Notes on Relativity and Cosmology - Physics Department, UCSB

9.1. INVESTIGATING THE SCHWARZSCHILD METRIC 229
Recall the relation (equation 7.16) is
( ∫
∆τ b
b
= exp
∆τ a
a
α(s)
c 2 ds )
. (9.4)
Here, α(s) is the acceleration of a static clock relative to a freely falling clock at
s, and s measures distance. To compare this with our formula above, we want
to take a = s and b = ∞. Taking the ln of both sides gives us
ln
( ) ∫ τ(s) s
=
τ ∞ ∞
Now, taking a derivative with respect to s we find:
α(s)
c 2
α(s)
ds. (9.5)
c2 = − d ds ln ( τ(s)
τ ∞
)
. (9.6)
Now, it is important to know what exactly s measures in this formula. Recall
that when we derived this result we were interested in the actual physical height
of a tower. As a result, this s describes proper distance, say, above the surface
of the earth.
On the other hand, equation (9.3) is given in terms of r which, it turns out,
does not describe proper distance. To see this, let’s think about the proper
distance ds along a radial line with dt = dθ = dφ = 0. In this case, we have
ds 2 =
dr2
1−R s/r
dr
, or ds = √ , and
1−Rs/r
dr
ds = √ 1 − R s /r. (9.7)
However we can deal with this by using the chain rule:
α = c 2 d ( ) ( ) ( )
τ(s) dr d τ(r)
ds ln = c 2
τ ∞ ds dr ln . (9.8)
τ ∞
Going through the calculation yields:
α = c 2√ 1 − R s /r d dr ln√ 1 − R s /r
Note that for r ≫ R s , we have α ∼ c2 2
result.
= c 2√ 1 − R s /r 1 d
2 dr ln(1 − R s/r)
= c2 √ 1 +R s
1 − Rs /r
2 1 − R s /r r 2
c 2
=
2 √ R s
1 − R s /r r 2 . (9.9)
R s
r 2 = MG
r 2 . This is exactly Newton’s