The Schwarzschild Solution

In November 1915 Einstein published the field equations of general relativity — ten coupled, non-linear partial differential equations for the metric. He suspected they might never be solved exactly. Within weeks he was proved wonderfully wrong. Karl Schwarzschild, a German astronomer serving on the Russian front of the First World War, computing between artillery calculations and already ill with the disease that would kill him months later, found the exact metric outside a spherical mass. It is the single most important solution in the theory, and it hands us the orbit of Mercury, the bending of starlight, gravitational redshift — and black holes.

This page is about that one solution: the geometry surrounding any spherical, non-rotating mass — a star, a planet, or a collapsed object with nothing left to hold it up. We read the metric, extract the event horizon where the geometry does something extraordinary, and meet the black hole that lives inside.

The metric around a mass

Solve the vacuum field equations outside a spherical mass M, demand that far away spacetime becomes flat, and there is only one answer (Birkhoff's theorem). In the usual coordinates (t, r, \theta, \phi) it is the Schwarzschild metric:

ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2\,dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2\left(d\theta^2 + \sin^2\theta\,d\phi^2\right),

where the whole influence of the mass is bundled into a single length, the Schwarzschild radius:

Look at the time coefficient g_{tt} = -(1 - r_s/r)c^2. A clock at rest at radius r ticks proper time d\tau = \sqrt{1 - r_s/r}\; dt — slower the deeper it sits. That is gravitational time dilation, now exact rather than the weak-field approximation gh/c^2. As r \to r_s the factor \sqrt{1 - r_s/r} \to 0: time, seen from far away, freezes. Slide inward on the graph to watch it happen.

The event horizon

Two radii make the metric misbehave. At r = 0 the curvature genuinely blows up — a real singularity. But at r = r_s something subtler happens: g_{tt} \to 0 and g_{rr} \to \infty. For a star like the Sun this is harmless, because r_s = 3\ \text{km} lies deep inside the Sun, where the vacuum solution doesn't apply. But if a mass is compressed until its whole body fits within its own r_s, that surface becomes an event horizon, and the object is a black hole.

The horizon is not a wall — an infalling astronaut sails through r = r_s feeling nothing special locally (the equivalence principle again: free fall is free fall). It is a one-way causal boundary. Inside r_s the roles of t and r effectively swap: decreasing r becomes as unavoidable as the forward march of time. Every future-pointing path leads to r = 0. Not even light can climb back out — hence black.

The anatomy of a black hole

The Schwarzschild geometry has a clean set of special radii, all measured in units of r_s. Reveal them one shell at a time.

In 2019 the Event Horizon Telescope imaged the shadow of the supermassive black hole in galaxy M87 — a dark disk about 2.6\,r_s across ringed by light bent around it — and in 2022 did the same for Sagittarius A* at the centre of our own galaxy. Schwarzschild's wartime calculation, once thought a mathematical curiosity, is now something we take pictures of.

Yes — and the apparent freeze is a trick of the light, not the fate of the faller. From far away you watch an astronaut fall toward the horizon and see them slow, redden, and dim, asymptotically approaching r_s but seemingly never crossing: the last photons they emit just outside the horizon take longer and longer to climb out, stretching the image in time and wavelength until it fades to black. But in the astronaut's own frame — their proper time — nothing of the sort happens. They cross r = r_s in a finite, unremarkable interval and reach the singularity moments later. The two stories don't contradict; they are two different coordinates on the same events. Coordinate time t runs to infinity at the horizon, but proper time \tau does not. It is one of relativity's sharpest lessons: "when does it happen" has no observer-free answer.

The most common Schwarzschild confusion is treating r = r_s as if it were a dangerous solid shell. It is not. Three corrections. First, the blow-up of the metric at r = r_s is a coordinate singularity, an artefact of the Schwarzschild coordinates — switch to Eddington–Finkelstein or Kruskal coordinates and the horizon is perfectly smooth. The curvature (the tensor R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}) is finite there. Second, the real singularity is at r = 0, where curvature genuinely diverges and known physics ends. Third, an astronaut crossing the horizon of a large black hole feels nothing locally — no bump, no wall — because free fall is locally indistinguishable from floating in empty space. What is special about the horizon is global and causal: it is the surface past which no signal can return, not a place where the local laws of physics break.