Spacetime and the Invariant Interval

The two postulates force a single geometric object on us: a four-dimensional spacetime with an indefinite metric whose interval is shared by all inertial observers. This page builds that arena — events, worldlines, the interval, proper time, light cones, and causal structure — which the Lorentz transformations then act on.

Natural units $c=1$ are used freely; the metric is mostly-minus, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Events and coordinates

An event is a point in spacetime, labeled in an inertial frame by

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

A particle traces a worldline $x^{\mu }(\lambda )$. The history of a clock, observer, or signal is a worldline; events are the atoms of every relativistic statement.

The invariant interval

The central invariant between two events is the spacetime interval

$$\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2,$$

with differential form

$$d{s}^2=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2={\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }.$$

That $\Delta s^2$ is the same in every inertial frame is the content of the second postulate: a light pulse satisfies $\Delta s^2=0$ in all frames, and requiring this quadratic form be preserved (up to scale) for all events forces full invariance. The transformations preserving ${\eta }_{\mu \nu }$ are exactly the Lorentz transformations.

Why the interval is invariant

The full invariance follows from the postulates in three steps.

Step 1 — Null cones agree. The constancy of $c$ means a light pulse has $\Delta s^2=0$ in every inertial frame. So whenever $\Delta s^2=0$, then $\Delta s^{\prime 2}=0$ too: the vanishing of the quadratic form is frame-independent.

Step 2 — Two quadratic forms with the same zeros are proportional. With linear transformations (homogeneity of spacetime), $\Delta s^{\prime 2}$ is a quadratic form in the same coordinates. Two quadratic forms that vanish on exactly the same cone must be proportional, so $$\Delta s^{\prime 2}=K(\mathbf{v})\Delta s^2,$$ where the factor $K$ can depend only on the relative speed (isotropy forbids dependence on direction).

Step 3 — Reciprocity forces $K=1$. Apply the boost from $S$ to $S^{\prime }$ and back: $\Delta s^2=K(\mathbf{v})K(-\mathbf{v})\Delta s^2$. Isotropy gives $K(\mathbf{v})=K(-\mathbf{v})$, so $K^2=1$ and $K=+1$ (continuity rules out $-1$). Hence $\Delta s^{\prime 2}=\Delta s^2$: the interval is invariant for all events, not just null ones. The transformations preserving ${\eta }_{\mu \nu }$ are exactly the Lorentz transformations.

This is one of two equivalent directions: invariance characterizes the boost, or the boost implies invariance. The reverse — assume the boost, verify $\Delta s^2$ unchanged in one line — is shown in the boost derivation. Neither is more fundamental; ${\Lambda }^{\mathsf{T}}\eta \Lambda =\eta$ is the single statement, read both ways.

Causal classification

The sign of $\Delta s^2$ is frame-independent and fixes causal relations:

Only timelike- or null-separated events can be causally ordered; for spacelike pairs no signal $\le c$ connects them, so different frames disagree on which came first — without paradox, since neither can influence the other.

The light cone

The set of null rays through an event $p$ forms its light cone, splitting spacetime into:

• the future (timelike, $t>0$): events $p$ can influence;
• the past (timelike, $t<0$): events that can influence $p$;
• the elsewhere (spacelike): causally disconnected.

graph TD
  A[future cone] --- P((event p))
  P --- B[past cone]
  P --- E[elsewhere / spacelike]

The light cone is frame-independent and is the geometric statement of causality.

Proper time and proper length

Along a timelike worldline define proper time by

$$c^{2}d{\tau }^2=d{s}^2\Rightarrow d\tau =dt\sqrt{1-v^2/c^2}=dt/\gamma ,$$

the time read by a clock carried along the line — a Lorentz scalar. The elapsed proper time between two events is $\tau =\int d\tau$, maximized by the inertial (straight) worldline; this is the geometric root of the twin paradox. The spacelike analog, proper length, is the length in the object's rest frame.

Minkowski diagrams

A 2D spacetime ($ct$ vs $x$) plot is a Minkowski diagram: light cones are $45^{\circ }$ lines; a boosted frame's axes tilt symmetrically toward the light line, encoding relativity of simultaneity. These diagrams resolve paradoxes pictorially (see Paradoxes).

Next

With the arena fixed, Lorentz transformations give the explicit frame change, and Kinematics reads off dilation, contraction, and Doppler. Tensor machinery on this space is in four-vectors and tensors.