# Aberration redshift

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#### Aberration redshift

Aberration redshift is an effect that exists under Nineteenth Century Newtonian theory, and which predicts effects qualitatively indistinguishable from the transverse redshift effects of special relativity.

Verification of the transverse redshift effect of special relativity was commonly cited as one of the major proofs of the theory, since earlier theories supposedly predicted no visible redshift in a body moving along a line at right angles to the viewing path (where "right-angles" is defined in the experimenter's frame).

## Concept and SR argument

Imagine that we draw two lines on the floor of our laboratory, meeting at a cross. We build a straight-line section of particle accelerator tube along one of the lines, and we train a sensitive spectrometer along the other line, to view radiation emitted by particles moving past at 90 degrees.

If we assume that the speed of light is globally fixed in the lab frame, then we expect the longitudinal Doppler shift to be freq'/freq = c/(c+v), and the transverse redshift to be freq'/freq=1 (or, "no effect").

In practice we find that here is a redshift, that the shift sends the visible frequency towards zero as v tends to c, and that this puts an upper limit of cBACKGROUND other maximum speed that we can achieve with a particle accelerator using direct acceleration. From the point of view of a moving particle, special relativity says that the signals from the accelerator could behind and beside the particle have energies that drop to zero as v approaches c, so the coupling efficiency of the coils has a lightspeed limit.

The argument is then that since we "know" that the speed of light is globally fixed w.r.t. the observer, the transverse redshift effect wouldn't happen without special relativity's time-dilation effect, and if it was not for special relativity, particle accelerators would not work like this.

This argument appears to be compelling, and is cited by authorities such as GR-testing expert Clifford Will, but is, unfortunately, wrong.

## The transverse redshift effect under Newtonian theory

If we assumed the Newtonian momentum laws, we get shift and contraction relationships that are nominally "redder and shorter" then those of SR, by an additional Lorentz factor. With these equations, the recession redshift is freq'/freq = (c-v)/c, and the transverse redshift is freq'/freq=1-v^2/c^2 . Both relationships appear in C19th ballistic emission theory, but are not exclusively owned by it.

The longitudinal prediction is commonplace, but the transverse prediction is somewhat obscure. We can obtain it by calculating the recession redshift under NM, calculating the effect of aberration (SR and NM use the same aberration formula), appreciating that the ray that enters the equipment at 90 degreesLAB has been angled forwards by aberration effects, calculate the original slightly rearward-aimed angle of the ray, and then calculated the original recession redshift component present in the slightly rearward-aimed ray.

This exercise tells us that the ray that is seen at 90degrees in the lab should be redshifted not by the Lorentz factor, but by the square of the Lorentz factor. Redshifts in this situation are very much NOT unique to special relativity!

## Aberration redshifts within special relativity

This Newtonian equivalent of the SR transverse effect can be proven to have precisely the same associated phenomenology as special relativity, and has to produce the same result regardless of which arbitrarily-chosen frame we use to do our calculations.

If we return to the original SR scenario, we can argue that under SR, we can say that we are stationary wrt the propagation frame of light, there is zero propagation-based transverse Doppler effect, the particle moving wrt our chosen frame can be declared to be Lorentz time-dilated, and therefore we see a Lorentz redshift.

However, we can also do the calculations by assuming that the speed of light is fixed in the particle frame. We then expect to see the previously-mentioned Lorentz-squared aberration redshift effect, but since we are "moving" wrt the selected lightspeed frame, it is our own reference clocks that are time-dilated. This means that we see things happening faster by the Lorentz ratio, which cancels one part of the Lorentz-squared redshift leaving just a single Lorentz redshift.

Both SR calculations produce precisely the same physical outcome (just as they are supposed to!). However, for this SR magic to work, the same physical observed transverse redshift MUST be interpretable either as the result of SR time-dilation or as the result of NM aberration partially cancelled by inverse SR time dilation - for SR to be correct, there cannot be any qualitative physical phenomenological difference between the effect predicted by NM and the effect predicted by SR – we can only tell the two predictions apart by their magnitude for a given nominal velocity, and this comparison is complicated by the fact that, starting from agreed theory-neutral inputs such as energy and momentum, NM and SR will tend to assign different nominal velocities to the same situation.

## Aberration redshift in Einstein's 1905 paper

Einstein's electrodynamics paper, section 7: Theory of Doppler's Principle and of Aberration provides a Doppler equation from the perspective of a "moving" observer:

$\nu' = \nu \frac{1- \cos \phi \cdot v/c}{\sqrt{( 1 - v^2 / c^2}}$

Ignoring the fancy vector stuff, later down the page Einstein says that for and angle of 90° ($\pi / 2$), $\cos \phi = - v / c$, so:

$\nu' = \nu \frac{1- v^2 / c^2 }{\sqrt{( 1 - v^2 / c^2)}}$

, where the top part of the right hand side is a Lorentz-squared redshift (aberration redshift), and the lower part is a partly-cancelling Lorentz blueshift due to the observer's own nominal SR time-dilation.

While it might seem slightly perverse to write the equation this way (using the assumption that the speed of light is fixed for someone other than the observer), this method simplifies the trigonometry when want an equation that will work for any angle.