Playground
Schwarzschild radius as a disk whose size scales with mass (log). Shows r_s = 2GM/c^2 with M in solar masses.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Schwarzschild Radiusoutput Event horizon radius | m | L | 0 – 10000000000 | |
| Mass Total mass of the object | kg | M | 1 – 1e+40 | |
| Gravitational Constant Newton's constant (fixed) | m^3 kg^-1 s^-2 | M^-1 L^3 T^-2 | 6.674e-11 – 6.674e-11 | |
| Speed of Light Speed of light in vacuum (fixed) | m/s | LT^-1 | 299792458 – 299792458 |
Deep dive
Derivation
Set Newtonian escape speed v_esc = sqrt(2GM/r) equal to c: c^2 = 2GM/r_s → r_s = 2GM/c^2. The same radius drops out of the metric component g_tt = 0 in the Schwarzschild solution of general relativity.
Experimental verification
Event Horizon Telescope imaging of M87* and Sgr A* (2019, 2022) confirmed shadow sizes consistent with 2GM/c^2 to within 10%.
Common misconceptions
- r_s is not a physical surface — free-falling observers feel nothing crossing it
- Newtonian derivation gives the correct answer only by coincidence
- Schwarzschild radius grows linearly with M, not with sqrt(M)
Real-world applications
- Black hole mass estimation from EHT images
- LIGO merger waveform templates
- Accretion disk physics
- Primordial black hole searches
Worked examples
Earth as a black hole
Given:
- M:
- 5.972e+24
- G:
- 6.674e-11
- c:
- 299792458
Find: r_s
Solution
r_s = 2 × 6.674e-11 × 5.972e24 / (299792458)^2 ≈ 8.87e-3 m ≈ 8.87 mm.
Sgr A* supermassive
Given:
- M:
- 8.54e+36
- G:
- 6.674e-11
- c:
- 299792458
Find: r_s
Solution
M ≈ 4.3e6 M_sun → r_s ≈ 1.27e10 m ≈ 0.085 AU.
Scenarios
What if…
- scenario:
- What if the mass rotates?
- answer:
- Use the Kerr metric: horizon shrinks as spin increases, down to GM/c^2 at maximal spin.
- scenario:
- What if you fall through r_s?
- answer:
- Tidal effects dominate locally; an outside observer sees you frozen at the horizon due to infinite redshift.
- scenario:
- What if G were 10× larger?
- answer:
- Every black hole's horizon would be 10× bigger, and stars would collapse far more easily.
Limiting cases
- condition:
- M → 0
- result:
- r_s → 0
- explanation:
- No mass, no horizon.
- condition:
- M → M_sun
- result:
- r_s ≈ 2.95 km
- explanation:
- The Sun would fit inside 3 km to become a black hole.
- condition:
- M → 10^9 M_sun
- result:
- r_s ≈ 3e12 m
- explanation:
- Supermassive black hole horizon ≈ 20 AU.
Context
Karl Schwarzschild · 1916
Schwarzschild derived the first exact solution of Einstein's field equations while serving on the Russian front during WWI.
Hook
What radius would Earth need to be crushed to, to become a black hole?
With M = 5.972e24 kg, compute r_s = 2GM/c^2 ≈ 8.87 mm.
Dimensions:
- lhs:
- r_s → [L]
- rhs:
- 2GM/c^2 → [M^-1 L^3 T^-2]·[M]/[L^2 T^-2] = [L]
- check:
- Both sides length. ✓
Validity: Valid for a non-rotating, uncharged spherically symmetric mass (Schwarzschild metric). Rotating masses use Kerr radius.