Notes on Relativity and Cosmology - Physics Department, UCSB

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218 CHAPTER 8. GENERAL RELATIVITY AND CURVED SPACETIME here) to look for new effects is close to the sun. One might therefore start by considering the orbit of Mercury. Actually, there is an interesting story about Mercury and its orbit. Astronomers had been tracking the motion of the planets for hundreds of years. Ever since Newton, they had been comparing these motions to what Newton’s law of gravity predicted. The agreement was incredible. In the early 1800’s, they had found small discrepancies (30 seconds of arc in 10 years) in the motion of Uranus. For awhile people thought that Newton’s law of gravity might not be exactly right. However, someone then had the idea that maybe there were other objects out there whose gravity affected Uranus. They used Newton’s law of gravity to predict the existence of new planets: Neptune, and later Pluto. They could even tell astronomers where to look for Neptune within about a degree of angle on the sky. However, there was one discrepancy with Newton’s laws that the astronomers could not explain. This was the ‘precession’ of Mercury’s orbit. The point is that, if there were nothing else around, Newton’s law of gravity would say that Mercury would move in a perfect ellipse around the sun, retracing its path over, and over, and over... Mercury Sun Of course, there are small tugs on Mercury by the other planets that modify this behavior. However, the astronomers knew how to account for these effects. Their results seemed to say that, even if the other planets and such were not around, Mercury would do a sort of spiral dance around the sun, following a path that looks more like this: Ellipse does not close Mercury Sun Here, I have drawn the ellipse itself as rotating (a.k.a. ‘precessing’) about the sun. After all known effects had been taken into account, astronomers found that Mercury’s orbit precessed by an extra 43 seconds of arc per century. This is certainly not very much, but the astronomers already understood all of the other
8.5. EXPERIMENTAL VERIFICATION OF GR 219 planets to a much higher accuracy. So, what was going wrong with Mercury? Most astronomers thought that it must be due to some sort of gas or dust surrounding the Sun (a big ‘solar atmosphere’) that was somehow affecting Mercury’s orbit. However, Einstein knew that his new theory of gravity would predict a precession of Mercury’s orbit for two reasons. First, he predicted a slightly stronger gravitational field (since the energy in the gravitational field itself acts as a source of gravity). Second, in Einstein’s theory, space itself is curved and this effect will also make the ellipse precess (though, since the velocity of mercury is small, this effect turns out to be much smaller than the one due to the stronger gravitational field). The number that Einstein calculated from his theory was 43 seconds of arc per century. That is, his prediction agreed with the experimental data to better than 1%! Clearly, Einstein was thrilled. This was big news. However, it would have been even bigger news if Einstein had predicted this result before it had been measured. Physicists are always skeptical of just explaining known effects. After all, maybe the scientist (intentionally or not) fudged the numbers or the theory to get the desired result? So, physicists tend not to really believe a theory until it predicts something new that is then verified by experiments. This is the same sort of idea as in double blind medical trials, where even the researchers don’t know what effect they want a given pill to have on a patient! 8.5.2 The Bending of Starlight Luckily, Einstein had an idea for such an effect and now had enough confidence in his theory to push it through. The point is that, as we have discussed, light will fall in a gravitational field. For example, a laser beam fired horizontally across the classroom will be closer to the ground on the side where it hits the far wall it was when it left the laser. Similarly, a ray of light that goes skimming past a massive object (like the sun) will fall a bit toward the sun. The net effect is that this light ray is bent. Suppose that the ray of light comes from a star. What this means in the end is that, when the Sun is close to the line connecting us with the star, the star appears to be in a slightly different place than when the Sun is not close to that light ray. For a light ray that just skims the surface of the Sun, the effect is about .875 seconds of arc. star appears to be here actual star Sun earth actual path ‘falls’ toward the Sun
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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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12 CONTENTS way you deal with almos
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14 CONTENTS but this does not mean
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16 CONTENTS 0.3 Coursework and Grad
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18 CONTENTS d) include references t
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20 CONTENTS However, your challenge
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22 CONTENTS 0.4 Some Suggestions fo
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24 CONTENTS Course Calendar The fol
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26 CHAPTER 1. SPACE, TIME, AND NEWT
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44 CHAPTER 2. MAXWELL, E&M, AND THE
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56 CHAPTER 3. EINSTEIN AND INERTIAL
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86 CHAPTER 4. MINKOWSKIAN GEOMETRY
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144 CHAPTER 6. DYNAMICS: ENERGY AND
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272 CHAPTER 9. BLACK HOLES r = 0 r
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280 CHAPTER 9. BLACK HOLES r = R s
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282 CHAPTER 9. BLACK HOLES (c) What
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284 CHAPTER 10. COSMOLOGY Well, the
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294 CHAPTER 10. COSMOLOGY Since P =
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