Appearance
6.1 — The Crisis: A Wave With No Medium
Sit on a train moving at a steady 100 km/h and throw a ball straight up. It comes back into your hand. Pour a drink and it goes into the glass. Nothing about the physics inside the carriage tells you the train is moving at all — only looking out of the window does.
That observation is older than Newton. Galileo wrote it up in 1632 using a ship's cabin, with fish in a bowl, butterflies flying about and water dripping from a bottle, and concluded that no mechanical experiment performed inside can detect uniform motion. It is called the principle of relativity, and it was uncontroversial for two hundred and fifty years.
Then Maxwell's equations arrived and quietly contradicted it.
Galilean relativity, and why it is reasonable
Two observers, one on the ground and one on the train moving at speed v along the x direction. If they synchronise their origins at t = 0, the coordinates of an event relate as:
x' = x - vt, \qquad y' = y, \qquad z' = z, \qquad t' = t
These are the Galilean transformations, and the last one is the one that will turn out to be false: time is the same for everyone.
Differentiate the first with respect to time and you get velocity addition:
u' = u - v
Throw a ball forward at 20 m/s on a train doing 30 m/s, and someone on the ground measures 50 m/s. Everyone's intuition agrees, and every measurement anyone had ever made agreed too.
Differentiate again and the acceleration comes out the same for both:
a' = a
This is the key point. Newton's second law is F = ma, and since force depends on separations and accelerations, both of which are unchanged, the laws of mechanics look identical in every uniformly moving frame. There is no experiment that picks out a preferred one. Absolute velocity is not measurable; only relative velocity is.
Frames in which Newton's first law holds — an object with no force on it moves in a straight line at constant speed — are called inertial frames. Galilean relativity says all inertial frames are equivalent for mechanics.
Maxwell breaks it
Chapter 4.7 derived from Maxwell's equations that electromagnetic waves travel at:
c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00\times10^{8}\ \text{m/s}
Now ask the question that took forty years to answer properly: relative to what?
The equations do not say. They contain \mu_0 and \varepsilon_0, which are properties of empty space, and they produce a definite speed with no reference frame attached. Under a Galilean transformation, that speed should become c - v for someone chasing the light. But if you transform Maxwell's equations into a moving frame using x' = x - vt, the equations change form — they pick up extra terms and are no longer Maxwell's equations.
So one of three things had to be true:
- Maxwell's equations are wrong, and the correct ones are Galilean-invariant.
- Maxwell's equations are right in one special frame only, and the principle of relativity fails for electromagnetism.
- The Galilean transformation is wrong.
Almost everybody in 1890 chose option 2, and it was a perfectly sensible choice.
The ether
Every wave known to physics travelled in something. Sound needs air. Water waves need water. A wave on a string needs the string. A wave is a disturbance of a medium, and the wave speed is a property of that medium — v = \sqrt{T/\mu} for a string (Chapter 2.3), v = \sqrt{\gamma P/\rho} for air.
So light must have a medium too, and the constants \mu_0 and \varepsilon_0 must be its mechanical properties. It was named the luminiferous ether, and Maxwell's equations were understood to hold in the frame at rest with respect to it.
That resolves the problem neatly. There is a preferred frame — the ether's — and light travels at c in that frame and at c \pm v for anyone moving through it. Relativity survives for mechanics and fails for light, which is fine, because now light gives you a way to measure your absolute motion.
The ether had to have absurd properties, and everyone knew it. It must fill all of space, including the space between the stars. It must be perfectly transparent and offer no resistance to planetary motion — the Earth has orbited for billions of years without slowing. Yet it must be extraordinarily stiff, because wave speed goes as the square root of stiffness over density, and 3\times10^8 m/s demands a rigidity far exceeding steel. A rigid solid filling all space that planets pass through without friction is not a comfortable object.
More decisively, light is a transverse wave (Chapter 4.7), and transverse waves propagate only in solids — a fluid has no shear strength to restore a sideways displacement. So the ether had to be a solid.
Nobody liked it. Everybody accepted it, because the alternative was worse: a wave with nothing waving.
The Michelson–Morley experiment
If the Earth moves through the ether, then there is an "ether wind" blowing past us at the Earth's orbital speed, about 30 km/s. Light travelling with or against that wind should go at c + v or c - v.
The trouble is that v/c = 30{,}000/3\times10^8 = 10^{-4}, so the effect on a one-way measurement is one part in ten thousand — and any one-way measurement of light speed requires two clocks synchronised to a precision nobody had.
Albert Michelson's solution, refined with Edward Morley in 1887, was to compare two round trips at right angles using interference, which as Chapter 5.3 showed is an extraordinarily sensitive comparator.
The prediction, derived
Let the apparatus move at speed v through the ether, with arm 1 along the motion and arm 2 across it. Each arm has length L.
Arm 1, along the motion. Going out, the light chases a mirror that is running away, so its closing speed is c - v. Coming back, the mirror is approaching, so the closing speed is c + v:
t_1 = \frac{L}{c-v}+\frac{L}{c+v} = L\cdot\frac{(c+v)+(c-v)}{(c-v)(c+v)} = \frac{2Lc}{c^2-v^2}
t_1 = \frac{2L}{c}\cdot\frac{1}{1-v^2/c^2}
Arm 2, across the motion. Here the light must be aimed slightly upstream to arrive at the mirror, because the mirror moves sideways while the light is in flight. In the ether frame the light travels along the hypotenuse of a triangle: while the light covers ct/2 one way, the apparatus has moved vt/2 sideways, and the arm length L is the perpendicular. By Pythagoras:
\left(\frac{ct_2}{2}\right)^2 = L^2 + \left(\frac{vt_2}{2}\right)^2
\frac{t_2^2}{4}(c^2-v^2) = L^2 \quad\Longrightarrow\quad t_2 = \frac{2L}{\sqrt{c^2-v^2}} = \frac{2L}{c}\cdot\frac{1}{\sqrt{1-v^2/c^2}}
The two differ. Write \beta = v/c:
t_1 = \frac{2L}{c}\cdot\frac{1}{1-\beta^2}, \qquad t_2 = \frac{2L}{c}\cdot\frac{1}{\sqrt{1-\beta^2}}
\Delta t = t_1 - t_2 = \frac{2L}{c}\left[\frac{1}{1-\beta^2}-\frac{1}{\sqrt{1-\beta^2}}\right]
For small \beta, expand using the binomial approximation (1-x)^{-1} \approx 1+x and (1-x)^{-1/2} \approx 1+x/2:
\Delta t \approx \frac{2L}{c}\left[(1+\beta^2) - \left(1+\frac{\beta^2}{2}\right)\right] = \frac{2L}{c}\cdot\frac{\beta^2}{2} = \frac{Lv^2}{c^3}
Then rotate the apparatus 90°. The arms swap roles, so the difference reverses sign, and the total change in path difference is twice this. In wavelengths, the fringe shift is:
N = \frac{2\Delta t\, c}{\lambda} = \frac{2Lv^2}{\lambda c^2}
The numbers
Michelson and Morley used multiple reflections to make the effective arm length L = 11 m, floated the whole apparatus on a bath of mercury so it could be rotated smoothly without strain, and used sodium light at \lambda = 589 nm.
N = \frac{2(11)(3\times10^{4})^2}{(589\times10^{-9})(3\times10^{8})^2} = \frac{2(11)(9\times10^{8})}{(589\times10^{-9})(9\times10^{16})}
N = \frac{1.98\times10^{10}}{5.30\times10^{10}} = 0.37
A shift of about four tenths of a fringe was expected. Their apparatus could detect one hundredth of a fringe — forty times better than needed.
The result
Nothing. No shift at any orientation, at any time of day, at any time of year.
They repeated it over months, so that the Earth's orbital velocity pointed in every direction relative to the apparatus. Any ether wind of any origin should have shown up at some point. The measured shift never exceeded their noise floor, which they placed at about 0.01 fringes, corresponding to an ether wind under 8 km/s. Later versions pushed the limit to under 1 km/s, and modern versions using optical cavities have constrained the anisotropy of light speed to about one part in 10^{18}.
It is the most famous null result in the history of physics. Michelson considered it a failure and spent decades trying again.
The attempted rescues
Nobody abandoned the ether immediately. Three explanations were proposed, and following what happened to each is the clearest way to see why Einstein's answer was needed.
Ether drag. Perhaps the Earth carries a layer of ether with it, so there is no wind at the surface. This is killed by stellar aberration, an effect James Bradley discovered in 1728: the apparent position of every star shifts back and forth over the year by about 20 arcseconds, because the Earth's motion changes the angle at which starlight must be caught — exactly as you must tilt an umbrella forward when walking through vertical rain. If the Earth dragged the ether along, the light would be dragged with it and there would be no aberration. There is.
Emission theory. Perhaps light travels at c relative to its source, like a bullet from a gun, so no medium is needed. This was Walter Ritz's proposal, and it explains Michelson–Morley perfectly, since the source is on the apparatus. It is killed by binary stars: in a two-star system, one star is approaching us while the other recedes, so their light would travel at different speeds and arrive out of sequence, and closely orbiting binaries at large distances would appear to do impossible things — being in several places at once, or orbiting backwards. De Sitter pointed this out in 1913 and no such distortion is seen. It is also killed directly by measurements of gamma rays from fast-moving pion decays, which travel at c regardless of the source speed.
Length contraction. George FitzGerald in 1889 and Hendrik Lorentz in 1892 independently proposed that objects moving through the ether physically shrink along the direction of motion, by exactly the factor:
L = L_0\sqrt{1-v^2/c^2}
Put that into the arm-1 calculation and the shortening exactly compensates the longer travel time, so t_1 = t_2 and there is no fringe shift.
This is not wrong. It is the right formula. What was wrong was the reasoning behind it. Lorentz proposed it as a real mechanical effect: the ether wind squashes the electrical forces holding matter together, so objects genuinely compress. It was a hypothesis introduced for the sole purpose of explaining one null result, with no independent support, and it left the ether unmeasurable in principle — a frame that exists and can never be detected.
Lorentz went further and worked out the complete set of coordinate transformations that leave Maxwell's equations unchanged, including a modified time coordinate he called "local time" and regarded as a mathematical convenience with no physical meaning. He had the Lorentz transformation of Chapter 6.2 in his hands by 1904 and did not believe it described reality.
Henri Poincaré came closer still, writing in 1904 that the laws of physics should be the same for all observers and that no experiment could detect absolute motion. He called it "the principle of relativity" and named the transformations after Lorentz.
All the pieces were on the table by 1904. What was missing was somebody willing to say that time itself is what changes.
The other inconsistency: the one Einstein actually cared about
Michelson–Morley is the experiment every account leads with, and Einstein's 1905 paper does not mention it. What his paper opens with is something quite different and more revealing — the asymmetry noted at the end of Chapter 4.6.
Take a magnet and a coil of wire. Move the magnet towards the coil and a current flows. Move the coil towards the magnet at the same relative speed and the same current flows. Same measurement, same number, every time.
But the textbook explanation is different in the two cases:
Magnet moving: the changing magnetic field creates an electric field (Faraday's law), and that electric field pushes the charges in the wire.
Coil moving: there is no electric field anywhere; instead the charges in the wire are moving through a magnetic field and feel the q\vec{v}\times\vec{B} force.
Two entirely different mechanisms, two different intermediate quantities — an electric field exists in one story and not in the other — and an identical result. Einstein's first sentence says this asymmetry "does not appear to be inherent in the phenomena".
His reading: if the two descriptions are equally valid and only the relative motion is observable, then the split between "electric field" and "magnetic field" is not absolute. It depends on who is looking. And if that is true, the frame-dependence goes much deeper than anyone had assumed.
Where this leaves us
The situation in 1904, laid out plainly:
- Mechanics obeys the principle of relativity and the Galilean transformation.
- Electromagnetism obeys the principle of relativity experimentally, and the Lorentz transformation mathematically.
- The two transformations disagree. They cannot both describe how coordinates work.
- The ether, invented to fix this, has never been detected and appears to be undetectable in principle.
Something had to give, and the options were:
Keep Galileo, fix Maxwell. Nobody could construct a Galilean-invariant electromagnetism that reproduced the experiments. Every attempt failed.
Keep Maxwell, abandon relativity. This is the ether position, and it requires a preferred frame that no experiment can find. Physics does not usually tolerate that for long.
Keep both, abandon the Galilean transformation. Which means abandoning t' = t. Which means abandoning the idea that time is the same for everyone.
That third option sounds like the most extreme of the three and it is the one that turned out to be right. Its cost is the notion of absolute simultaneity, and its payment is that everything else — Maxwell, mechanics, the principle of relativity, and every experiment ever done — falls into place at once.
What the next chapter fixes
Einstein started from two postulates and nothing else: the laws of physics are the same in every inertial frame, and the speed of light in vacuum is the same for every observer regardless of how the source or the observer is moving. The second is not a consequence of the first; it is a separate and far stranger assumption, and it flatly contradicts velocity addition. Chapter 6.2 takes those two statements and derives the Lorentz transformation from them with no other input — no ether, no contraction hypothesis, no mechanical model — and then recovers the Galilean transformation as what the Lorentz transformation becomes when everything is moving slowly compared with light.