Notes on Relativity and Cosmology - Physics Department, UCSB

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192 CHAPTER 7. RELATIVITY AND THE GRAVITATIONAL FIELD Down t=3s A B C D 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?
Chapter 8 General Relativity and Curved Spacetime Read Einstein, ch. 23-29, Appendices 3-5 In chapter 7 we saw that we could use the equivalence principle to calculate the effects of a gravitational field over a finite distance by carefully patching together local inertial frames. If we are very, very careful, we can calculate the effects of any gravitational field in this way. However, this approach turns out to be a real mess. Consider for example the case where the gravitational field changes with time. Then, it is not enough just to patch together local inertial frames at different positions. One must make a quilt of them at different places as well as at different times! y ff31 ff32 ff33 ff21 ff22 ff23 ff11 ff12 ff13 As you might guess, this process becomes even more complicated if we consider all 3+1 dimensions. One then finds that clocks at different locations in the gravitational field may not agree about simultaneity even if the gravitational 193 x
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Notes on Relativit
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4 CONTENTS 2.3 Homework Problems .
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6 CONTENTS 8 General Relativity and
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8 CONTENTS
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10 CONTENTS While I have worked to
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26 CHAPTER 1. SPACE, TIME, AND NEWT
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56 CHAPTER 3. EINSTEIN AND INERTIAL
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284 CHAPTER 10. COSMOLOGY Well, the
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Notes on Relativity and Cosmology - Physics Department, UCSB

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