Why Was Michelson Surprised?
Maxwell's theory of electromagnetism, first published in 1865, suggests that light propagates as a wave with a fixed characteristic speed. It was commonly assumed that this speed must be evaluated with respect to the rest frame of a substantial medium, sometimes called the luminiferous ether. To determine the rest frame of this ether, Maxwell proposed various experiments to measure the speed of light relative to the Earth, which presumably is in motion with respect to the ether. Some of his proposals involved observations of Jupiter's moons, while others were based on terrestrial measurements of the permittivity of the vacuum, and so on. However, none of these experiments, to the extent that they could be performed, showed any variation in the speed of light between relatively moving systems of inertial coordinates.
Albert Michelson (1852-1931) read Maxwell's great treatise on electromagnetism, and was intrigued by the idea of detecting the Earth's motion through the (supposed) ether by measuring variations in the speed of light. He reasoned that the time required for light to make a round trip back and forth along a measuring bar of length L, assuming the bar is moving axially with speed v relative to the ether, is
so the first-order effect cancels out, but Michelson developed techniques of interferometry which were precise enough to detect variations of the second order in v/c, and he fully expected to observe such variations when he pointed the device in various directions. Hence he was surprised to find that the experiment, performed in 1887 with his co-worker Edward Morley, revealed no such variations, i.e., the speed of light was apparently the same in all direction with respect to the inertial coordinates of his laboratory, regardless of the direction of light travel.
Michelson's null result also came as a surprise to Hendrik Lorentz (1853-1928), one of the leading theorists in the field of electrodynamics at the time. Lorentz was initially unable to account for the apparent absence of second-order variations in the speed of light, but he eventually developed a theory incorporating both time dilation and length contraction (first suggested by George Fitzgerald in 1892) to match the experimental results. (The unsuccessful efforts to detect the Earth's motion through the purported ether also provided some of the impetus for Einstein's special theory relativity in 1905, which entails the same time dilation and length contraction effects contained in Lorentz's theory, albeit with a different interpretation.)
Lorentz had been laboring to develop a constructive theory, by determining the detailed physical processes that give material objects their shapes and sizes, all assuming a Galilean background of space and time, i.e., a spacetime consisting of the Cartesian product of a three-dimensional Euclidean space manifold with a one-dimensional time manifold. At that time (around 1900), the only known force with any appreciable strength on a small scale was the electromagnetic force, which Lorentz believed was propagated at the invariant speed c with respect to an absolutely motionless background medium (the luminiferous ether). From this standpoint it was not unreasonable to hypothesize that the structure of all (non-ether) material entities was established and enforced by signals that propagate at the absolute speed c with respect to the ether. For example, an elementary particle at rest in the ether would be expected to have a spherical shape, based on the idea that a wave emanating from the geometric center of the particle would expand spherically until reaching the radius of the particle, where we can imagine that it is reflected, and the reflected wave then contracts spherically back to a point (like a spatial filter) and re-expands on the next cycle. This is illustrated by the left-hand cycle below. (Only two spatial dimensions are shown; in full 4-dimensional spacetime each shell is a sphere).
If the particle is moving relative to the putative ether, then obviously the absolute shape must change from a sphere to an ellipsoid, as illustrated by the right-hand figure above. Of course, the spatial size of the particle with respect to the ether rest frame coordinates is just the intersection of a horizontal time slice with the shaft swept out by these shells. It's also clear that, for any given characteristic particle, since there is no motion relative to the ether in the transverse direction, the size in the transverse direction is unaffected by the motion. Thus the widths of the shells in the "y" direction in the above figure are equal.
On this basis, what can we infer about the dimensions of moving objects? The figure below shows side and top views of one cycle of a stationary and a moving particle (with motions referenced to the rest frame of the putative ether).
It's understood that these represent the same characteristic particle, so the transverse size is the same. The right-hand particle is moving with a speed v in the positive x direction. In each case the geometric center of the particle is moving from point A to point B. The characteristic radius of the particle shell has been taken as unity.
In order to make the transverse sizes of the shells equal, the enclosed areas of the cross-sectional side views must be equal. Thus, light emanating from point A of the moving particle extends a distance 1/l to the left and a distance l to the right, where l is a constant function of v. Specifically, we must have
The leading edge of the shaft swept out by the moving shell crosses the x axis at a distance l(1-v) from the center point A, which implies that the object's instantaneous spatial extent from the center to the leading edge is only
Likewise it's easy to see that the elapsed time (according to the putative ether rest frame coordinates) for one cycle of the moving particle, i.e., from point A to point B, is simply
compared with an elapsed time of 2 for the same particle at rest.
Hence we unavoidably arrive at Fitzgerald's length contraction and Lorentz's time dilation for objects in motion with respect to the x,y,t coordinates, provided only that all characteristic spatial and temporal intervals associated with physical entities are enforced by signals that propagate at the fixed speed c = 1 with respect to these coordinates.
Lorentz's writings on this subject include references to what he called the "theorem of corresponding states". He noticed that there is a system of space and time coordinates, x',y',t', in terms of which the right-hand particle in the above figure is at rest, and the physical description of the moving particle corresponds precisely to the description of the left-hand particle with respect to its rest frame coordinates (i.e., the ether frame). The transformation between these two coordinate systems is easily shown to be
Noting that the speed c = 1 is preserved under this transformation, we see that the x',y',t' coordinate system could just as well be regarded as the rest frame of the ether. Indeed, if we just algebraically invert this transformation we find
In other words, the relationship between these two systems is reciprocal and symmetrical, and they are moving with speed -v and +v relative to each other. Thus if all physically meaningful substantial intervals of time and space are ultimately enforced by signals propagating at a fixed speed c with respect to one particular "absolute" system of space and time coordinates, then it automatically follows that there exist infinitely many systems of space and time coordinates, related according to the above transformation, all of which have the same fixed speed c, and each of which is equally suitable to be considered the "absolute" system of reference. Henri Poincare (1854-1912) noted that the so-called Lorentz transformations constitute a group that maintains the invariance of all quantities of the form (dt)2 - (dx)2 - (dy)2 - (dz)2, just as the group of translations and rotations in Euclidean space maintains the invariance of all quantities of the form (dx)2 + (dy)2 + (dz)2. Just as there is no intrinsically distinguished position or orientation in Euclidean space, there is no intrinsically distinguished rest frame in Minkowskian spacetime, so this naturally leads to Einstein's relativistic view of spacetime.
It's interesting that, despite the unavoidability of these conclusions from the basic premises that were held by many of the physicists from Maxwell to Lorentz (i.e., the electromagnetic world view), few if any researchers anticipated Michelson's null result. There are, I think, two reasons for this. First, Maxwell's equations are linear, so it was always clear that they could never yield stable particles. Hence there must be other forces at work, fundamentally different from electromagnetism, in order to achieve stable particles. This realization obviously undermines the cogency of Lorentz's molecular force hypothesis. (Indeed, it was this very realization that prompted Einstein to seek a more general phenomenological principle, not directly tied to Maxwell's equations, but compatible with them.) Second, and perhaps more importantly, Lorentz's theorem of corresponding states rested on another hypothesis, namely, the hypothesis that the inertial masses of material objects transform (longitudinally and transversely) in accord with the same expressions as do electromagnetic forces. There is no a priori reason - aside from the principle of relativity - to suppose that this hypothesis is true. In fact, the combination of this inertial hypothesis with the molecular force hypothesis is tantamount to the assumption of relativity, i.e., the assumption that electromagnetism and mechanics are covariant with respect to the same family of velocity transformations. The reason people were surprised by this is that it entailed a modification of either Maxwell's equations or Newton's laws, both of which seemed satisfactory within their respective areas of applicability. (See Lorentz's Assumptions for a more detailed discussion.)
Incidentally, the projections of the points A and B in the above drawing are the foci of the projected ellipse. Also, the semi-latus rectum of this ellipse is the factor (1-v2)1/2, and the eccentricity of the ellipse is v. To see this, we need only take the locus of points comprising the shell of the particle with respect to the co-moving coordinates
and apply the (inverted) Lorentz transformation, which gives
Since sin(q)2 + cos(q)2 = 1, the second two equations give the projected locus
which is the equation of an ellipse with the stated semi-latus rectum and eccentricity, with one focus at (0, 0) and the other at (2v/(1-v2)1/2, 0) The polar equation of this ellipse is
The angle f in this equation is different from the angle q in the equations for x and y, because q is evaluated with respect to the co-moving x',y',t' coordinates, whereas f is evaluated with respect to the x,y,t coordinates. Since (x)2 + (y)2 = (r)2, these angles are related by the equation
Expanding the squares and simplifying, this gives
This is equivalent to the well-known aberration formula, which can be seen by expanding the left hand side and solving for cos(q), which gives
