The Need for Relativity

Learning Objectives¶

By the end of this chapter, you should be able to:

• Explain why late-nineteenth-century physics appeared essentially complete, and identify the handful of unresolved puzzles that undid that picture.

• State the principle of relativity as it already existed in Newtonian mechanics, and explain why it does not, by itself, forbid a preferred frame for light.

• State the Galilean transformation and Galilean velocity-addition rule, and explain what they predict for the speed of light.

• Describe the luminiferous-ether hypothesis and the reasoning behind the Michelson–Morley experiment, and compute the fringe shift it predicted.

• Explain why the null result of the Michelson–Morley experiment could not be explained away within Newtonian mechanics and a stationary ether.

• Explain why the Kennedy–Thorndike experiment constrains length-only explanations of the null results and, together with independent clock tests, supports the linked space-and-time effects of Lorentz invariance.

• Describe the Fizeau experiment and explain why its result was equally difficult to accommodate within a simple ether picture.

• State Einstein’s two postulates of special relativity and explain why they are jointly incompatible with Galilean relativity.

Introduction¶

By the 1890s, many physicists believed their subject was nearly finished. Newtonian mechanics predicted the motion of planets, projectiles, and machinery with extraordinary precision; Maxwell’s equations had unified electricity, magnetism, and optics into a single theory of the electromagnetic field and correctly predicted that light is an electromagnetic wave. A famous (if perhaps apocryphal) remark attributed to physicists of the era held that all that remained was to measure constants to more decimal places.

That confidence concealed a handful of loose threads. This chapter follows one of them: Maxwell’s equations predict a definite speed for light, $c \approx 3.00\times10^{8}\ \text{m/s}$, but they do not say relative to what that speed is measured. Every other wave known to nineteenth-century physics — sound, water waves, waves on a string — is a disturbance in a material medium, and its speed is fixed relative to that medium, not relative to an observer. It was natural to assume light must be the same: a wave in some all-pervading substance, the luminiferous ether, with $c$ being its speed relative to the ether alone. Observers moving through the ether should then measure a different speed for light, just as a swimmer’s speed relative to the shore depends on whether they swim with or against a current. Two experiments designed to detect this effect — one that stubbornly refused to find it, and one that found something stranger still — are where this chapter begins, before turning to the radical reinterpretation Einstein proposed for both results at once.

1.1The Ether Hypothesis and Michelson–Morley¶

The Principle of Relativity, Before Einstein¶

It is worth being precise about what was, and was not, already understood before 1905. Galileo had already argued, nearly three centuries earlier, that uniform motion is undetectable from the inside. In his Dialogue Concerning the Two Chief World Systems (1632), he imagined an observer sealed below the deck of a smoothly sailing ship, with no portholes: butterflies fly about the cabin, fish swim in a bowl, drops fall from a bottle into a jar beneath it. Galileo’s point was that no mechanical experiment performed in that sealed cabin can tell you whether the ship is at rest in harbor or cruising at constant velocity across a calm sea. The butterflies do not pile up against the stern wall; the drops fall straight down into the jar exactly as they would in harbor. Only changes in the ship’s velocity — a lurch, a turn — are detectable from inside, because those involve acceleration, and acceleration is not relative in the same way.

This is the principle of relativity applied to mechanics alone: the laws of mechanics take the same form in every inertial (non-accelerating) reference frame, so no mechanical experiment, performed entirely inside a closed laboratory, can determine that laboratory’s velocity — only changes in velocity are detectable. Formally, this is exactly the content of the Galilean transformation worked out below: Newton’s second law, $F = ma$, keeps the same form under $x' = x - vt$ for any constant $v$, because acceleration is unchanged by the transformation ($a' = a$, since $v$ is constant). Nothing about this principle was controversial in 1890; it was already a working assumption of every mechanic and astronomer.

What was not yet settled was whether this principle extended beyond mechanics — in particular, to light and electromagnetism. Maxwell’s equations, unlike Newton’s laws, single out a specific speed $c$ for electromagnetic waves, and by the reasoning of the Introduction above, a wave’s speed is normally measured relative to its medium, not relative to an arbitrary observer. If light truly is a disturbance in a material ether, then Maxwell’s equations should hold their simplest form only in the ether’s own rest frame, and an observer moving relative to the ether — such as anyone standing on the surface of a planet orbiting the Sun — should be able to detect that motion optically, even though no mechanical experiment inside a closed lab could reveal it. The question the rest of this chapter pursues is empirical: can such motion actually be detected?

Galilean Relativity and the Speed of Light¶

Newtonian mechanics is built on a specific rule for relating measurements made by observers moving relative to one another. Consider two reference frames, $S$ and $S'$, with $S'$ moving at constant velocity $v$ along the $x$-axis of $S$, and with their origins coinciding at $t = 0$. The Galilean transformation relates the coordinates of an event as measured in each frame:

The last equation encodes an assumption so natural to everyday experience that it usually goes unstated: time flows identically for every observer, regardless of how they are moving. Differentiating the transformation for $x$ gives the familiar classical velocity-addition rule. If an object moves with velocity $u_x$ in frame $S$, its velocity in $S'$ is

This is the rule that lets you add your walking speed to a moving walkway, or a thrown ball’s speed to the speed of the train from which it is thrown. Applied to light, it makes a sharp prediction: if light travels at speed $c$ relative to the ether, and Earth moves through the ether at orbital speed $v \approx 3\times 10^4\ \text{m/s}$, then an observer on Earth measuring light traveling in the same direction as Earth’s motion should get $c - v$, and light traveling in the opposite direction should give $c + v$. The effect is small — about one part in 10⁴ — but with an interferometer sensitive enough to detect a shift in the interference pattern of order that size, it should be measurable.

The assumption hiding inside that argument is worth making explicit, because every wave known to nineteenth-century physics encouraged it. A wave is a disturbance of something, and that something fixes the frame in which the wave has its textbook speed: Figure 1.1 runs one oscillating source as a ripple on water, as a sound wave in air, and as light. On the first two screens the medium can be put on display directly — the water surface, and the individual air molecules being pushed back and forth — and it is relative to that medium that the disturbance travels at its stated speed. The ether hypothesis is the assumption that the third screen works like the first two, and that the medium there is merely one nobody had yet managed to detect.

The Michelson–Morley Experiment¶

Albert Michelson (and later Michelson working with Edward Morley) built exactly such an interferometer. Its principle is to split a beam of light into two perpendicular paths, reflect each off a mirror, recombine the beams, and observe the resulting interference pattern. If Earth moves through the ether, the round-trip travel time along the path parallel to that motion should differ slightly from the round-trip time along the perpendicular path, because a “cross-stream” trip and an “upstream–downstream” trip through a moving medium take different times even when the two paths have equal length — the same effect that makes a boat crossing a river directly and returning take less time than travelling the same distance upstream and back. The predicted difference is small, but the interferometer was sensitive enough to detect a shift of a small fraction of a fringe, and the apparatus was mounted on a stone slab floating on mercury so that it could be rotated smoothly to swap the roles of the two arms.

Computing the Predicted Fringe Shift¶

The prediction can be made quantitative. Let both interferometer arms have length $L$, with the apparatus moving at speed $v$ through the ether, and suppose (for the moment) that one arm lies along the direction of motion and the other perpendicular to it. Light traveling down the parallel arm and back takes

since it travels at $c-v$ relative to the apparatus going one way (against the ether wind) and $c+v$ coming back. Light traveling down the perpendicular arm and back must be aimed slightly upstream (in the ether frame) to compensate for the sideways drift of the apparatus, so that its actual path through the ether is the hypotenuse of a right triangle with leg $L$; a short calculation (directly analogous to the light-clock argument of Chapter 2) gives

For $v \ll c$, both expressions can be expanded with the binomial approximation, $(1-x)^{-1} \approx 1+x$ and $(1-x)^{-1/2} \approx 1 + x/2$:

Rotating the apparatus by $90°$ swaps the roles of the two arms, doubling the effect, so the predicted shift in the interference pattern, measured in units of fringes (one fringe corresponding to one extra wavelength $\lambda$ of path-length difference), is

Plugging in the actual 1887 apparatus — effective arm length $L \approx 11\ \text{m}$ (achieved by multiple reflections), sodium light $\lambda \approx 590\ \text{nm}$, and Earth’s orbital speed $v \approx 3.0\times10^4\ \text{m/s}$ — gives

comfortably above the apparatus’s sensitivity of about 0.01 fringe — roughly a factor of 37 in signal-to-noise. This is the calculation, worked in reverse from the design specifications, that told Michelson and Morley their apparatus was more than sensitive enough for the job.

The Null Result¶

The prediction was unambiguous: rotating the apparatus should shift the fringe pattern by an amount corresponding to Earth’s motion through the ether, roughly 0.4 fringe in the original 1887 apparatus — well above its sensitivity of about 0.01 fringe. The experiment was repeated at different times of day and during different seasons. If a stationary ether existed, Earth’s rotation and orbital motion would continually change the direction—and generally also the magnitude—of the laboratory’s velocity relative to it. A null result at one moment might therefore reflect an unfavorable orientation or an accidentally small projected velocity, but persistent null results could not. No shift of the predicted size was ever observed, in any orientation, at any time of year. The measured shift was consistent with zero and far smaller than the effect Galilean relativity plus a stationary ether required. Later, more sensitive repetitions of the experiment (through the twentieth century, using lasers and eventually rotating optical cavities) have pushed the upper bound on any such effect down by many more orders of magnitude, with the same result: none.

The beam-splitting layout that makes this comparison possible is shown in Figure 1.2.

What a fraction of a fringe looks like, and how little it takes to produce one, can be checked directly in Figure 1.3. Translating one mirror by half a wavelength moves the pattern by one full fringe, so the 0.37 fringe that Michelson and Morley were hunting corresponds to an optical path difference of $0.37\lambda \approx 0.22\ \mu\text{m}$, and to a mirror motion of half that — small, but on this display an unmistakable movement of the whole pattern rather than a subtle change of shape. That is why the null result was so hard to argue with.

1.2Further Null Results: Kennedy–Thorndike and Fizeau¶

A Second Null Result: The Kennedy–Thorndike Experiment¶

Michelson–Morley compared equal perpendicular arms while rotating the apparatus. In 1932, Roy Kennedy and Edward Thorndike used unequal arms and searched for changes as Earth’s velocity relative to a hypothetical preferred frame varied. Their null result closed an important loophole in explanations based on length contraction alone and supported the linked transformation of lengths and clock rates required by Lorentz invariance.

Why unequal arms provide a different test

The surviving photograph in Figure 1.4 shows the scale and mechanical character of the original apparatus described above.

One gap in the Michelson–Morley result deserves attention before moving on, because the way it was closed illustrates how tightly interlocking the eventual relativistic explanation would have to be. The Michelson–Morley experiment used two arms of equal length, and looked only for a fringe shift as the whole apparatus was rotated. This design has a subtle limitation: a length contraction (the Fitzgerald–Lorentz contraction, discussed below in “Attempts to Save the Ether”) of exactly the right amount would make $t_\parallel = t_\perp$ for any direction of the apparatus relative to the ether, since whichever arm happens to be aligned with the direction of motion at a given moment contracts by just enough to keep $t_\parallel$ equal to $t_\perp$ regardless of orientation — so rotating the equal-armed apparatus can never distinguish “no ether effects at all” from “an ether effect perfectly cancelled by length contraction alone.” It says nothing about whether the round-trip time itself depends on velocity through the ether, only about whether the two perpendicular round-trip times remain equal to each other.

Roy Kennedy and Edward Thorndike closed this gap in 1932 with a modified apparatus using two arms of substantially unequal length, $L_1 \ne L_2$. With unequal arms, length contraction alone can no longer guarantee $t_1 = t_2$ at all times of year: even if each arm individually contracts as $L\sqrt{1-v^2/c^2}$, the resulting difference $t_1 - t_2$ still depends on the apparatus’s instantaneous speed $v$ through the hypothetical ether — a speed that changes over the course of a year as Earth’s orbital velocity vector changes direction (and, if the solar system itself moves through the ether, also has a component that does not average to zero). A length-contraction-only patch, with no accompanying change in the rate at which clocks run, predicts a fringe shift that slowly drifts as the year progresses. None was observed, to a precision even finer than Michelson and Morley’s original bound.

The Kennedy–Thorndike null result therefore rules out length contraction as a stand-alone patch within this class of ether models. More precisely, it constrains how lengths and the frequency of the light source transform with velocity; interpreted together with Michelson–Morley and independent clock-comparison experiments, it supports the linked length and time effects predicted by the Lorentz transformation. It does not, by itself, constitute a direct measurement of time dilation or uniquely derive the full transformation. This is a useful methodological lesson that recurs throughout physics: a single null result can usually be explained by more than one patch, and it typically takes a family of independent experiments, each closing off a different escape route, to establish a genuinely new theory.

A Complication: The Fizeau Experiment¶

The ether’s troubles did not begin with Michelson and Morley. Decades earlier, in 1851, Hippolyte Fizeau performed a related but distinct experiment: rather than looking for an effect of Earth’s motion through vacuum, he measured the speed of light inside a moving medium — water flowing rapidly through a pipe. Splitting a light beam so that one part traveled with the flow and the other against it, then recombining them interferometrically, Fizeau could measure how much the water’s motion changed the light’s speed compared to light in stationary water, $c/n$ (where $n\approx1.33$ is water’s refractive index).

The naive, purely mechanical expectation — light as a stream of particles, or the water fully “dragging” the ether along with it, either of which would give ordinary Galilean addition — predicts a shift of exactly $v$, the full speed of the flowing water:

To first order in the water speed $v$, Fizeau measured something different, and reproducibly so:

The water changes the measured speed of light, but only partially at this order — by a fraction $f = 1 - 1/n^2$ of its own speed, not the full speed $v$ predicted by simple addition, and not zero either. Augustin-Jean Fresnel had predicted this coefficient in 1818 using an ether model in which a transparent medium partially entrained the ether. The prediction matched Fizeau’s measurement, but its physical interpretation required a specialized rule for the interaction between matter and ether.

Fizeau’s result sat alongside the Michelson–Morley null result as a second, independent puzzle: whatever the ether was doing, it was not doing anything as simple as being either fully dragged along by matter or entirely undisturbed by it. As will become clear in Chapter 2, once relativistic velocity addition replaces the Galilean rule, Fresnel’s fractional drag coefficient emerges automatically — with no separate assumption about entrainment at all — as the leading term of $u_x = (u_x' + v)/(1 + u_x'v/c^2)$ evaluated at $u_x' = c/n$. What looked like an ad hoc patch to the ether model instead emerges, to first order in $v/c$, from relativistic kinematics.

Fizeau is shown in a late-nineteenth-century portrait in Figure 1.5.

The three possibilities are summarized visually in Figure 1.6: no drag, full Galilean drag, and the intermediate result actually observed.

1.3Toward Einstein’s Postulates¶

Attempts to Save the Ether¶

The null result did not immediately convince anyone to abandon the ether; instead, it triggered a series of increasingly strained patches:

• Ether drag. Perhaps Earth drags the nearby ether along with it, so light near Earth’s surface shows no relative motion. The simplest version—complete, uniform drag—conflicts with stellar aberration, the small annual shift in the apparent position of stars (about 20.5 arcseconds, first measured by James Bradley in 1727). In the falling-rain analogy, complete local drag would make the incoming light share Earth’s sideways motion and would remove the need to tilt the telescope, contrary to observation. Historically, more elaborate drag models were proposed to recover aberration, so aberration alone did not dispose of every possible ether theory; rather, it forced those theories to become substantially more complicated.

• The Fitzgerald–Lorentz contraction. George FitzGerald and, independently, Hendrik Lorentz proposed that objects moving through the ether physically contract along their direction of motion by just the factor needed to make the two arms of the interferometer take equal time, hiding the effect. Concretely, if a rod of ether-frame length $L_0$ oriented along the direction of motion instead has length $L_0\sqrt{1-v^2/c^2}$, then $t_\parallel$ recomputed with this shortened arm length equals $t_\perp$ exactly, and no fringe shift results, for any $v$. Lorentz went further, introducing a fictitious “local time” that varied with position in exactly the combination $t - vx/c^2$ that would, decades later, turn out to be the correct relativistic time coordinate. Taken on its own, however, this is an ad hoc fix — a contraction (and a local time) invented for the sole purpose of explaining away one null result, with the ether itself still doing the conceptual work of defining a single true rest frame. It turns out to contain a genuine piece of the truth, but only once it is derived from a deeper principle rather than posited to save a failing hypothesis, which is what Einstein did in 1905 (Chapter 2).

• Emission theories. Perhaps light simply travels at speed $c$ relative to its source, with no ether needed at all — a proposal in the spirit of a bullet’s speed depending on the gun that fired it. De Sitter argued in 1913 that a simple version of this theory would badly distort observations of binary stars: light emitted while a star approached Earth would arrive faster than light emitted while it receded, potentially scrambling the apparent orbit. Such distortions are not observed, providing important evidence against simple ballistic theories. The historical argument was not conclusive by itself, because interactions with intervening matter might erase a source-dependent speed before the light reached Earth; laboratory and astronomical tests taken together provide the stronger case that light’s vacuum speed is independent of its source.

Worked Example: Estimating the Aberration Angle¶

The stellar-aberration bullet above asserts a value of about 20.5 arcseconds; it is worth seeing where a number of that size comes from, since the same reasoning about combining velocities recurs throughout this chapter. Consider a star directly “overhead,” so that in the Sun’s rest frame its light arrives travelling straight down (speed $c$, along $-y$) at a telescope. In the frame of Earth, which moves at orbital speed $v \approx 3.0\times10^4\ \text{m/s}$ in some direction in the $x$-$y$ plane perpendicular to the line of sight, ordinary (Galilean) velocity subtraction gives the light’s velocity components as measured on Earth: still $-c$ along $y$, but now also $-v$ along $x$, exactly as raindrops falling straight down acquire an apparent forward velocity component for a runner moving beneath them. The apparent direction of arrival is tilted from vertical by an angle $\theta$ with

Since this is a very small angle, $\theta \approx 1.0\times10^{-4}\ \text{rad}$. Converting to arcseconds (using $1\ \text{rad} = 206{,}265''$),

in close agreement with the historically measured value of $20.5''$. Because Earth’s velocity vector sweeps around in a full circle over the course of a year (always perpendicular to the instantaneous direction to a star near the pole of the ecliptic), each such star traces out a small circle of this same angular radius on the sky annually — exactly the periodic wobble Bradley detected in 1727, long before anyone suspected the ether was in trouble.

Each patch could account for the existing data but only by adding a new, unmotivated assumption, and no single patch accounted for both puzzles — the Michelson–Morley null result and the Fizeau partial-drag coefficient — without contradicting itself. What was needed was not another patch but a new starting point.

Einstein’s Postulates¶

In 1905, Einstein proposed to stop patching Galilean relativity and instead take the null result at face value, elevating it (together with the principle of relativity already implicit in mechanics, discussed above) to a postulate. Special relativity rests on two statements:

1. The principle of relativity. The laws of physics take the same form in all inertial reference frames. No experiment performed entirely within a closed laboratory can distinguish one inertial frame from another.

2. The constancy of the speed of light. The speed of light in vacuum has the same value $c$ in every inertial reference frame, independent of the motion of the source or the observer.

The first postulate was not new — as Galileo’s ship illustrates, it already held for Newtonian mechanics, where no mechanical experiment can detect uniform motion (this is why you cannot feel the constant-velocity phase of a smooth flight, and why the butterflies in Galileo’s cabin fly about undisturbed). Einstein’s step was to insist that this postulate apply to all of physics, electromagnetism included, so that no experiment of any kind — mechanical or optical — can distinguish inertial frames; there is no preferred, ether-defined rest frame for anything, including light. The second postulate is the one that clashes head-on with Galilean relativity: if $u_x' = u_x - v$ held for light as it does for baseballs, two observers in relative motion could not both measure the same speed $c$ for the same light beam. Taken together, the postulates require that time and space themselves — not just the “aether wind” — behave differently than Galilean relativity assumes.

This is a genuinely radical move. Rather than asking “what medium is light waving in?”, Einstein asked what kinematics — what set of rules for relating space and time coordinates between observers — is consistent with light having the same speed for everyone. The answer is not the Galilean transformation but the Lorentz transformation, and working out its consequences — the relativity of simultaneity, time dilation, length contraction, and (as noted above) the Fizeau drag coefficient as a special case of velocity addition — is the subject of Chapter 2.

Einstein was not working in a vacuum: Lorentz had already written down transformation equations of essentially the same mathematical form (motivated, as noted above, by fitting the Michelson–Morley and Fizeau data rather than by a physical principle), and Henri Poincaré, in the same year, independently emphasized that the principle of relativity should be regarded as an exact law of nature rather than an approximate consequence of ether dynamics, and showed that the Lorentz transformations form a mathematical group — a property essential for consistency, since applying two velocity boosts in succession must itself be equivalent to some single valid transformation. What distinguished Einstein’s paper was not the equations themselves but their derivation: rather than reverse-engineering a transformation to fit the data, he started from the two postulates and showed that the Lorentz transformation is the unique kinematic consequence of taking them seriously, with time dilation, length contraction, and the relativity of simultaneity following as necessary theorems rather than independently adjustable assumptions.

It is worth appreciating just how much this single change in starting point accomplishes. The Michelson–Morley null result, stellar aberration, binary-star observations, and the Fizeau drag coefficient were separate empirical constraints that simple ether and emission models struggled to accommodate together. Relativistic kinematics provides a common framework for all four, without assigning light to a material medium or introducing a separate ether parameter for each result.

1.4Summary¶

• The principle of relativity — that no mechanical experiment inside a closed laboratory can detect uniform motion — already applied to Newtonian mechanics before 1905 (Galileo’s ship). What was not settled was whether it extended to optics and electromagnetism, where Maxwell’s equations single out a specific speed $c$ naturally interpreted as a speed relative to a medium, the luminiferous ether.

• The Galilean transformation and its associated velocity-addition rule, $u_x' = u_x - v$, predict that an observer moving through the ether should measure a speed of light different from $c$, by an amount of order Earth’s orbital speed; the predicted Michelson–Morley fringe shift is $\Delta N = 2Lv^2/(\lambda c^2) \approx 0.4$ fringe for the 1887 apparatus.

• The Michelson–Morley interferometer was sensitive enough to detect this predicted shift but found none, at any time of year or orientation: the speed of light showed no dependence on the observer’s motion through the presumed ether.

• The Kennedy–Thorndike experiment, using unequal interferometer arms, constrained length-contraction-only explanations. Together with Michelson–Morley and independent clock tests, its null result supports the linked length and time effects of Lorentz invariance.

• The Fizeau experiment, measuring the speed of light in flowing water, found to first order a partial drag coefficient, $f = 1-1/n^2$ — neither the full entrainment nor the zero entrainment that simple ether pictures would predict. Relativistic velocity addition produces this coefficient naturally at first order in $v/c$.

• Proposed rescues of the ether hypothesis faced additional constraints: stellar aberration conflicts with simple complete-drag models, binary-star observations weigh against simple emission theories, and length contraction introduced solely to hide the Michelson–Morley effect was an incomplete patch. More elaborate models could evade individual objections, but only at the cost of additional assumptions.

• Einstein’s 1905 postulates take the null result at face value: the laws of physics (postulate 1) and the speed of light in particular (postulate 2) are the same in every inertial frame. Together these are incompatible with the Galilean transformation and require a new kinematics, developed in Chapter 2, which reproduces all of the results above — Michelson–Morley, aberration, binary-star timing, and the Fizeau coefficient — without any adjustable ether parameters.

1.5Problems¶

Solution to Exercise 1.2 #

For the cross-river trip, the swimmer must devote a velocity component $v$ upstream to cancel the current. If $u_y$ is the shore-frame component across the river, then

$$u^2=v^2+u_y^2, \qquad u_y=\sqrt{u^2-v^2}.$$

(1.17)

Each crossing takes $L/u_y$, so

$$t_\perp=2\frac{L}{\sqrt{u^2-v^2}}.$$

(1.18)

For the upstream--downstream trip, the two shore-frame speeds are $u-v$ and $u+v$. Thus

$$t_\parallel=\frac{L}{u-v}+\frac{L}{u+v} =\frac{L[(u+v)+(u-v)]}{(u-v)(u+v)} =\frac{2Lu}{u^2-v^2}.$$

(1.19)

Their ratio is

$$\frac{t_\parallel}{t_\perp} =\frac{2Lu/(u^2-v^2)}{2L/\sqrt{u^2-v^2}} =\frac{u}{\sqrt{u^2-v^2}}>1 \quad (v>0).$$

(1.20)

Therefore, the upstream--downstream trip takes longer than the cross-river trip; this is precisely the classical ether prediction that the interferometer arm parallel to an ether wind has a longer round-trip light time than the perpendicular arm.

Solution to Exercise 1.7 #

The stated upper bound is

$$\Delta N_{\max}=(0.4\ \text{fringe})(10^{-5})=4.0\times10^{-6}\ \text{fringe}.$$

(1.26)

For a fixed apparatus, $\Delta N\propto v^2$. Thus, comparing this bound with the original prediction at $v_\oplus=3.0\times10^4\ \text{m/s}$ gives

$$\frac{\Delta N_{\max}}{0.4\ \text{fringe}} =\left(\frac{v_{\max}}{v_\oplus}\right)^2 =10^{-5},$$

(1.27)

and hence

$$v_{\max}=(3.0\times10^4\ \text{m/s})\sqrt{10^{-5}} =9.5\times10^1\ \text{m/s}.$$

(1.28)

Therefore, under the deliberately incorrect ether interpretation, the residual ether speed would be below about $95\ \text{m/s}$, which is about $3.2\times10^2$ times smaller than Earth’s orbital speed.