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4月9日 考試時限（分鐘）：110分鐘（9:10 - 12:00） 試題 : General Relativity midterm examination April 9, 2018 [1] Determine the proper times measured by the reunion twins in the twin paradox, assuming the star is 15 light years away from earth and the space ship is accel- erated/decelerated with g to reach the constant speed at 0.6c or zero. You need to derive the form for a proper constant acceleration motion and apply a rough numerical estimation for the above case. [2] The Greisen-Zatsepin-Kuzmin limit (GZK limit) is a theoretical upper limit on the energy of extragalatic protons observed on earth. The limit is ~ 5 x 10^{19} eV and is set by the interactions of the protons with the microwave background radiation, pγCMB -> Δ+, over long distances. The mass of proton and Δ+ are 938 MeV/c2 and 1232 MeV/c2, respectively. Take the rms thermal energy ~ 0.04 eV at 300K and use the relativistic mechanics to determine this threshold energy of proton for pγCMB -> Δ+. You need to specify the moving direction of γCMB w.r.t. the incident proton in your calculation. [3] Show explicitly that the Lorentz inner product, A^μB_μ= AtBt - AxBx - AyBy - AzBz, of two four-vectors A and B is invarient under the Lorentz boost along the x axis with a speed v. Note that the above diagonal (1,-1,-1,-1) feature of the symmetric metric tensor g_{μν} is valid only in a flat Minkowski space. [4] Find the metric tensor of a spherical polar coordinate for a flat three-dimensional space. The coordinate bases are denoted by dx^1 = dr, dx^2 = dθ and dx^3 = dφ. Use your favorite definition of the Christoffel symbols of the second kind associated with the covarient derivative on a manifold to determine Γ^{2}_{12} and Γ^{3}_{12}. [5] A particle in a gravitational field under free fall will follow a time-like geodesic path. Derive the geodesic equation on a curved space-time. You need to use x^μ to denote the coordinate and Γ^{λ}_{μν} for the Christoffel symbol of the metric. --

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