AcademyRelativity
Academy
Length Contraction
Level 1 - Physics topic page in Relativity.
Principle
An object moving relative to an observer is measured shorter along the direction of motion.
Notation
\(L_0\)
proper length measured in the object's rest frame
\(\mathrm{m}\)
\(L\)
length measured in a frame where the object moves
\(\mathrm{m}\)
\(v\)
relative speed
\(\mathrm{m\,s^{-1}}\)
\(\gamma\)
Lorentz factor
\(\Delta x\)
endpoint separation in the measuring frame
\(\mathrm{m}\)
Method
Derivation 1: Measure both ends at once
A moving length is the distance between the two ends measured at the same time in the observer's frame.
Simultaneous endpoint condition
\[\Delta t=0\]
Measured moving length
\[L=|\Delta x|\]
Derivation 2: Relate to proper length
The rest frame of the object measures the proper length. Transforming the simultaneous endpoint events gives the contracted length.
Length contraction
\[L=\frac{L_0}{\gamma}\]
Proper length
\[L_0=\gamma L\]
Derivation 3: Keep direction clear
Only the component parallel to relative motion contracts. Transverse dimensions are unchanged.
Rules
Contracted length
\[L=\frac{L_0}{\gamma}\]
Lorentz factor
\[\gamma=\frac{1}{\sqrt{1-v^2/c^2}}\]
Proper length
\[L_0=\gamma L\]
Examples
Question
A rod has proper length
\[5.00\,\mathrm m\]
and moves at \[0.600c\]
along its length. Find its measured length.Answer
\[\gamma=1.25\]
\[L=\frac{L_0}{\gamma}=\frac{5.00}{1.25}=4.00\,\mathrm m\]
Checks
- Proper length is measured in the object's rest frame.
- Moving length is measured using simultaneous endpoint positions in the observer's frame.
- Only lengths parallel to the relative velocity contract.
- A contracted length is never greater than the proper length.