Schwarzschild Metric
Introduction
The Schwarzschild metric is a solution to Einstein's field equations of general relativity that describes the gravitational field outside a spherical mass, provided that the electric charge and angular momentum are zero. Named after Karl Schwarzschild, a German physicist who first solved the Einstein field equations in 1916, the Schwarzschild metric has been fundamental in the development of the modern understanding of black holes.
Derivation
The derivation of the Schwarzschild metric involves solving the Einstein field equations for a static, spherically symmetric mass distribution. The metric is expressed in Schwarzschild coordinates, which are a type of spherical coordinates adapted to the symmetry of the problem.
Schwarzschild Radius
The Schwarzschild radius is a characteristic radius associated with every mass. It is the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape speed from the surface of the sphere would equal the speed of light. If a stellar remnant's mass is within its Schwarzschild radius, a black hole is formed.
Black Holes
The Schwarzschild metric is most famously used to describe a black hole, a region of space from which nothing, not even light, can escape. The boundary of this region is called the event horizon, and it corresponds to the Schwarzschild radius of the black hole.
Tests and Predictions
The Schwarzschild metric has been tested extensively and has made several predictions that have been confirmed by observations. These include the precession of the perihelion of Mercury and the deflection of light by the Sun, both of which were confirmed and are now considered classic tests of general relativity.
Limitations and Extensions
While the Schwarzschild metric provides a precise description of the spacetime outside a spherical, non-rotating mass, it is not applicable to rotating or charged masses. Extensions to the Schwarzschild solution to include these cases are known as the Kerr and Reissner-Nordström metrics, respectively.