Simple Nature - Light and Matter

h / 1. A ray of light is emittedupward from the floor of theelevator. The elevator acceleratesupward. 2. By the time thelight is detected at the ceiling, theelevator has changed its velocity,so the light is detected with aDoppler shift.i / Pound and Rebka at thetop and bottom of the tower.than one down at its foot.To calculate this effect, we make use of the fact that the gravitationalfield in the area around the mountain is equivalent to anacceleration. Suppose we’re in an elevator accelerating upward withacceleration a, and we shoot a ray of light from the floor up towardthe ceiling, at height h. The time ∆t it takes the light ray to getto the ceiling is about h/c, and by the time the light ray reachesthe ceiling, the elevator has sped up by v = a∆t = ah/c, so we’llsee a red-shift in the ray’s frequency. Since v is small compared toc, we don’t need to use the fancy Doppler shift equation from subsection7.2.8; we can just approximate the Doppler shift factor as1 − v/c ≈ 1 − ah/c 2 . By the equivalence principle, we should expectthat if a ray of light starts out low down and then rises up througha gravitational field g, its frequency will be Doppler shifted by a factorof 1 − gh/c 2 . This effect was observed in a famous experimentcarried out by Pound and Rebka in 1959. Gamma-rays were emittedat the bottom of a 22.5-meter tower at Harvard and detected atthe top with the Doppler shift predicted by general relativity. (Seeproblem 25.)In the mountain-valley experiment, the frequency of the clockin the valley therefore appears to be running too slowly by a factorof 1 − gh/c 2 when it is compared via radio with the clock at thetop of the mountain. We conclude that time runs more slowly whenone is lower down in a gravitational field, and the slow-down factorbetween two points is given by 1 − gh/c 2 , where h is the differencein height.We have built up a picture of light rays interacting with gravity.To confirm that this make sense, recall that we have alreadyobserved in subsection 7.3.3 and in problem 11 on p. 439 that lighthas momentum. The equivalence principle says that whatever hasinertia must also participate in gravitational interactions. Thereforelight waves must have weight, and must lose energy when they risethrough a gravitational field.Local flatnessThe noneuclidean nature of spacetime produces effects that growin proportion to the area of the region being considered. Interpretingsuch effects as evidence of curvature, we see that this connectsnaturally to the idea that curvature is undetectable from close up.For example, the curvature of the earth’s surface is not normallynoticeable to us in everyday life. Locally, the earth’s surface is flat,and the same is true for spacetime.Local flatness turns out to be another way of stating the equivalenceprinciple. In a variation on the alien-abduction story, supposethat you regain consciousness aboard the flying saucer andfind yourself weightless. If the equivalence principle holds, then428 Chapter 7 Relativity