The Schwarzschild Solution

The Schwarzschild solution (Karl Schwarzschild, 1916) is the first and most important exact solution of the Einstein field equations: the vacuum gravitational field outside a static, spherically symmetric mass. It governs the gravity of any nearly spherical, slowly rotating body — the Sun, the Earth, a non-rotating star — and, taken to its extreme, describes a non-rotating black hole. Every classical test of general relativity is a geodesic computation in this metric.

We use $c = 1$ , keep $G$ explicit, mostly-plus signature, and write $M$ for the mass, $d Ω^{2} = d θ^{2} + sin^{2} θ d φ^{2}$ .

The metric

Solving $R_{μν} = 0$ (the vacuum field equations, $T_{μν} = 0$ , $Λ = 0$ ) subject to static spherical symmetry and asymptotic flatness gives the unique result

$d s^{2} = - (1 - \frac{2 GM}{r}) d t^{2} + (1 - \frac{2 GM}{r})^{- 1} d r^{2} + r^{2} d Ω^{2} .$

The single parameter $M$ is the mass, identified by matching to the Newtonian limit $g_{00} \to - (1 + 2Φ)$ with $Φ = - GM / r$ at large $r$ . As $r \to \infty$ the metric approaches Minkowski: the field is asymptotically flat.

The characteristic length is the Schwarzschild radius

$r_{s} = 2 GM,$

where $g_{tt}$ vanishes and $g_{rr}$ diverges. For the Sun $r_{s} \approx 3$ km; for the Earth $\approx 9$ mm — far inside the body, where the vacuum solution does not apply, so no horizon forms for ordinary stars.

Birkhoff's theorem

Birkhoff's theorem (1923) sharpens the result: any spherically symmetric vacuum solution is necessarily static and equal to Schwarzschild. Two consequences:

Uniqueness. The exterior field of any spherically symmetric body — even a collapsing or pulsating one — is exactly Schwarzschild with its total mass $M$ . The interior dynamics leave no exterior imprint.
No monopole radiation. A spherically symmetric pulsating or collapsing star emits no gravitational waves — the GR analogue of the electromagnetic statement that a spherically symmetric charge distribution does not radiate. Gravitational radiation requires a time-varying quadrupole, not monopole, moment.

Birkhoff is the gravitational counterpart of the Newtonian shell theorem, and it means the spacetime inside a spherical cavity is flat.

The classical tests

Timelike and null geodesics in this metric yield the four experimental pillars of general relativity. The static and rotational symmetries provide two Killing vectors, giving conserved energy $E$ and angular momentum $L$ per unit mass, which reduce the geodesic equations to a one-dimensional problem with an effective potential

$\frac{1}{2} \overset{r}{˙}^{2} + V_{eff} (r) = \frac{1}{2} E^{2}, V_{eff} (r) = \frac{1}{2} (1 - \frac{2 GM}{r}) (1 + \frac{L ^{2}}{r ^{2}}) .$

The extra $- GM L^{2} / r^{3}$ term (absent in Newton) drives the relativistic corrections.

Test	Effect	Measured
Perihelion precession	orbits precess by $Δ φ = \frac{6 π GM}{a ( 1 - e ^{2} )}$ per revolution — $4 3^{''}$ /century for Mercury	resolved the long-standing Mercury anomaly
Light deflection	a ray grazing mass $M$ bends by $Δ φ = \frac{4 GM}{b}$ — $1.7 5^{''}$ at the solar limb, twice the Newtonian value	Eddington 1919; VLBI to $\sim 1 0^{- 4}$
Gravitational redshift	$\frac{ν _{\infty}}{ν _{emit}} = 1 - 2 GM / r$ ; light climbing out is reddened	Pound–Rebka 1959; optical clocks
Shapiro delay	radar signals passing the Sun are delayed by the extra light-travel time in the well	Cassini to $\sim 1 0^{- 5}$

The factor-of-two in light deflection over the naive Newtonian value is the signature success: it comes from the spatial curvature ( $g_{rr}$ ), which a scalar theory of gravity misses, and it is what Eddington's 1919 eclipse expedition confirmed.

Innermost stable circular orbit

The relativistic potential has a further feature with no Newtonian analog: below $r = 6 GM$ there are no stable circular orbits. This innermost stable circular orbit (ISCO) sets the inner edge of accretion disks around black holes and fixes the efficiency of energy release in accreting systems — up to $\sim 6%$ of rest mass for Schwarzschild.

The two "singularities"

The metric misbehaves at $r = r_{s} = 2 GM$ and at $r = 0$ . These are fundamentally different:

$r = 2 GM$ — a coordinate singularity. Although $g_{tt} \to 0$ and $g_{rr} \to \infty$ , curvature scalars such as the Kretschmann scalar $K = R_{μν ρ σ} R^{μν ρ σ} = 48 G^{2} M^{2} / r^{6}$ are finite there. Nothing physical diverges; the blow-up is an artifact of the Schwarzschild time coordinate, which becomes singular the way lines of longitude do at the poles. Better coordinates (Eddington–Finkelstein, Kruskal–Szekeres) extend smoothly across it.
$r = 0$ — a genuine curvature singularity. Here $K \to \infty$ : tidal forces diverge, geodesics end, and the classical theory breaks down. This is a real, coordinate-independent singularity, of the kind the singularity theorems guarantee.

The event horizon and the black hole

The surface $r = 2 GM$ is the event horizon. Its physical character emerges once the coordinate defect is removed:

One-way membrane. Inside $r < 2 GM$ the coefficients $g_{tt}$ and $g_{rr}$ swap sign: $r$ becomes timelike and $t$ spacelike. Decreasing $r$ becomes as inevitable as advancing in time — every future-directed worldline hits $r = 0$ . Nothing, not even light, escapes: the horizon is a null surface, a boundary of no return.
Infinite redshift. To a distant observer, a probe falling in appears to slow and freeze at the horizon, its light infinitely redshifted; in the probe's own proper time it crosses in finite time and reaches $r = 0$ shortly after.

graph LR
  A["r > 2GM<br/>exterior:<br/>signals escape"] -->|cross horizon| B["r = 2GM<br/>event horizon<br/>(null, one-way)"]
  B --> C["r < 2GM<br/>interior:<br/>r timelike,<br/>all paths → r=0"]
  C --> D["r = 0<br/>curvature<br/>singularity"]

A body compressed within its Schwarzschild radius (a sufficiently massive collapsing stellar core) forms such a black hole. Direct evidence is now abundant: the Event Horizon Telescope images of M87* (2019) and Sgr A* (2022), stellar orbits around the Milky Way's central mass, and the gravitational-wave signals of merging black holes seen by LIGO/Virgo.

The full maximal extension (Kruskal diagram, white hole, second asymptotic region), the rotating generalization, and the thermodynamics of horizons are developed in the Kerr solution and the planned blackholes/ pages.

Interior and collapse

Inside the matter, $T_{μν} \neq = 0$ and Schwarzschild is replaced by an interior solution matched to the exterior at the surface. The simplest — a uniform-density perfect fluid star — yields the Tolman–Oppenheimer–Volkoff (TOV) equation for hydrostatic equilibrium. It predicts a maximum mass beyond which no static configuration exists: pressure cannot halt collapse, and the star must form a black hole. This is the physical origin of black holes as the endpoint of massive stellar collapse (Oppenheimer–Snyder, 1939).

Summary

Schwarzschild is the unique static, spherically symmetric vacuum solution; parameter $M$ ; Schwarzschild radius $r_{s} = 2 GM$ .
Birkhoff's theorem: any spherical vacuum field is Schwarzschild ⇒ exterior fixed by total mass, and no monopole gravitational radiation.
Geodesics give the four classical tests (perihelion precession, light deflection $= 2 \times$ Newton, redshift, Shapiro delay) and the ISCO at $6 GM$ .
$r = 2 GM$ is a removable coordinate singularity (finite Kretschmann scalar) — the event horizon, a one-way null surface; $r = 0$ is a genuine curvature singularity.
A mass within its Schwarzschild radius is a black hole, the endpoint of collapse (TOV limit), now observed directly (EHT, LIGO/Virgo).

The Kerr solution adds rotation — frame dragging, the ergosphere, and the no-hair theorem — and FLRW turns from local sources to the whole universe. Horizon thermodynamics (the four laws, Hawking radiation) is planned in blackholes/.

youyuanwu