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Stellar aberration (derivation from Lorentz transformation)
Stellar aberration is an astronomical phenomenon "which produces an apparent motion of celestial objects". It can be proven mathematically that stellar aberration is due to the change of the astronomer's inertial frame of reference. The formula is derived with the use of Lorentz transformation of the star's ''coordinates''.
As the astronomer John Herschel has already explained in 1844, the stellar aberration does not depend on the relative velocity of the star towards Earth. Otherwise eclipsing binary stars would appear to be separated, in stark contrast to observation: both stars are rotating with high speed —and ever changing and different velocity vectors— around each other, but they appear as ''one'' spot all the time.
==Stellar aberration is only due to the change of the astronomer's inertial frame of reference==
In the year 1926 the astrophysicist Robert Emden published the article ''Aberration und Relativitätstheorie'' in the journal ''Naturwissenschaften''.〔 (14. Jahrgang, Heft 16)〕 In this article he states that the direction of a light ray isn't influenced by the motion of the star or by the motion of Earth.〔R. Emden's (rhetorical) question "How will the direction of the light rays of stars incident upon Earth be influenced by the motion of Earth and the motion (not the location) of the star at the time of emission?" is answered by him with "Not at all."〕 At that time, the opponents of the special theory of relativity reasoned that the theory must be flawed, because it would state that the stellar aberration would depend on the relative velocity of the star — which would be in contradiction to observation — and R. Emden's article explains that the special theory of relativity does not predict this. Today, the special theory of relativity isn't contested anymore but there are still articles that suggest that the aberration would depend on the relative velocity of the star.〔See for example equation (4) in 〕
Although a (relativistic) velocity-addition formula can be used to explain stellar aberration, (see Aberration of light), another (relativistic) explanation using only the Lorentz transformation is also possible, as will be demonstrated. This derivation only uses the star's ''coordinates'' at the time of emission, and therefore has the ''formal'' advantage there is no place for the relative velocity of the star towards the astronomer and therefore it is evident that the observed position doesn't depend on the star's velocity — provided that the resultant change of position is much smaller than the distance between star and Earth.〔I.e. the stars can rotate very fast like eclipsing binary stars as long as the change in position is much smaller than the star-earth distance.〕 The observed position of the star wouldn't depend on Earth's motion either, if the astronomer could use the same inertial frame of reference all the time. But of course that is technically impossible,〔It's easy to (technically) compensate the deviation of the ''direction'' of the axes due to Earth's rotation and earth's orbit but it almost impossible to (technically) compensate a velocity in the order of several 1000 m/s. This can only be achieved with help of mathematics.〕 the astronomer uses his current rest frame and these current rest frames are different at different times as Earth orbits around the Sun. It is mathematically convenient to declare the position of star in a rest frame of the Sun (more exactly: the center of mass of the Solar System) as the "real" position and that the difference to this "real" position derives form the "aberration".〔Of course every inertial frame of reference is physically equal, but the rest system of the barycenter has the mathematical advantage, that it's the ''mean'' frame of reference and since the time of Galileo Galilei it's the preferred frame of reference〕
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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