---
title: Minkowski Spacetime and the Interval
module: Spacetime and the Lorentz Group
moduleNumber: 2
lessonNumber: 1
order: 201
summary: >
  The Lorentz transformation of the foundations module is repackaged as the
  geometry of a four-dimensional space whose invariant is not a distance but the
  spacetime interval. Events, worldlines, and the metric signature define a
  causal structure that every observer shares. Proper time is the length of a
  timelike worldline, and the twin paradox becomes the statement that a straight
  worldline accumulates the most proper time.
topics: [Spacetime and the Lorentz Group]
draft: false
sources:
  - book: Taylor & Wheeler
    ref: "Spacetime Physics, Ch. 1 — The Interval; Ch. 5 — Trekking Through Spacetime"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §1 — Special Relativity and Flat Spacetime"
---

The [Lorentz transformation](/relativity/foundations/lorentz-transformation-spacetime)
mixes space and time coordinates so that the speed $c$ is the same in every
inertial frame. Read algebraically, it is a rule for converting one observer's
numbers into another's. Read geometrically, it is a symmetry of a single
four-dimensional object — spacetime — that leaves one quantity fixed. That
quantity is the interval, and building the rest of the theory on it turns
coordinate bookkeeping into geometry. This lesson sets up the arena; the tensor
language that follows in the next lessons is the calculus that runs on it.

## Events, coordinates, and worldlines

The primitive object is the **event**: a point in space at an instant of time, a
single "here-now" with no extent and no duration. An inertial observer labels
each event with four numbers, one time and three space,

$$
x^\mu = (x^0, x^1, x^2, x^3) = (ct, x, y, z), \qquad \mu = 0,1,2,3.
$$

The time coordinate is carried as $x^0 = ct$ so that all four components have
the dimension of length. The index sits upstairs for a reason that the next
lesson makes precise; for now it is a label running over the four coordinates.
The collection of all events is **Minkowski spacetime**, written $\mathbb{R}^{1,3}$
to record that one coordinate plays a different role from the other three.

> **Definition (Worldline).** The continuous succession of events occupied by a
> particle is its worldline, a curve $x^\mu(\lambda)$ through spacetime
> parametrized by any monotonic parameter $\lambda$. A particle at rest traces a
> line parallel to the time axis; a particle in uniform motion traces a straight
> line tilted toward the light direction; an accelerating particle traces a
> curved worldline.

A worldline is the complete history of a particle, not a trajectory swept out in
time. Nothing "moves along" it — the whole curve exists at once, and different
observers slice it into space-at-an-instant differently. A light pulse from the
origin traces the surface $c^2t^2 = x^2 + y^2 + z^2$, which in a diagram with one
space axis suppressed is a pair of lines at $45^\circ$, and in two space
dimensions is a cone.

## The invariant interval

Two nearby events separated by coordinate differences $\d x^\mu = (c\,\d t, \d x,
\d y, \d z)$ have a **spacetime interval** built from a specific quadratic
combination. Adopting the signature convention used throughout the general-theory
modules,

$$
\d s^2 = -c^2\,\d t^2 + \d x^2 + \d y^2 + \d z^2.
$$

> **Convention (Signature and units).** From here on the metric signature is
> $(-,+,+,+)$: the time term enters $\d s^2$ with a minus sign, so timelike
> separations have $\d s^2 < 0$ and spacelike separations have $\d s^2 > 0$. This
> is the opposite sign convention from the foundations module, which wrote
> $(\Delta s)^2 = (c\,\Delta t)^2 - (\Delta x)^2$; the physics is identical and
> only the overall sign of the labeling differs. In dynamical and geometric
> contexts later modules also use geometrized units $c = G = 1$, restored
> explicitly wherever a number is computed.

The content of the Lorentz transformation is that $\d s^2$ takes the same value
in every inertial frame. Substituting the boost $x' = \gamma(x - vt)$,
$t' = \gamma(t - vx/c^2)$ into $-c^2\,\d t'^2 + \d x'^2$ and expanding, the cross
terms cancel and the $\gamma^2(1 - \beta^2) = 1$ identity collapses the result to
$-c^2\,\d t^2 + \d x^2$. The transverse terms are untouched. Hence

$$
-c^2\,\d t'^2 + \d x'^2 + \d y'^2 + \d z'^2 = -c^2\,\d t^2 + \d x^2 + \d y^2 + \d z^2.
$$

Individual coordinate differences are frame-dependent — this is time dilation and
length contraction — but this one combination is not. The interval is to
spacetime what squared distance is to Euclidean space, with the single sign
difference in the time slot carrying all of relativity.

$$
% caption: Two frames assign different coordinates to the same pair of events A
% and B, yet the interval AB computed from either set of coordinates lands on the
% same invariant hyperbola.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % S axes
  \draw[->, black] (-0.3,0) -- (5.0,0) node[right] {x};
  \draw[->, black] (0,-0.3) -- (0,4.6) node[above] {ct};
  % light line
  \draw[black, dashed] (0,0) -- (4.3,4.3);
  % S' axes beta = 0.45
  \draw[->, acc, thick] (0,0) -- (1.85,4.1) node[above] {ct'};
  \draw[->, acc, thick] (0,0) -- (4.1,1.85) node[right] {x'};
  % invariant hyperbola (ct)^2 - x^2 = 4 : ct = sqrt(4 + x^2)
  \draw[black!70, thick] (0,2) .. controls (0.9,2.05) and (1.7,2.35) .. (2.3,2.75)
     .. controls (2.9,3.15) and (3.5,3.7) .. (3.9,4.1);
  % event A on ct axis, event B where hyperbola meets ct' axis
  \fill[black!75] (0,2) circle (1.7pt);
  \node[anchor=east] at (-0.05,2) {A};
  \fill (1.0,2.22) circle (1.7pt);
  \node[anchor=south west] at (1.05,2.25) {B};
  \node[black!70, anchor=west] at (2.6,3.15) {same interval};
\end{tikzpicture}
$$

## Causal structure

The sign of the interval between two events is invariant, so it sorts pairs of
events into three classes that all observers agree on. For two events separated
by $\Delta t$ in time and $\Delta r = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$
in space, with $\Delta s^2 = -c^2\Delta t^2 + \Delta r^2$:

- **Timelike** ($\Delta s^2 < 0$): $c\,\abs{\Delta t} > \Delta r$. A signal
  slower than light can travel between the two events, and every observer agrees
  on their time order. A massive particle can have both events on its worldline.
- **Null** or **lightlike** ($\Delta s^2 = 0$): $c\,\abs{\Delta t} = \Delta r$.
  Only a light signal connects them. The two events lie on each other's light
  cone.
- **Spacelike** ($\Delta s^2 > 0$): $c\,\abs{\Delta t} < \Delta r$. No signal at
  or below $c$ connects them. Their time order is frame-dependent, so neither can
  be a cause of the other.

The set of events null-separated from a given event $O$ forms $O$'s **light
cone**, and it partitions spacetime into the absolute future (timelike, later),
the absolute past (timelike, earlier), and the elsewhere (spacelike). Because the
partition is built from the invariant $\Delta s^2$, it is the same partition for
every inertial observer.

$$
% caption: The light cone of event O partitions spacetime. Timelike-separated
% events lie inside the cone with a frame-independent time order; spacelike events
% lie outside in the elsewhere, where the time order depends on the observer.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[acc!10] (0,0) -- (2.5,2.5) -- (-2.5,2.5) -- cycle;
  \fill[acc!10] (0,0) -- (2.5,-2.5) -- (-2.5,-2.5) -- cycle;
  \draw[acc, thick] (-2.8,-2.8) -- (2.8,2.8);
  \draw[acc, thick] (-2.8,2.8) -- (2.8,-2.8);
  \draw[->, black] (-3.1,0) -- (3.1,0) node[right] {x};
  \draw[->, black] (0,-3.1) -- (0,3.3) node[above] {ct};
  \fill[black] (0,0) circle (2pt);
  \node[anchor=north east] at (-0.02,-0.02) {O};
  \node at (0,1.9) {future};
  \node at (0,-1.95) {past};
  \node[black] at (2.05,0.5) {elsewhere};
  \node[black] at (-2.05,0.5) {elsewhere};
  % a timelike and a spacelike sample event
  \fill[black!75] (0.5,1.7) circle (1.6pt);
  \node[black!75, anchor=west] at (0.6,1.55) {timelike};
  \fill[black] (2.0,0.9) circle (1.6pt);
  \node[black, anchor=south] at (2.0,1.0) {spacelike};
\end{tikzpicture}
$$

That the causal partition is invariant is the reason relativity never lets an
effect precede its cause even while it lets observers disagree about
simultaneity. If $B$ lies in the future light cone of $A$, no boost can carry $B$
outside it; the ordering "$A$ before $B$" is absolute exactly for the pairs that
could be cause and effect.

## Proper time as the length of a worldline

Along a timelike worldline $\d s^2 < 0$, and the natural positive quantity is the
**proper time**, the time read by a clock carried along the worldline. Define it
by

$$
c^2\,\d\tau^2 = -\d s^2 = c^2\,\d t^2 - \d x^2 - \d y^2 - \d z^2.
$$

Dividing through by $\d t^2$ and writing the coordinate speed as
$v^2 = (\d x^2 + \d y^2 + \d z^2)/\d t^2$,

$$
\d\tau = \d t\,\sqrt{1 - \frac{v^2}{c^2}} = \frac{\d t}{\gamma}.
$$

This reproduces time dilation, now read off the geometry: the proper time
elapsed on a worldline is its Minkowski arc length (up to the factor $c$), and
because $\gamma \ge 1$, a moving clock's proper time advances more slowly than
coordinate time. Integrating along a worldline from event $\mathcal{A}$ to event
$\mathcal{B}$,

$$
\tau_{\mathcal{A}\mathcal{B}} = \int_{\mathcal{A}}^{\mathcal{B}} \d\tau
= \int_{t_\mathcal{A}}^{t_\mathcal{B}} \sqrt{1 - \frac{v^2(t)}{c^2}}\; \d t.
$$

Proper time is a property of the worldline itself, independent of the coordinates
used to compute the integral, because $\d\tau$ is built from the invariant
$\d s^2$. Two worldlines connecting the same pair of events generally accumulate
different proper times.

$$
% caption: Proper time accumulates as the Minkowski arc length of a worldline.
% The curved worldline is sampled into short timelike segments; each contributes
% its own d-tau, and the total is the reading of a clock carried along the path.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (4.6,0) node[right] {x};
  \draw[->, black] (0,-0.3) -- (0,4.6) node[above] {ct};
  % curved worldline
  \draw[acc, very thick] (0.4,0.2) .. controls (0.5,1.3) and (1.4,1.8) ..
     (2.0,2.4) .. controls (2.6,3.0) and (2.7,3.9) .. (2.5,4.3);
  % tick marks of equal proper time along it
  \fill (0.44,0.7) circle (1.3pt);
  \fill (0.75,1.5) circle (1.3pt);
  \fill (1.35,2.05) circle (1.3pt);
  \fill (2.05,2.55) circle (1.3pt);
  \fill (2.62,3.25) circle (1.3pt);
  \fill (2.62,4.0) circle (1.3pt);
  \node[anchor=west] at (2.7,4.2) {worldline};
  \node[black, anchor=west] at (1.5,1.4) {equal-tau ticks};
  \draw[black, ->] (1.9,1.5) -- (1.35,1.95);
\end{tikzpicture}
$$

## The twin paradox as extremal proper time

The [twin paradox](/relativity/foundations/time-dilation-length-contraction) has
a one-line geometric statement. Take two timelike-separated events, departure
$\mathcal{A}$ and reunion $\mathcal{B}$. One twin stays inertial: a straight
worldline from $\mathcal{A}$ to $\mathcal{B}$. The other travels out and back: a
bent worldline through the same endpoints. The proper times differ because the
worldlines differ, and the inertial one is longer.

> **Theorem (Longest proper time on a straight worldline).** Among all timelike
> worldlines connecting two timelike-separated events, the straight (inertial)
> worldline accumulates the greatest proper time. Any worldline with a kink or
> curvature reads less.

> **Proof.** Work in the inertial frame in which the straight worldline is at
> rest, so its endpoints are at the same spatial point and its proper time is the
> coordinate time difference $\Delta t$. Any other worldline has $v \ne 0$ over
> part of its extent, so its proper time is
> $$
> \tau = \int \sqrt{1 - v^2/c^2}\; \d t \le \int \d t = \Delta t,
> $$
> with equality only if $v = 0$ throughout, i.e. only for the straight worldline.
> Hence every bent worldline has $\tau < \Delta t$. The inequality runs opposite
> to the Euclidean triangle inequality: in Minkowski geometry the straight path
> between timelike-separated events is the longest, not the shortest.

The traveling twin, whose worldline bends at the turnaround, returns younger. No
paradox survives once the two histories are recognized as different curves
between the same endpoints; asking "which twin really moved" is asking which
curve is straight, and only one of them is. The asymmetry is the turnaround,
where the traveling twin's frame is non-inertial.

$$
% caption: Twin paradox as a length comparison. The straight worldline SR of the
% stay-at-home twin and the bent worldline SM-MR of the traveler share the events
% S and R; the straight one carries more proper time, so its twin ages more.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.3,0) -- (3.6,0) node[right] {x};
  \draw[->, black] (0,-0.3) -- (0,5.0) node[above] {ct};
  % stay-at-home: straight up
  \draw[acc, very thick] (0.4,0.3) -- (0.4,4.5);
  \node[anchor=east] at (0.35,2.4) {stay};
  % traveler: out to M and back
  \draw[black, very thick, dashed] (0.4,0.3) -- (2.8,2.4) -- (0.4,4.5);
  \node[black, anchor=west] at (2.85,2.4) {turnaround};
  \fill[black] (0.4,0.3) circle (2pt);
  \node[anchor=north east] at (0.42,0.28) {S};
  \fill[black] (0.4,4.5) circle (2pt);
  \node[anchor=south east] at (0.42,4.55) {R};
  \fill[black] (2.8,2.4) circle (1.8pt);
  \node[black, anchor=north west] at (2.85,2.2) {M};
\end{tikzpicture}
$$

The straight worldline is the flat-spacetime instance of a **geodesic**, the
extremal-proper-time curve between two events. In flat spacetime the geodesic is
a straight line and it maximizes proper time; the curved-spacetime modules
generalize the extremal principle, and free fall in a gravitational field turns
out to be exactly geodesic motion — the longest-proper-time worldline through a
curved geometry.

## The metric as a measuring convention

The interval assigns a number to every pair of nearby events, and that
assignment is the **metric** of Minkowski spacetime. In the coordinates
$x^\mu = (ct, x, y, z)$ it is the diagonal array

$$
\eta_{\mu\nu} = \operatorname{diag}(-1, +1, +1, +1),
\qquad
\d s^2 = \eta_{\mu\nu}\,\d x^\mu\,\d x^\nu,
$$

with the summation over repeated indices anticipated. The metric is what
converts coordinate differences into the physical, frame-independent interval;
it is simultaneously the clock (for timelike directions, via $\d\tau$) and the
ruler (for spacelike directions, via proper length). On the light cone it reads
zero — light-cone directions have no proper time and no proper length, which is
why one cannot ride alongside a light beam and watch it stand still.

$$
% caption: The metric reads a clock along timelike directions and a ruler along
% spacelike directions; on the null cone it returns zero, so light-cone directions
% carry neither proper time nor proper length.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, dashed] (-2.6,-2.6) -- (2.6,2.6);
  \draw[black, dashed] (-2.6,2.6) -- (2.6,-2.6);
  \draw[->, black] (-3.0,0) -- (3.0,0) node[right] {x};
  \draw[->, black] (0,-2.9) -- (0,3.1) node[above] {ct};
  \fill[black] (0,0) circle (1.8pt);
  % timelike arrow
  \draw[acc, very thick, ->] (0,0) -- (0.5,2.4);
  \node[anchor=west] at (0.45,2.0) {clock};
  % spacelike arrow
  \draw[black, very thick, ->] (0,0) -- (2.4,0.5);
  \node[black, anchor=south] at (2.1,0.55) {ruler};
  % null arrow
  \draw[black, very thick, ->] (0,0) -- (1.9,1.9);
  \node[black, anchor=west] at (1.55,1.35) {null: zero};
\end{tikzpicture}
$$

The array $\eta_{\mu\nu}$ is the object that the next lesson promotes to a
tensor, giving the rules for raising and lowering indices, forming invariant
scalar products, and defining the four-velocity and four-acceleration that carry
relativistic dynamics. Everything downstream is bookkeeping organized so that
the invariance of $\d s^2$ is automatic.

[^tw-interval]: **Taylor & Wheeler**, _Spacetime Physics_, Ch. 1 — the interval as the invariant of spacetime, the light cone and causal classification, and proper time as the wristwatch time read along a worldline; Ch. 5 develops the worldline picture and the extremal character of the straight timelike path.

[^carroll-flat]: **Carroll**, _Lecture Notes on General Relativity_, §1 — Minkowski spacetime, the metric $\eta_{\mu\nu} = \operatorname{diag}(-1,+1,+1,+1)$, the invariant interval, and the causal structure of the light cone in the mostly-plus signature.