A black hole with zero charge Q = 0 and no angular momentum J = 0. The exterior solution for such a black hole is known as the Schwarzschild solution (or Schwarzschild metric), and is an exact unique solution to the Einstein field equations of general relativity for the general static isotropic metric (i.e., the most general metric tensor that can represent a static isotropic gravitational field),

In 1915, when Einstein first proposed them, the Einstein field equations appeared so complicated that he did not believe that a solution would ever be found. He was therefore quite surprised when, only a year later, Karl Schwarzschild(1916) discovered one by making the assumption of spherical symmetry.
In empty space, the Einstein field equations become

where is the Ricci tensor.Reading off ,, and from the static isotropic metric (1) gives

so if Also

so





But as ,the metric tenor approaches the Minkowski metric, so


Plugging this into and gives


So we only have to make ,then and by (4),


Now, at great distance,

where the gravitational potential is

Here, M is the mass of the black hole, G is the gravitational constant, and c is the speed of light.Note that it is very common to omit all factors of c (or equivalently, to set c = 1) in the equations of general relativity.Although slightly confusing, the c = 1 convention allows equations to be written more concisely, and no information is actually lost since the missing factors of c can always be unambiguously inserted by dimensional analysis.Combining (16) and (17) gives the constant in (15) as , so


and the metric in standard form is therefore

This is the Schwarzschild solution in standard form.

The radius at which the metric becomes singular is

known as the Schwarzschild radius.
The Killing vector fields for the Schwarzschild solution are ,,,and .
An exact solution turns out to also be possible for a spherical body with constant density; see Schwarzschild black hole--constant density.