The Postulates of Special Relativity

Classical mechanics already contains a principle of relativity. Newton's second law $\vec{F}=m\mathbf{d}\vec{v}/\mathbf{d}t$ holds in any inertial frame, a frame in which a body free of forces moves at constant velocity, and it takes the same form in every frame moving at constant velocity relative to one of them. No mechanical experiment picks out a preferred inertial frame; only relative velocities of frames are measurable. Electromagnetism broke that symmetry. Maxwell's equations fix a single speed $c=1/\sqrt{{\mu }_0{\epsilon }_0}\approx 3.00\times 10^{8}m/s$ for electromagnetic waves, and they do not keep their form under the coordinate change that leaves Newton's laws invariant. Either electromagnetism has a preferred frame or the coordinate change is wrong. Special relativity is the second answer.¹

Galilean relativity

Two inertial frames $S$ and $S^{\prime }$ with parallel axes, $S^{\prime }$ moving at speed $v$ along the shared $x$ axis and coinciding with $S$ at $t=0$, are related by the Galilean transformation:

Differentiating gives the classical velocity addition $u_x^{\prime }=u_x-v$, and a second derivative gives $a^{\prime }=a$, since $v$ is constant. Acceleration is invariant, so $\vec{F}=m\vec{a}$ reads identically in both frames.

The crucial feature of the Galilean transformation is the last equation, $t^{\prime }=t$: time is absolute, shared by all frames, and the same clock reading labels an event everywhere. Two events simultaneous in $S$ are simultaneous in $S^{\prime }$, and a rod's length is the same in every frame. Both statements fail once the transformation is corrected.

Electromagnetism cannot tolerate the velocity rule $u_x^{\prime }=u_x-v$. If light moves at $c$ along $x$ in $S$, the Galilean rule predicts $c-v$ in $S^{\prime }$. Maxwell's equations, applied in $S^{\prime }$, still predict $c$. The two cannot both be right.

The ether and Maxwell's speed

Every wave known to nineteenth-century physics propagated in a medium, and its speed was fixed relative to that medium: sound in air, ripples on water, vibrations along a string. Light was assumed to be no exception. The medium, called the ether, was supposed to fill all space, including the interior of matter, and to be the frame in which Maxwell's $c$ is the wave speed. In any other frame the measured speed of light would follow the Galilean rule, ranging between $c-v$ and $c+v$ depending on direction.

Earth orbits the Sun at about $30km/s$, so relative to the ether its speed is at least that. The fractional change in a light-speed measurement caused by this motion is of order $v^2/c^2\approx 10^{-8}$, so detecting Earth's motion through the ether requires an accuracy near one part in $10^8$. Direct time-of-flight measurements of light fell far short. A difference measurement using interference could reach it.

The Michelson-Morley experiment

Michelson's interferometer splits a beam in two, sends the halves down perpendicular arms of equal length $L$, reflects each off a mirror, and recombines them. The recombined beams interfere, producing a pattern of bright and dark fringes. A difference in travel time between the two arms shifts the pattern; rotating the apparatus by $90^{\circ }$ swaps the roles of the arms and doubles the shift.

The travel times follow from the boat-crossing-a-river analogy. Let the apparatus move at speed $v$ through the ether along arm 2. For the beam along the direction of motion (arm 2, out and back),

For the transverse beam (arm 1) the light must aim slightly upstream to return to the splitter, giving an effective speed $\sqrt{c^2-v^2}$:

The difference is second order in $v/c$ but nonzero,

and rotating the apparatus $90^{\circ }$ changes the sign of $\Delta t$, so the observable phase change corresponds to $2\Delta t$. With arm length $L\approx 11\mathrm{m}$, wavelength $\lambda =590\mathrm{nm}$, and $v=30km/s$, the expected fringe shift on rotation is

about forty times the minimum shift the instrument could resolve.

The 1887 experiment saw a shift of at most $0.01$ fringe, consistent with zero. Repetitions over the following decades lowered the bound on Earth's speed through the ether to well under $1km/s$; laser versions reach $15m/s$. No motion through the ether has ever been detected.

Einstein's postulates

Einstein reached the same conclusion from theory rather than from the null result, which he barely referenced. He raised the classical principle of relativity to cover all of physics and added the invariance of $c$ as a separate postulate.²

The second postulate makes the Michelson-Morley null result automatic: if $c$ is the same in every direction in every inertial frame, there is no ether wind to detect, and rotating the interferometer changes nothing. Recent bounds from gamma-ray bursts confirm the speed of light is independent of source motion to one part in $10^{20}$.

The two postulates together force a stronger statement than either alone. Take a source $S$ and two observers, $R_1$ at rest relative to $S$ and $R_2$ moving toward $S$ at speed $v$. Observer $R_1$ measures the light speed $c$. By the first postulate, the configuration in which $R_2$ moves toward a stationary source is physically identical to one in which $R_2$ is at rest and the source moves toward $R_2$. By the second postulate, a moving source still emits light at $c$. So $R_2$ also measures $c$, not $c+v$. Every inertial observer measures the same speed for the same light.

Events, observers, and clock synchronization

Making these ideas precise requires care with the words event and observer.

• Event. Something that happens at a definite point in space and a definite instant in time: a lightning strike, a collision, a flash. An event is not owned by any frame; different observers assign it different coordinates.
• Observer. Not a single person with a single clock, but a lattice of measuring rods filling the frame with a synchronized clock at every intersection. Each event is recorded by the clock nearest to it, so an observer never has to wait for light to arrive before assigning a time.

Synchronization within one frame uses the second postulate. Start a reference clock at $t=0$ and let it emit a spherical flash. A clock at distance $d$ is preset to read $d/c$ and is started when the flash arrives. Every clock in the frame is then synchronized to the reference. This procedure works within a single frame; it does not synchronize clocks across frames in relative motion, and that failure is the heart of what follows.

Relativity of simultaneity

Einstein's train illustrates it. A train moves at speed $v$ past a platform. Lightning strikes the front and back of the train, scorching both train and platform, and the strikes are simultaneous in the platform frame $S$: an observer $C$ at the platform midpoint receives both flashes at once. An observer $C^{\prime }$ at the train's midpoint is moving toward the front strike and away from the back one. The flash from the front reaches $C^{\prime }$ first. Since $C^{\prime }$ sits exactly halfway between the two scorch marks on the train, $C^{\prime }$ must conclude the front strike happened before the back strike. Both observers are right within their own frames; simultaneity is not absolute.

The same argument shows that clocks synchronized in one frame are not synchronized in another. In the platform frame the train's clocks are set so that, moving from rear to front, the leading clock reads behind the trailing one. This offset is not an error; it is what the platform frame measures for a correctly synchronized set of train clocks. Quantitatively the leading clock lags by $v\Delta x/c^2$ over a separation $\Delta x$, a relation the next lesson derives from the Lorentz transformation.

Consequences of the postulates

The Galilean assumptions that survived from Newton do not survive the second postulate. The table records which classical certainties become frame-dependent.

Each entry in the right column is a consequence, not a new assumption. The two postulates are the whole input. The Lorentz transformation converts them into the exact relations connecting coordinates in two frames, and from those relations time dilation and length contraction follow directly.