Chapter 5: Length Contraction
Just as time dilates for moving observers, lengths contract in the direction of motion. A moving object is measured to be shorter than when at rest. This is not an optical illusionâit's a real consequence of how space and time are intertwined in special relativity.
The Length Contraction Formula
\( L = \frac{L_0}{\gamma} = L_0 \sqrt{1 - \frac{v^2}{c^2}} \)
where Lâ is the proper length (length in the object's rest frame)
Proper Length (Lâ)
- ⢠Length measured when object is at rest relative to observer
- ⢠The "rest length" of the object
- ⢠Always the longest measurement
- ⢠Invariant quantity everyone can agree on
Contracted Length (L)
- ⢠Length measured when object moves relative to observer
- ⢠Only in the direction of motion
- ⢠Always shorter than proper length
- ⢠L â 0 as v â c
Important: Contraction only occurs along the direction of motion. Perpendicular dimensions remain unchanged. A moving sphere becomes an oblate spheroidâsquashed in the direction of travel.
Deriving Length Contraction
Length contraction follows directly from time dilation plus the principle of relativity.
Thought Experiment: Measuring a Moving Rod
Setup: A rod of proper length Lâ moves at velocity v. An observer measures its length by recording when the front and back pass a single point.
In the rod's rest frame:
The observer moves at velocity v past the rod. Time to pass: \( \Delta t' = L_0/v \) (using the rod's clocks).
In the observer's frame:
The rod's clocks are dilated by factor Îł, so: \( \Delta t = \Delta t'/\gamma = L_0/(\gamma v) \)
The observed length:
\( L = v \cdot \Delta t = v \cdot \frac{L_0}{\gamma v} = \frac{L_0}{\gamma} \)
The derivation shows that length contraction and time dilation are two sides of the same coinâboth arise from the geometry of spacetime.
Numerical Examples
| Speed (v/c) | Îł | L/Lâ | 1m rod appears as |
|---|---|---|---|
| 0.1c | 1.005 | 0.995 | 99.5 cm |
| 0.5c | 1.155 | 0.866 | 86.6 cm |
| 0.8c | 1.667 | 0.600 | 60.0 cm |
| 0.9c | 2.294 | 0.436 | 43.6 cm |
| 0.99c | 7.089 | 0.141 | 14.1 cm |
| 0.999c | 22.37 | 0.045 | 4.5 cm |
Worked Example: Cosmic Ray Muon (Again!)
Remember the muons created at altitude 10 km? From the muon's perspective, it's at rest and Earth is rushing toward it at v = 0.9999c. The 10 km distance contracts to:
\( L = 10 \text{ km} \times \sqrt{1 - 0.9999^2} = 10 \text{ km} \times 0.014 = 140 \text{ m} \)
At 0.9999c, the muon crosses 140 m in about 0.5 Îźsâwell within its 1.5 Îźs half-life!
Is Length Contraction "Real"?
This is one of the most misunderstood aspects of relativity. Length contraction is not:
- âAn optical illusion or visual distortion
- âA measurement error or instrumental artifact
- âA physical compression of the object
Length contraction IS:
- âA consequence of how space and time measurements relate in different frames
- âCompletely symmetricâeach observer sees the other's meter sticks as shortened
- âConnected to the relativity of simultaneity
The key insight: Different observers don't just measure different valuesâthey're measuring genuinely different things. Measuring "length" requires marking the positions of both endsat the same time. But simultaneity is relative! So the "length" itself depends on the observer.
The Barn-Pole Paradox
A famous puzzle that illustrates how length contraction really works:
Setup
- ⢠A pole of proper length 10 m
- ⢠A barn of proper length 8 m (with doors at both ends)
- ⢠A runner carries the pole at v = 0.866c (γ = 2)
Barn's Perspective
The moving pole contracts to 10/2 = 5 m. It easily fits inside the 8 m barn! Both doors can be closed simultaneously with the pole inside.
Runner's Perspective
The barn contracts to 8/2 = 4 m while the pole is still 10 m. The pole doesn't fit! How can both doors be closed?
Resolution
The two doors don't close "simultaneously" in both frames! In the barn frame, they close at the same time. In the runner's frame, the front door closes first (while the back of the pole is still outside), then the back door closes (after the front of the pole has exited). No contradictionâjust the relativity of simultaneity in action.
What Would You Actually See?
Interestingly, length contraction is NOT what you would see with your eyes or a camera. The visual appearance of a fast-moving object is more complex due to:
1. Light Travel Time (Penrose-Terrell Effect)
Light from different parts of an object reaches your eye at different times. A sphere moving past you would still look spherical, but rotated! The contraction is hidden by this optical effect.
2. Relativistic Doppler Effect
Colors would shiftâblue ahead (approaching), red behind (receding). At extreme speeds, visible light could shift to UV or infrared.
3. Relativistic Beaming
Light intensity is concentrated in the direction of motion. The object would appear much brighter when approaching and dimmer when receding.
Length contraction is the "true" geometric effect in spacetime. What you see is complicated by the finite speed of light reaching your eyes from different parts of the object.