---
title: The Postulates of Special Relativity
module: Foundations of Relativity
moduleNumber: 1
lessonNumber: 1
order: 101
summary: >
  Newton's laws are the same in every inertial frame, but Maxwell's are not:
  the equations of electromagnetism single out one speed, c, and the
  nineteenth century read that as the speed of light relative to a medium, the
  ether. The Michelson-Morley experiment looked for Earth's motion through
  that medium and found nothing. Einstein's two postulates replace the ether,
  and their first consequence is that simultaneity is frame-dependent.
topics: [Foundations of Relativity]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 1 — Relativity I; §1-1 The Experimental Basis of Relativity"
  - book: Tipler & Llewellyn
    ref: "§1-2 Einstein's Postulates; Relativity of Simultaneity"
---

Classical mechanics already contains a principle of relativity. Newton's second
law $\vec{F} = m\,\d\vec{v}/\d t$ 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 $c = 1/\sqrt{\mu_0 \varepsilon_0}
\approx 3.00 \times 10^8\ \mathrm{m/s}$ 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.[^tl-exp]

## Galilean relativity

Two inertial frames $S$ and $S'$ with parallel axes, $S'$ moving at speed $v$
along the shared $x$ axis and coinciding with $S$ at $t = 0$, are related by the
**Galilean transformation**:

$$
x' = x - vt, \qquad y' = y, \qquad z' = z, \qquad t' = t.
$$

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

> **Definition (Inertial frame).** A reference frame in which the law of
> inertia holds: a body subject to no net force moves with constant velocity.
> Any frame moving at constant velocity relative to an inertial frame is itself
> inertial, and Newton's laws are invariant across all of them.

The crucial feature of the Galilean transformation is the last equation,
$t' = t$: time is absolute, shared by all frames, and the same clock reading
labels an event everywhere. Two events simultaneous in $S$ are simultaneous in
$S'$, 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 $u'_x = u_x - v$.
If light moves at $c$ along $x$ in $S$, the Galilean rule predicts
$c - v$ in $S'$. Maxwell's equations, applied in $S'$, 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 $30\ \mathrm{km/s}$, 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.

$$
% caption: The Michelson interferometer. A half-silvered plate at A splits the
% beam; the two halves reflect from mirrors M1 and M2 and recombine at the
% telescope, where their phase difference sets the fringe pattern.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % source
  \node[draw, minimum width=13mm, minimum height=8mm, align=center] (src) at (0,0) {source};
  % beam splitter
  \node[draw, acc, rotate=45, minimum width=10mm, minimum height=1mm, fill=acc!12] (bs) at (3,0) {};
  \node[anchor=north] at (3,-0.55) {A};
  % mirror 2 (right)
  \node[draw, minimum width=1.5mm, minimum height=12mm, fill=black!12] (m2) at (6.4,0) {};
  \node[anchor=west] at (6.55,0) {M2};
  % mirror 1 (top)
  \node[draw, minimum width=12mm, minimum height=1.5mm, fill=black!12] (m1) at (3,3.2) {};
  \node[anchor=south] at (3,3.35) {M1};
  % telescope (bottom)
  \node[draw, minimum width=13mm, minimum height=8mm, align=center] (tel) at (3,-3.2) {telescope};
  % beams
  \draw[->, acc, thick] (src.east) -- (bs);
  \draw[->, acc, thick] (bs) -- (m2) node[midway, above] {arm 2, length L};
  \draw[->, acc, thick] (bs) -- (m1) node[midway, right] {arm 1, length L};
  \draw[->, acc, thick] (bs) -- (tel);
  % motion arrow
  \draw[->, black, thick] (5.6,-1.9) -- (6.9,-1.9) node[midway, below] {v (ether wind)};
\end{tikzpicture}
$$

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),

$$
t_2 = \frac{L}{c-v} + \frac{L}{c+v} = \frac{2L}{c}\,\frac{1}{1 - v^2/c^2}
\approx \frac{2L}{c}\parens{1 + \frac{v^2}{c^2}}.
$$

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

$$
t_1 = \frac{2L}{\sqrt{c^2 - v^2}} \approx \frac{2L}{c}\parens{1 + \frac{v^2}{2c^2}}.
$$

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

$$
\Delta t = t_2 - t_1 \approx \frac{L v^2}{c^3},
$$

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 $v = 30\
\mathrm{km/s}$, the expected fringe shift on rotation is

$$
\Delta N = \frac{2 L v^2}{\lambda c^2} \approx 0.4\ \text{fringe},
$$

about forty times the minimum shift the instrument could resolve.

> **Example (What the interferometer should have seen).** Substituting the
> numbers: $v^2/c^2 = (10^{-4})^2 = 10^{-8}$, so $2Lv^2/c^2 = 2(11)(10^{-8}) =
> 2.2 \times 10^{-7}\ \mathrm{m}$; dividing by $\lambda = 5.9 \times 10^{-7}\
> \mathrm{m}$ gives $\Delta N \approx 0.37$. The apparatus resolved $0.01$
> fringe. The predicted effect was well inside its reach.

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 $1\ \mathrm{km/s}$; laser versions reach
$15\ \mathrm{m/s}$. No motion through the ether has ever been detected.

$$
% caption: Predicted versus observed fringe shift on a ninety-degree rotation.
% The ether model predicts about 0.4 fringe; the measurement is consistent with
% zero, well below the instrument's 0.01-fringe resolution.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axis
  \draw[->, black] (0,0) -- (0,3.4) node[above] {fringe shift};
  \draw[black] (0,0) -- (5.6,0);
  % predicted bar
  \fill[acc!14] (0.7,0) rectangle (1.9,3.0);
  \draw[acc] (0.7,0) rectangle (1.9,3.0);
  \node[anchor=north, align=center] at (1.3,-0.15) {ether\\prediction};
  \node[acc, anchor=south] at (1.3,3.0) {0.4};
  % observed bar
  \fill[black] (3.7,0) rectangle (4.9,0.08);
  \draw[black] (3.7,0) rectangle (4.9,0.08);
  \node[anchor=north, align=center] at (4.3,-0.15) {observed};
  \node[black, anchor=south] at (4.3,0.12) {0.01};
  % resolution line
  \draw[black, dashed] (0.3,0.6) -- (5.3,0.6) node[right, black] {resolution};
\end{tikzpicture}
$$

## 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.[^tl-post]

> **Postulate (Principle of relativity).** The laws of physics are the same in
> all inertial reference frames. No experiment of any kind distinguishes one
> inertial frame from another; absolute motion cannot be detected.

> **Postulate (Constancy of the speed of light).** The speed of light in vacuum
> has the value $c$, the same in all inertial frames, independent of the motion
> of the source.

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.

$$
% caption: A flash emitted at the coincident origins expands as a sphere of
% radius ct in frame S and, by the second postulate, as a sphere of radius ct
% centered on the S-prime origin in frame S-prime. Each frame sees itself at
% the center.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % frame S
  \draw[acc, thick] (0,0) circle (1.7);
  \fill[acc] (0,0) circle (2pt);
  \node[acc, anchor=north east] at (0,0) {O};
  \draw[acc, ->] (0,0) -- (30:1.7) node[midway, above, sloped] {ct};
  \node[anchor=south] at (0,1.85) {frame S: sphere about O};
  % frame S'
  \begin{scope}[xshift=5.4cm]
    \draw[black, thick] (0,0) circle (1.7);
    \fill[black] (0,0) circle (2pt);
    \node[black, anchor=north east] at (0,0) {O'};
    \draw[black, ->] (0,0) -- (30:1.7) node[midway, above, sloped] {ct'};
    \node[anchor=south] at (0,1.85) {frame S': sphere about O'};
  \end{scope}
\end{tikzpicture}
$$

## 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

> **Definition (Relativity of simultaneity).** Two spatially separated events
> that are simultaneous in one inertial frame are, in general, not simultaneous
> in another frame moving relative to the first. Two events are defined to be
> simultaneous in a frame if light signals from them reach the midpoint,
> measured by a local clock there, at the same instant.

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
$C'$ at the train's midpoint is moving toward the front strike and away from the
back one. The flash from the front reaches $C'$ first. Since $C'$ sits exactly
halfway between the two scorch marks on the train, $C'$ must conclude the front
strike happened before the back strike. Both observers are right within their
own frames; simultaneity is not absolute.

$$
% caption: Lightning strikes front (A) and rear (B) of a train moving right at
% speed v. In the platform frame the strikes are simultaneous, reaching the
% platform midpoint C together. The train midpoint C-prime moves toward A, so
% the front flash reaches it first and the two strikes are not simultaneous for
% the train.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- top panel: strikes simultaneous in the platform frame S ---
  \draw[black] (0,2.6) -- (8.3,2.6);
  \node[anchor=east] at (-0.05,2.6) {S};
  \draw[acc, thick] (1.2,2.6) rectangle (6.8,3.4);
  \node[acc] at (6.5,3.0) {A};
  \node[acc] at (1.5,3.0) {B};
  \node at (4,3.0) {C'};
  \node[black] at (4,2.35) {C};
  \fill[acc] (6.8,2.6) circle (2.4pt);
  \fill[acc] (1.2,2.6) circle (2.4pt);
  \draw[->, black] (7.2,3.75) -- (8.0,3.75) node[right] {v};
  \node[anchor=south] at (4,3.5) {strikes at once in S};
  % --- bottom panel: train shifted right, front pulse reaches C-prime sooner ---
  \draw[black] (0,0) -- (8.3,0);
  \node[anchor=east] at (-0.05,0) {S};
  \draw[acc, thick] (2.1,0) rectangle (7.7,0.8);
  \node[acc] at (7.4,0.4) {A};
  \node[acc] at (2.4,0.4) {B};
  \node at (4.9,0.4) {C'};
  % scorch marks fixed on the platform at the original strike positions
  \fill[acc] (6.8,0) circle (2.4pt);
  \fill[acc] (1.2,0) circle (2.4pt);
  % light pulses converging on the train midpoint C-prime at x = 4.9
  \draw[acc, dashed, ->] (1.2,0) -- (4.72,0.35);
  \draw[acc, dashed, ->] (6.8,0) -- (5.08,0.35);
  \node[anchor=north, align=center] at (4,-0.25) {front pulse reaches C' sooner};
\end{tikzpicture}
$$

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.

| Quantity | Galilean status | Relativistic status |
| --- | --- | --- |
| Speed of light | $c \pm v$, direction-dependent | $c$ in every inertial frame |
| Simultaneity of separated events | absolute | frame-dependent |
| Time interval between two events | absolute ($t' = t$) | frame-dependent |
| Length of a moving rod | absolute | frame-dependent |
| Synchronization of a clock set | frame-independent | frame-dependent |
| Acceleration | invariant | not invariant |

Each entry in the right column is a consequence, not a new assumption. The two
postulates are the whole input. The
[Lorentz transformation](/relativity/foundations/lorentz-transformation-spacetime)
converts them into the exact relations connecting coordinates in two frames,
and from those relations
[time dilation and length contraction](/relativity/foundations/time-dilation-length-contraction)
follow directly.

[^tl-exp]: **Tipler & Llewellyn**, _Modern Physics_, §1-1 — The Experimental Basis of Relativity: Galilean invariance of Newton's laws, the non-invariance of Maxwell's equations, the ether hypothesis, and the Michelson-Morley interferometer with its boat-race analogy and null result.
[^tl-post]: **Tipler & Llewellyn**, _Modern Physics_, §1-2 — Einstein's Postulates: the principle of relativity and the constancy of $c$, events and observers, clock synchronization, and the relativity of simultaneity via the lightning-and-train example.