REL-27

Interval ds²

1+1 Minkowski interval.

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GoverningInterval ds²

Governing equation

ds^2=c^2 dt^2-dx^2

$dx \in \left[0, 5\right]$

where

dt: dt (ns)
dx: dx (m)
ds2: Interval ds² (m²)

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

ds^2=c^2 dt^2-dx^2

$dx \in \left[0, 5\right]$

$dx = 0$ $ds2 = 0.09$

Along the curve

②

Domain of the variable — blue

dx_{\min} = 0,\quad dx_{\max} = 5

$dx \in \left[0, 5\right]$

$dx = 0$ $dx = 0$

Along the curve

③

Response curve — green

ds^2=c^2 dt^2-dx^2

$dx \in \left[0, 5\right]$

$dx = 0$ $ds2 = 0.09$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}ds2}{\mathrm{d}dx}

$dx \in \left[0, 5\right]$

$dx = 0.031$ $ds2^{\prime} = -0.063$

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-27 — Interval ds²) is the form associated with Interval ds². Working symbols: $dt$ , $dx$ $\rightarrow$ $ds2$ . 1+1 Minkowski interval. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

ds2

from

dt

dx

in Relativity via

ds^2=c^2 dt^2-dx^2

1+1 Minkowski interval. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

dt = 1.000\,\mathrm{ns}

dx = 0.200\,\mathrm{m}

, the governing relation

ds^2=c^2 dt^2-dx^2

yields

ds2 = 0.050\,\mathrm{m^{2}}

. 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

dt dt1 nsdx dx0.2 m

Outputs

Interval ds² ds20.050 m²

Reading speed

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Narration of this film

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

1+1 Minkowski interval. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube