Notes on Relativity and Cosmology - Physics Department, UCSB

228 CHAPTER 9. BLACK HOLES
of gravity.
9.1.1 Gravitational Time Dilation from the Metric
Suppose we want to calculate how clocks run in this gravitational field. This
has to do with proper time dτ, so we should remember that dτ 2 = −ds 2 . For
the Schwarzschild metric we have:
dτ 2 = −ds 2 =
(
1 − R )
s
dt 2 − dr2
r 1 − Rs
r
− r 2 (dθ 2 + sin 2 θdφ 2 ). (9.1)
The Schwarzschild metric describes any spherically symmetric gravitational field
in the region outside of all the matter. So, for example, it gives the gravitational
field outside of the earth. In using the Schwarzschild metric, remember that
R s = 2MG/c 2 .
Let’s think about a clock that just sits in one place above the earth. It does not
move toward or away from the earth, and it does not go around the earth. It
just ‘hovers.’ Perhaps it sits in a tower, or is in some rocket ship whose engine
is tuned in just the right way to keep it from going either up or down. Such a
clock is called a static clock since, from it’s point of view, the gravitational field
does not change with time.
Consider the worldline of this clock through spacetime. Along this worldline,
what is dr? How about dθ and dφ?
Since r, θ, and φ do not change, we have dr = dθ = dφ = 0. So, on our clock’s
worldline we have just: dτ 2 = ( )
1 − Rs
r dt 2 . That is,
√
dτ = 1 − R s
dt. (9.2)
r
Note that if the clock is at r = ∞ then the square √ root factor is equal to 1. So,
we might write dτ ∞ = dt. In other words, dτ = 1 − Rs
r dτ ∞, or,
√
∆τ
= 1 − R s
∆τ ∞ r . (9.3)
As saw before, clocks higher up run faster. Now, however, the answer seems to
take a somewhat simpler form than it did back in section 7.4.2, when we were
using only the Newtonian approximation.
9.1.2 Corrections to Newton’s Law
Note that the Schwarzschild geometry is a time independent gravitational field.
This means that we can use our results from section 7.4.2 to relate the rate at
which various clocks run to the acceleration of freely falling observers. In other
words, we can use this to compute the corrections to Newton’s law of gravity.