Length Contraction

Equivalent forms

L = L_0 \sqrt{1 - v^2/c^2}

Space itself compresses along the direction of motion — a direct consequence of the invariance of c.

Symbol	Name	SI unit	Dimension	Range
$L$	Contracted Lengthoutput Length measured in observer frame	m	L	0 – 100
$L0$	Proper Length Length in the object's rest frame	m	L	0.1 – 100
$v$	Velocity Relative velocity of the object	m/s	LT^-1	0 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$L$	m	cm	ft
$L0$	m	cm	ft
$v$	m/s	cm/s	ft/s

Where it holds

Applies only to lengths measured along the direction of relative motion. Transverse dimensions are unchanged.

Dimensional analysis

L \to [L]

L_{0}/\gamma \to [L]/[1] = [L]

Both sides length.

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Discovery

George FitzGerald & Hendrik Lorentz · 1889

FitzGerald proposed it in 1889 to explain the Michelson–Morley null result; Lorentz gave it a formal transformation in 1892.

Try this

A 100 m rocket flies past you at 0.8c — how long does it look?

Compute L = L0/gamma with L0 = 100 m and v = 0.8c, giving gamma = 5/3 and L = 60 m.

Research status: stable

Real-world applications

Relativistic heavy-ion collisions (nuclei appear as pancakes)
Muon atmospheric depth calculations
Synchrotron radiation geometry

Common misconceptions

Contraction is not an optical illusion — it is a measured property
Only the dimension along motion is contracted; perpendicular dimensions are unchanged
The rod is not physically crushed; it is a geometric projection effect

Experimental verification

Directly inferred from muon flux measurements (the muon 'sees' the atmosphere contracted) and indirectly from relativistic collider kinematics.

Derivation

To measure a moving rod, the observer records the positions of its endpoints simultaneously.

Using the Lorentz transformation

x' = \gamma (x - vt)

, simultaneous measurements in the lab frame give

L = L_{0}/\gamma

.

Limiting cases

v \to 0

⟶

L \to L_{0}

No contraction at rest.

v \to c

⟶

L \to 0

An object approaching c appears infinitely flattened.

v = 0.866c

⟶

L = L_{0}/2

Half the proper length at

\gamma = 2

.

What if…

What if the rod is perpendicular to motion?

No contraction — only the parallel component is affected.

What if two observers pass a barn with a ladder?

The ladder paradox: both see the other as shorter; simultaneity of events resolves the contradiction.

What if the object is rotating?

Different parts have different velocities; the shape transforms into a complex contracted form (Terrell rotation).

1

Starship flyby

Given ·

L0:: 100
v:: 239833966

Find · L

Steps

$\beta = 0.8 \to \gamma = 1/\sqrt{1 - 0.64} = 1/0.6 = 5/3$
$L = L_{0}/\gamma = 100/(5/3) = 60\,\mathrm{m}$

Result ·

\gamma = 5/3 \to L = 100/1.667 = 60\,\mathrm{m}

.

2

Muon pathlength

Given ·

L0:: 10000
v:: 296794533

Find · L

Steps

$\gamma \approx 7.09$
$L = 10000/7.09 \approx 1410\,\mathrm{m}$

Result · The 10 km atmosphere appears as 1410 m in the muon's frame.

Lorentz Factor→Time Dilation→Relativistic Momentum→