Notes on Relativity and Cosmology - Physics Department, UCSB

7.4. GOING BEYOND LOCALITY 185
Still, in a uniform gravitational field with g(l) = g the integral in (7.15) is easy
to do and we get just:
∆τ l
∆τ 0
= e gl/c2 . (7.17)
In this case, the difference in clock rates grows exponentially with distance.
The other interesting case to consider is something that describes (to a good
approximation) the gravitational field near the earth. We have seen that Newton’s
law of gravity is a pretty good description of gravity near the earth, so we
should be able to use the Newtonian form of the gravitational field:
g = m EG
r 2 , (7.18)
where r is the distance from the center of the earth. This means that we can
use dr in place of dl in (7.15). Let us refer to the radius of the earth as r 0 .
For this case, it is convenient to compare the rate at which some clock runs at
radius r to the rate at which a clock runs on the earth’s surface (i.e., at r = r 0 ).
Since ∫ r 2
r 1
r −2 dr = r1 −1 − r2 −1 , we have
∫
∆τ(r) r
[
∆τ(r 0 ) = exp( m E G
r 0
c 2 r 2 dr) = exp mE G
c 2
( 1
r 0
− 1 r
)]
. (7.19)
Here, it is interesting to note that the r dependence drops out as r → ∞, so
that the gravitational time dilation factor between the earth’s surface (at r 0 )
and infinity is actually finite. The result is
∆τ(∞)
∆τ(r 0 ) = e m E G
r 0 c 2 . (7.20)
So, time is passing more slowly for us here on earth than it would be if we were
living far out in space..... By how much? Well, we just need to put in some
numbers for the earth. We have
m E
= 6 × 10 24 kg,
G = 6 × 10 −11 Nm 2 /kg 2 ,
r 0 = 6 × 10 6 m. (7.21)
Putting all of this into the above formula gives a factor of about e 2 3 ×10−10 .
Now, how big is this? Well, here it is useful to use the Taylor series expansion
e x = 1 + x + small corrections for small x. We then have
∆τ(∞)
∆τ(r 0 ) ≈ 1 + 2 3 × 10−10 . (7.22)
This means that time passes more slowly for us than it does far away by roughly
one part in 10 10 , or, one part in ten billion! This is an incredibly small amount