REL-30

Twin proper time

Inertial twin ageing.

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GoverningTwin proper time

Governing equation

\tau=t/\gamma

$beta \in \left[0.1, 0.99\right]$

where

t: t (y)
beta: beta (—)
tau: Twin proper time (y)

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

\tau=t/\gamma

$beta \in \left[0.1, 0.99\right]$

$beta = 0.1$ $tau = 9.95$

Along the curve

②

Domain of the variable — blue

beta_{\min} = 0.1,\quad beta_{\max} = 0.99

$beta \in \left[0.1, 0.99\right]$

$beta = 0.1$ $beta = 0.1$

Along the curve

③

Response curve — green

\tau=t/\gamma

$beta \in \left[0.1, 0.99\right]$

$beta = 0.1$ $tau = 9.95$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}tau}{\mathrm{d}beta}

$beta \in \left[0.1, 0.99\right]$

$beta = 0.106$ $tau^{\prime} = -1.062$

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-30 — Twin proper time) is the form associated with Twin proper time. Working symbols: $t$ , $beta$ $\rightarrow$ $tau$ . Inertial twin ageing. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

tau

from

t

beta

in Relativity via

\tau=t/\gamma

Inertial twin ageing. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

t = 10.000\,\mathrm{y}

beta = 0.800\,\mathrm{—}

, the governing relation

\tau=t/\gamma

yields

tau = 6.000\,\mathrm{y}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

t t10 ybeta beta0.8 —

Outputs

Twin proper time tau6.000 y

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Inertial twin ageing. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube