# Lorentz Transformation and the Aberration of Starlight

The aberration of starlight is generally accepted as constituting a direct, observational evidence for the validity of the Lorentz transformation formulae. Usually, the line of arguing is as follows:

If the rest frame of the star is the unprimed system and the rest frame of the observer the primed system and both move relative to each other with velocity v in the direction of their x-axes, then the corresponding Lorentz- transformation is (c= speed of light):

x'=γ(x-vt)
y'=y
z'=z
t'=γ(t-vx/c2)   ,

with

γ= 1/[1-(v/c)2]   .

At t=t'=0 the origin of the two systems shall coincide and at this moment a light signal be emitted from the star along the z- axis. According to the above formulae this yields the following trajectory in the observer's frame:

x'=-γvt
z'=z=ct
t'=γt

or

x'=-vt'
z'=ct'/γ   .

Thus, the light ray appears to be rotated compared to the unprimed frame by the well known 'tilt angle' α with

tanα= (-x')/z' =γv/c   .

However, as an isotropically emitting body, the star obviously also sends a light ray along the coordinates

x=vt
y=0
z=ct/γ   ,

which, as one easily finds by inserting these values into the original equations, is observed in the primed system as travelling along the z'- axis.
The star is therefore observed exactly at the same apparent point where the light was emitted (the original and all other rays would miss the observer (located on the z'-axis) altogether). As the only consequence of the Lorentz transformation, the star would appear to be rotated by the angle α; one would need in fact an additional Lorentz- transformation at the observer in order to have an aberration effect.

However, even a rotation of the star should not occur if the speed of light is truly independent of any motion of source or observer (see my page Speed of Light). And obviously no aberration of star positions either. The latter could only arise if the speed of light is merely constant in the longitudinal direction (i.e. along the direction of observation) but not transverse to it. Another possibility would be that the light signal couples to the matter and/or the magnetic field in interplanetary space (relative to which the earth is moving), but this is questionable as the interplanetary medium is quite variable in its velocity, so the aberration should show some variability as well.

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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