- #31
Bartolomeo
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1977ub said:
That goes straight from Einstein 1905 paper. He doesn't mention that the observer is non inertial. According to Einstein, from the point of view of a moving observer a clock at rest „is ticking faster“. Since rotating observer can never be „at rest“, he always measures that a clock in the center is ticking faster. But, Einstein's calculation works perfectly well for inertial observer also.
Here it is:
https://www.fourmilab.ch/etexts/einstein/specrel/www/ - § 7. Theory of Doppler's Principle and of Aberration
From the equation for ##\omega'## it follows that if an observer is moving with velocity ##v## relatively to an infinitely distant source of light of frequency ##\nu##, in such a way that the connecting line “source-observer” makes the angle ##\varphi## with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency of the light perceived by the observer is given by the equation
$$\nu= \nu' \frac {(1-\cos\varphi \cdot v/c)}{\sqrt {1-v^2/c^2}}$$
We see that, in contrast with the customary view, when ##v=-c, \nu'=\infty##
It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.
So, in Transverse condition ray of light from the source moves at right angle to direction of motion of the observer. If observer rotates, it always comes at right angle to direction of its motion. Rotating observer always moves at tangential to wavefront. In classical case there is no Doppler effect. So, ## \cos \pi/2 = 0## and purely Transverse effect is seen
$$\nu= \frac {\nu'}{\sqrt {1-v^2/c^2}}$$
According to celebrated Einstein's 1905 paper, both rotating and the inertial observer (who momentarily coincides with the rotating one) will see, that clock in the center of the circumference is ticking faster.
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