KAHIBARO
Table of Contents
Seeing Length in Different Frames
Length contraction is the relativistic effect in which an object moving relative to an observer is measured to be shorter along the direction of motion than it is in the object's own rest frame. This effect does not mean that the object is physically crushed by some force. It means that different observers, moving relative to one another, do not agree on the measured length of the same moving object.
The key idea is that a length measurement requires knowing the positions of both ends of the object at the same time in the observer's frame. Because simultaneity depends on the frame, the measured length can change from one observer to another.
Proper Length
The longest length associated with an object is its proper length, often written as $L_0$. This is the length measured in the frame where the object is at rest.
If the object moves with speed $v$ relative to an observer, then that observer measures a shorter length $L$ in the direction of motion:
$$
L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
$$
where $c$ is the speed of light.
Proper length $L_0$ is measured in the rest frame of the object.
Length contraction occurs only along the direction of motion.
The observed length is
$$
L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
$$
so $L < L_0$ whenever $v \neq 0$.
It is often useful to write this using the Lorentz factor $\gamma$:
$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
Then the contraction formula becomes
$$
L = \frac{L_0}{\gamma}
$$
Direction Matters
Only the dimension parallel to the relative motion contracts. Dimensions perpendicular to the motion do not change.
For example, if a spaceship moves horizontally, its measured length from nose to tail is contracted for a stationary observer, but its height and width are unchanged.
| Dimension of object | Relative to motion | Measured change |
|---|---|---|
| Front to back | Parallel | Contracts |
| Side to side | Perpendicular | No change |
| Top to bottom | Perpendicular | No change |
Why Simultaneous Measurement Is Necessary
To measure the length of a moving object, an observer must record the position of its front and back at the same instant in that observer's frame. If the two positions are not taken simultaneously, the object moves between measurements, and the result is not its length.
This is why length contraction is deeply connected to relativity of simultaneity. Different frames disagree about which events happen at the same time, so they can disagree about the length of a moving object.
A Simple Example
Suppose a train has proper length
$$
L_0 = 100 \ \text{m}
$$
and moves at
$$
v = 0.80c
$$
Then
$$
L = 100\sqrt{1 - 0.80^2}
= 100\sqrt{1 - 0.64}
= 100\sqrt{0.36}
= 100(0.60)
= 60 \ \text{m}
$$
So an observer who sees the train moving at $0.80c$ measures its length as $60 \ \text{m}$.
Physical Interpretation
Length contraction is not an illusion caused by bad eyesight, and it is not simply a visual effect from light travel time. It is a real difference in measurement between inertial frames.
In the rest frame of the object, nothing unusual happens. The object keeps its proper length. In another frame where the object is moving, the measured distance between its ends, taken simultaneously in that frame, is smaller.
An object never contracts in its own rest frame.
Each inertial observer measures normal rulers in their own frame.
Contraction is a comparison between measurements made in different frames.
Extreme Speeds
At everyday speeds, the factor $\sqrt{1 - v^2/c^2}$ is extremely close to $1$, so length contraction is too small to notice. At speeds close to the speed of light, the effect becomes large.
| Speed | Factor $\sqrt{1 - v^2/c^2}$ | Measured length $L/L_0$ |
|---|---|---|
| $0.10c$ | $0.995$ | $0.995$ |
| $0.50c$ | $0.866$ | $0.866$ |
| $0.80c$ | $0.600$ | $0.600$ |
| $0.90c$ | $0.436$ | $0.436$ |
| $0.99c$ | $0.141$ | $0.141$ |
This table shows that the contraction becomes important only when $v$ is a substantial fraction of $c$.
Spacecraft Illustration
Common Misunderstanding
A common mistake is to think that every observer sees the other's ruler as absolutely shorter in a contradictory way. In relativity, this is not a contradiction because each observer makes the measurement using simultaneity defined in their own frame. The measurement procedure itself depends on the frame.
Another common mistake is to apply length contraction to any distance. The formula applies to the length of an object or separation between points that are at rest in one frame. One must identify clearly which frame measures the proper length.
Summary Formula
Length contraction relates the proper length of an object to the length measured in a frame where it moves:
$$
L = L_0 \sqrt{1 - \frac{v^2}{c^2}} = \frac{L_0}{\gamma}
$$
Use length contraction only when:
- $L_0$ is the length in the object's rest frame.
- $L$ is the length measured in a frame where the object moves.
- The contraction is only along the direction of motion.
- The two ends are measured simultaneously in the observer's frame.
This effect is one of the central results of special relativity and becomes important whenever speeds are close to the speed of light.