14.19:
Schwarzschild Radius and Event Horizon
An imaginary projectile launched from the Sun's surface would require a velocity of 618 km/s to escape the Sun's gravitational field.
Now, suppose the Sun's radius is reduced by half, keeping its mass constant. Then its escape velocity would increase to 873 km/s. If the radius is decreased further, eventually to 2.9 km, the escape velocity will equal the speed of light.
Recall that the escape velocity of a projectile depends on the mass and radius of the object from which it is launched.
In general, the radius at which any mass has an escape velocity equal to the speed of light is called the Schwarzschild radius. Therefore, Schwarzschild radius equals twice the product of gravitational constant and the object's mass divided by the square of light's speed.
Any spherical non-rotating object having a radius smaller than its Schwarzschild radius is called a black hole.
The spherical surface surrounding the black hole, at Schwarzschild radius, is known as the event horizon, inside which the escape velocity is greater than the speed of light.
14.19:
Schwarzschild Radius and Event Horizon
No object with a finite mass can travel faster than the speed of light in a vacuum. This fact has an interesting consequence in the domain of extremely high gravitational fields.
The minimum speed required to launch a projectile from the surface of an object to which it is gravitationally bound so that it eventually escapes the object’s gravitational field is called the escape velocity. The escape velocity is independent of the mass of the object. Merging the idea of escape velocity with the maximum speed with which matter can travel leads to the concepts of the Schwarzschild radius, black holes, and event horizons.
If an object is sufficiently dense, it collapses upon itself and is surrounded by an event horizon from which nothing can escape. The name for such an object, "black hole," was coined by astronomer John Wheeler in 1969. It refers to the fact that even light cannot escape such an object. Karl Schwarzschild was the first person to note this phenomenon in 1916, but at that time, it was considered to be a mathematical curiosity.
Although a black hole cannot be directly observed because light does not escape its event horizon, there is overwhelming evidence that black holes exist in the universe. Their effects on nearby stars and infalling matter are typically interpreted by making astronomical observations.
This text is adapted from Openstax, University Physics Volume 1, Section 13.7: Einstein's Theory of Gravity.