シュワルツシルト計量

以下のコードは、シュワルツシルト計量のリーマン曲率テンソルを計算しています。

-- Riemann curvature tensor of Schwarzschild metric def x := [| t, r, θ, φ |] -- Schwarzschild metric def g_i_j := [| [| '(c^2 * r - 2 * G * M) / (c^2 * r), 0, 0, 0 |] , [| 0, (-1) / ('(c^2 * r - 2 * G * M) / (c^2 * r)), 0, 0 |] , [| 0, 0, -r^2, 0 |] , [| 0, 0, 0, -r^2 * (sin θ)^2 |] |] def g~i~j := M.inverse g_#_# assertEqual "Metric tensor g_1_#" g_1_# [| '(c^2 * r - 2 * G * M) / (c^2 * r), 0, 0, 0 |]_# assertEqual "Metric tensor g_2_#" g_2_# [| 0, (-1) / ('(c^2 * r - 2 * G * M) / (c^2 * r)), 0, 0 |]_# assertEqual "Metric tensor g_3_#" g_3_# [| 0, 0, -r^2, 0 |]_# assertEqual "Metric tensor g_4_#" g_4_# [| 0, 0, 0, -r^2 * (sin θ)^2 |]_# -- Christoffel symbols def Γ_i_j_k := (1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i) def Γ~i_j_k := withSymbols [m] g~i~m . Γ_m_j_k assertEqual "Christoffel symbols of the second kind Γ~1_#_#" Γ~1_#_# [| [| 0, G * M / (c^2 * r^2 - 2 * G * M * r), 0, 0 |] , [| G * M / (c^2 * r^2 - 2 * G * M * r), 0, 0, 0 |] , [| 0, 0, 0, 0 |] , [| 0, 0, 0, 0 |] |]_#_# assertEqual "Christoffel symbols of the second kind Γ~2_#_#" Γ~2_#_# [| [| G * M * (c^2 * r - 2 * G * M) / (c^4 * r^4), 0, 0, 0 |] , [| 0, -1 * G * M / (c^2 * r^2 - 2 * G * M * r), 0, 0 |] , [| 0, 0, -1 * r + 2 * G * M / c^2, 0 |] , [| 0, 0, 0, (-1 * r + 2 * G * M / c^2) * (sin θ)^2 |] |]_#_# assertEqual "Christoffel symbols of the second kind Γ~3_#_#" Γ~3_#_# [| [| 0, 0, 0, 0 |] , [| 0, 0, 1 / r, 0 |] , [| 0, 1 / r, 0, 0 |] , [| 0, 0, 0, -1 * sin θ * cos θ |] |]_#_# assertEqual "Christoffel symbols of the second kind Γ~4_#_#" Γ~4_#_# [| [| 0, 0, 0, 0 |] , [| 0, 0, 0, 1 / r |] , [| 0, 0, 0, (cos θ) / (sin θ) |] , [| 0, 1 / r, (cos θ) / (sin θ), 0 |] |]_#_# -- Riemann curvature def R~i_j_k_l := withSymbols [m] expandAll (∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l) -- Ricci curvature (should be 0 for Schwarzschild in vacuum) def Ric_i_j := withSymbols [m] sum (contract R~m_i_m_j) -- The Schwarzschild metric is a vacuum solution: Ric = 0 assertEqual "Ricci curvature Ric_1_#" Ric_1_# [| 0, 0, 0, 0 |]_# assertEqual "Ricci curvature Ric_2_#" Ric_2_# [| 0, 0, 0, 0 |]_# assertEqual "Ricci curvature Ric_3_#" Ric_3_# [| 0, 0, 0, 0 |]_# assertEqual "Ricci curvature Ric_4_#" Ric_4_# [| 0, 0, 0, 0 |]_#

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