REL-04

Length contraction

L = L0 / γ along the boost.

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Special relativityFitzGerald / Lorentz

Governing equation

L=L_0/\gamma=L_0\sqrt{1-v^2/c^2}

where

L_0: Proper length (m)
v/c: Speed in c (—)
L: Contracted length (m)

Lecture brief

Historical brief

Einstein’s 1905 Lorentz kinematics and 1915 field equation, then Schwarzschild (1916) and Hawking temperature, recast time, mass and gravity. The lab computes dilation, E=mc² and horizon scales. This sheet (REL-04 — Length contraction) is the form associated with FitzGerald / Lorentz. Working symbols: $L_0$ , $v/c$ $\rightarrow$ $L$ . Simultaneity of the two ends of a rod in the lab frame shortens the measured length by γ, the Lorentz–FitzGerald contraction.

Purpose

Purpose: compute

L

from

L_0

v/c

in Relativity via

L=L_0/\gamma=L_0\sqrt{1-v^2/c^2}

L = L0 / γ along the boost. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

L_0 = 1.000\,\mathrm{m}

v/c = 0.800\,\mathrm{—}

, the governing relation

L=L_0/\gamma=L_0\sqrt{1-v^2/c^2}

yields

L = 0.6000\,\mathrm{m}

. Proper length L0 along v. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Proper length L_01 mSpeed in c v/c0.8 —

Outputs

Contracted length L0.6000 m

Reading speed

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Narration of this film

Proper length L0 along v.

Simultaneity of the two ends of a rod in the lab frame shortens the measured length by γ, the Lorentz–FitzGerald contraction.

Reading speed

Watch on YouTube