[試題] 102-2 陳義裕 相對論 期中考
課程名稱︰相對論
課程性質︰選修
課程教師︰陳義裕
開課學院：理學院
開課系所︰物理所
考試日期（年月日）︰2014/04/14
考試時限（分鐘）：170
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(40 pts) The lorentz transformation in one-dimension assumes the standard
symmetric form
t' = γ(t - vx),
x' = γ(x - vt),
in "natural units," that is, time is measured in units of length, so that
the speed of light is unity. The lab frame observer at the origin of the
unprimed frame sends out the light signals in the positive x-direction at the
respective time t_1 = 0 and t_2 = Δt. The promed observer who always sits at
the origin of the primed frame receives the signals at the two respective
time t_1' and t_2'.
a)(15 pts) Please compute Δt'≡ t_2' - t_1' in terms of Δt and v.
b)(5 pts) Describe in words how we may use above result to derive the
relativistic Doppler effect. (Noderivation is required.)
c)(5 pts) A particle has an instantaneous velocity u as measured by the
unprimed frame. Please use the Lorentz transformation to find its velocity u'
measured by the primed frame.
d)(5 pts) If this particle experiences an instantaneous acceleration a relative
to the unprimed frame, please use the Lorentz transformation to find its
acceleration a' relative to the primed frame.
e)(10 pts) The relativistic momentum p of a particle of mass m and velocity v
is
mu
p ≡ ──────
√1 - u^2
and force F is defined as F ≡ dp/dt. If the particle is experiencing a force
as measured by a lab observer, then please show that one measures a force
which has exactly the same magnitude F in the co-moving frame of the
particle.
2.(10 pts) A rod is being accelerated from rest to a high speed along the axial
direction of the rod. It is known that the rod is being accelerated in such a
way that its length remains the same in the co-moving frame of the rod.
Please explain why a lab observer will observe the rod to be shrinking
whenever the rod is being accelerated forward.
3.(15 pts) Assume that the pole and the barn have the same proper length in the
"barn-pole paradox." Denote the two ends of the pole by A and B, while the
two barn doors are denoted by C and D,respectively. (See Fig. 1 below.)
Define three events as follows
E1: B passes through D.
E2: A passes through C, and simultaneously sends a small rock back
towards B.
E3: The rock hits B.
a)(3 pts) Write down the temporal order of the three events in the barn frame.
b)(3 pts) Write down the temporal order of the three events in the pole frame.
c)(5 pts) Use the Lorentz transformation to prove your claim of Part (b).
—→
d)(2 pts) Is the 4-vector E1E2 space-like of time like? (No derivation is
needed.)
—→
e)(2 pts) Is the 4-vector E2E3 space-like of time like? (No derivation is
needed.)
4.(25 pts) The Rindler coordinates of a three-dimensional spacetime is given by
ds^2 = -(1 + aX)^2 dT^2 + dX^2 + dY^2,
where a is the uniform acceleration of some "fiducial" observer Fiona located
at (X,Y) = (0,0). In the above, T is in fact the proper time of Fiona,
whereas X and Y are her spatial coordinates.
a)(5 pts) Consider a stationary observer Stan located at (X,Y). Please express
his proper time τ in terms of Y, X and Y.
b)(5 pts) Please express the 4-velocity of Stan as some suitable linear
→ → → →
combinations of e_T, e_X and e_Y. Here , the e's refer to the coordinate
basis.
→
c)(5 pts) If the 4-acceleration of Fiona is denoted by a , please compute the
4-acceleration experienced by Stan.
d)(10 pts) It is known that the transformation between the lab frame
coordinates (t,x,y) and (T,X,Y) is
t = (1/a + X) sinhaT ,
x = (1/a + X) coshaT - 1/a,
y = Y.
Please show that a light ray has a trajectory that is a part of a circle in
the (X,Y) coordinates.
5.(10 pts) Suppose we use the coordinates (θ,φ) to describe the position
→
vector r ≡ (x,y) of a two-dimensional Euclidean space via
→
r = (coshθcosφ, sinhθsinφ).
Please compute the metric coefficients f, g and h in
ds^2 = f dθ^2 + g dθd φ + h dφ^2.
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