{ "cells": [ { "cell_type": "markdown", "id": "ca283802", "metadata": {}, "source": [ "# Riemann Curvature of the FLRW Metric\n", "\n", "The Friedmann--Lemaître--Robertson--Walker geometry models a homogeneous,\n", "isotropic universe. In comoving coordinates $(w,r,\\theta,\\phi)$ and units\n", "$c=1$,\n", "\n", "$$\n", "ds^2=-dw^2+a(w)^2\\left(\n", " \\frac{dr^2}{1-Kr^2}+r^2d\\theta^2+r^2\\sin^2\\theta\\,d\\phi^2\n", "\\right).\n", "$$\n", "\n", "The scale factor $a(w)$ is left as an arbitrary symbolic function and $K$\n", "is the constant spatial-curvature parameter.\n" ] }, { "cell_type": "markdown", "id": "b0ce9139", "metadata": {}, "source": [ "## Metric data\n", "\n", "Write $W(r)=(1-Kr^2)^{-1}$. The inverse metric is entered explicitly so\n", "that selected connection and curvature components stay responsive.\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "ba06e181", "metadata": {}, "outputs": [], "source": [ "declare symbol w, r, θ, φ, K: MathValue\n", "\n", "def x : Vector MathValue := [| w, r, θ, φ |]\n", "def a : MathValue := function (w)\n", "\n", "def W (r: MathValue) : MathValue := 1 / `(1 - K * r^2)\n", "\n", "def g_i_j : Matrix MathValue :=\n", " [| [| -1, 0, 0, 0 |]\n", " , [| 0, a^2 * W r, 0, 0 |]\n", " , [| 0, 0, a^2 * r^2, 0 |]\n", " , [| 0, 0, 0, a^2 * r^2 * (sin θ)^2 |]\n", " |]_i_j\n", "\n", "def g~i~j : Matrix MathValue :=\n", " [| [| -1, 0, 0, 0 |]\n", " , [| 0, 1 / (a^2 * W r), 0, 0 |]\n", " , [| 0, 0, 1 / (a^2 * r^2), 0 |]\n", " , [| 0, 0, 0, 1 / (a^2 * r^2 * (sin θ)^2) |]\n", " |]~i~j\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "727f0130", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$(-K r^{2} + 1)^{-1}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "W r\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "04031550", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\begin{pmatrix} -1 & 0 & 0 & 0 \\\\ 0 & (-K r^{2} + 1)^{-1} a^{2} & 0 & 0 \\\\ 0 & 0 & a^{2} r^{2} & 0 \\\\ 0 & 0 & 0 & \\sin(θ)^{2} a^{2} r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g_#_#\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "38ce36de", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\begin{pmatrix} -1 & 0 & 0 & 0 \\\\ 0 & (-K r^{2} + 1) a^{-2} & 0 & 0 \\\\ 0 & 0 & a^{-2} r^{-2} & 0 \\\\ 0 & 0 & 0 & \\sin(θ)^{-2} a^{-2} r^{-2} \\\\ \\end{pmatrix}_{\\;\\;}^{\\#\\#}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g~#~#\n" ] }, { "cell_type": "markdown", "id": "12794dc1", "metadata": {}, "source": [ "## Expansion enters the connection\n", "\n", "Time derivatives of the spatial metric generate Christoffel symbols\n", "proportional to $a'(w)$. In particular, radial expansion appears in both\n", "$\\Gamma^w{}_{rr}$ and $\\Gamma^r{}_{wr}$.\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "d2dd34a3", "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": "8cdf4e37", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$(-K r^{2} + 1)^{-1} a \\frac{\\partial a}{\\partial 1}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Γ~1_2_2\n" ] }, { "cell_type": "code", "execution_count": 7, "id": "e6adce4a", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$a^{-1} \\frac{\\partial a}{\\partial 1}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Γ~2_1_2\n" ] }, { "cell_type": "markdown", "id": "0dd35c18", "metadata": {}, "source": [ "## Riemann tensor and contractions\n", "\n", "Components mixing time and space measure the acceleration of the scale\n", "factor; purely spatial components combine $(a')^2$ with $K$. The complete\n", "contractions are\n", "\n", "$$\n", "\\operatorname{Ric}_{ij}=R^m{}_{imj},\\qquad\n", "\\mathcal R=g^{ij}\\operatorname{Ric}_{ij}.\n", "$$\n" ] }, { "cell_type": "code", "execution_count": 8, "id": "14b7867d", "metadata": {}, "outputs": [], "source": [ "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 scalarCurvature : MathValue := withSymbols [i, j]\n", " expandAll' (g~i~j . Ric_i_j)\n" ] }, { "cell_type": "code", "execution_count": 9, "id": "acfcba0a", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$(-K r^{2} + 1)^{-1} a \\frac{\\partial^2 a}{\\partial 1^2}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "R~1_2_1_2\n" ] }, { "cell_type": "code", "execution_count": 10, "id": "923fe1f1", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\frac{\\partial a}{\\partial 1}^{2} r^{2} + K r^{2}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "R~2_3_2_3\n" ] }, { "cell_type": "markdown", "id": "b16ed4a7", "metadata": {}, "source": [ "## Scalar curvature and interpretation\n", "\n", "With this convention, the expected scalar is\n", "\n", "$$\n", "\\mathcal R\n", " =6\\left(\\frac{a''(w)}{a(w)}+\n", " \\frac{a'(w)^2+K}{a(w)^2}\\right)\n", " =\\frac{6\\bigl(a''a+(a')^2+K\\bigr)}{a^2}.\n", "$$\n", "\n", "The selected Riemann components expose the two geometric ingredients:\n", "accelerated expansion and curvature of spatial slices. Evaluating\n", "`scalarCurvature` asks the CAS to form and simplify the full contraction\n", "for an arbitrary function $a$ and can take several minutes, so it is\n", "deliberately a definition rather than an automatic output cell.\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 }