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Mathematics for Physics

192 Proper Length

The length of an object measured in its own rest frame, which is the maximum length any observer attributes to that object.

Rest-frame length. Proper length is the distance between two points measured in the frame where those points are at rest. A rest frame is the reference frame in which the object has zero velocity. This principle is used to fix the intrinsic length of an object.

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

The length contraction formula is

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

where

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

Simultaneous endpoint marking. Measuring the length of a moving object requires marking both ends 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.

Invariance of transverse lengths. Lengths perpendicular to the relative motion are unchanged. This principle is used to leave the transverse size of a moving object the same.

Non-invariance of spatial separation. A purely spatial distance is not an invariant, unlike the spacetime interval. A spacetime interval is the observer-independent four-dimensional squared separation of two events. This principle is used to separate frame-dependent lengths from invariant geometry.

Note: Also called \(L_{0}\).

192.1 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. — rest-frame length; \(L=\dfrac{L_{0}}{\gamma}\); simultaneous ends.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — contraction along the motion; unchanged transverse lengths.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — simultaneity of endpoints; spatial distance is not invariant.
  5. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — contracted length of a moving rod.
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — spatial separation versus the invariant interval.