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The origins of length contraction: I. The FitzGerald-Lorentz deformation hypothesis

One of the widespread confusions concerning the history of the 1887 Michelson-Morley experiment has to do with the initial explanation of this celebrated null result due independently to FitzGerald and Lorentz. In neither case was a strict, longitudinal length contraction hypothesis invoked, as is commonly supposed. Lorentz postulated, particularly in 1895, any one of a certain family of possible deformation effects for rigid bodies in motion, including purely transverse alteration, and expansion as well as contraction; FitzGerald may well have had the same family in mind. A careful analysis of the Michelson-Morley experiment (which reveals a number of serious inadequacies in many text-book treatments) indeed shows that strict contraction is not required.


Introduction
It is well-known that in providing a derivation of the inertial coordinate Einstein showed in his celebrated 1905 relativity paper 1 that what Henri Poincaré had dubbed the "Lorentz transformations" lead to a longitudinal contraction for rigid bodies in motion. It is still widely believed that such an effect-if not its interpretation-corresponds precisely to that involved in the FitzGerald-Lorentz 'contraction' hypothesis which had appeared over a decade earlier.
But this assumption is not strictly correct. The discrepancy between Einstein's relativistic contraction prediction and the original Lorentz hypothesis has already been noted, as we shall see, in the work of a handful of commentators; in the case of FitzGerald the discrepancy is probably the same but it is almost never acknowledged in modern treatments. It is our intention here to spell out the nature and significance of this discrepancy in more detail than has hitherto been done. The motivation for this study is the belief that the transition from the original FitzGerald-Lorentz hypothesis to the correct form of the contraction prediction in 1905 (to be discussed in a subsequent paper) was an important and under-rated development in our understanding of the nature of kinematics.
The FitzGerald-Lorentz (FL) hypothesis was of course the result of a somewhat desperate attempt to reconcile the null result of the 1887 Michelson-Morley (MM) experiment with the hitherto successful Fresnel-Lorentz theory of a stationary luminiferous ether, a medium through which the earth is assumed to move with unappreciable drag. The MM experiment is rightly regarded today as one of the turning points in physics, and although it is discussed widely in textbooks, it is remarkable how much confusion still surrounds its structure and meaning. In order then to understand the FL hypothesis, it is necessary first to go over some welltrodden ground; sections 2 and 3 below are designed to show what the 1887 null result does and does not imply. In particular it is shown in section 3 that in the context of a theory of light in which the light-speed is independent of the speed of the source, a certain motion-induced deformation of rigid bodies, of which contraction is a special case, is required. (A common misunderstanding about the role of the Galilean law of addition, or transformation of velocities, in the analysis of the MM experiment is also exposed.) In section 4, this result is incorporated into a more systematic account of what we call MM kinematics, which is found to be consistent with the 1949 review by Robertson of the experimental underpinnings of relativistic kinematics.
In sections 5 and 6, the history of the independent responses of both FitzGerald and Lorentz to the MM experiment is reviewed, and it is argued that in neither case was a purely (longitudinal) contraction effect being proposed originally.
Nor was the FL deformation hypothesis as artificial or ad hoc as it was (and is) often portrayed. In both cases, a plausible dynamical underpinning of the hypothesis was proposed, as is seen in sections 7 and 8. But the subtle, yet important difference between these dynamical arguments, each relying on an analogy with the effect of motion on electrostatic forces, is discussed in a subsequent paper, as is the history of the eventual shift from FL deformation to relativistic contraction.

The non-relativistic analysis of the Michelson-Morley experiment.
The keystone assumption in the ether theory of light was that relative to the inertial rest-frame of the ether, the 'two-way' (or 'round-trip') light-speed in vacuo is a 'constant' c, that is to say, independent of the motion of the source of light and isotropic. It was originally expected, on the basis of Galilean kinematics, that the twoway light-speeds (i.e both ray and phase speeds 2 ) relative to an inertial frame moving with respect to the ether-such as that defined by the earth over sufficiently short periods of time-would likewise be source-speed independent but not isotropic. The 1887 experiment of Michelson and Morley was designed to detect the predicted anisotropy of the ray speed-the first attempt, following Michelson's similar but inconclusive effort in 1881, to detect a second-order optical effect of the ether wind. 3 (The null results in all previous first-order ether wind experiments were explained by appeal to the Fresnel drag coefficient for light-speeds inside moving transparent media, which H. A. Lorentz showed to be a consequence of his dynamical theory of the stationary ether. 4 ) Famously, the shift in the interference pattern that was expected to appear in the process of rotating the Michelson optical interferometer through 90°w as not observed.
Let S represent the inertial rest frame of the ether, and S' the inertial rest frame of the earth-bound laboratory. Relative to S the two-way light-speed is assumed to be isotropic, as we have seen. Thus we may naturally adopt for S the Einstein convention for synchronising distant clocks, which has the effect of rendering isotropic the oneway speed of light in that frame. For the moment we shall leave unspecified the corresponding convention in S'-nothing in this and the next section depends on it.
Initially at rest relative to S' is the MM interferometer, one of whose two, perpendicular arms (call it A) points in the direction of the motion of S' relative to S.
As usual, we shall suppose that the coordinate systems adapted to S and S' are in the 'standard configuration', with the arm A pointing in direction of the the positive x-, x'axes. The second arm B lies parallel to the y'-axis. With a view to later discussion, we analyse the general case in which the rest lengths of the two arms, L A ' and L B ' , measured of course relative to the laboratory frame S', are not equal.
Standard non-relativistic treatments of the experiment in the literature divide into two kinds: those which (like the 1887 MM account) calculate delay times etc. relative, in the first instance, to the ether rest frame S, and those which, by appeal to the Galilean rule for transformation of (ray) velocities, deal from the outset with such time intervals defined relative to the lab frame S'. We shall for the most part rehearse the first kind, for reasons that become clear shortly.
Relative to S, the speed of light in both directions in A is c. This claim follows immediately from the above assumption of the 'constancy' of the two-way light-speed relative to S, and the further Fresnel-Lorentz assumption that the interferometer does not drag the ether with it while in motion. A simple calculation shows that the time taken for a pulse of light to leave the beam splitter (semi-transparent mirror) at the join of the two arms and return to it via arm A is, relative to S, where L A is the length of arm A relative to S and γ is the now familiar 'Lorentz' factor (1 -v 2 /c 2 ) -1/2 .
In calculating the corresponding traverse time along arm B, it is normally assumed that the (ray) velocities in this case have components u x = v, u y = ±(c 2 -v 2 ) 1 /2 . Let us call this assumption the "B-claim". It now follows that where L B is the length of B relative to S. 5 (It should be noticed that both T A and T B are by definition measured by pairs of synchronised, separated clocks at rest relative to S; whereas their primed counterparts are measured by a single clock at rest relative to the interferometer.) Consider now the delay time based on these expressions: We wish to compare this delay ∆ with that occurring after the interferometer is rotated by 90˚, so that arm A comes to rest pointing in the positive y' direction, B in the negative x' direction. Similar reasoning yields for the new delay time ∆ rot (assuming the isotropy of space in S) where L A rot is the length relative to S of the rotated arm A, etc. 6 (Such at least is the standard claim, but as we shall see, it seems that (3) and (4) can not both be exact expressions without a re-adjustment of the angle of the beam-splitter during rotation.) Notice that so far, no assumptions inconsistent with relativity theory have appeared. This is in contrast with the second type of common analysis mentioned above, in which a derivation is given of the expressions for pre-and post-rotation delay times defined from the start relative to the lab frame. As the speed of light is no longer expected to be isotropic relative to this frame, the speeds in the A and B arms are now provided by way of the Galilean rule for transforming ray velocities. The expressions for the delay times are identical to (3), (4) but in the primed variables. We shall return to this type of analysis and its potential misconstrual in the next section.
The eye-piece, or telescope, attached to the interferometer received, throughout the rotation, a superposition of two monochromatic beams of light emerging from the beam-splitter that were identical up to a certain phase difference; the mirrors were arranged to produce straight fringes. Suppose n and n rot are the (frame-independent) number of periods of the light waves associated with the time delay in traversing the two arms, before and after rotation respectively. According to Michelson and Morley, and many subsequent commentators, the phase shift due to 90˚ rotation is then (in units measured by the distance between the fringes) where λ is the known wavelength of the sodium light.
Assuming that the orbital velocity of the earth relative to the sun coincides with the velocity v of the laboratory relative to the ether, then for L' ~ 11m (the length obtained in the MM interferometer by means of multiple reflections in each arm), the prediction was ∆n ~ 0.4 fringe. Michelson and Morley reported that the "actual displacement was certainly less than the twentieth part of this, and probably less than the fortieth part." But before we analyse the implications of this 'null' result, we should finish the story about what ought to be the exact non-relativistic prediction for the experiment.
First, note that equation (5) seems to be mixing quantities that are defined relative to different frames. The wavelength of light is always defined relative to the rest frame of the actual equipment-specifically the detector-used by the observer, and should be denoted ′ . Thus it is best to understand the quantity (∆ − ∆ rot ) in (5) as the relative time delay defined also with respect to the lab frame S′. Under the nonrelativistic assumption of zero time dilation, it is of course determined by the difference between the delay times in (3) and (4). But we face further problems if we insist on treating (5) as exact: we are now left with the (phase) velocity c defined relative to the wrong frame, and the rest wavelength of the sodium light is apparently being treated as isotropic relative to S′. This last assumption is inconsistent with the fact that the rest frequency of light is frame-independent, combined with the fact that the phase velocity is not isotropic relative to the lab frame S according to  8 ; what is significant for our purposes is that it coincides to second order with (2) and hence yields (6).
Third, we note that in a detailed and provocative analysis of the MM experiment published in 1994, M. Mamone Capria and F. Pambianco claimed that like its original MM counterpart, (2) above is also correct only to second order! This new analysis relied on a systematic application of Huygens' principle to the case of reflection of plane light waves off moving mirrors, and was based on the assumption that the beam splitter is inclined at exactly 45˚ to the direction of the incoming beam. It was demonstrated that T B is given by 9 which to second order coincides with (2). The analysis showed moreover that the paths of the two rays from the beam splitter to the telescope do not coincide, and in particular do not have the same optical length. When this detail is taken into account, the new, exact total delay time is where T A is given by (1), T B is given by (7), and F is the distance from the x-axis to the telescope 10 . The discrepancy between this delay time and the standard one given by (3) (where T B is given by (2)) is, happily, only in third and higher order terms.
Returning to the standard account based on (2) and (3), it was argued above that the effect of the 90˚ rotation of the interferometer is simply to exchange the role of arms A and B in calculating the delay times. Mamone Capria and Pambianco showed, again under the exact 45˚ tilt angle assumption, that such symmetry again only strictly holds to second order, and then only when the assumption 11 The upshot is that whereas the standard fringe shift (6) survives to second order in this new non-relativistic analysis, the exact treatment therein looks considerably more complicated than its standard counterpart.
Is it right? Yes, but not in the sense of impugning the exact validity of (2).
This expression was based only on the B-claim above, which in turn is based on the assumption that the transverse ray in B passes after reflection back through the midpoint of the beam splitter. In practice, the beam splitter is adjusted so as to maximise the definition of the interference fringes in the telescope. And although occasionally it is stated 12 in instances of the standard non-relativistic analysis in the literature that the (frame-independent) angle of tilt of the beam splitter is 45˚, given the ether wind supposedly blowing through the apparatus it should not be expected to be exactly so 13 .

Mamone Capria and Pambianco in fact
showed what the precise tilt angle must be (sin -1 (2 -v 2 /c 2 ) -1/2 ) for the simple formula (2) to hold exactly, and they anticipated the objection that the beam splitter could be adjusted to achieve this angle. They claimed that in this case a further tiny adjustment is required for (4) to hold exactly, and not just to second order. 14 Indeed, it is not hard to show that according to the nonrelativistic analysis, the 90˚ rotation does not exactly preserve the 'ideal' angle of tilt of the beam splitter.
3. Deformation-the general explanation of the null result.
Let us now return to our previous considerations for the case where L A ' is not necessarily equal to L B ', and where Galilean kinematics are not assumed to hold. We define the longitudinal and transverse 'deformation factors' relative to the ether frame S: where L A(B) is the length of arm A(B) relative to S, when the interferometer is hypothetically at rest relative to that frame. (These deformation factors are expected to depend on v and of course to tend to unity in the limit v→0.) It follows from the isotropy of space relative to S that such rest lengths do not depend on the orientation of the interferometer. If it is further assumed that any length alterations due to motion hold for all the rigid bodies (including rulers) in the laboratory as well as for stone slab on which the interferometer is mounted, then rest-lengths will be invariant, or frame-independent, i.e.: In particular, it follows from the two assumptions above that L A ' and L B ' also do not depend on the orientation of the interferometer.
Now the exact absence of any fringe shift under rotation (when not assuming L A ' = L B ' = L') would require in the standard analysis that the pre-and post-rotation delay times given in (3) and (4) are equal. It follows in this case, using (9) and (10), Even if it is accepted that (3) and (4) cannot both be exact, an explanation of the null MM result ensues if the dimensions relative to S of any rigid body undergo an anisotropic change as a result of motion, which is consistent up to second order with (12) and hence with: Note that if such shape deformation is to occur, then the angle of tilt of the beam splitter will no longer be frame independent. We have mentioned that the wavetheoretical analysis of Mamone Capria and Pambiano fixes the value of the tilt angle to be sin -1 (2 -v 2 /c 2 ) -1/2 relative to the ether frame S in order that the transverse ray in B passes back through the mid-point of the beam splitter-in other words to ensure the exact validity of (2). Unsurprisingly, it is also the angle given by applying the anisotropic deformation (12) directly to the semi-transparent mirror itself, when the angle relative to the lab frame S′ is exactly 45˚! 15 Several further points must be stressed at this stage. First, such a motioninduced distortion or deformation (12) is, of course, more general than the purely longitudinal contraction effect we are familiar with in special relativity theory (which follows from (12) when C T = 1). The deformation may actually involve expansion and/or contraction effects (i.e. it is not required that C L , C T be less than 1), but it cannot vanish in toto. Second, without appeal to the relativity principle, there is no reason to expect that the deformation factors will be frame-independent (as will be become evident below). Third, the existence of such an anisotropic deformation defined relative to S does not invalidate the assumption implicitly made by Michelson and Morley that relative to the laboratory frame S' the lengths of the two interferometer arms were equal before and after rotation. Fourth, the explanation above of the MM null result involving rigid-body deformation does not depend on prior knowledge of the form of the coordinate transformations between the frames S and S', nor even on the associated rule for transformation of ray velocities. However, the existence of such a universal deformation will of course act as a constraint on the form of the coordinate transformations, as is seen in the next section.
Very few textbook treatments of the MM experiment recognise anything other than a purely longitudinal contraction as an explanation of the null result (the 1994 treatment by Mills being a notable recent exception 16 ). However, the failure in several treatments to recognise the third and fourth points above is arguably even more alarming. Mention was made in the previous section of a common textbook analysis of the MM experiment making use of the Galilean law for transformation of ray velocities (for both longitudinal and transverse components) in order to calculate directly the delay times ∆', (∆ rot )' as defined relative to the laboratory frame S', assuming as always light-speed constancy in S. Given this law, in the case where the arms of the interferometer are of equal length the primed delay times are readily shown to be: where L' is taken to be invariant under rotation. The prediction (14) is obviously not consistent with the null result. It is usually now argued 17 that the implicit MM claim that each arm has the same length relative to S' must be abandoned, and replaced with the statement that, both before and after rotation, the length of the arm pointing in the direction of the ether wind must be shortened to γ -1 L'.

Michelson-Morley kinematics.
Given the isotropy of the 'relative 3-space' in S, we can write the linear (standard configuration) coordinate transformations between S and S' quite generally as: where u x is the x-component of the velocity of some body as measured relative to S, and u x ' is the x'-component of the velocity of the same body relative to S', etc.
Now the null result of the MM experiment demonstrates that the two-way speed of light is isotropic relative to the laboratory frame S'. 20 But it does not establish that this speed is numerically equal to c, the two-way light-speed in S. From the linearity of the transformations (15) it follows from the constancy of the light-speed in S that the two-way light-speed in S' is also independent of the speed of the source.
Given that it is also isotropic, it is natural to adopt in S', just as in S, the Einstein convention for synchronising clocks, which renders the one-way light-speed isotropic in S'.
Let us denote by c' the two-way light-speed in S'. Assuming Einstein synchrony in S', and hence using (16a) with u The relationship between C L , C T , is fixed by (12), so combining this result with (15), (17) and (18), we arrive at the following form of the coordinate transformations consistent with the MM null result: where |v| < c. The temporal transformation (19d) can be rewritten as  (19) is not fixed by convention, but experiment: it will affect in particular the measurable degree to which rods deform in motion. We have from (19a, b) that We shall have more to say later about the experimental significance of k; in the meantime we note that unless k is an even function of v, the deformation effects will be anisotropic relative to S.
(ii) In the derivation of (19) on the basis of the MM result, no appeal is made to the relativity principle. Indeed, in general the deformation effects, say, defined for motion relative to arbitrary frames may differ considerably from those defined for motion relative to S and exhibited in (21) and (22). For example, given the choice k = γ, it follows from (21) that C L = γ -2 defined relative to S, but inverting (19) shows that C L = 1 relative to S' when the 'moving' rod is at rest relative to S, even when c' = c. (iv) It was noted in the previous section that the Galilean rule of transformation of velocities leads to (14), which is inconsistent with the null MM result. The reader will also see from (16a,b) that even if we choose the synchrony convention in S' in such a fashion that = 0 (which we must do if we want Galilean kinematics to emerge, and which does not affect the contraction factors defined relative to S) the MM result C L ≠ C T rules out consistency with the Galilean rule.

The FitzGerald deformation hypothesis
In 1889, the American journal Science published in its May 2 nd issue a brief letter by the Irish physicist George Francis FitzGerald 26  In the second paper, to which we return later, FitzGerald's claim is taken to be that the  39  Lorentz's 1892 paper initially gives for the pre-rotation delay times in arms A and B of the Michelson interferometer the expressions (in our notation) Lorentz attributes these results to Maxwell. Although expressions (23), (24) are the second-order versions of our equations (1), (2) for L A = L B = L, it is initially ambiguous in the context of Lorentz's thinking as to which of the two frames S, S' the times and lengths in (23) and (24) refer. Such measures were of course taken to be frame-independent, giving rise to a predicted delay which is proportional, as we have seen, to the shift in the interference pattern originally expected under a 90˚ rotation of the apparatus.
In order now to account for the null result of the MM experiment, Lorentz replaces L in (25) by (1 -)L. It is now clear that L in (23), (24) is the rest length of both arms A and B relative to the ether frame S, and that = (v) is the decrease in the longitudinal length, defined relative to S, of a rod of unit rest length, produced by the motion v. (We note again that if such a length change affects all the rigid matter in the laboratory as well as the arms of the interferometer, as Lorentz clearly believed, then we can equate the rest length L above with L', i.e. the rest length of the arms relative to the laboratory frame S', as in equation (11) above.) We must regard then the expressions (25), (26) and (27) as referring in the first instance to spatiotemporal measures defined relative to the privileged ether frame S. Ignoring terms of the order αv/c 2 , Lorentz noted that the delay ∆ effectively vanishes (and "with it the whole difficulty") if α = v 2 / 2c 2 . Observe that again the entire argument need make no appeal to the Galilean law of transformation of ray velocites: all that is involved is the constancy of the light-speed relative to S and the possibility of length contraction. Nor did Lorentz explicitly evoke the Galilean law. Nonetheless, however appealing the possibility is that he avoided it-and the attendant difficulties outlined in sections 3 and 4 above-in his implicit thinking, the issue cannot be regarded as settled. 45 We can at any rate now write Lorentz's solution in terms of our previous notation: C L ~ (1 -v 2 / 2c 2 ), C T ~ 1. Yet Lorentz was aware in 1892 that other deformation effects are also consistent with the MM experiment; in particular he mentioned that a purely transverse increase in the dimensions of bodies "would answer the purpose equally well". It was later in his 1895 book 46 on electromagnetic and optical phenomena associated with moving bodies (often referred to simply as the 'Versuch') that the point was brought out most systematically. Here, Lorentz replaces the 1892 contraction factor by a combination of the longitudinal factor C L = (1 + δ) and the transverse factor C T = (1 + ε). He claims now that the vanishing of the delay ∆ requires εδ ~ v 2 / 2c 2 (26) which is consistent with our condition (13). Lorentz stresses that "the value of one of the quantities δ and ε would remain undetermined". As well as the 1892 values δ = -v 2 / 2c 2 , ε = 0, he explicity refers to the possibilities δ = 0, ε = v 2 / 2c 2 (transverse expansion), and δ = -v 2 / 4c 2 , ε = v 2 / 4c 2 (longitudinal contraction and transverse expansion), though clearly these are not the only possibilities. The only question is, how? Hunt took FitzGerald's suggestion as an indication that he thought that intermolecular forces might be electromagnetic. But even though, as Hunt stresses, FitzGerald regarded all physical forces as having their origin in disturbances of a single ether, it is not clear that Hunt's extrapolation is correct; no explicit statement to this effect seems ever to have been made by FitzGerald. 53 (It has also been stressed by Andrew Warwick that such a statement is hard to reconcile with Lodge's 1893 account of FitzGerald's hypothesis 54 .) The formulas Heaviside (and later J.J. Thomson) derived for the electric and magnetic field strengths surrounding a uniformly moving charged body are precisely those obtained more simply by assuming the validity of Maxwell's equations relative to the rest frame of the charge and using the known transformation properties of the fields-i.e. by doing it the modern way. 55 The formula for the electric field is: where E is evaluated at a point with displacement r from the centre of the charged body and θ is the angle between r and the direction of motion. Hunt  Returning finally to the early years, it seems to have been Lorentz's belief both in 1892 and 1895 that a collection of charged particles held in equilibrium under electrostatic forces (insofar as this makes sense) undergoes a solely longitudinal contraction, according to the (relativistic) factors C L = γ -1 , C T = 1. The validity of the strong version of the molecular forces hypothesis would therefore yield (to second order) the values δ = -v 2 / 2c 2 , ε = 0. But Lorentz had already warned in 1892 65 : One may not of course attach much importance to this result; the application to molecular forces of what has found to hold for electric forces is too venturesome for that.
Furthermore, by 1899, it had become apparent to him that the strong version of the electrostatic analogy -or rather his construal of it-would after all allow for the more general deformation factors C L = (kγ) -1 , C T = k -1 , with k unknown, consistent (to second order) with his relation (28) above. Lorentz wrote in 1899 66 : We need hardly remark that for the real transformation produced by translatory motion, the factor [k] should have a definite value. I see however no means to determine it.

Final remarks
Lorentz would attempt to show systematically in 1904 that the value of the scale factor k is unity, and consequently that the effect of motion on rigid bodies is a purely longitudinal contraction. But the argument was convoluted and later not even