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）︰2014/06/16
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. (25 pts)
It is known that the Riemann curvature tensor satisfies the folowing Bianchi
identity
Ｒ ＋ Ｒ ＋ Ｒ ＝ 0 (1)
αβμν;γ βγμν;α γαμν;β
Here, ";" maens covariant derivative. The Ricci tensor is defined to be
β
Ｒ ≡ Ｒ ,where contraction on the index β has been made, with
αμ αβμ
Einstein's summation convention implied. We also define the Ricci scalar
α
to be Ｒ ≡ Ｒ .
α
(a) (10 pts) By repeatedly contracting the index pairs (β,ν) and then (α,μ),
please show that α 1
Ｒ － ─ Ｒ ＝ 0
γ ;α 2 ;γ
αβ αβ 1 αβ
(b) (5 pts) Show that the Einstein tensor Ｇ ≡ Ｒ － ─ ｇ Ｒ has
2
zero divergence.
(c) (5 pts) One version of Einstein's field equation reads
αβ 1 αβ αβ αβ
Ｒ － ─ ｇ Ｒ ＋ ｇ Λ ＝ 8πＧＴ (2)
2
Please use it to show that one can recast it into
αβ αβ 1 αβ μ αβ
Ｒ ＝ 8πＧ (Ｔ － ─ ｇ Ｔ ) ＋ ｇ Λ.
2 μ
(d) (5 pts) When we look at Eqn.2 we see that Einstein's field equation is
constructed in such a way that it can always make the energy-momentum
αβ
Ｔ divergence-free (in the full four-dimensional spacetime, not just
αβ
in 3-space). Physically what does it means that the energy-momentum Ｔ
is divergence-free？
2. (35 pts)
ｊ
It is known that a curve ｑ (λ) making
λ
final 1 2 ．1 ．2
∫ Ｌ(ｑ ,ｑ ,...,ｑ ,ｑ ,...) dλ
0
an extremum automatically satisfies the Euler-Lagrange equation
d ΔＬ ΔＬ
─ ( ── ) ＝ ── (註：Δ為偏導數)
dλ ．j j
Δｑ Δｑ
．j j
where ｑ means dｑ / dλ. We now consider the Lagrangian associated with
the Schwarzschild metric
╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴
∕
∕ ．2
∕ 2ＧＭ ．2 ｒ 2．2
Ｌ ≡ ∕ (1 － ───) ｔ － ────── － ｒ ψ
∕ ｒ 2ＧＭ
√ (1 － ───)
ｒ
(a) (20 pts) Please show that
2 dψ
ｌ ≡ ｒ ── ＝ constant, (3)
dτ
2ＧＭ dｔ
Ｅ ≡ (1 － ───) ── ＝ another constant, (4)
ｒ dτ
2
d ｒ ＧＭ dψ 2
── ＝ － ─── ＋ (ｒ － 3ＧＭ) (──) (5)
2 2 dτ
dτ ｒ
In the above, dτ is the proper time defined by
╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴╴
∕
∕ 2
∕ 2ＧＭ 2 dｒ 2 2
dτ ≡ ∕ (1 － ───) dｔ － ────── － ｒ dψ
∕ ｒ 2ＧＭ
√ (1 － ───)
ｒ
(b) (5 pts) A particle initially at rest at infinity falls radially inward.
Please derive its coordinate r as a function of τ.
(c) (10 pts) Find the coordinate speed dｒ/ dｔ as a function of the coordinate
radius r, then find the radius r which makes │dｒ/ dｔ│ a maximum.
3. (25 pts)
We work with freely falling circular orbits in the Schwarzschild geometry in
this problem. You can freely use Equations 3, 4, and 5.
(a) (5 pts) If a freely falling circular orbit is described by ψ＝ωｔ for
some constant ω, then please show that the orbit must have a coordinate
radius r satisfying
ＧＭ 2
─── ＝ ω ｒ
2
ｒ
(b) (5 pts) It is claimed that no freely falling circular orbits can exist when
the radius r is too small. Please find this smallest admissible radius ｒ_c.
(c) (5 pts) Does (b) mean that it is impossible to move uniformly on a circular
orbit with a radius r satisfying ｒ_s ＜ ｒ ＜ ｒ_c ？ Here ｒ_s ≡ 2ＧＭ
is the Schwarzschild radius. No deviation needed for this problem. Please
just state your reasoning.
(d) (10 pts) It is said that freely falling circular orbits are un stable if
the radius r is too small. Please find the radius of this smallest possible
stable circular orbit.
4. (15 pts)
Using spherical coordinates for this problem, a latitude circle θ＝ constant
on a sphere of radius Ｒ can be parameterized by the angular variable ψ, with
→ ︿
ψ in [0, 2π]. The outward normal Ｎ at a point on the curve is simply ｅ_r.
→ ︿
The unit tangent vector Ｔ along the curve is simply ｅ_ψ, and clearly
→ → → ︿
Ｂ ≡ Ｎ ×Ｔ is simply －ｅ_θ.
→
→ dＴ
(a) (5 pts) Show that Ｂ‧─── ＝ cosθ
dψ
(b) (5 pts) Starting from the point ψ＝ 0 on the curve, if we
→
parallel-transport the tangent vector Ｔ at this point along he curve, we
→* →*
will obtain a vector field Ｔ . Explain why Ｔ appears to rotate clockwise
→
with respect to the local tangent vector Ｔ by the amount ψcosθ.
In particular, when we complete one circuit, so that ψ increases from 0
→*
to 2π, then Ｔ now has precessed by the angle 2πcosθ clockwise with
→*
respect to the starting vector Ｔ_0.
→*
(c) (5 pts) Draw schematically the vector field Ｔ for θ ＝ vey small,
that is, near the north pole.