "cell_type": "code", "execution_count": 8, "id": "5f547b8f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\frac{-1}{12}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Γ_2_2_4\n" ] }, { "cell_type": "code", "execution_count": 9, "id": "f5b5e2a0", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\frac{-1}{12}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Γ_2_4_2\n" ] }, { "cell_type": "markdown", "id": "6ac949ef", "metadata": {}, "source": [ "## Interpretation and evaluation strategy\n", "\n", "The two sampled first-kind symbols come from the simple $g_{22}=p/6$\n", "entry and show the expected symmetry $\\Gamma_{ijk}=\\Gamma_{ikj}$. The\n", "complete test is `einsteinResidual_#_#`; mathematically it is the zero\n", "matrix, so the scalar curvature is $5\\cdot4=20$.\n", "\n", "Expanding all 25 residual entries involves the inverse coupled block and\n", "many rational simplifications. It is intentionally not an automatic\n", "output cell: evaluate individual entries such as\n", "`einsteinResidual_2_2` when investigating the identity interactively.\n" ] } ], "metadata": { "kernelspec": { "display_name": "Egison", "language": "egison", "name": "egison" }, "language_info": { "codemirror_mode": "egison", "file_extension": ".egi", "mimetype": "text/x-egison", "name": "egison" } }, "nbformat": 4, "nbformat_minor": 5 }