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.