⚫ Schwarzschild Radius Calculator
Calculate the event horizon radius of a black hole from its mass
Stellar black holes: 3-50 M☉ | Intermediate: 10²-10⁵ M☉ | Supermassive: 10⁶-10¹⁰ M☉
How to Use This Calculator
Enter Black Hole Mass
Input the mass of the black hole in solar masses (M☉). For stellar black holes, typical values range from 3-50 solar masses. For supermassive black holes at galaxy centers, values can be millions to billions of solar masses (e.g., Sagittarius A* has about 4 million solar masses).
Calculate Radius
Click "Calculate" to determine the Schwarzschild radius. This is the radius of the event horizon - the boundary beyond which nothing can escape. The result is shown in kilometers, meters, and solar radii for comparison.
Understand the Result
The Schwarzschild radius is directly proportional to mass. For a 10 solar mass black hole, the event horizon is about 30 km. For Earth-mass black hole (hypothetical), it would be only about 9 mm. For a supermassive black hole with 4 million solar masses, it's about 12 million km.
Formula
R_s = 2GM / c²
Where:
- R_s = Schwarzschild radius (event horizon) (meters)
- G = Gravitational constant = 6.67430 × 10⁻¹¹ m³kg⁻¹s⁻²
- M = Black hole mass (kg)
- c = Speed of light = 299,792,458 m/s
Example Calculation: 10 Solar Mass Black Hole
Given:
- Mass: M = 10 M☉ = 10 × 1.989 × 10³⁰ kg = 1.989 × 10³¹ kg
- G = 6.67430 × 10⁻¹¹ m³kg⁻¹s⁻²
- c = 2.998 × 10⁸ m/s
Calculation:
R_s = 2GM / c²
R_s = 2 × 6.67430×10⁻¹¹ × 1.989×10³¹ / (2.998×10⁸)²
R_s = 2.654×10²¹ / 8.988×10¹⁶
R_s = 29,530 m = 29.5 km
A 10 solar mass black hole has an event horizon radius of about 30 km.
Example Calculation: Sagittarius A* (Supermassive Black Hole)
Given:
- Mass: M = 4.1 × 10⁶ M☉ ≈ 8.15 × 10³⁶ kg
Calculation:
R_s = 2GM / c²
R_s = 2 × 6.674×10⁻¹¹ × 8.15×10³⁶ / (2.998×10⁸)²
R_s ≈ 1.21 × 10¹⁰ m = 12.1 million km
Sagittarius A*, the supermassive black hole at our galaxy's center, has an event horizon about 12 million km in radius - about 17 times the Sun's radius!
Example Calculation: Earth-Mass Black Hole (Hypothetical)
Given:
- Mass: M = 5.972 × 10²⁴ kg (Earth's mass)
Calculation:
R_s = 2 × 6.674×10⁻¹¹ × 5.972×10²⁴ / (2.998×10⁸)²
R_s ≈ 8.87 × 10⁻³ m = 8.87 mm
If Earth were compressed into a black hole, its event horizon would be only about 9 millimeters - smaller than a ping pong ball!
Key Insights:
- Schwarzschild radius is directly proportional to mass: R_s ∝ M
- For solar mass: R_s ≈ 2.95 km × (M/M☉)
- Doubling mass doubles the event horizon radius
- Density decreases with size: ρ ∝ M / R³ ∝ 1/M²
- Larger black holes have lower average density
About the Schwarzschild Radius Calculator
The Schwarzschild Radius Calculator determines the event horizon radius of a black hole from its mass. Named after Karl Schwarzschild, who found this solution to Einstein's field equations in 1916, the Schwarzschild radius defines the boundary beyond which nothing, not even light, can escape. This is the fundamental size scale of black holes and a key concept in general relativity.
When to Use This Calculator
- Black Hole Physics: Understand the size and properties of black holes
- General Relativity: Learn about the Schwarzschild metric and event horizons
- Astrophysics Research: Calculate event horizon sizes for observed black holes
- Educational Purposes: Understand black hole physics and general relativity
- Cosmology Studies: Explore supermassive black holes at galaxy centers
Why Use Our Calculator?
- ✅ Accurate Formula: Uses the exact Schwarzschild radius formula from general relativity
- ✅ Multiple Units: Shows results in kilometers, meters, and solar radii
- ✅ Educational Tool: Learn about black holes and event horizons
- ✅ Real Physics: Based on Einstein's general theory of relativity
- ✅ Free to Use: No registration required
- ✅ Mobile Friendly: Works on all devices
Understanding the Schwarzschild Radius
The Schwarzschild radius defines the event horizon:
- Event Horizon: The boundary beyond which nothing can escape
- Point of No Return: Once crossed, escape is impossible
- Light Speed: At the event horizon, escape velocity equals the speed of light
- Time Dilation: Time appears to stop at the event horizon (from outside perspective)
- Mass Dependent: R_s = 2GM/c² - proportional to mass
Historical Context
- 1916: Karl Schwarzschild found the first exact solution to Einstein's field equations
- Schwarzschild Metric: Describes spacetime around a non-rotating, uncharged black hole
- Event Horizon: The radius where the metric becomes singular
- Black Hole Name: Term "black hole" coined by John Wheeler in 1967
- Modern Understanding: Schwarzschild solution describes static black holes
Properties of the Schwarzschild Radius
- Proportional to Mass: R_s = 2GM/c² means doubling mass doubles radius
- Density Decreases: Average density ρ ∝ M/R³ ∝ 1/M² - larger black holes are less dense
- Solar Mass Scale: 1 M☉ black hole has R_s ≈ 2.95 km
- Supermassive: 10⁹ M☉ black hole has R_s ≈ 2.95 × 10⁹ km ≈ 20 AU
- Minimum Size: For a given mass, this is the smallest possible radius
Real-World Applications
- Stellar Black Holes: Typically 3-50 M☉ with R_s ≈ 9-150 km
- Sagittarius A*: ~4 million M☉ with R_s ≈ 12 million km
- M87 Black Hole: ~6.5 billion M☉ with R_s ≈ 19 billion km
- Event Horizon Telescope: Images the shadow of the event horizon
- Gravitational Waves: Black hole mergers create ripples in spacetime
Limitations and Considerations
- Non-Rotating: Schwarzschild solution assumes no rotation (Kerr metric for rotating)
- Uncharged: Assumes no electric charge (Reissner-Nordström for charged)
- Isolated: Assumes black hole is alone (no external matter)
- Classical: Quantum effects become important near the event horizon
- Inside Event Horizon: The Schwarzschild solution breaks down at the singularity
Tips for Using This Calculator
- Remember that Schwarzschild radius is proportional to mass - double the mass, double the radius
- For stellar mass black holes (3-50 M☉), event horizons are typically 9-150 km
- For supermassive black holes, event horizons can be millions to billions of kilometers
- If Earth were a black hole, its event horizon would be only about 9 mm
- The Schwarzschild solution describes non-rotating black holes - real black holes typically rotate
Frequently Asked Questions
What is the Schwarzschild radius?
The Schwarzschild radius is the radius of the event horizon of a non-rotating, uncharged black hole. It's defined as R_s = 2GM/c², where G is the gravitational constant, M is the mass, and c is the speed of light. It represents the boundary beyond which nothing, not even light, can escape.
What happens at the Schwarzschild radius?
At the Schwarzschild radius (event horizon), the escape velocity equals the speed of light. From an outside observer's perspective, time appears to stop, and light from objects at the event horizon is infinitely redshifted. Once something crosses this boundary, it cannot escape - it's the "point of no return."
How big is the Schwarzschild radius for different masses?
The Schwarzschild radius is proportional to mass. For 1 solar mass, R_s ≈ 2.95 km. For a 10 solar mass black hole, R_s ≈ 29.5 km. For Earth's mass (hypothetically), R_s ≈ 8.87 mm. For Sagittarius A* (4 million solar masses), R_s ≈ 12 million km. For the supermassive black hole in M87 (6.5 billion solar masses), R_s ≈ 19 billion km.
Can we see the event horizon?
We can't see the event horizon directly (it's black!), but we can observe its effects. The Event Horizon Telescope (EHT) imaged the "shadow" of the event horizon of the M87 black hole in 2019. The shadow is about 2.5 times larger than the actual event horizon due to gravitational lensing. We can also observe matter falling into black holes and the effects on nearby stars and gas.
Is the Schwarzschild radius the same as the black hole's size?
The Schwarzschild radius defines the event horizon, which is the boundary of the black hole. However, inside the event horizon, there's a singularity at the center (where density becomes infinite according to general relativity). The "size" of a black hole typically refers to the Schwarzschild radius, though the actual matter is compressed to a point (or near-point) at the center.
What is the difference between Schwarzschild and Kerr black holes?
The Schwarzschild solution describes non-rotating, uncharged black holes. Real black holes typically rotate, described by the Kerr metric. Rotating black holes have two horizons (outer event horizon and inner Cauchy horizon) and an ergosphere. The Schwarzschild radius is still relevant as it gives the scale, but the actual event horizon shape depends on rotation. Most astrophysical black holes are better described by the Kerr metric.