The Schwarzschild Metric

The Einstein field equations are ten coupled nonlinear partial differential equations for the metric, and no general solution exists. Progress comes from symmetry: assuming enough symmetry reduces the equations to ordinary differential equations that can be integrated in closed form. The most consequential such case is the vacuum outside a static, spherically symmetric mass, worked out by Karl Schwarzschild within months of the theory's publication. The result describes the exterior of any non-rotating star or planet and, taken to its own limit, the simplest black hole. This lesson derives the metric, reads the meaning of its coordinates, and establishes the two theorems that make it the reference geometry for the solar-system tests.

The static spherically symmetric ansatz

Two symmetry assumptions constrain the metric before any field equation is solved.

Staticity: the geometry is time-independent and time-reversal symmetric. There is a timelike coordinate $t$ under which no metric component depends on $t$ , and no cross term $d t d x^{i}$ mixes time with space (that would pick out a direction of time, as a rotating body does).
Spherical symmetry: the geometry is invariant under rotations. The metric is built from the rotationally invariant combinations, and the angular part is proportional to the metric of a unit 2-sphere, $d Ω^{2} = d θ^{2} + sin^{2} θ d ϕ^{2}$ .

These reduce the metric to two unknown functions of the radial coordinate alone. Writing them as exponentials to keep the signature manifest,

d s^{2} = - e^{2 α (r)} c^{2} d t^{2} + e^{2 β (r)} d r^{2} + r^{2} d Ω^{2} .

The radial coordinate $r$ is defined geometrically, not as a distance from the center: a sphere at fixed $t$ and $r$ has area $4 π r^{2}$ , exactly as in flat space, because the angular block is $r^{2} d Ω^{2}$ . This is the areal radius. The proper radial distance between two such spheres is $\int e^{β} d r$ , which is not $Δ r$ once $β \neq = 0$ . Fixing the area rather than the distance is the choice that keeps the algebra tractable.

The static spherically symmetric setup: nested spheres of areal radius r surround the central mass M. Each sphere has area 4 pi r squared, but the proper distance between neighbouring spheres exceeds their coordinate difference.

The vacuum field equations

Outside all matter the stress–energy tensor vanishes, so the Einstein equation $G_{μν} = \frac{8 π G}{c ^{4}} T_{μν}$ reduces to $G_{μν} = 0$ . Taking the trace shows the Ricci scalar vanishes, and the equation collapses to the vacuum equation

R_{μν} = 0.

Computing the Christoffel symbols of the ansatz and contracting them into the Ricci tensor gives three independent nonzero components, $R_{tt}$ , $R_{r r}$ , and $R_{θ θ}$ (the $ϕϕ$ component repeats $θ θ$ up to $sin^{2} θ$ ). Two combinations do the work. The first is $e^{2 (β - α)} R_{tt} + R_{r r} = 0$ , which reduces to

α^{'} (r) + β^{'} (r) = 0 ⟹ α (r) = - β (r) + const .

The constant shifts by a rescaling of $t$ ; absorbing it fixes $α = - β$ , so $e^{2 α} = e^{- 2 β}$ . The time and radial coefficients are reciprocals. The second equation is $R_{θ θ} = 0$ , which with $α = - β$ becomes

e^{2 α} (2 r α^{'} + 1) = 1 ⟺ \frac{d}{d r} (r e^{2 α}) = 1.

Integrating gives $r e^{2 α} = r + C$ for a constant $C$ , hence

e^{2 α (r)} = 1 + \frac{C}{r} .

The constant $C$ is not fixed by the field equations, which are local; it is fixed by matching to the physical situation far away. In the weak-field limit the time–time metric component reproduces the Newtonian potential through $g_{tt} = - (1 + 2Φ/ c^{2})$ , derived in the Newtonian limit of the geodesic equation. With $Φ = - GM / r$ for a body of mass $M$ , matching $e^{2 α} = 1 + C / r$ to $1 - 2 GM / (c^{2} r)$ identifies $C = - 2 GM / c^{2}$ .

The Schwarzschild metric

Assembling the pieces gives the Schwarzschild line element,

d s^{2} = - (1 - \frac{2 GM}{c ^{2} r}) c^{2} d t^{2} + (1 - \frac{2 GM}{c ^{2} r})^{- 1} d r^{2} + r^{2} (d θ^{2} + sin^{2} θ d ϕ^{2}) .

One combination of constants recurs and is given its own symbol, the Schwarzschild radius

r_{s} = \frac{2 GM}{c ^{2}},

so that $g_{tt} = - (1 - r_{s} / r) c^{2}$ and $g_{r r} = (1 - r_{s} / r)^{- 1}$ . For the Sun, $r_{s} \approx 2.95 km$ ; for the Earth, $r_{s} \approx 8.9 mm$ . Both sit deep inside the body, where the vacuum solution does not apply, so no horizon exists for ordinary stars. The metric is exact everywhere in the vacuum exterior, not a weak-field approximation.

The coordinates carry specific meanings that repay stating precisely, because much of the physics of the geometry is the difference between coordinate quantities and locally measured ones.

$t$ is the time read by a clock at rest infinitely far from the mass, where the metric is flat. A static clock at finite $r$ reads proper time $d τ = 1 - r_{s} / r d t < d t$ : it runs slow relative to $t$ , the gravitational time dilation.
$r$ is the areal radius, fixed by the sphere's area $4 π r^{2}$ , not a proper distance from the center.
$θ, ϕ$ are ordinary polar angles on each sphere.

The metric coefficients against radius, in units of the Schwarzschild radius. The solid curve is the time part,

- g_{tt} / c^{2} = 1 - r_{s} / r

, rising from zero at

r_{s}

toward one far away; the dotted curve is the radial part,

g_{r r} = (1 - r_{s} / r)^{- 1}

, diverging at

r_{s}

and falling toward one. Both approach the flat value one at large

r

The far field and the Newtonian limit

Far from the mass, $r ≫ r_{s}$ , expand the coefficients in the small ratio $r_{s} / r$ :

g_{tt} = - (1 - \frac{r _{s}}{r}) c^{2} = - c^{2} (1 - \frac{2 GM}{c ^{2} r}) = - c^{2} (1 + \frac{2Φ}{c ^{2}}),

with $Φ = - GM / r$ the Newtonian potential. A slowly moving particle in this geometry follows the geodesic equation, whose spatial part reduces to $\ddot{r} = - \nablaΦ = - \frac{GM}{r ^{2}} \overset{r}{^}$ , Newton's law of gravity. The Schwarzschild metric contains Newtonian gravity as its weak-field slow-motion limit; the corrections that distinguish the two are of order $r_{s} / r$ , tiny in the solar system ( $r_{s} / r \sim 1 0^{- 8}$ at the Sun's surface) and responsible for the small deviations the classical tests measure.

The spatial geometry and Flamm's paraboloid

The curvature of space alone, separate from the time dilation, shows in a slice of constant $t$ through the equatorial plane $θ = π /2$ :

d s^{2} = (1 - \frac{r _{s}}{r})^{- 1} d r^{2} + r^{2} d ϕ^{2} .

This two-dimensional geometry is curved: circles of circumference $2 π r$ are separated by proper radial distances larger than $Δ r$ . It can be visualized by embedding it as a surface $z (r)$ in ordinary flat three-space, where the induced metric $d s^{2} = (1 + z^{'2}) d r^{2} + r^{2} d ϕ^{2}$ matches the slice when

1 + z^{'} (r)^{2} = (1 - \frac{r _{s}}{r})^{- 1} ⟹ z^{'} (r) = \frac{r _{s}}{r - r _{s}} .

Integrating gives Flamm's paraboloid,

z (r) = 2 r_{s} (r - r_{s}), equivalently r = r_{s} + \frac{z ^{2}}{4 r _{s}},

a parabola of revolution. Far away it flattens to a plane (Euclidean space); near $r_{s}$ its slope becomes vertical, the geometric picture of $g_{r r}$ diverging. The funnel is a picture of the spatial curvature only — it says nothing about the time dilation, which is the larger effect for slow orbits and is carried entirely by $g_{tt}$ .

Flamm's paraboloid embeds the equatorial spatial slice of the Schwarzschild geometry. The throat at r-s flattens to a plane far from the mass; the vertical slope near r-s is the geometric form of g-rr diverging.

Birkhoff's theorem

The derivation assumed a static geometry, but that assumption can be dropped. If the metric is only required to be spherically symmetric, allowing the two functions to depend on both $t$ and $r$ , the vacuum equations force the $t$ -dependence to cancel: the solution is necessarily static and equals the Schwarzschild metric.

Two consequences follow.

A pulsating spherical star has a static exterior. A spherically symmetric star that oscillates or collapses radially produces exactly the same external field at all times, set by its total mass $M$ . It cannot emit spherically symmetric gravitational radiation, the gravitational analog of the fact that a spherically pulsating charge distribution emits no electromagnetic radiation.
The interior is shielded. Inside a spherical shell of matter the vacuum solution with no enclosed mass is the flat metric ( $M = 0$ , so $r_{s} = 0$ ), the relativistic version of Newton's shell theorem: no gravitational field inside a spherical shell.

Birkhoff's theorem: a spherically symmetric star that pulsates or collapses (dashed radii) leaves its exterior field unchanged, fixed only by the total mass M. The exterior geometry is Schwarzschild at all times.

The uniqueness is what makes the Schwarzschild metric the reference solution for the solar system: to the accuracy at which the Sun is spherical and non-rotating, its external geometry is Schwarzschild with $M = M_{⊙}$ , and every planetary orbit and light ray is a geodesic of that one metric. The next lesson uses the two Killing symmetries of this metric, time-translation and rotation, to reduce geodesic motion to a one-dimensional problem in an effective potential.