Length contraction is another fundamental consequence of the Lorentz transformations in Special Relativity. It describes how the length of an object depends on the relative motion of the observer.
In particular, an object moving at constant velocity relative to an observer will appear contracted in the direction of motion, compared to its length measured in its own rest frame.
We derive length contraction using the Lorentz transformations and a simple measurement procedure.
Consider a rigid rod moving at speed v v x x S S S ′ S ′
Let:
The proper length of the rod (its length in its own rest frame S ′ S ′ L 0 L 0
The contracted length of the rod, as measured in frame S S L L
Our goal is to relate L L L 0 L 0
In frame S S simultaneously — at the same time t t
Let the two ends of the rod have x x x 1 x 1 x 2 x 2 S S
Then the measured length is:
L = x 2 − x 1 L = x 2 − x 1
In contrast, in the rest frame S ′ S ′ x 1 ′ x 1 ′ x 2 ′ x 2 ′
L 0 = x 2 ′ − x 1 ′ L 0 = x 2 ′ − x 1 ′
The Lorentz transformations between the two frames are:
x ′ = γ ( x − v t ) t ′ = γ ( t − v x c 2 ) x ′ t ′ = γ ( x − v t ) = γ ( t − c 2 v x )
where:
γ = 1 1 − v 2 c 2 γ = 1 − c 2 v 2 1
Apply this to both ends of the rod:
x 1 ′ = γ ( x 1 − v t ) x 2 ′ = γ ( x 2 − v t ) x 1 ′ x 2 ′ = γ ( x 1 − v t ) = γ ( x 2 − v t )
Now compute the difference:
x 2 ′ − x 1 ′ = γ ( x 2 − v t ) − γ ( x 1 − v t ) = γ ( x 2 − x 1 ) x 2 ′ − x 1 ′ = γ ( x 2 − v t ) − γ ( x 1 − v t ) = γ ( x 2 − x 1 )
Therefore:
L 0 = γ L L 0 = γ L
Rearranging:
L = L 0 γ = L 0 1 − v 2 c 2 L = γ L 0 = L 0 1 − c 2 v 2
In frame S S x 1 x 1 x 2 x 2 at the same time t t .
In frame S ′ S ′ t 1 ′ t 1 ′ t 2 ′ t 2 ′
L 0 L 0 proper length : the length measured in the rest frame of the object.L L v v
Conclusion : Moving objects appear contracted along the direction of motion by a factor 1 / γ 1 / γ
Length contraction arises directly from the Lorentz transformations.
An object moving at speed v v contracted by a factor 1 / γ 1 / γ
Example Calculation
Suppose an AGN jet component has a proper length L 0 = 1 L 0 = 1 v = 0.99 c v = 0 . 9 9 c
Compute γ γ
γ = 1 1 − ( 0.99 ) 2 ≈ 7.09 γ = 1 − ( 0 . 9 9 ) 2 1 ≈ 7 . 0 9
Then:
L = 1 7.09 ly ≈ 0.141 ly L = 7 . 0 9 1 ly ≈ 0 . 1 4 1 ly
The observed length of the moving jet component is only about 14% of its proper length.
Rindler, W. Introduction to Special Relativity
Jackson, J.D. Classical Electrodynamics , Chapter 11
Urry & Padovani (1995), Unified Schemes for Radio-Loud AGN