Event Horizon — Physica

Equivalent forms

r_H = \frac{2GM}{c^2}

A_H = 4\pi r_H^2 = \frac{16\pi G^2 M^2}{c^4}

A purely causal boundary fixed by mass alone: its area, not its 'surface', is what obeys a thermodynamic law (the second law of black-hole mechanics).

Symbol	Name	SI unit	Dimension	Range
$rH$	Horizon Radiusoutput Radius of the event horizon	m	L	0 – 10000000000000
$M$	Mass Mass enclosed by the horizon	kg	M	100000000000000000000 – 1e+40
$G$	Gravitational Constant Newton's constant (fixed)	m^3 kg^-1 s^-2	M^-1 L^3 T^-2	6.674e-11 – 6.674e-11
$c$	Speed of Light Speed of light in vacuum (fixed)	m/s	LT^-1	299792458 – 299792458

Dimensional analysis

r_H -> [L]

2GM/c^2 ->

[M^-1\,\mathrm{L}^3\,\mathrm{T}^-2][M]/[L^2\,\mathrm{T}^-2] = [L]

Length.

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Discovery

John Michell / David Finkelstein · 1958

Michell imagined 'dark stars' in 1783, but only Finkelstein's 1958 coordinates revealed the Schwarzschild surface as a one-way causal membrane rather than a physical singularity.

Try this

Why can light leave a star but never leave a black hole of the same mass?

Because the star's surface sits outside r_H = 2GM/c^2, while a black hole's mass lies inside it — at r_H the escape speed reaches c and the surface becomes one-way.

Research status: stable

Real-world applications

Black-hole shadow imaging
Black-hole thermodynamics and the information paradox
Merger ringdown 'no-hair' tests

Common misconceptions

Nothing special happens locally at the horizon — no wall, no fire, no infinite force for a large black hole
The horizon is a global, not local, feature: you can only know you crossed it in hindsight
The horizon can only grow (classically) — Hawking's area theorem — the way entropy does

Experimental verification

The Event Horizon Telescope resolved the bright photon ring surrounding the dark shadow of M87* (2019) and Sgr A* (2022); the shadow diameter

is \sim 2.6\,\mathrm{r}_H

, matching general relativity. Gravitational-wave ringdowns from LIGO/Virgo mergers match horizon-forming black holes.

Derivation

The horizon is where the Schwarzschild time-metric factor vanishes:

1 - 2GM/(rc^2) = 0

->

r_H = 2GM/c^2

.

Equivalently, it is the outermost null surface from which outgoing light rays have zero radial velocity.

The Newtonian escape-velocity condition

v_esc = \sqrt{2GM/r} = c

gives the same radius by coincidence.

Limiting cases

M = M_sun

⟶

r_H \approx 2.95\,\mathrm{km}

A solar-mass black hole horizon is city-sized.

M = 4.3e_{6} M_sun (Sgr

A*)⟶

r_H \approx 1.3e10\,\mathrm{m}

Our galaxy's central black hole horizon spans

\sim 0.08\,\mathrm{AU}

.

M -> 0 (evaporation)⟶ r_H -> 0Hawking evaporation shrinks the horizon; the final flash is when r_H reaches the Planck scale.

What if…

What if two black holes merge?

The total horizon AREA can only increase (Hawking's theorem), even as mass is radiated as gravitational waves — the black-hole analogue of the second law.

What if you fall in?

You cross the horizon in finite proper time feeling nothing (for a big hole), but a distant observer sees your image redshift and freeze at r_H forever.

1

Horizon of the Sun's mass

Given ·

M:: 1.989e+30
G:: 6.674e-11
c:: 299792458

Find · r_H

Steps

$2GM = 2*6.674e-11*1.989e30 = 2.655e20$
$c^2 = 8.988e16$
$r_H = 2.655e20/8.988e16 = 2.95e_{3} m$

Result ·

r_H = 2*6.674e-11*1.989e30/(2.998e_{8})^2 = 2.95e_{3} m \approx 2.95\,\mathrm{km}

.

Schwarzschild Radius→Schwarzschild Metric→Hawking Temperature→Gravitational Time Dilation→