Spacetime Interval

Equivalent forms

ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2

s^2 = \eta_{\mu\nu} \Delta x^\mu \Delta x^\nu

A single sign flip in the Pythagorean theorem turns Euclidean space into Minkowski spacetime — the geometric heart of relativity.

Symbol	Name	SI unit	Dimension	Range
$s2$	Invariant Interval Squaredoutput Lorentz-invariant squared interval between two events	m^2	L^2	-1000000000000000000 – 1000000000000000000
$Dt$	Time Separation Coordinate-time difference between two events	s	T	0 – 10
$Dx$	Spatial Separation Spatial distance between two events	m	L	0 – 1000000000
$c$	Speed of Light Speed of light in vacuum	m/s	LT^-1	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$s2$	m^2	cm^2	ft^2
$Dt$	s	s	s
$Dx$	m	cm	ft
$c$	m/s	cm/s	ft/s

Where it holds

Valid in flat (Minkowski) spacetime; in curved spacetime replace

\eta

with

g_\mu \nu

.

Dimensional analysis

s^2 \to [L^2]

[LT^-1]^2 [T^2] + [L^2] = [L^2]

Both terms have units of length squared.

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Discovery

Hermann Minkowski · 1908

Minkowski reformulated Einstein's 1905 relativity geometrically in his Cologne lecture: 'Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows.'

Try this

Two flashes happen 3 light-seconds apart and 5 seconds apart in time — what does every observer agree on?

The invariant interval is s^2 = -c^2 Δt^2 + Δx^2 = -16 c^2 (light-seconds)^2 — timelike and negative; all inertial observers compute the same value.

Research status: stable

Real-world applications

Foundations of GR (line element of any metric)
Light cone causal structure in cosmology
GPS path integration
Gravitational wave detector analysis

Common misconceptions

s^2 can be negative — that's allowed and physical
Sign conventions differ: some texts use (+,-,-,-) instead of (-,+,+,+)
Proper time tau satisfies c^2 dtau^ $2 = -ds^2$ (timelike)

Experimental verification

Tests of Lorentz invariance with atomic clocks (Hughes-Drever experiments), high-energy cosmic-ray observations, and Michelson-Morley-type tests bound deviations to < 10^-17.

Derivation

Postulate (i) invariance of c and (ii) homogeneity/isotropy.

For a light pulse

\Delta x = c \Delta t

so -

c^2 \Delta t^2 + \Delta x^2 = 0

in every frame.

Linearity then forces this quadratic form to be the unique invariant under Lorentz boosts.

The interval s^2 is preserved exactly as |r|^2 is preserved under rotations.

Limiting cases

s^2 < 0⟶ TimelikeEvents causally connected; some observer sees them at the same location.

s^2 = 0

⟶ Lightlike (null)Events connected by a light ray; lie on each other's light cone.

s^2 > 0⟶ SpacelikeEvents causally disconnected; some observer sees them as simultaneous.

What if…

What if I observed the events from a different frame?

$\Delta t$ $and \Delta x$ individually change, but s^2 stays exactly the same — that's the whole point.

What if s^2 > 0 between two events?

No signal (including light) can connect them — they are causally disconnected.

What if we work in curved spacetime?

Replace $\eta _\mu \nu$ with $g_\mu \nu (x)$ ; the interval becomes $ds^2 = g_\mu \nu dx^\mu dx^\nu$ .

1

Two flashes

Given ·

Dt:: 5
Dx:: 900000000

Find · s2

Steps

$c^2 \Delta t^2 = (3e_{8})^2 * 25 = 2.25e18$
$\Delta x^2 = (9e_{8})^2 = 8.1e17$
$s^2 = -2.25e18 + 8.1e17 \approx -1.44e18\,\mathrm{m}^2$

Result ·

s^2 = -(3e_{8})^2 * 25 + (9e_{8})^2 = -2.25e18 + 8.1e17 \approx -1.44e18\,\mathrm{m}^2

— timelike.

2

Lightlike check

Given ·

Dt:: 1
Dx:: 299792458

Find · s2

Steps

$c^2 * 1 = 8.99e16$
$\Delta x^2 = (299792458)^2 = 8.99e16$
$s^2 = -8.99e16 + 8.99e16 = 0$

Result ·

s^2 = -c^2 + \Delta x^2 = 0

exactly — events on a light cone.

Lorentz Transformation→Lorentz Factor→Proper Time→Time Dilation→