[试题] 102-2 陈义裕 相对论 期末考

楼主: jeff20414 (乌哩)   2014-06-19 20:09:18
课程名称︰相对论
课程性质︰选修
课程教师︰陈义裕
开课学院:理学院
开课系所︰物理所
考试日期(年月日)︰2014/06/16
考试时限(分钟):170
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
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)

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
─ ( ── ) = ── (注:Δ为偏导数)
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 - ───)

(a) (20 pts) Please show that
2 dψ
l ≡ r ── = constant, (3)

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 - ───)

(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

(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

(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.

Links booklink

Contact Us: admin [ a t ] ucptt.com