{ "cells": [ { "cell_type": "markdown", "id": "c20860aa", "metadata": {}, "source": [ "# シュワルツシルト計量のリーマン曲率\n", "\n", "静止した球対称質量$M$の外部で、次のようにおきます。\n", "\n", "$$\n", "A(r)=1-\\frac{2GM}{c^2r}.\n", "$$\n", "\n", "符号$(+---)$、座標$(t,r,\\theta,\\phi)$を用いると、線素は\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", "この計量はリッチ平坦ですが、リーマン平坦ではありません。曲率の\n", "全成分を計算する高負荷な展開を行わず、いくつかの成分を選んで\n", "この違いを確認します。\n" ] }, { "cell_type": "markdown", "id": "459657af", "metadata": {}, "source": [ "## 局所フレーム、計量、逆計量\n", "\n", "この座標表示は、$r>0$かつ$A=0$でない領域を覆います。曲面\n", "$r=2GM/c^2$はこの座標表示における座標地平面ですが、$r=0$の\n", "特異性は曲率不変量によって検出されます。\n", "$\\eta=\\operatorname{diag}(1,-1,-1,-1)$である局所Lorentzフレームでは、\n", "座標接ベクトルの成分は\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は計量や逆計量を成分ごとに入力するのではなく、\n", "$g_{ij}=\\eta(e_i,e_j)$として計量を求め、その計量から逆計量を計算します。\n" ] }, { "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" ] }, { "cell_type": "code", "execution_count": 2, "id": "b9c12b39", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$(c^{2} r - 2 G M) c^{-2} r^{-1}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "A\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "f690f1c6", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\begin{pmatrix} (c^{2} r - 2 G M) c^{-2} r^{-1} & 0 & 0 & 0 \\\\ 0 & -(c^{2} r - 2 G M)^{-1} c^{2} r & 0 & 0 \\\\ 0 & 0 & -r^{2} & 0 \\\\ 0 & 0 & 0 & -\\sin(θ)^{2} r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g_#_#\n" ] }, { "cell_type": "code", "execution_count": 4, "id": "ad569a81", "metadata": {}, "outputs": [ { "data": { "text/html": [ "$\\begin{pmatrix} (c^{2} r - 2 G M)^{-1} c^{2} r & 0 & 0 & 0 \\\\ 0 & -(c^{2} r - 2 G M) c^{-2} r^{-1} & 0 & 0 \\\\ 0 & 0 & -r^{-2} & 0 \\\\ 0 & 0 & 0 & -\\sin(θ)^{-2} r^{-2} \\\\ \\end{pmatrix}_{\\;\\;}^{\\#\\#}$" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g~#~#\n" ] }, { "cell_type": "markdown", "id": "014e4807", "metadata": {}, "source": [ "## レヴィ・チヴィタ接続\n", "\n", "接続を表として入力するのではなく、計量から計算します。\n", "$\\Gamma^\\theta{}_{r\\theta}=1/r$や\n", "$\\Gamma^\\phi{}_{\\theta\\phi}=\\cot\\theta$などの角度方向の成分は、\n", "特に簡潔な検算になります。\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": [ "## リーマンテンソルとリッチテンソル\n", "\n", "球面のNotebookと同じ符号規約を用います。$M\\ne0$のとき、角度成分\n", "$R^\\theta{}_{\\phi\\theta\\phi}$は0ではありません。平坦な球座標計量では、\n", "その動径方向と角度方向の項が打ち消し合います。リッチ曲率は\n", "$R^m{}_{imj}$を縮約したものです。\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": [ "## 物理的・幾何学的な意味\n", "\n", "リーマンテンソルの成分は潮汐曲率を捉えます。一方、リッチテンソルの\n", "成分は、真空アインシュタイン方程式が要請するとおり0に簡約されます。\n", "座標に依存しないクレッチマン不変量は\n", "\n", "$$\n", "R_{abcd}R^{abcd}=\\frac{48G^2M^2}{c^4r^6}.\n", "$$\n", "\n", "です。したがって、曲率はシュワルツシルト地平面では有限ですが、\n", "$r=0$では発散します。4つの添字の全組合せを展開すると対話的なデモには\n", "不必要なほど計算量が大きくなるため、不変量の完全な縮約は出力セルで\n", "計算せず、式として示しています。\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 }