REL-06

Schwarzschild radius

Rs = 2 G M / c².

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Black holesSchwarzschild 1916

Governing equation

R_s=\dfrac{2GM}{c^2}

$M \in \left[0.1, 1\times 10^{9}\right]$

where

M: Mass (suns) (M_\odot)
R_s: 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=\dfrac{2GM}{c^2}

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 0.1$ $R_s = 0.295$

Along the curve

②

Domain of the variable — blue

M_{\min} = 0.1,\quad M_{\max} = 1\times 10^{9}

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 0.1$ $M = 0.1$

Along the curve

③

Response curve — green

R_s=\dfrac{2GM}{c^2}

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 0.1$ $R_s = 0.295$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}R_s}{\mathrm{d}M}

$M \in \left[0.1, 1\times 10^{9}\right]$

$M = 6.29\times 10^{6}$ $R_s^{\prime} = 2.953$

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-06 — Schwarzschild radius) is the form associated with Schwarzschild 1916. Working symbols: $M$ $\rightarrow$ $R_s$ . Schwarzschild's vacuum solution of Einstein's equation has a coordinate singularity at r = 2GM/c², the event horizon of a non-rotating black hole.

Purpose

Purpose: compute

R_s

from

M

in Relativity via

R_s=\dfrac{2GM}{c^2}

Rs = 2 G M / c². 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 = 10.000\,\mathrm{M_odot}

, the governing relation

R_s=\dfrac{2GM}{c^2}

yields

R_s = 29.533\,\mathrm{km}

. Non-rotating uncharged mass. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass (suns) M10 M_\odot

Outputs

Schwarzschild radius R_s29.533 km

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Narration of this film

Non-rotating uncharged mass.

Schwarzschild's vacuum solution of Einstein's equation has a coordinate singularity at r = 2GM/c², the event horizon of a non-rotating black hole.

Reading speed

Watch on YouTube