Escape velocity and the Schwarzschild radius are both concepts related to the gravitational field around a massive object, particularly in the context of general relativity and black holes.
The Schwarzschild radius (Rs) is a characteristic radius associated with a non-rotating, spherically symmetric black hole. It is defined as:
Rs = (2GM) / c^2
Where: Rs is the Schwarzschild radius, G is the gravitational constant, M is the mass of the object, and c is the speed of light in a vacuum.
Escape velocity (Ve) is the minimum velocity required for an object to escape the gravitational pull of a massive body. It is influenced by the mass and size of the object, as well as the gravitational field it is in. For a non-rotating object like a black hole, the escape velocity at a distance r from its center can be calculated using:
Ve = √((2GM) / r)
The relationship between the Schwarzschild radius and escape velocity arises when considering an object that has a radius smaller than the Schwarzschild radius. If the radius of the object is smaller than Rs, then its escape velocity at the surface (r = R) would be greater than the speed of light (c), making it impossible for anything, including light, to escape its gravitational pull. This condition defines the event horizon of a black hole.
In summary, the Schwarzschild radius is a measure of the size of a non-rotating black hole, while the escape velocity is the minimum velocity required to escape the gravitational pull of a massive object. If the radius of an object is smaller than the Schwarzschild radius, it becomes a black hole, and its escape velocity at the event horizon exceeds the speed of light.