AcademyGravitation
Academy
Black Holes
Level 1 - Physics topic page in Gravitation.
Principle
Black-hole formation is tied to compactness: when the escape speed from a body's surface reaches the speed of light, the corresponding horizon scale is the Schwarzschild radius.
Notation
\(M\)
mass of the compact body
\(\mathrm{kg}\)
\(r\)
distance from the center
\(\mathrm{m}\)
\(r_s\)
Schwarzschild radius
\(\mathrm{m}\)
\(v_{\mathrm{esc}}\)
escape speed
\(\mathrm{m\,s^{-1}}\)
\(G\)
gravitational constant
\(\mathrm{N\,m^{2}\,kg^{-2}}\)
\(c\)
speed of light in vacuum
\(\mathrm{m\,s^{-1}}\)
Method
Derivation 1: Build the escape-speed condition
Escape requires enough kinetic energy to climb out of the gravitational potential well. The threshold case arrives at infinity with zero remaining speed.
Threshold energy at launch
\[\frac12 mv_{\mathrm{esc}}^2-\frac{GMm}{r}=0\]
Solve for escape speed
\[v_{\mathrm{esc}}=\sqrt{\frac{2GM}{r}}\]
Derivation 2: Define the horizon scale
If the threshold escape speed reaches \(c\), even light cannot escape outward from that radius. Setting the escape formula equal to \(c\) defines the characteristic radius.
Set the threshold to light speed
\[c=\sqrt{\frac{2GM}{r_s}}\]
Square both sides
\[c^2=\frac{2GM}{r_s}\]
Schwarzschild radius
\[r_s=\frac{2GM}{c^2}\]
Rules
These are the compact results from the method above.
Escape speed
\[v_{\mathrm{esc}}=\sqrt{\frac{2GM}{r}}\]
Schwarzschild radius
\[r_s=\frac{2GM}{c^2}\]
Black-hole condition
\[r\le r_s\]
Examples
Question
Find the Schwarzschild radius of an object with mass
\[8.0\times10^{30}\,\mathrm{kg}\]
Use \[c=3.00\times10^8\,\mathrm{m\,s^{-1}}\]
Answer
\[r_s=\frac{2GM}{c^2}=\frac{2(6.67\times10^{-11})(8.0\times10^{30})}{(3.00\times10^8)^2}=1.19\times10^4\,\mathrm{m}\]
So the Schwarzschild radius is about \[11.9\,\mathrm{km}\]
Checks
- The event horizon is a radius scale, not a solid surface.
- A larger mass does not guarantee a black hole; the mass must be compressed inside \(r_s\).
- Escape speed depends on both mass and radius, so compactness matters.
- The Schwarzschild radius grows linearly with mass.