Schwarzschild Solution from Spherical Vacuum

Derivation

Schwarzschild Solution from Spherical Vacuum

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Statement

Assuming a static, spherically symmetric spacetime, the vacuum Einstein equations \( R_{\mu\nu}=0 \) admit a unique one-parameter family of solutions, the Schwarzschild metric \( ds^2 = -\left(1-\frac{2GM}{c^2 r}\right)c^2\,dt^2 + \left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2 + r^2\left(d\theta^2 + \sin^2\theta\,d\phi^2\right) \), and by Birkhoff's theorem the staticity assumption is redundant: any spherically symmetric vacuum region is necessarily a piece of this static geometry.

Why it matters

The Schwarzschild metric is the exterior gravitational field of every non-rotating, uncharged mass — a star, a planet, or a black hole — and it underwrites the classical tests of general relativity: perihelion precession, light bending, gravitational redshift, and the Shapiro delay. It is the first exact solution of the field equations ever found (Schwarzschild, 1916) and the template for all later black-hole physics.

Birkhoff's theorem is the relativistic analogue of Newton's shell theorem, but sharper: it forbids a pulsating spherical star from radiating gravitational waves, and it guarantees that the interior of a spherical cavity is flat. Spherical symmetry alone freezes the exterior into a static form, so monopole gravitational radiation does not exist.

Assumptions

Vacuum exterior: \( T_{\mu\nu}=0 \), hence \( R_{\mu\nu}=0 \).With matter present the right-hand side is \( 8\pi G/c^4\,T_{\mu\nu} \) and the metric functions couple to pressure and density (interior/TOV solution), not to the exterior form derived here. Spherical symmetry: the isometry group contains \( SO(3) \) acting on spheres of area \( 4\pi r^2 \).Dropping it admits axisymmetric rotating solutions (Kerr) or anisotropic geometries; the areal-radius coordinate \( r \) and the two-function ansatz both fail. Areal (Schwarzschild) radial coordinate: \( r \) is defined so that a sphere of symmetry has area \( 4\pi r^2 \).This gauge fixes the angular block to \( r^2 d\Omega^2 \); a different radial coordinate (isotropic, tortoise) reshuffles \( g_{tt},g_{rr} \) but leaves the invariant geometry unchanged. Asymptotic flatness: \( g_{\mu\nu}\to\eta_{\mu\nu} \) as \( r\to\infty \).Needed to fix the integration constant that rescales \( t \) and to match the Newtonian potential; without it a constant conformal factor on \( dt^2 \) remains undetermined.

Derivation

\[ ds^2 = -e^{2\alpha(r)}\,c^2 dt^2 + e^{2\beta(r)}\,dr^2 + r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right) \]

Most general static (no \( t \)-dependence, no \( dt\,dr \) cross term), spherically symmetric metric in areal gauge; exponentials keep the metric signature manifest for all real \( \alpha,\beta \). A

\[ \Gamma^{r}_{tt}=e^{2(\alpha-\beta)}\alpha' c^2,\quad \Gamma^{t}_{tr}=\alpha',\quad \Gamma^{r}_{rr}=\beta',\quad \Gamma^{\theta}_{r\theta}=\Gamma^{\phi}_{r\phi}=\frac1r,\quad \Gamma^{r}_{\theta\theta}=-r e^{-2\beta} \]

Christoffel symbols \( \Gamma^{\lambda}_{\mu\nu}=\tfrac12 g^{\lambda\sigma}(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}) \); primes denote \( d/dr \). Only \( r \)-derivatives survive by staticity and diagonality. B

\[ R_{tt}=e^{2(\alpha-\beta)}c^2\!\left[\alpha''+\alpha'^2-\alpha'\beta'+\frac{2\alpha'}{r}\right],\qquad R_{rr}=-\left[\alpha''+\alpha'^2-\alpha'\beta'-\frac{2\beta'}{r}\right] \]

Contract the Riemann tensor, \( R_{\mu\nu}=\partial_\lambda\Gamma^{\lambda}_{\mu\nu}-\partial_\nu\Gamma^{\lambda}_{\mu\lambda}+\Gamma^{\lambda}_{\lambda\sigma}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\nu\sigma}\Gamma^{\sigma}_{\mu\lambda} \). C

\[ R_{\theta\theta}=1-e^{-2\beta}\left[1+r(\alpha'-\beta')\right],\qquad R_{\phi\phi}=\sin^2\theta\;R_{\theta\theta} \]

Angular Ricci components; \( R_{\phi\phi} \) is fixed by \( R_{\theta\theta} \) through the \( \sin^2\theta \) of the round metric, so it carries no new information. C

\[ \frac{e^{-2(\alpha-\beta)}}{c^2}R_{tt}+R_{rr}=\frac{2}{r}\left(\alpha'+\beta'\right)=0 \;\Longrightarrow\; \alpha'+\beta'=0 \]

Vacuum sets every \( R_{\mu\nu}=0 \); this particular linear combination cancels the second-derivative terms, isolating a first-order relation. B

\[ \alpha(r)+\beta(r)=\text{const}\;\xrightarrow{\;t\to e^{-\text{const}}t\;}\;\beta=-\alpha \]

Integrate; the additive constant is a constant rescaling of \( t \), removed by demanding asymptotic flatness \( \alpha,\beta\to0 \). A

\[ R_{\theta\theta}=0:\quad 1-e^{2\alpha}\left(1+2r\alpha'\right)=0 \]

Substitute \( \beta=-\alpha \) (so \( e^{-2\beta}=e^{2\alpha} \) and \( \alpha'-\beta'=2\alpha' \)) into Step 4. B

\[ \text{Let } f\equiv e^{2\alpha}:\quad f+rf'=\frac{d}{dr}\!\left(rf\right)=1 \;\Longrightarrow\; rf=r-r_s \]

Recognise \( e^{2\alpha}(1+2r\alpha')=f+rf' \) as an exact derivative; integrate once, \( r_s \) the integration constant. B

\[ f(r)=e^{2\alpha}=1-\frac{r_s}{r},\qquad e^{2\beta}=f^{-1}=\left(1-\frac{r_s}{r}\right)^{-1} \]

Divide by \( r \); the \( R_{tt}=R_{rr}=0 \) equations are then satisfied identically, so no further constraint arises (the system is consistent, not over-determined). B

\[ g_{tt}=-c^2\!\left(1-\frac{r_s}{r}\right)\;\xrightarrow{\,r\to\infty\,}\;-c^2\!\left(1+\frac{2\Phi}{c^2}\right),\;\;\Phi=-\frac{GM}{r}\;\Longrightarrow\; r_s=\frac{2GM}{c^2} \]

Match the weak-field limit \( g_{tt}=-c^2(1+2\Phi/c^2) \) of the geodesic equation to Newtonian gravity to identify the constant. A

\[ \text{Drop staticity: }\alpha,\beta\to\alpha(t,r),\beta(t,r).\quad R_{tr}=\frac{2}{r}\,\dot\beta=0\;\Longrightarrow\;\beta=\beta(r) \]

Birkhoff step: with time dependence allowed a mixed component \( R_{tr} \) appears; vacuum forces \( \partial_t\beta=0 \). C

\[ \alpha'+\beta'=0\Rightarrow \alpha(t,r)=-\beta(r)+h(t);\quad t\to\!\int e^{h(t)}dt\;\text{removes }h(t) \]

The same combination as Step 5 still holds; the surviving arbitrary function of \( t \) is pure gauge, absorbed by redefining the time coordinate. The metric is therefore static — Birkhoff's theorem. C

Result

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

Reading. Outside any spherical mass \( M \) the geometry depends on a single parameter, the Schwarzschild radius \( r_s=2GM/c^2 \). Time runs slower and radial rulers stretch as \( r\to r_s \); at \( r=r_s \) the coordinates \( t,r \) break down (a coordinate, not curvature, singularity), while \( r=0 \) is a true curvature singularity where \( R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12r_s^2/r^6\to\infty \). Birkhoff guarantees this holds even while the source pulsates spherically.

Units check. \( [GM/c^2 r] = \dfrac{(\mathrm{m^3\,kg^{-1}\,s^{-2}})(\mathrm{kg})}{(\mathrm{m^2\,s^{-2}})(\mathrm{m})} = \dfrac{\mathrm{m^3\,s^{-2}}}{\mathrm{m^3\,s^{-2}}} \) is dimensionless, so \( g_{tt} \) carries the \( c^2 \) and \( [ds^2]=\mathrm{m^2} \); \( r_s=2GM/c^2 \) has units of metres. Consistent.

Limiting cases

\( r\gg r_s \): \( g_{tt}\approx -c^2(1-2GM/c^2r) \), reproducing the Newtonian potential \( \Phi=-GM/r \) and the inverse-square force.
\( M\to0 \) or \( r\to\infty \): \( ds^2\to -c^2dt^2+dr^2+r^2d\Omega^2 \), flat Minkowski space in spherical coordinates.
\( r\to r_s^+ \): \( g_{tt}\to0 \), \( g_{rr}\to\infty \) — infinite gravitational redshift and coordinate freezing at the horizon; regular in Eddington–Finkelstein or Kruskal coordinates.
Weak field, slow motion: geodesics reduce to \( \ddot{\vec r}=-\nabla\Phi \), recovering Newtonian orbital mechanics with the leading GR correction \( \sim r_s/r \).

Breaks when

Interior of the source (\( T_{\mu\nu}\neq0 \)): inside a star the vacuum equations no longer apply; one must solve the Tolman–Oppenheimer–Volkoff equations, and \( g_{tt},g_{rr} \) are set by the pressure and density profile, not by \( 1-r_s/r \).
Rotation or charge: angular momentum breaks spherical symmetry, giving the Kerr metric (frame dragging, ergosphere); net charge adds a stress-energy term giving Reissner–Nordström. Birkhoff does not extend to either.
Cosmological constant or non-asymptotic-flatness: with \( \Lambda\neq0 \) the vacuum equation is \( R_{\mu\nu}=\Lambda g_{\mu\nu} \), yielding Schwarzschild–de Sitter \( f=1-r_s/r-\Lambda r^2/3 \); the pure Schwarzschild form fails at large \( r \).
At the curvature singularity \( r=0 \): tidal invariants diverge, classical general relativity ceases to be predictive, and a quantum theory of gravity is required.

Failure modes

Treating \( r=r_s \) as a physical singularity. Computing \( R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12r_s^2/r^6 \) shows it is finite at \( r_s \); the blow-up of \( g_{rr} \) is a coordinate artifact.
Interpreting \( r \) as a radial distance. \( r \) is the areal radius (area\( =4\pi r^2 \)); proper radial distance is \( \int (1-r_s/r)^{-1/2}dr \), which exceeds \( \Delta r \).
Forgetting to rescale \( t \) in Step 6. Leaving the constant \( \alpha+\beta \) unfixed produces a spurious constant factor on \( dt^2 \) and a metric that is not asymptotically flat.
Claiming a pulsating star radiates gravitationally. Birkhoff forbids monopole radiation; students often expect the field to "wobble" with the source.
Dropping the \( \sin^2\theta \) and treating \( R_{\phi\phi}=0 \) as independent. It is \( \sin^2\theta\,R_{\theta\theta} \); using it as a fourth equation over-counts constraints.
Using coordinate time \( t \) as a clock rate. Proper time is \( d\tau=\sqrt{1-r_s/r}\,dt \); redshift factors get inverted if this is missed.

Discussion

The derivation is remarkable for how little freedom the field equations leave. Two unknown functions \( \alpha(r),\beta(r) \) are pinned by essentially two independent vacuum equations — one first-order relation forcing \( \alpha=-\beta \), and one integrable equation for \( f=e^{2\alpha} \) — so the entire exterior geometry collapses to a single constant \( r_s \), identified with mass by Newtonian matching. This rigidity is the content of "no hair" in its simplest form: a static spherical vacuum has exactly one label.

Birkhoff's theorem elevates this from a convenient ansatz to a physical law. Because spherical symmetry alone (without assuming staticity) forces the metric to be static, the gravitational field outside a radially oscillating star is frozen and cannot carry energy away — the lowest multipole of gravitational radiation is quadrupole, not monopole or dipole. This is why a collapsing spherical dust cloud produces no gravitational waves, and why the field just outside its surface is Schwarzschild throughout the collapse.

Geometrically, the theorem follows because sphericity supplies a preferred areal radius \( r \), and the \( tr \)-component of the vacuum equation forces the \( g_{rr} \) function to be time-independent; the residual time dependence in \( g_{tt} \) is then a pure reparametrisation of \( t \). The deeper statement is that \( \partial_t \) emerges as a Killing vector that was not assumed — the vacuum equations manufacture their own timelike symmetry. Outside the horizon this Killing vector is timelike (staticity); inside \( r_s \) it becomes spacelike, which is the coordinate-invariant reason the roles of \( t \) and \( r \) swap and why crossing the horizon is causally one-way.

Common misconceptions. The Schwarzschild solution is often called "the metric of a black hole," but it is equally the exterior of the Sun or Earth; only when the entire mass lies within \( r_s \) does an event horizon become physical. Likewise, \( r_s \) is not where gravity becomes strong for ordinary bodies — for the Sun \( r_s\approx3 \) km sits deep inside, where the vacuum solution does not even apply.

Worked examples

\[ \text{Schwarzschild radius of the Sun: } r_s=\frac{2GM_\odot}{c^2} \]

Symbolic form first; insert \( G=6.674\times10^{-11}\ \mathrm{m^3kg^{-1}s^{-2}} \), \( M_\odot=1.989\times10^{30}\ \mathrm{kg} \), \( c=2.998\times10^{8}\ \mathrm{m\,s^{-1}} \). A

\[ r_s=\frac{2(6.674\times10^{-11})(1.989\times10^{30})}{(2.998\times10^{8})^2}=\frac{2.655\times10^{20}}{8.988\times10^{16}}\ \mathrm{m} \]

Numerator \( \mathrm{m^3\,s^{-2}} \), denominator \( \mathrm{m^2\,s^{-2}} \), quotient in metres. A

\[ r_s^{\odot}\approx 2.95\times10^{3}\ \mathrm{m}\approx 2.95\ \mathrm{km} \]

Reading. The Sun's true radius (\( 6.96\times10^{5}\ \mathrm{km} \)) dwarfs \( r_s \) by a factor \( \sim2\times10^{5} \), so its exterior field is only weakly curved and no horizon exists.

\[ \text{Gravitational redshift of light leaving the Sun's surface: } 1+z=\frac{1}{\sqrt{1-r_s/R_\odot}} \]

Photon frequency scales as \( \sqrt{-g_{tt}}=\sqrt{1-r_s/r} \) between emitter at \( R_\odot \) and observer at infinity. B

\[ \frac{r_s}{R_\odot}=\frac{2.95\times10^{3}}{6.96\times10^{8}}=4.24\times10^{-6},\qquad z\approx\frac{r_s}{2R_\odot} \]

Since \( r_s/R_\odot\ll1 \), Taylor-expand \( (1-x)^{-1/2}\approx1+x/2 \). B

\[ z\approx 2.12\times10^{-6},\qquad \frac{\Delta\lambda}{\lambda}\approx 2.1\ \mathrm{ppm} \]

Reading. A \( 500\ \mathrm{nm} \) line is redshifted by \( \Delta\lambda\approx1.1\times10^{-3}\ \mathrm{nm} \), a shift of about \( 0.64\ \mathrm{km\,s^{-1}} \) in velocity units — measured and confirmed in the solar spectrum.

Problems

Compute the Schwarzschild radius of the Earth (\( M_\oplus=5.97\times10^{24}\ \mathrm{kg} \)) and compare it to the Earth's radius \( 6.37\times10^{6}\ \mathrm{m} \).
Solution
\( r_s=2GM_\oplus/c^2 = 2(6.674\times10^{-11})(5.97\times10^{24})/(8.988\times10^{16}) = 7.97\times10^{14}/8.988\times10^{16}\approx 8.9\times10^{-3}\ \mathrm{m}\approx 8.9\ \mathrm{mm} \). The ratio \( r_s/R_\oplus\approx1.4\times10^{-9} \): Earth's gravity is extraordinarily weak in relativistic terms.
Show that the Kretschmann scalar \( K=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12r_s^2/r^6 \) is finite at \( r=r_s \) but diverges at \( r=0 \), and evaluate \( K \) at the Sun's horizon radius.
Solution
At \( r=r_s \), \( K=12r_s^2/r_s^6=12/r_s^4 \), finite — so \( r=r_s \) is a coordinate singularity. As \( r\to0 \), \( K\sim r^{-6}\to\infty \), a genuine curvature singularity. For the Sun, \( r_s=2.95\times10^{3}\ \mathrm{m} \): \( K=12/(2.95\times10^{3})^4=12/7.57\times10^{13}\approx1.6\times10^{-13}\ \mathrm{m^{-4}} \).
A clock sits at \( r=4r_s \) outside a black hole. Find its proper time rate relative to a distant observer, \( d\tau/dt \).
Solution
\( d\tau/dt=\sqrt{1-r_s/r}=\sqrt{1-1/4}=\sqrt{3}/2\approx0.866 \). The local clock ticks at about \( 86.6\% \) of the distant rate; over one distant day it advances \( 20.8 \) hours.
Compute the proper radial distance between \( r=2r_s \) and \( r=3r_s \), \( \ell=\int_{2r_s}^{3r_s}(1-r_s/r)^{-1/2}dr \), and compare it to the coordinate difference \( r_s \).
Solution
Substitute \( r=r_s u \): \( \ell=r_s\int_{2}^{3}\sqrt{u/(u-1)}\,du \). The antiderivative is \( \sqrt{u(u-1)}+\ln\!\left(\sqrt{u}+\sqrt{u-1}\right) \); evaluating from \( u=2 \) to \( u=3 \) gives \( \left[\sqrt6+\ln(\sqrt3+\sqrt2)\right]-\left[\sqrt2+\ln(\sqrt2+1)\right]\approx(2.449+1.147)-(1.414+0.881)\approx1.30 \). Thus \( \ell\approx1.30\,r_s \), about \( 30\% \) larger than the coordinate gap \( r_s \): space is radially stretched near the horizon.
Include a cosmological constant: solve \( R_{\mu\nu}=\Lambda g_{\mu\nu} \) with the same static spherical ansatz and show \( f(r)=1-r_s/r-\Lambda r^2/3 \). Identify the two horizons for small \( \Lambda>0 \).
Solution
Repeating Steps 5–8, \( \alpha=-\beta \) still holds, and \( R_{\theta\theta}=\Lambda g_{\theta\theta} \) gives \( (rf)'=1-\Lambda r^2 \), so \( rf=r-r_s-\Lambda r^3/3 \), i.e. \( f=1-r_s/r-\Lambda r^2/3 \) (Schwarzschild–de Sitter). Setting \( f=0 \): for small \( \Lambda \) there is a black-hole horizon near \( r\approx r_s \) and a cosmological horizon near \( r\approx\sqrt{3/\Lambda} \). The pure Schwarzschild form is recovered as \( \Lambda\to0 \).