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.