Notes on Relativity and Cosmology - Physics Department, UCSB

9.7. HOMEWORK PROBLEMS 277
9.7 Homework Problems
Fun with the Schwarzschild Metric: In the first few problems below, you
will explore the Schwarzschild metric – the metric that describes the gravitational
field (that is, the shape of spacetime) outside of any spherically symmetric
distribution of matter. Thus, this same metric describes the shape of spacetime
outside of round planets, round starts, round neutron stars, and round black
holes!!! The only differences between these objects are 1) the value of the mass
M and 2) how close you can get to the object before you hit the surface of the
matter (so that the Schwarzschild metric no longer applies).
In it’s full glory, the Schwarzschild metric is:
(
ds 2 = − 1 − R )
s
dt 2 + dr2
r 1 − Rs
r
where R S = 2MG/c 2 .
+ r 2 dθ 2 + r 2 sin 2 θ dφ 2 , (9.48)
Here, θ and φ are coordinates on a sphere as shown below. θ is similar to
latitude – it is the angle between your position on the sphere and the north
pole – thus, θ goes from zero (at the north pole) to π at the south pole. φ is
like longitude, running from 0 to 2π around the sphere. These are the standard
spherical coordinates used by physicists; however, they are ‘backwards’ from the
coordinates used in many calculus classes (that is, θ and φ have been switched).
The point p below has coordinates θ, φ:
North Pole
p
θ
φ
1. Consider a static clock at some value of r.
(a) In terms of the radial position r, how fast does a static clock run
relative to a clock far away (at r = ∞)? Hint: calculate the proper
time dτ measured by such a clock.
(b) Suppose that you are standing on the surface of the Sun. How fast
does your clock run relative to a clock far away (at r = ∞)? (i.e.,
calculate the relevant number.) Note that M sun = 2 × 10 30 kg, G =
6.7 × 10 −11 Nm 2 /kg 2 , and c = 3 × 10 8 m/s. Also, at the surface of
the sun, r = 7 × 10 8 m.