Notes on Relativity and Cosmology - Physics Department, UCSB

192 CHAPTER 7. RELATIVITY AND THE GRAVITATIONAL FIELD
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t=3s
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t=1s
x=0 x=5m x=10m
7-4. Due to the effects of General Relativity (and also due to the fact that
the earth is not completely round) the effective Newtonian gravitational
field of the earth is not exactly GM/r 2 . Other terms contribute, and these
must be considered by the designers of the Global Positioning System, as
they effect the rate at which the clocks run on the GPS satellites. A more
accurate model of the earth’s gravitational field is
g = GM/r 2 + a/r 3 + b/r 4 (7.23)
in terms of the distance r away from the center of the earth. Here, a and
b are certain constants having to do with the exact shape of the earth.
Using this model, compute the ratio between the rate of ticking of a clock
(A) at distance r A from the center and a clock (B) at a distance r B from
the center. Express the answer in terms of the constants G, M, r A , r B , a
and b.
Note: This problem is really just a calculus problem. In section 7.4 we
discussed equation (7.16) which tells us the relationship between the rates
at which clocks run at different places l in a gravitational field. When this
gravitational field is produced by a round object like the earth, it is natural
to use the radial distance r from the center of the earth as our coordinate
l. So, the problem above just consists of performing the corresponding
integral for the specified function g(r). Note: In (7.16) we used l as
the distance variable. In this problem I have used r. Just replace
l in (7.16) by r to use that formula here.
7-5. Use the mass and radius of the earth as given in section 7.4 to calculate
how much faster a clock in Denver runs than does a clock in Washington,
D.C. Denver is 1600m higher than Washington, D.C.
7-6. Of course, the earth also spins on its axis, so that neither city in 7-5 is
standing still... Estimate the relative velocity between the two cities, and
use this to estimate the size of the time dilation effect you would expect
from special relativity. Do we need to consider this effect as well?