Hong Life CalculatorsUnderstanding the Schwarzschild radius and time dilation calculator — concept and principle
The Schwarzschild radius is the size to which a mass must be compressed to become a black hole: the radius of its event horizon if it does not rotate and has no charge. This calculator gives that radius for a mass in solar masses and the gravitational time dilation for a clock held at rest at a chosen distance from the centre.
How it is calculated
- Schwarzschild radius: r_s = 2GM ÷ c², with G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻², c = 299,792,458 m/s and one solar mass = 1.98847 × 10³⁰ kg.
- Clock rate at r: dτ/dt = √(1 − r_s/r), the ratio of a static clock's ticking to that of a clock far away. If r ≤ r_s, the calculator reports "inside horizon".
Inputs: M in solar masses, r in km from the centre.
The principle
Karl Schwarzschild found the first exact solution of Einstein's field equations in 1916, for the space-time outside a spherical, non-rotating mass. In it, the factor (1 − r_s/r) appears in front of time. Far away the factor is 1, and as r approaches r_s it falls to 0, so a clock hovering near the horizon runs very slowly when seen from far away. A useful classical check: the escape speed √(2GM/r) equals c exactly at r = 2GM/c², the same radius, although the true meaning is relativistic. r_s grows in proportion to mass, about 2.95 km per solar mass.
Worked example
- Defaults: M = 10 M☉, r = 100 km.
- r_s = 2 × 6.6743e-11 × 1.98847e31 ÷ (299,792,458)² = 29,533.39 m = 29.53339382 km.
- r_s/r = 29.533 ÷ 100 = 0.295334; 1 − 0.295334 = 0.704666.
- Clock rate = √0.704666 = 0.8394439003: one second for the hovering clock lasts about 1.19 seconds for a distant observer.
| Mass (solar masses) | Schwarzschild radius (km) |
|---|---|
| 2.5 | 7.383348455 |
| 5 | 14.76669691 |
| 7.5 | 22.15004537 |
| 10 | 29.53339382 |
| 12.5 | 36.91674228 |
| 15 | 44.30009073 |
| 20 | 59.06678764 |
How to read the result
A value near 1 means negligible dilation; on Earth's surface it differs from 1 by less than one part in a billion, yet GPS still corrects for it. At r = 1.5 r_s (the photon sphere) the rate is about 0.577, and at 3 r_s (the innermost stable circular orbit for a non-spinning hole) about 0.816. The Sun's r_s is about 2.95 km and Earth's about 9 mm.
Common mistakes
- Entering r in metres or in units of r_s instead of km.
- Applying the static formula to an orbiting or falling clock, which also has velocity time dilation.
Frequently asked questions
Would the Earth become a black hole if compressed?
Only if squeezed to about 9 mm in radius, which no known natural process can do.
Is the event horizon a physical surface?
No. It is a boundary in space-time from which light cannot escape; a falling observer notices nothing special locally when crossing a large black hole's horizon.
How big is the black hole at the Milky Way's centre?
Sagittarius A* has about 4 million solar masses, giving r_s of roughly 12 million km.
Sources: Schwarzschild K., Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (1916) · Misner C.W., Thorne K.S. and Wheeler J.A., Gravitation, W. H. Freeman, 1973 · CODATA 2018 recommended values of the fundamental physical constants (NIST)