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Concept
Schwarzschild radius
Summary
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with any quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.
The Schwarzschild radius is given as
where G is the gravitational constant, M is the object mass, and c is the speed of light.
In 1916, Karl Schwarzschild obtained the exact solution to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass (see Schwarzschild metric). The solution contained terms of the form and , which becomes singular at and respectively. The has come to be known as the Schwarzschild radius. The physical significance of these singularities was debated for decades. It was found that the one at is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that was used; while the one at is a spacetime singularity and cannot be removed. The Schwarzschild radius is nonetheless a physically relevant quantity, as noted above and below.
This expression had previously been calculated, using Newtonian mechanics, as the radius of a spherically symmetric body at which the escape velocity was equal to the speed of light. It had been identified in the 18th century by John Michell and Pierre-Simon Laplace.
The Schwarzschild radius of an object is proportional to its mass. Accordingly, the Sun has a Schwarzschild radius of approximately , whereas Earth's is only about and the Moon's is about .
Derivation of the Schwarzschild solution
Any object whose radius is smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body (a rotating black hole operates slightly differently).
Official source
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