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I derived an equation for the relativistic Doppler effect in the course of a routine problem: the frequency change of a photon moving in the $x$ direction, when considered a frame S' which has a relative velocity $u$ in the $x$ direction compared to another frame S is by a factor: $\sqrt{\frac{1-\beta}{1+\beta}}$ where $\beta=\frac{u}{c}$.

However, I noticed this is the same factor as the difference in "ticking" between clocks: considered clock C in inertial frame S which reads $t_1$ and sends out a radio signal. This is picked up by clock C' in inertial frame S' which has a relative velocity $u$ with S, at time $t_2'$. I previously showed $t_1=t_2'\sqrt{\frac{1-\beta}{1+\beta}}$.

What is the connection please?

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You can think of the Doppler effect as something involving frequencies, or their reciprocals [their periods].

Imagine the emissions of a radio signal as a periodic process.

With meeting event O, think of
the interval $Ot_1$ as the emission period and
the interval $Ot_2$ as the reception period. As you found, there is a relationship between these periods, which you can also express in terms of their frequencies.

By the way, you might be interested in a way of developing special relativity emphasizing the Doppler Effect. It's called the Bondi k-calculus.

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Be careful not to confuse the relativistic Doppler effect with the time dilation.
The longitudinal relativistic Doppler effect as measured by a stationary observer (receiver) against a moving frame (source) experiences a lower frequency of the light emitted when the source is moving away and a higher frequency when instead it is approaching. A peculiarity of SR (special relativity) is the transverse Doppler effect when the source and the receiver are at the closest approach. In that case the frequency measured by the observer is lower by the Lorentz factor $\gamma = 1 / \sqrt{1 - v^2/c^2}$.
The time dilation measured by the observer is always the same, independently of whether the source is moving away or approaching, and it is given by the Lorentz factor.

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  • $\begingroup$ Minor detail: when the source and the receiver ARE at the closest approach, measured frequency will be $\gamma$ times higher. This a case of moving observer and a source is "at rest". Due to dilation of his own clock the observer sees, that the clock at rest is ticking $\gamma$ times faster. Light emitted at points of closest would be $\gamma$ times redshifted. In this case the source moves in the observers frame and emits light pulse at oblique angle backward. The light pulse approaches the observer at right angle. The observer measures that sources clock slows down $\gamma$ times. $\endgroup$ – Albert Mar 21 '18 at 18:16

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