REL-03

Time dilation

Δt = γ Δτ for a moving clock.

Reading speed

Special relativityEinstein 1905

Governing equation

\Delta t=\gamma\Delta\tau

$v/c \in \left[0, 0.999\right]$

where

\Delta\tau: Proper time (µs)
v/c: Speed in c (—)
\Delta t: Lab time (µs)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\Delta t=\gamma\Delta\tau

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $\Delta t = 2.2$

Along the curve

②

Domain of the variable — blue

v/c_{\min} = 0,\quad v/c_{\max} = 0.999

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $v/c = 0$

Along the curve

③

Response curve — green

\Delta t=\gamma\Delta\tau

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $\Delta t = 2.2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta t}{\mathrm{d}v/c}

$v/c \in \left[0, 0.999\right]$

$v/c = 0.006$ $\Delta t^{\prime} = 0.014$

Along the curve

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-03 — Time dilation) is the form associated with Einstein 1905. Working symbols: $\Delta\tau$ , $v/c$ $\rightarrow$ $\Delta t$ . A clock at rest in a boosted frame ticks proper time Δτ; the lab interval is stretched by γ, confirmed by muon lifetime and GPS.

Purpose

Purpose: compute

\Delta t

from

\Delta\tau

v/c

in Relativity via

\Delta t=\gamma\Delta\tau

Δt = γ Δτ for a moving clock. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\Delta\tau = 2.200\,\mathrm{\mu s}

v/c = 0.980\,\mathrm{—}

, the governing relation

\Delta t=\gamma\Delta\tau

yields

\Delta t = 11.055\,\mathrm{\mu s}

. Inertial motion, enter v/c and proper time. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Proper time \Delta\tau2.2 µsSpeed in c v/c0.98 —

Outputs

Lab time \Delta t11.055 µs

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Narration of this film

Inertial motion, enter v/c and proper time.

A clock at rest in a boosted frame ticks proper time Δτ; the lab interval is stretched by γ, confirmed by muon lifetime and GPS.

Reading speed

Watch on YouTube