Experiments: Light, the Aether, and Time
The kinematics of special relativity is not a philosophical preference; it is forced by measurement. This chapter sets out the experimental base: the aether the nineteenth century postulated as the medium of light, the conflict between that hypothesis and the principle of relativity, and the experiments whose outcomes single out the Lorentz transformations of Lorentz Transformations over Galilean kinematics.
Foundations
The vacuum and the aether
The aether hypothesis
Every wave known before the nineteenth century was a disturbance of something: sound of air, ripples of water, elastic waves of a solid. Each has a speed fixed with respect to its medium, and each obeys Galilean velocity addition — sound travels downwind faster than upwind, by exactly the wind speed. When Maxwell's equations (The Maxwell Equations) yielded wave solutions travelling at a speed \(c=1/\sqrt{\varepsilon_{0}\mu_{0}}\) fixed by two laboratory constants, the analogy seemed to settle the matter: light too must be a disturbance of a medium, the luminiferous aether, and \(c\) must be its speed relative to that medium.
The hypothesis was not idle. It made the aether's mechanical properties calculable and alarming: to carry a transverse wave at \(c\) it had to be more rigid than steel, yet offer no measurable resistance to the planets. That tension was tolerated because the alternative — a wave in nothing — seemed worse.
The conflict with the principle of relativity
The aether hypothesis contradicts Postulate 38.1. If light travels at \(c\) relative to the aether, then in a laboratory moving at velocity \(\vect{v}\) through it the speed of light must be \(c-\vect{v}\) in the direction of motion and \(c+\vect{v}\) against it. Measuring that anisotropy would determine \(\vect{v}\) by an experiment performed entirely inside the laboratory, and the aether's rest frame would be a physically preferred inertial frame — absolute space, restored by the back door.
So the aether is testable, and the test is quantitative. The Earth's orbital speed is \(v_{\oplus}\approx29.8\,\mathrm{km}/\mathrm{s}\), so \(\beta_{\oplus}\approx10^{-4}\); even if the Solar System happened to be at rest in the aether at one moment, six months later the Earth's velocity would have reversed and the anisotropy would be \(2\beta_{\oplus}\). A first-order effect is out of reach of an ordinary optical measurement, but a round-trip interference measurement is sensitive at order \(\beta_{\oplus}^{2}\approx10^{-8}\) — and interferometry can resolve that. This is the experiment Michelson and Morley performed.
The finite speed of light
That light travels at all — rather than arriving instantaneously — is itself an experimental result, and the first quantitative one. Timing the eclipses of Jupiter's moon Io, R{ø}mer found in 1676 that they run late when the Earth is receding from Jupiter and early when it approaches, by an amount consistent with a finite propagation speed [Roemer:1676]. He gave the light-crossing time of the Earth's orbit rather than a speed; with the modern orbital diameter it corresponds to a \(c\) within about \(25\,\mathrm{\%}\) of the value now fixed by definition. The relevant point for this part is that \(c\) is finite, so that the question of what it is finite with respect to can be asked at all.
Michelson and Morley: the search for the aether wind (1887)
Tests Equation (37.1).
Apparatus
A Michelson interferometer (Figure 37.1) mounted on a sandstone slab about \(1.5\,\mathrm{m}\) square, floated on a bath of mercury so that the whole instrument could be rotated continuously without flexure or interruption. Multiple reflections folded each arm to an optical path of about \(11\,\mathrm{m}\), which the authors quote as \(2\times10^{7}\) wavelengths of yellow light. A sodium lamp supplied the source; the fringes were viewed through a telescope carried around with the slab.
Procedure
Monochromatic light is divided at a half-silvered plate into two beams travelling along perpendicular arms, retroreflected, and recombined to interfere. If the apparatus moves at speed \(v\) through an aether, the round-trip times along the two arms differ, and rotating the instrument by \(90\,^\circ\) exchanges the arms and reverses the difference. The observable is therefore not a fringe position — which nothing calibrates — but the shift of the fringes as the instrument turns, repeated at different times of day and at different seasons so that the Earth's rotation and orbital motion sweep the laboratory velocity through the sky.
Observations and data
For arms of equal length \(L\), expansion of the two round-trip times to second order in \(\beta=v/c\) gives a time difference \(\Delta t\simeq L\beta^{2}/c\), hence an expected fringe shift on rotation of
which for \(L=11\,\mathrm{m}\), \(\lambda=590\,\mathrm{nm}\) and \(\beta=\beta_{\oplus}\) is about \(0.4\) of a fringe — some forty times the resolution of the instrument.
The observed displacement was, in the authors' words, “certainly less than the twentieth part of this, and probably less than the fortieth part”:
Translated into a velocity through Equation (37.1), they concluded that “the relative velocity of the earth and the ether is probably less than one sixth the earth's orbital velocity, and certainly less than one-fourth.” The paper states the bound only as that fraction and prints no figure in kilometres per second; with \(v_{\oplus}=29.8\,\mathrm{km}/\mathrm{s}\) the two limits correspond to about \(5\,\mathrm{km}/\mathrm{s}\) and \(7.5\,\mathrm{km}/\mathrm{s}\), but those numbers are derived here and are not Michelson and Morley's.
Interpretation
The predicted effect was absent. Contraction hypotheses of the Fitz\-Gerald–Lorentz type could be arranged to cancel it, at the cost of a new postulate whose only function was to hide the aether; Kennedy and Thorndike later closed that escape with an unequal-arm version of the experiment, whose null result constrains the velocity dependence that length contraction alone cannot explain away [Kennedy:1932]. The economical reading is the one Einstein took: there is no aether wind because there is no aether frame, and the speed of light is the same in every inertial frame — Postulate 38.2.
Primary references
[Michelson:1887]; the unequal-arm follow-up [Kennedy:1932].
![[figure: st-michelson-morley.pdf]](/images/treatise/de/figure/st-michelson-morley.svg)
The Michelson interferometer of Section 37.3. Light is divided at the beam splitter, travels the two perpendicular arms of lengths \(L_{1}\) and \(L_{2}\), is retroreflected at \(M_{1}\) and \(M_{2}\), and recombines at the detector. An aether wind of speed \(v\) would make the two round-trip times differ at order \(\left(v/c\right)^{2}\); rotating the whole instrument through \(90\,^\circ\) exchanges the arms and doubles the predicted fringe shift.
Fizeau: light dragged by moving water (1851)
Tests Phenomenon 38.18.
Apparatus
Two parallel tubes, each \(1.487\,\mathrm{m}\) long with an interior bore of \(5.3\,\mathrm{mm}\), through which water was driven at \(7.069\,\mathrm{m}/\mathrm{s}\). A beam of light was split so that one part traversed both tubes with the current and the other against it, the two being recombined to interfere.
Procedure
The water flow is started and stopped and the fringe displacement recorded. Sending each beam through both tubes — once with the flow and once against — doubles the effect and cancels systematic differences between the two optical paths.
Observations and data
Fizeau reports fringe displacements only. From nineteen sufficiently concurring observations the average was
against the \(0.40\) predicted by Fresnel's partial-drag hypothesis. No uncertainty is quoted anywhere in the note, and no numerical drag coefficient is printed; the values often attributed to Fizeau are back-derivations by later authors.
Interpretation
Complete dragging of the aether by the water would give a displacement proportional to the full water speed; no dragging at all would give none. The measurement sits between the two, near Fresnel's coefficient \(1-1/n^{2}\), which is \(0.434\) for water at \(n=1.333\). As an aether result this is incoherent — the medium must be dragged partially, by an amount depending on the colour of the light. As a relativistic result it is immediate: the composition law Equation (38.29) applied to \(u'=c/n\) gives exactly \(c/n+v\left(1-1/n^{2}\right)\) to first order, with no free parameter and no medium to drag. The derivation is Example 38.20.
Primary references
[Fizeau:1851]; the relativistic reading is Einstein's [Einstein:1905a].
Cavity resonators: the modern Michelson–Morley (1979–2015)
Tests Equation (37.1). Assuming Postulate 38.2.
Apparatus
Two orthogonal electromagnetic resonators — laser-stabilized optical cavities [Brillet:1979] [Herrmann:2009], or cryogenic sapphire microwave oscillators [Nagel:2015] — mounted on a turntable and rotated continuously, with their beat frequency recorded.
Procedure
A cavity's resonant frequency depends on its optical length; an anisotropy in the speed of light would make the beat frequency of two perpendicular cavities vary as the pair rotates, at twice the rotation frequency and with sidereal modulation as the Earth turns. The measurement is the amplitude of that modulation, integrated over months.
Observations and data
The searched-for fractional anisotropy is bounded at
with the rotating optical cavities of [Herrmann:2009] reaching the \(10^{-17}\) level and the rotating cryogenic microwave experiment of [Nagel:2015] the \(10^{-18}\) level in the corresponding Lorentz-violating coefficients.
Interpretation
Michelson and Morley bounded the anisotropy near \(10^{-8}\) in \(\beta^{2}\); a century of refinement has improved that by ten orders of magnitude and found nothing. Postulate 38.2 is, on present evidence, one of the most stringently tested statements in physics. The modern experiments are usually reported not as bounds on an aether velocity — nobody expects one — but as bounds on the coefficients of a general Lorentz-violating extension of the Standard Model, which is the form in which a null result remains informative (What We Observe but Do Not Understand).
Primary references
[Brillet:1979] [Herrmann:2009] [Nagel:2015].
The muon lifetime
The experiments above concern the propagation of light. That the time of a moving system is affected as well is shown most directly by unstable particles, whose decay is a clock carried along with them. Muons created by cosmic-ray primaries at an altitude of some \(15\,\mathrm{km}\) have a proper lifetime near \(2.2\,\mu\mathrm{s}\), so even at the speed of light they should travel only about \(660\,\mathrm{m}\) before decaying, and essentially none should reach the ground. Very many do.
This is the subject of Experiment: Time Dilation and Relativistic Kinematics, where the measurements of Rossi and Hall [Rossi:1941], Frisch and Smith [Frisch:1963] and the CERN storage ring [Bailey:1977] are reported in full, once the time-dilation formula Equation (38.27) they test has been derived.
What the experiments establish
Taken together the results above fix the kinematics uniquely. The Michelson–Morley null result and its modern descendants say the two-way speed of light is isotropic to \(10^{-18}\), so no preferred frame is detectable optically. Fizeau's measurement says velocities do not add in the Galilean way, and fixes the first-order correction. Kennedy and Thorndike say the effect cannot be removed by contraction alone. The muon experiments say that moving clocks run slow by the factor \(\gamma\), not by some other function of speed.
What survives is a kinematics in which \(c\) is invariant and the transformation between inertial frames is not Galilean. Lorentz Transformations shows that these two requirements have essentially one solution.