REL-31

Aberration

Headlight effect.

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GoverningAberration

Governing equation

\cos\theta'=\frac{\cos\theta+\beta}{1+\beta\cos\theta}

$th \in \left[0, 180\right]$

where

th: th (deg)
beta: beta (—)
tp: Aberration (deg)

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

\cos\theta'=\frac{\cos\theta+\beta}{1+\beta\cos\theta}

$th \in \left[0, 180\right]$

$th = 0$ $tp = 0$

Along the curve

②

Domain of the variable — blue

th_{\min} = 0,\quad th_{\max} = 180

$th \in \left[0, 180\right]$

$th = 0$ $th = 0$

Along the curve

③

Response curve — green

\cos\theta'=\frac{\cos\theta+\beta}{1+\beta\cos\theta}

$th \in \left[0, 180\right]$

$th = 0$ $tp = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}tp}{\mathrm{d}th}

$th \in \left[0, 180\right]$

$th = 1.132$ $tp^{\prime} = 0.734$

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-31 — Aberration) is the form associated with Aberration. Working symbols: $th$ , $beta$ $\rightarrow$ $tp$ . Headlight effect. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

tp

from

th

beta

in Relativity via

\cos\theta'=\frac{\cos\theta+\beta}{1+\beta\cos\theta}

Headlight effect. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

th = 30.000\,\mathrm{deg}

beta = 0.300\,\mathrm{—}

, the governing relation

\cos\theta'=\frac{\cos\theta+\beta}{1+\beta\cos\theta}

yields

tp = 22.247\,\mathrm{deg}

. 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

th th30 degbeta beta0.3 —

Outputs

Aberration tp22.247 deg

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Narration of this film

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

Headlight effect. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube