# Fitzgerald contraction

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Fitzgerald contraction, or Fitzgerald-Lorentz contraction, or Lorentz contraction is the hypothesis that a "moving" body should contract in length, in order to avoid generating effects that would violate the principle of relativity. The idea is attributed to George Fitzgerld in 1889, and again, independently to H.A. Lorentz in 1904. The concept survives in Albert Einstein's special theory of relativity, although in Einstein's theory, the body's assumed contraction is not referenced to an absolute aether, but to any legal inertial frame of reference that we choose to use for our calculations (usually that of a specified observer).

## Origin of the concept

In the middle-to-late Nineteenth Century, improvements in experimental accuracy and the upheaval in Newtonian theory in around 1800 led to a proliferation of aether models, and an enthusiasm for testing their predictions, sometimes with contradictory results. It was a matter of debate whether the reference medium for the propagation of light, which dictated its speed, was absolute, or whether the light-carrying medium was partially or fully dragged along by moving bodies. If there was an absolute background reference-aether for light, then if we built a clock whose "ticks" were light bouncing between two mirrors, then if if the mirrors were moving, light ought to take longer to make a round-tip than if the mirrors were stationary. If the Earth equipment was then constantly circling the Earth's axis, and the Earth was also sweeping out a large ellipse around the Sun, then at some times these two sets of motion would act against each other, and at other times they would reinforce each other, causing the clock to tick at different rates that would vary with the time of day and the season.

This was not observed to happen.

in 1889, George Francis FitzGerald (1851-1901) proposed an ad hoc hypothesis that could be used to explain the fact that the Michelson-Morley interferometer equipment hadn't reported these variations – what if any body that was moving with respect to the absolute aether contracted in length, with the shorter pathlength allowing the clock to tick faster, by exactly the right amount to cancel out the motion-slowing effect? That would mean that no matter how quickly or slowly we moved with respect to the absolute aether, our clocks would still tick at the same rate.

## Documentation

Historical C20th accounts tend to say that Fitgerald never wrote down his idea, so that no record existed of his claim to priority except a student's notes for a lecture that he once gave, but that when told about FitzGerald's earlier use of the idea, Lorentz graciously accepted FitzGerald's priority anyway (Fitzgerald had died just a few years before Lorentz's 1904 paper, so the two of them probably wouldn't have had a chance to discuss the matter).

However, thanks to archival dititisatio nprojects and the wonder of the internet, we now know that FitzGerald did get the idea into print in his lifetime, in a "Letter to the Editor" in "Science".

## 1889: Fitzgerald's letter to Science

#### The Ether and the Earth's Atmosphere

I have read with much interest Messrs. Michelson and Morley's wonderfully delicate experiment attempting to decide the important question as to how far the ether is carried along by the earth. Their result seems opposed to other experiments showing that the ether in the air can be carried along only to an inappreciable extent. I would suggest that almost the only hypothesis that can reconcile this opposition is that the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocity to that of light. We know that electrical forces are affected by the motion of the electrified bodies relative to the ether, and it seems a not improbable supposition that the molecular forces are affected by the motion, and that the size of a body alters consequently. It would be very important if secular experiments on electrical attractions between permanently electrified bodies, such as in a very delicate quadrant electrometer, were instituted in some of the equatorial parts of the earth to observer whether there is any diurnial and annual variation of attraction, – diurnial due to the rotation of the earth being added and subtracted form its orbital velocity;, and annual similarly for its orbital velocity and the motion of the solar system.

— , GEO. FRAS, FITZGERALD, Dublin, May, , Letters to the Editor, Science, , 17 May 1889

## Calculations

Fitzgerald was assuming that the measuring equipment was moving w.r.t. light's absolute propagation frame, so if two mirrors were placed one lightsecond apart, a signal with speed $c$ aimed with the aether flow would travel at $c+s$ m/s, and take $c/(c+s)$ seconds to make the journey, while a signal aimed against the aether flow would have a slower speed of $c-s$, and take the longer time $c/(c-s)$ seconds to reach its destination.

In the presence of an "aether wind", where $s\gt0$, the expected time for a round trip would therefore not be two seconds, but $c/(c+s) + c/(c-s) = 2c/(c^2 - s^2), = 1/(1- s^2 / c^2)$.

This explains Fitzgerald's conclusion that if the distance between the mirrors shorted by a factor related to ${(s/c)}^2$, we would not be able to use an interferometer to measure our absolute speed.