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Mathematics for Physics

158 Contracted Length

The length of an object measured by an observer moving relative to it, always shorter than the proper length along the motion.

Simultaneous endpoint measurement. The contracted length is the distance between the two endpoints recorded at the same coordinate time in the observer’s frame. Simultaneity is the condition that those two events share one time coordinate. This principle is used to define \(L\) as an operational measurement, not as a mechanical squeeze.

Relativistic shortening along the motion. That measured length is shorter than the proper length along the motion. Proper length is the distance between the endpoints in the object’s rest frame. This principle is used to compute the shortened size of a moving object.

The contracted length is

\[ L = L_{0}\sqrt{1 - \dfrac{v^{2}}{c^{2}}} \]

where

  • \(L\) is the contracted length.
  • \(L_{0}\) is the proper length.
  • \(v\) is the relative speed.
  • \(c\) is the speed of light.

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

Invariance of transverse dimensions. The shortening is only along the motion. Transverse dimensions are axes at right angles to the relative velocity. This principle is used to leave height and width unchanged.

Note: Also called \(L\). Also called Lorentz-FitzGerald contracted length.

158.1 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. — simultaneous endpoints; \(L=L_{0}\sqrt{1-\dfrac{v^{2}}{c^{2}}}\); reciprocal contraction.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — contracted length along the motion; 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. — operational definition of moving length; reciprocity.