Notes on Relativity and Cosmology - Physics Department, UCSB

9.7. HOMEWORK PROBLEMS 279
just give you the answer: an object in a circular orbit at r moves around
with an angular velocity of
dφ
dt = √
Rs
2r 3 . (9.49)
That √is, it moves around the planet, star, or black hole so that φ =
R
φ 0 + t s
2r
. In particular, this means that, along its orbit we have,
3
dφ = dt
√
Rs
2r 3 . (9.50)
(a) You can now use this to calculate the relationship between the proper
time τ measured by the clock in orbit and the time t (which, you may
recall [see problem 1], is the proper time measured by a static clock
far away). Consider a clock in a circular orbit (i.e., at constant r)
around the equator (constant θ, θ = π/2). You can use the above
equation and the Schwarzschild metric to relate dτ directly to dt.
[Remember, dτ 2 = −ds 2 .] Solve the resulting equation to express τ
in terms of t.
(b) If your calculations above are correct, something interesting should
happen at r = 3R S /2. What happens to the relationship of τ and t
there?
(c) [extra credit] Do you know how to interpret the result you found
in (3b)? [Hint: what happens to dτ 2 for r < 3R S /2]?
4. Suppose that you are tossed out of a space ship outside of a black hole
and that you fall in. To answer the questions below, use the fact that the
spacetime near the horizon of a black hole is just like the region of flat
spacetime near an acceleration horizon.
(a) Recall that, near the black hole horizon, a spacetime diagram (in a
certain freely falling frame) looks like this: