REL-28

Rapidity

Additive boost parameter.

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GoverningRapidity

Governing equation

\phi=\mathrm{artanh}\\,\beta

$beta \in \left[0.01, 0.99\right]$

where

beta: beta (—)
phi: Rapidity (—)

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

\phi=\mathrm{artanh}\\,\beta

$beta \in \left[0.01, 0.99\right]$

$beta = 0.01$ $phi = 0.01$

Along the curve

②

Domain of the variable — blue

beta_{\min} = 0.01,\quad beta_{\max} = 0.99

$beta \in \left[0.01, 0.99\right]$

$beta = 0.01$ $beta = 0.01$

Along the curve

③

Response curve — green

\phi=\mathrm{artanh}\\,\beta

$beta \in \left[0.01, 0.99\right]$

$beta = 0.01$ $phi = 0.01$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}phi}{\mathrm{d}beta}

$beta \in \left[0.01, 0.99\right]$

$beta = 0.016$ $phi^{\prime} = 1$

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-28 — Rapidity) is the form associated with Rapidity. Working symbols: $beta$ $\rightarrow$ $phi$ . Additive boost parameter. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

phi

from

beta

in Relativity via

\phi=\mathrm{artanh}\\,\beta

Additive boost parameter. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

beta = 0.600\,\mathrm{—}

, the governing relation

\phi=\mathrm{artanh}\\,\beta

yields

phi = 0.693\,\mathrm{—}

. 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

beta beta0.6 —

Outputs

Rapidity phi0.693 —

Reading speed

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Narration of this film

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

Additive boost parameter. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube