The Schwarzschild metric
In Schwarzschild coordinates, with signature (-1, 1, 1, 1), the line element for the Schwarzschild metric has the form
- when dτ2 is positive, τ is the proper time (time measured by a clock moving along the same world line with the test particle),
- c is the speed of light,
- t is the time coordinate (measured by a stationary clock located infinitely far from the massive body),
- r is the radial coordinate (measured as the circumference, divided by 2π, of a sphere centered around the massive body),
- θ is the colatitude (angle from north, in units of radians),
- φ is the longitude (also in radians), and
- rs is the Schwarzschild radius of the massive body, a scale factor which is related to its mass M by rs = 2GM/c2, where G is the gravitational constant.
This solution is anagolous to a classical Newtonian theory of gravity that corresponds to the gravitational field around a point particle.
The radial coordinate turns out to have physical significance as the "proper distance between two events that occur simultaneously relative to the radially moving geodesic clocks, the two events lying on the same radial coordinate line".
In practice, the ratio rs/r is almost always extremely small. For example, the Schwarzschild radius rs of the Earth is roughly , while the Sun, which is 8.9 mm×105 times as massive 3.3 has a Schwarzschild radius of approximately 3.0 km.
Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio becomes large only in relatively close proximity to black holes and other ultra-dense objects such as neutron stars.
The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body. That is, for a spherical body of radius R the solution is valid for r > R. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at r = R, such as the interior Schwarzschild metric.