Schwarzschild Metric - The Schwarzschild Metric

The Schwarzschild Metric

See also: Deriving the Schwarzschild solution

In Schwarzschild coordinates, the Schwarzschild metric has the form:

where:

• τ is the proper time (time measured by a clock moving with the particle) in seconds,
• c is the speed of light in meters per second,
• t is the time coordinate (measured by a stationary clock at infinity) in seconds,
• r is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters,
• θ is the colatitude (angle from North) in radians,
• φ is the longitude in radians, and
• r_s is the Schwarzschild radius (in meters) of the massive body, which is related to its mass M by r_s = 2GM/c2, where G is the gravitational constant.

The analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle.

In practice, the ratio r_s/r is almost always extremely small. For example, the Schwarzschild radius r_s of the Earth is roughly 8.9 millimeters (0.35 in), while the sun, which is 3.3×105 times as massive has a Schwarzschild radius of approximately 3.0 km (1.9 mi). A satellite in a geosynchronous orbit has a radius r that is roughly four billion times larger than the earth's Schwarzschild radius at 42,164 km (26,199 mi). Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close 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.