Playground
A rod of proper length L0 shown alongside its contracted length L in the lab frame. Slider controls v/c.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Contracted Lengthoutput Length measured in observer frame | m | L | 0 – 100 | |
| Proper Length Length in the object's rest frame | m | L | 0.1 – 100 | |
| Velocity Relative velocity of the object | m/s | LT^-1 | 0 – 299792458 |
Deep dive
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 = L0/gamma.
Experimental verification
Directly inferred from muon flux measurements (the muon 'sees' the atmosphere contracted) and indirectly from relativistic collider kinematics.
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
Real-world applications
- Relativistic heavy-ion collisions (nuclei appear as pancakes)
- Muon atmospheric depth calculations
- Synchrotron radiation geometry
Worked examples
Starship flyby
Given:
- L0:
- 100
- v:
- 239833966
Find: L
Solution
gamma = 5/3 → L = 100/1.667 = 60 m.
Muon pathlength
Given:
- L0:
- 10000
- v:
- 296794533
Find: L
Solution
The 10 km atmosphere appears as 1410 m in the muon's frame.
Scenarios
What if…
- scenario:
- What if the rod is perpendicular to motion?
- answer:
- No contraction — only the parallel component is affected.
- scenario:
- What if two observers pass a barn with a ladder?
- answer:
- The ladder paradox: both see the other as shorter; simultaneity of events resolves the contradiction.
- scenario:
- What if the object is rotating?
- answer:
- Different parts have different velocities; the shape transforms into a complex contracted form (Terrell rotation).
Limiting cases
- condition:
- v → 0
- result:
- L → L0
- explanation:
- No contraction at rest.
- condition:
- v → c
- result:
- L → 0
- explanation:
- An object approaching c appears infinitely flattened.
- condition:
- v = 0.866c
- result:
- L = L0/2
- explanation:
- Half the proper length at gamma = 2.
Context
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.
Hook
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.
Dimensions:
- lhs:
- L → [L]
- rhs:
- L0/gamma → [L]/[1] = [L]
- check:
- Both sides length. ✓
Validity: Applies only to lengths measured along the direction of relative motion. Transverse dimensions are unchanged.