---
title: The Schwarzschild Metric
module: The Schwarzschild Solution
moduleNumber: 6
lessonNumber: 1
order: 601
summary: >
  The first exact solution of Einstein's equation follows from two assumptions,
  staticity and spherical symmetry, imposed on the vacuum outside a mass. Solving
  the vacuum field equations fixes two metric functions and produces the
  Schwarzschild geometry, whose one length scale is the Schwarzschild radius
  $r_s = 2GM/c^2$. Birkhoff's theorem shows this is the only spherical vacuum,
  and the far field reduces to Newtonian gravity.
topics: [The Schwarzschild Solution]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 9 — The Geometry Outside a Spherical Star"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §7 — The Schwarzschild Solution and Black Holes"
---

The [Einstein field equations](/relativity/curved-spacetime/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,
  $\d\Omega^2 = \d\theta^2 + \sin^2\theta\,\d\phi^2$.

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

$$
\d s^2 = -e^{2\alpha(r)}\,c^2\,\d t^2 + e^{2\beta(r)}\,\d r^2 + r^2\,\d\Omega^2 .
$$

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

$$
% caption: The static spherically symmetric setup: nested spheres of areal radius
% r surround the central mass M. Each sphere has area 4 pi r squared, but the
% proper distance between neighbouring spheres exceeds their coordinate difference.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw (0,0) circle (0.55);
  \node at (0,0) {M};
  \draw[black] (0,0) circle (1.5);
  \draw[black] (0,0) circle (2.4);
  \draw[black] (0,0) circle (3.2);
  % radial ray with tick marks
  \draw[->, acc, thick] (0,0) -- (3.5,0) node[right, black] {r};
  \foreach \x in {1.5,2.4,3.2} \fill (\x,0) circle (1.4pt);
  \node[black, anchor=south] at (1.06,1.06) {area $=$ 4 pi r sq};
  \draw[<->, black] (1.5,-0.28) -- (2.4,-0.28);
  \node[black, anchor=north] at (1.95,-0.3) {proper gap};
\end{tikzpicture}
$$

## The vacuum field equations

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

$$
R_{\mu\nu} = 0 .
$$

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

$$
\alpha'(r) + \beta'(r) = 0 \quad\Longrightarrow\quad \alpha(r) = -\beta(r) + \text{const}.
$$

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

$$
e^{2\alpha}\bigl(2r\alpha' + 1\bigr) = 1
\quad\Longleftrightarrow\quad
\frac{\d}{\d r}\!\left(r\,e^{2\alpha}\right) = 1 .
$$

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

$$
e^{2\alpha(r)} = 1 + \frac{C}{r} .
$$

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
$g_{tt} = -(1 + 2\Phi/c^2)$, derived in the
[Newtonian limit](/relativity/curved-spacetime/geodesics-and-the-geodesic-equation)
of the geodesic equation. With $\Phi = -GM/r$ for a body of mass $M$, matching
$e^{2\alpha} = 1 + C/r$ to $1 - 2GM/(c^2 r)$ identifies $C = -2GM/c^2$.

## The Schwarzschild metric

Assembling the pieces gives the Schwarzschild line element,

$$
\d s^2 = -\left(1 - \frac{2GM}{c^2 r}\right)c^2\,\d t^2
       + \left(1 - \frac{2GM}{c^2 r}\right)^{-1}\d r^2
       + r^2\bigl(\d\theta^2 + \sin^2\theta\,\d\phi^2\bigr) .
$$

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

$$
r_s = \frac{2GM}{c^2},
$$

so that $g_{tt} = -(1 - r_s/r)c^2$ and $g_{rr} = (1 - r_s/r)^{-1}$. For the Sun,
$r_s \approx 2.95\ \text{km}$; for the Earth, $r_s \approx 8.9\ \text{mm}$. 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.

> **Definition (Schwarzschild radius).** The length $r_s = 2GM/c^2$ is the radius
> at which the metric coefficient $g_{tt}$ vanishes and $g_{rr}$ diverges. A body
> compressed within $r_s$ is a black hole with an event horizon at $r = r_s$; for
> any body larger than $r_s$ the surface intervenes and the coordinate value
> $r = r_s$ lies inside the matter, where a different interior solution holds.

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
  $\d\tau = \sqrt{1 - r_s/r}\,\d t < \d t$: it runs slow relative to $t$, the
  gravitational time dilation.
- **$r$** is the areal radius, fixed by the sphere's area $4\pi r^2$, not a
  proper distance from the center.
- **$\theta, \phi$** are ordinary polar angles on each sphere.

$$
% caption: The metric coefficients against radius, in units of the Schwarzschild
% radius. The solid curve is the time part, $-g_{tt}/c^2 = 1 - r_s/r$, rising from
% zero at $r_s$ toward one far away; the dotted curve is the radial part,
% $g_{rr} = (1 - r_s/r)^{-1}$, diverging at $r_s$ and falling toward one. Both
% approach the flat value one at large $r$.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.2,0) -- (5.6,0) node[right] {r};
  \draw[->, black] (0,-0.2) -- (0,3.6) node[above] {metric value};
  % horizon line at r = r_s (x=0 corresponds to r=r_s; use x=r/rs-1 scaled)
  \draw[black, dashed] (0,0) -- (0,3.4);
  \node[black, anchor=south, rotate=90] at (-0.02,1.7) {r $=$ r$_s$};
  % asymptote y=1
  \draw[black, dashed] (0,1) -- (5.4,1);
  \node[black, anchor=west] at (4.7,1.18) {value one};
  % x = (r/rs - 1), so r/rs from 1..6 -> x 0..5
  % f = 1 - 1/(r/rs) ; at r/rs=1->0, 1.25->0.2, 1.5->0.333,2->0.5,3->0.667,4->0.75,6->0.833
  \draw[acc, very thick] plot coordinates
    {(0,0)(0.25,0.2)(0.5,0.333)(1,0.5)(2,0.667)(3,0.75)(5,0.833)};
  \node[acc, anchor=west] at (3.2,0.62) {time part};
  % g = 1/f = (r/rs)/(r/rs - 1); at 1.45->3.22,1.5->3,2->2,2.5->1.667,3->1.5,4->1.333,6->1.2
  \draw[black, very thick, densely dotted] plot coordinates
    {(0.45,3.22)(0.5,3)(1,2)(1.5,1.667)(2,1.5)(3,1.333)(5,1.2)};
  \node[black, anchor=west] at (1.7,2.5) {radial part};
\end{tikzpicture}
$$

## The far field and the Newtonian limit

Far from the mass, $r \gg r_s$, expand the coefficients in the small ratio
$r_s/r$:

$$
g_{tt} = -\left(1 - \frac{r_s}{r}\right)c^2
       = -c^2\left(1 - \frac{2GM}{c^2 r}\right)
       = -c^2\left(1 + \frac{2\Phi}{c^2}\right),
$$

with $\Phi = -GM/r$ the Newtonian potential. A slowly moving particle in this
geometry follows the geodesic equation, whose spatial part reduces to
$\ddot{\vec r} = -\nabla\Phi = -\tfrac{GM}{r^2}\hat r$, 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 ($r_s/r \sim 10^{-8}$ at the Sun's surface) and responsible for the
small deviations the classical tests measure.

> **Worked example (Redshift from the Sun's surface).** A photon emitted at the
> solar surface $r_\odot = 6.96\times10^8\ \text{m}$ and received far away is
> redshifted by the ratio of clock rates. With $r_s = 2.95\times10^3\ \text{m}$,
>
> $$
> \frac{\Delta\lambda}{\lambda} = \frac{1}{\sqrt{1 - r_s/r_\odot}} - 1
> \approx \frac{r_s}{2 r_\odot}
> = \frac{2.95\times10^3}{2 \times 6.96\times10^8}
> \approx 2.1\times10^{-6}.
> $$
>
> The fractional shift is about two parts per million, matching the measured
> gravitational redshift of solar spectral lines. The
> [redshift lesson](/relativity/tests-of-general-relativity/gravitational-redshift-and-shapiro-delay)
> derives the exact factor from $g_{tt}$.

## 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 $\theta = \pi/2$:

$$
\d s^2 = \left(1 - \frac{r_s}{r}\right)^{-1}\d r^2 + r^2\,\d\phi^2 .
$$

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

$$
1 + z'(r)^2 = \left(1 - \frac{r_s}{r}\right)^{-1}
\;\Longrightarrow\;
z'(r) = \sqrt{\frac{r_s}{r - r_s}} .
$$

Integrating gives **Flamm's paraboloid**,

$$
z(r) = 2\sqrt{r_s\,(r - r_s)}, \qquad\text{equivalently}\qquad r = r_s + \frac{z^2}{4 r_s},
$$

a parabola of revolution. Far away it flattens to a plane (Euclidean space); near
$r_s$ its slope becomes vertical, the geometric picture of $g_{rr}$ 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 $g_{tt}$.

$$
% caption: Flamm's paraboloid embeds the equatorial spatial slice of the
% Schwarzschild geometry. The throat at r-s flattens to a plane far from the mass;
% the vertical slope near r-s is the geometric form of g-rr diverging.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % z = 2 sqrt(rs (r - rs)) with rs=0.5; profile in the (r,z) plane, mirrored
  % r: 0.5..4 ; z=2 sqrt(0.5(r-0.5)) ; r=0.5->0,1->1,1.5->1.41,2->1.73,3->2.24,4->2.65
  % draw as a funnel: two profile curves and a few ellipse rings
  \draw[acc, very thick] plot coordinates
    {(0.5,0)(1,1)(1.5,1.41)(2,1.73)(3,2.24)(4,2.65)};
  \draw[acc, very thick] plot coordinates
    {(-0.5,0)(-1,1)(-1.5,1.41)(-2,1.73)(-3,2.24)(-4,2.65)};
  % rings (ellipses) at several heights
  \draw[black] (0,0) ellipse (0.5 and 0.16);
  \draw[black] (0,1) ellipse (1.0 and 0.30);
  \draw[black] (0,1.73) ellipse (2.0 and 0.55);
  \draw[black] (0,2.65) ellipse (4.0 and 1.0);
  \node[anchor=west] at (2.7,2.0) {far region};
  \node[black, anchor=north] at (0,-0.2) {throat at r$_s$};
\end{tikzpicture}
$$

## 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.

> **Theorem (Birkhoff).** Any spherically symmetric solution of the vacuum
> Einstein equations is static and is locally isometric to the Schwarzschild
> metric. The Schwarzschild geometry is the unique spherically symmetric vacuum,
> parametrized only by the mass $M$.

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.

$$
% caption: Birkhoff's theorem: a spherically symmetric star that pulsates or
% collapses (dashed radii) leaves its exterior field unchanged, fixed only by the
% total mass M. The exterior geometry is Schwarzschild at all times.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % pulsating star: two radii
  \fill[acc!12, draw=acc] (0,0) circle (0.9);
  \draw[acc, densely dotted] (0,0) circle (1.3);
  \draw[acc, densely dotted] (0,0) circle (0.55);
  \node at (0,0) {M};
  \draw[<->, black] (0.9,0.0) -- (1.3,0.0);
  \draw[<->, black] (-0.55,0.0) -- (-0.9,0.0);
  \node[acc, anchor=south] at (0,1.35) {pulsating star};
  % exterior field spheres unchanged
  \draw[black] (0,0) circle (2.1);
  \draw[black] (0,0) circle (2.7);
  \node[black, anchor=west] at (2.15,1.4) {exterior set by M};
  \draw[->, black] (2.0,1.25) -- (1.55,0.75);
\end{tikzpicture}
$$

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 $M = M_\odot$, and every planetary
orbit and light ray is a geodesic of that one metric. The
[next lesson](/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
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.

[^hartle-sph]: **Hartle**, _Gravity_, Ch. 9 — the geometry outside a spherical star, the Schwarzschild line element, the areal-radius interpretation of $r$, and the Newtonian correspondence $g_{tt} = -(1 + 2\Phi/c^2)$.

[^carroll-schw]: **Carroll**, _Lecture Notes on General Relativity_, §7 — the static spherically symmetric ansatz, integration of the vacuum equations $R_{\mu\nu} = 0$, the Schwarzschild radius, Birkhoff's theorem, and the embedding of the spatial slice.