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Mathematics for Physics

174 Length Contraction

The relativistic effect that an object’s length along its direction of motion is shorter for a moving observer than in the object’s rest frame, that is used to relate proper length to length in a moving frame.

Proper length as rest-frame baseline. Proper length is the distance between two points measured in the frame where those points are at rest. This principle is used to fix the baseline size of an object.

Relativistic shortening of moving lengths. In a frame where the object moves, the length along the motion is shorter than the proper length by the Lorentz factor. Length contraction is that shortening. This principle is used to compute the measured length of a moving object.

The length contraction formula is

\[ L = \dfrac{L_{0}}{\gamma} \]

where

  • \(L_{0}\) is the proper length.
  • \(L\) is the contracted length.
  • \(\gamma\) is the Lorentz factor.

Simultaneous endpoint measurement. Measuring a moving length requires recording both endpoints at the same coordinate time. Simultaneity is the condition that two events share one time coordinate in a given frame. This principle is used to show that contraction follows from the relativity of simultaneity, not from a mechanical squeeze.

Reciprocal contraction. Each of two observers in uniform relative motion measures the other’s rods as contracted. Reciprocity is that mutual shortening. This principle is used to keep neither observer in an absolute rest frame.

Invariance of transverse dimensions. Lengths perpendicular to the relative motion are unchanged. Transverse dimensions are axes at right angles to the relative velocity. This principle is used to leave height and width uncontracted.

Note: Also called Lorentz contraction.

174.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — proper length; simultaneous endpoints; reciprocal contraction.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(L=\dfrac{L_{0}}{\gamma}\); unchanged transverse lengths.
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — contracted length of a moving rod.
  5. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — reciprocal contraction; transverse invariance.