REL-26

Schwarzschild radius

Event-horizon scale.

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GoverningSchwarzschild radius

Governing equation

R_s=2GM/c^2

$M \in \left[0.1, 50\right]$

where

M: M (Msun)
Rs: Schwarzschild radius (km)

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

R_s=2GM/c^2

$M \in \left[0.1, 50\right]$

$M = 0.1$ $Rs = 0.295$

Along the curve

②

Domain of the variable — blue

M_{\min} = 0.1,\quad M_{\max} = 50

$M \in \left[0.1, 50\right]$

$M = 0.1$ $M = 0.1$

Along the curve

③

Response curve — green

R_s=2GM/c^2

$M \in \left[0.1, 50\right]$

$M = 0.1$ $Rs = 0.295$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Rs}{\mathrm{d}M}

$M \in \left[0.1, 50\right]$

$M = 0.414$ $Rs^{\prime} = 2.948$

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-26 — Schwarzschild radius) is the form associated with Schwarzschild radius. Working symbols: $M$ $\rightarrow$ $Rs$ . Event-horizon scale. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

Rs

from

M

in Relativity via

R_s=2GM/c^2

Event-horizon scale. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

M = 1.000\,\mathrm{Msun}

, the governing relation

R_s=2GM/c^2

yields

Rs = 2.948\,\mathrm{km}

. 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

M M1 Msun

Outputs

Schwarzschild radius Rs2.948 km

Reading speed

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Narration of this film

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

Event-horizon scale. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube