From Special Relativity to Feynman Diagrams.pdf

2 d τ 1 φ c 2 1 φ 2 c 2 377 since φ 2 0 we find

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2 d τ 1 φ c 2 = 1 φ 2 c 2 . (3.77) Since φ 2 < 0 we find dt 2 > dt 1 , that is time intervals are dilated in a gravitational field by a factor ( 1 φ 2 c 2 ). In the case of the terrestrial gravitational field we have: dt 2 = 1 + GM c 2 r d τ. (3.78) where M is the earth’s mass. In particular a same clock will tick at a different rate, depending on whether it is placed at sea level or at height h . Indeed, since φ 1 = − GM R ; φ 2 = − GM R + h , (3.79) we obtain: dt 2 = 1 φ 2 φ 1 c 2 dt 1 = 1 GM c 2 1 R 1 R + h dt 1 . (3.80) In general we may say that the more negative the gravitational potential is, (or the greater its absolute value is), the more dilated time intervals are. Because of the factor 1 / c 2 , these effects are generally small, however gravitational time dilation has been experimentally measured in various different situations. The first verification by a experiment on earth was performed by Pound and Rebka combining the gravitational dilation with Doppler effect in the emission and absorption of photons in Fe 57 . Further experimental evidence can be inferred from 25 As usual the value at infinity of φ is set equal to zero.
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88 3 The Equivalence Principle astrophysical observations (especially from the light spectra of white dwarfs). In this case we can safely set to zero the earth gravitational potential being much smaller in absolute value than the potential on the surface of a star. Thus, in this case, we have: dt 2 dt 1 = ν 1 ν 2 > 1 , (3.81) where ν denotes the light frequency and the suffix 1 and 2 are referred to the earth and to the star, respectively. The frequency of the emitted light is then higher than that observed on earth and we have a measurable shift towards the red. In the eighties further confirmations were gained by experiments with time signals sent to and from Viking 1 Mars lander and from time measurements using atomic clocks on airplanes; the clocks that traveled aboard the airplanes, upon return, were slightly faster than those on the ground. However, the most spectacular evidence of the gravitational red-shift is nowadays given by its technological application to GPS devices. A GPS gives the absolute position on the surface of the earth to within 5–10 m of precision; this requires the clock ticks on the GPS satellite to be known with an accuracy of 20–30 ns. Such an accuracy cannot be reached if we neglected the special and general relativity effects on time intervals . To compute these effects we first observe that the transmitting clock is subject to the special relativistic time dilation due to the satellite orbital speed, compared to an identical clock on the earth. From our discussion of time dilation in special relativity the clock on the satellite would run slower, compared to a clock on the earth, by the factor: 1 v 2 c 2 1 v 2 2 c 2 , (3.82) at first order in v 2 c 2 , since v , the satellite speed, is much smaller than c .
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