The Schwarzschild Solution: Gravity Around a Sphere
Just weeks after Einstein published his field equations in November 1915, Karl Schwarzschild found the first exact solution — from the trenches of World War I. The Schwarzschild metric describes the spacetime geometry outside any spherically symmetric, non-rotating, uncharged mass. It’s the starting point for understanding planets, stars, and black holes.
The metric
In Schwarzschild coordinates $(t, r, \theta, \phi)$:
$$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2 dt^2 + \frac{dr^2}{1 - r_s/r} + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2)$$
where $r_s = 2GM/c^2$ is the Schwarzschild radius. For the Sun, $r_s \approx 3$ km. For the Earth, $r_s \approx 9$ mm.
The last two terms ($r^2 d\Omega^2$) are just the geometry of a sphere — familiar from the metric on a globe. The physics is in the first two terms: the factors $(1 - r_s/r)$ that modify the flat-space coefficients.
Three classic predictions
1. Gravitational time dilation
The $g_{00}$ component gives the rate of clocks: $d\tau = \sqrt{1 - r_s/r} \, dt$. Clocks closer to the mass run slower. At the Earth’s surface vs infinity, the difference is about 7 × 10⁻¹⁰ — small but measurable. GPS satellites must correct for this every day.
2. Light bending
Light follows null geodesics ($ds^2 = 0$). Near a massive body, these geodesics curve. A light ray grazing the Sun is deflected by:
$$\delta\phi = \frac{4GM}{c^2 b} = \frac{2r_s}{b}$$
where $b$ is the closest approach distance. For the Sun: 1.75 arcseconds. Eddington confirmed this during the 1919 solar eclipse — the observation that made Einstein world-famous.
3. Orbital precession
In Newtonian gravity, orbits are closed ellipses. In GR, they precess — the ellipse slowly rotates. For Mercury:
$$\Delta\phi = \frac{6\pi G M}{c^2 a(1-e^2)} \approx 43 \text{ arcsec/century}$$
This 43 arcseconds per century had been an unexplained anomaly for decades. Einstein calculated it correctly in November 1915 — he said his heart “shuddered” when the number came out right.
A test particle orbits a central mass. In Newton’s gravity (blue), the orbit closes. In GR (orange), the orbit precesses — the ellipse rotates. Adjust the angular momentum to see different orbit shapes.
The Schwarzschild radius
As $r \to r_s$, the $g_{00}$ component goes to zero (clocks stop) and $g_{rr}$ diverges (radial distances blow up). At $r = r_s$, something dramatic happens — this is the event horizon. For ordinary stars and planets, $r_s$ is deep inside the object where the vacuum Schwarzschild solution doesn’t apply. But if a mass is compressed within its Schwarzschild radius, the horizon is real and nothing can escape from inside. That’s a black hole.
Birkhoff’s theorem
A remarkable uniqueness result: the Schwarzschild metric is the only spherically symmetric vacuum solution to Einstein’s equations. Any spherically symmetric mass — pulsating, collapsing, oscillating — produces the same exterior geometry. This is the GR analogue of Newton’s shell theorem.