About the Schwarzschild Radius
The Schwarzschild radius, named after physicist Karl Schwarzschild who derived it in 1916 as the first exact solution to Einstein's field equations, marks the boundary of a black hole's event horizon. Any mass compressed within its own Schwarzschild radius becomes a black hole — a region where the escape velocity exceeds the speed of light, so nothing, including light, can escape.
How the calculation works
Enter the object's mass and choose its unit (kilograms, solar masses, or Earth masses). The calculator converts the mass to kilograms and applies r = 2GM/c², where G ≈ 6.674×10⁻¹¹ N·m²/kg² is the gravitational constant and c = 299,792,458 m/s is the speed of light. It also reports the diameter (2r), the average density the mass would need in order to fit inside a sphere of that radius (ρ = M / (4/3 π r³)), and — if you enter the object's actual radius — how the real object compares to its own Schwarzschild radius.
Common mistakes
- Dropping the factor of 2: the formula is r = 2GM/c², not GM/c², which is a different (and unrelated) length scale.
- Assuming any object with a Schwarzschild radius is a black hole: every mass has a mathematical Schwarzschild radius, but it is only physically meaningful once the mass is actually compressed inside it — true for black holes, not for ordinary stars or planets.
- Mixing mass units: convert solar or Earth masses to kilograms consistently; a stray factor of 10³ or 10⁶ changes the radius by the same factor.
Real-world applications
- Astrophysicists use the Schwarzschild radius to estimate how much a collapsing stellar core must compress to form a black hole after a supernova.
- It sets natural length scales used throughout general relativity, such as the photon sphere (1.5 × r) and the innermost stable circular orbit (3 × r) around a non-rotating black hole.
- Gravitational-wave astronomers (e.g., LIGO/Virgo) use combined Schwarzschild radii of merging compact objects to sanity-check inferred masses.
- Popular comparisons — like "Earth would fit inside a 9 mm marble" or "the Sun would fit inside a small town" — come directly from this formula.
Frequently Asked Questions
What is the Schwarzschild radius?
The Schwarzschild radius is the radius of the event horizon that would surround a given mass if it were compressed into a perfect, non-rotating, uncharged sphere. At this radius the escape velocity equals the speed of light, so nothing — not even light — can escape from within it. It comes from Karl Schwarzschild's 1916 exact solution to Einstein's field equations.
What is the formula for the Schwarzschild radius?
r = 2GM / c², where G is the gravitational constant (about 6.674×10⁻¹¹ N·m²/kg²), M is the object's mass, and c is the speed of light (299,792,458 m/s). Doubling the mass doubles the Schwarzschild radius.
Does every object have a Schwarzschild radius, even if it is not a black hole?
Yes — the formula applies to any mass, but it only describes a real event horizon if the object is actually compressed within that radius. The Sun's Schwarzschild radius is about 2.95 km, yet its actual radius is roughly 696,000 km, so the Sun is nowhere near being a black hole.
How does the Schwarzschild radius relate to a black hole's size?
For a non-rotating (Schwarzschild) black hole, the event horizon is exactly a sphere of radius r = 2GM/c². Spinning (Kerr) black holes have a smaller, oblate event horizon, so this calculator gives the non-rotating idealization commonly used for order-of-magnitude estimates.
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |