REL-10

Time dilation

Δt = γ Δτ, γ = 1/√(1−β²), β = v/c.

Reading speed

RelativityLorentz

Governing equation

\Delta t=\gamma\Delta\tau,\quad\gamma=(1-v^2/c^2)^{-1/2}

where

v/c: Speed over c (—)
\Delta\tau: Proper time (ns)
\gamma: Lorentz factor (—)
\Delta t: Lab time (ns)

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-10 — Time dilation) is the form associated with Lorentz. Working symbols: $v/c$ , $\Delta\tau$ $\rightarrow$ $\gamma$ , $\Delta t$ . Moving clocks run slow as seen from the lab. Muons in the atmosphere are the textbook proof.

Purpose

Purpose: compute

\gamma

\Delta t

from

v/c

\Delta\tau

in Relativity via

\Delta t=\gamma\Delta\tau,\quad\gamma=(1-v^2/c^2)^{-1/2}

Δt = γ Δτ, γ = 1/√(1−β²), β = v/c. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

v/c = 0.600\,\mathrm{—}

\Delta\tau = 10.000\,\mathrm{ns}

, the governing relation

\Delta t=\gamma\Delta\tau,\quad\gamma=(1-v^2/c^2)^{-1/2}

yields

\gamma = 1.2500\,\mathrm{—}

\Delta t = 12.500\,\mathrm{ns}

. A moving clock, a lab clock, a γ stretch. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Speed over c v/c0.6 —Proper time \Delta\tau10 ns

Outputs

Lorentz factor \gamma1.2500 —
Lab time \Delta t12.500 ns

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A moving clock, a lab clock, a γ stretch.

Moving clocks run slow as seen from the lab. Muons in the atmosphere are the textbook proof.

Reading speed

Watch on YouTube