General Relativity Without Calculus by José Natário

Author:José Natário
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

5.2 Gravitational Redshift

The first thought experiment to which Einstein applied his equivalence principle was the following—suppose that Einstein, from an intermediate floor on the leaning tower of Pisa, sends a light signal with period T towards Galileo, who is on the top floor. What is the period measured by Galileo?

To answer this question let us suppose that both Einstein and Galileo jump from the tower as the signal is emitted. Then they would be in a free-falling frame, thus equivalent to an inertial frame (without gravity). So Galileo would measure the same period T for the light signal (as he would be at rest with respect to Einstein in this inertial frame). Let be the vertical distance from Einstein to Galileo, and g the gravitational acceleration. It takes the light signal a time to get to the top floor (as and in this time interval Galileo acquires a speed with respect to the tower (we are assuming that this speed is much smaller than the speed of light, so that this Newtonian formula is approximately correct; we are also assuming that the light signal’s period is much smaller than the time taken by the signal to reach the top floor, According to the Doppler effect formula for (see Exercise 5 in Chap. 2), we must have approximately

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