課程名稱︰相對論 課程性質︰選修 課程教師︰高湧泉 開課學院：理學院 開課系所︰物理學研究所 考試日期（年月日）︰2011/06/22 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. A particle of rest mass m1 and velocity v1 collides with a stationary particle of rest mass m2 and is absorbed by it. Find the rest mass m and velocity v of the resultant compound system. (15%) 2. Write down the Maxxwell equations in an arbitrary coordinate system. (Explain your notation.)(15%) 3. Consider a charged particle (charge q, mass m) in an electromagnetic field. Write down the equation for the particle's world line in an arbitrary coordinate.(15%) α ┌ ┐ α α β 4. Define the Riemann tensor R by ｜▽ ,▽ ｜V = R V βμν └ μ ν┘ βμν λ λ And Γ =Γ . Show that: μν μν ┌ ┐ α α β (a) ｜▽ ,▽ ｜W = - R W . └ μ ν┘ μνβ ┌ ┐ β (b) │▽ ,▽ ｜W = R W (15%) └ μ ν┘ α μνα β 5.(a)What is the so-called Birkhoff theorem? (b)What are the Killing vectors for the Schwarzschild geometry? (c)What does one mean when he/she says that the Schwarzschild surface(r=Rs) is only a coordinate singularity? (total 15%) 6.In simple qualitative terms, explain how gravitational lensing can give rise to "Einstein rings"?(15% 7.In the linearized Einstein equation, if we choose the so-called harmonic gauge (Lorenz gauge), we will get an equation analogous to the wave equation with sourcein Maxwell theory. (a)What is the Lorenz gauge in the linearized gravity? (b)What is the Linearized Einstein equation for weak source jn the Lorenz gauge? (c)Explain how the gravitational wave solution in free space contains only 2 polarization (helicity) modes? (total 15%) --

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