the Schwarzschild metric
/ SHVARTS-shilt /
Within months of Einstein publishing his field equations, and while serving on the Russian front in 1916, Karl Schwarzschild found their first exact solution: the spacetime outside a single, round, non-spinning mass. It describes the geometry surrounding the Sun, the Earth, or a black hole, wherever the source is spherical and the surroundings empty. It is the workhorse from which nearly every classic test and every textbook black hole is computed.
In the usual coordinates the line element is ds^2 = -(1 - r_s/r) c^2 dt^2 + (1 - r_s/r)^(-1) dr^2 + r^2 (dtheta^2 + sin^2(theta) dphi^2), where r_s = 2 G M / c^2 is the Schwarzschild radius set by the mass M. Far away (r much greater than r_s) it reduces to flat Minkowski spacetime, as it must. As you approach the mass, the factor (1 - r_s/r) shrinks the rate of clocks and stretches radial distances. Birkhoff's theorem guarantees this is the unique spherically symmetric vacuum solution, so even a pulsating star radiates no gravitational waves as long as it stays spherical, and the outside geometry stays exactly Schwarzschild.
This one metric powers the perihelion precession of Mercury, the bending of starlight, gravitational redshift, and the whole theory of non-rotating black holes. A crucial subtlety concerns its two apparent trouble spots. At r = r_s the metric components blow up, but this is only a coordinate singularity, an artifact of the chosen coordinates that a better chart removes cleanly; nothing physically catastrophic happens there. At r = 0, by contrast, the curvature itself diverges: that is a true, physical singularity where the theory breaks down.
For the Sun, r_s is about 3 km, tiny compared with the Sun's actual 700,000 km radius, so the (1 - r_s/r) factor differs from 1 by only a few parts in a million at the surface. That tiny departure is already enough to bend starlight and precess Mercury's orbit measurably.
Tiny for the Sun, yet enough to produce the classic tests.
The Schwarzschild coordinate r is not the distance from the center; it is defined so that a sphere at that r has area 4 pi r^2. Radial distances stretch, so the proper distance in to the center is larger than the coordinate difference suggests. Reading r as a naive radius is a classic beginner's trap.