課程名稱︰廣義相對論 課程性質︰天文物理所必修 課程教師︰王名儒 開課學院：理學院 開課系所︰物理系、物理所、天文物理所 考試日期（年月日）︰2018年5月21日 考試時限（分鐘）：110分鐘（9:10 - 12:00） 試題 : General Relativity midterm examination May 21, 2018 [1] The Riemann curvature tensor R^h ijk is defined by the following equation: R^h ijk = Γ^h ik,j - Γ^h ih,k + Γ^h aj Γ^a ik - Γ^h ak Γ^a ij where , denotes the partial derivative in the above equation and Γ^i jk = 1/2 g^hi (g hi,k + g hk,j - g jk,h) stands for the Christoffel symbol of the seco nd kind. For a 4-dimensional diagonal metric, ds^2 = A(dx^1)^2 + B(dx^2)^2 + C (dx^3)^2 + D(dx^4)^2. Show that 2R1213 = -A23 + αA2A3 + βA2B3 + γA3C2, wher e Ai denotes δA/δx^i, α stands for 1/2A, β for 1/2B, and γ for 1/2C. Note that R hijk is equal to g ha R^a ijk. [2] In the derivation of the Einstein field equation, we need the Bianchi iden tity: R hijk;l + R hikl;j + R hilj;k = 0 where R is the Riemann curvature tens or and ; denotes the covariant derivative. Prove the Bianchi identity. As a ti p, you can choose a geodesic coordinate in order to simplify your answer. [3] Write down the Schwarzchild solution of a static and spherical symmetric m etric tensor. You should use m = GM/c^2 to denote the mass ofthe spherical obj ect. What's the physical meaning or definition of the radial coordinate r? Fin d out the ruler distance and the radar distance between two objects at r1 and r2 witb identical θ and φ. You should state clearly about your choice of ref erence location to send and receive the radar signal. [4] Derive the functional form of the Flamms paraboloid with the extra dimensi on z to represent the Schwarzchild metric element in the radial r direction ge ometrically, i.e., (1 - 2m/r)^-1 dr^2 = dz^2 + dr^2. [5] Write down the so-called Lagrangian for a particle in free fall due to a s tatic and spherically symmetric object with mass m. You can assume its path is on the θ = π/2 plane. If this particle is a photon, what is its equation of path? You need to give a second order differential equation in terms of u = 1 /r and φ. --

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