This cheat sheet summarizes the physics of black holes, with a focus on the Schwarzschild radius and the event horizon. Students need it to connect Newtonian gravity, relativity, and observable effects such as time dilation and accretion. It is especially useful for comparing black holes with stars, planets, and other massive objects.
The reference keeps the most important equations, constants, and ideas in one printable place.
The central formula is the Schwarzschild radius, , which gives the event horizon radius for a non-rotating, uncharged black hole. Escape velocity reaches the speed of light when , making the event horizon a boundary beyond which light cannot escape. Gravitational time dilation near a Schwarzschild black hole is described by .
Key ideas include mass, radius, density, tidal forces, photon orbits, and the difference between the event horizon and the singularity.
Key Facts
- The Schwarzschild radius of a non-rotating, uncharged black hole is .
- Newtonian escape velocity is , and setting gives the Schwarzschild radius.
- The event horizon is located at , where the escape velocity equals the speed of light .
- For a Schwarzschild black hole, gravitational time dilation is for .
- The photon sphere of a Schwarzschild black hole is at , where light can orbit in an unstable circular path.
- The innermost stable circular orbit for matter around a Schwarzschild black hole is .
- The average density inside the event horizon can be estimated by , even though the real interior is not uniform.
- Tidal force effects increase when gravitational field strength changes rapidly with distance, approximately following .
Vocabulary
- Black hole
- A region of spacetime where gravity is so strong that nothing inside the event horizon can escape.
- Schwarzschild radius
- The radius of the event horizon for a non-rotating, uncharged black hole.
- Event horizon
- The boundary at beyond which light and matter cannot escape to distant observers.
- Singularity
- A predicted central point or region where classical general relativity gives infinite density and stops giving usable physical answers.
- Accretion disk
- A hot, rotating disk of gas and dust that can form around a compact object as matter spirals inward.
- Gravitational time dilation
- The slowing of time near a massive object compared with time far away, described near a Schwarzschild black hole by .
Common Mistakes to Avoid
- Confusing the event horizon with the singularity, because the event horizon is a boundary at while the singularity is predicted at the center.
- Using diameter instead of radius in , because the formula gives the radius from the center to the event horizon, not the full width.
- Forgetting to use SI units, because , must be in , and must be in .
- Thinking a black hole pulls harder than any object of the same mass at the same distance, because outside the event horizon the gravitational field depends mainly on mass and distance.
- Substituting into the time dilation formula as if it gives a normal finite result, because becomes at the event horizon for a distant observer.
Practice Questions
- 1 Calculate the Schwarzschild radius of a black hole with mass using , , and .
- 2 Find the escape velocity at from an object with mass using .
- 3 A spaceship hovers at from a Schwarzschild black hole. Use to find how much time a distant observer measures if the ship measures .
- 4 Explain why the Sun would not suddenly pull harder on Earth if it were replaced by a black hole with the same mass, assuming Earth stayed at the same orbital distance.
Understanding Black Holes & Schwarzschild Radius Reference
The radius linked to a black hole depends on mass, not on the size of the object before collapse. If Earth were compressed enough to become a black hole, its horizon would be only about the size of a marble. If the Sun became a black hole without changing its mass, its horizon would be about six kilometres across.
At distances much larger than the horizon, gravity from either object would pull in the same way as gravity from the original Earth or Sun. A black hole is not a cosmic vacuum cleaner. An orbiting planet can remain in orbit unless some process removes its energy.
The escape speed argument is useful because it gives the right horizon size, but it is not the full explanation. In general relativity, gravity is the curvature of space and time caused by mass. Near the horizon, every possible future path points inward once an object crosses it.
A falling astronaut does not strike a solid surface at the event horizon. In a sufficiently large black hole, they may notice nothing unusual at that exact location.
A distant observer receives light from the astronaut more slowly and at lower energy. The astronaut appears to fade and slow down because light has increasing difficulty climbing away from the strong gravitational field.
Matter outside a black hole can produce some of the brightest signals in astronomy. Gas usually has sideways motion, so it forms a flattened accretion disk instead of falling straight inward. Collisions and friction within the disk turn orbital energy into heat.
Very hot gas can emit visible light, ultraviolet light, and X rays. The hottest inner regions are especially important because they move fastest and feel the strongest gravity. Stable circular orbits do not exist arbitrarily close to a non-rotating black hole.
Gas that passes the innermost stable orbit tends to plunge inward rapidly. Some black holes launch narrow jets, but the jets are powered by magnetic fields and the surrounding disk, not by matter escaping from inside the horizon.
Tidal forces are different from the overall pull of gravity. They measure how much gravity changes from one end of an object to the other. Near a small black hole, this difference can become enormous before the horizon is reached.
It can stretch an object lengthwise and squeeze it sideways, an effect often called spaghettification. Near a supermassive black hole, the tidal difference at the horizon can be much weaker because the horizon is far larger. When studying this topic, keep radius, distance from the centre, and distance from the horizon separate.
Notice that the singularity is a prediction of the simple model, while the event horizon is a boundary in spacetime. Newtonian formulas build intuition, but general relativity gives the reliable description close to a black hole.