Schwarzschild Solution — Derivation
The first and most important exact solution to Einstein's field equations
In December 1915, just weeks after Einstein published the final form of his field equations, Karl Schwarzschild found the first exact solution while serving on the Eastern Front in World War I. This solution describes the unique spherically symmetric vacuum spacetime — the exterior gravitational field of any non-rotating, uncharged, spherical mass distribution.
The Spherically Symmetric Ansatz
We seek the most general static, spherically symmetric metric. Spherical symmetry demands that the angular part of the metric takes the form of the metric on S², while staticity requires no time-dependent components and no cross terms between time and spatial coordinates. The most general such line element is:
$$ds^2 = -e^{2\alpha(r)} c^2 \, dt^2 + e^{2\beta(r)} dr^2 + r^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right)$$
Here we have used the "areal radius" coordinate, meaning r is defined such that the area of a sphere at coordinate radius r is exactly 4πr². The functions α(r) and β(r) are to be determined by solving Einstein's vacuum field equations Rμν = 0.
Computing the Christoffel symbols for this metric yields the non-vanishing components:
$$\Gamma^t_{tr} = \alpha', \quad \Gamma^r_{tt} = \alpha' e^{2(\alpha - \beta)}, \quad \Gamma^r_{rr} = \beta'$$
$$\Gamma^r_{\theta\theta} = -r e^{-2\beta}, \quad \Gamma^r_{\phi\phi} = -r \sin^2\theta \, e^{-2\beta}$$
$$\Gamma^\theta_{r\theta} = \frac{1}{r}, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta, \quad \Gamma^\phi_{r\phi} = \frac{1}{r}, \quad \Gamma^\phi_{\theta\phi} = \cot\theta$$
where primes denote d/dr. From these, we compute the Ricci tensor components. The vacuum field equations Rμν = 0 give us a system of ordinary differential equations.
Solving the Vacuum Field Equations
The (t,t) and (r,r) components of the Ricci tensor yield:
$$R_{tt} = e^{2(\alpha-\beta)} \left[ \alpha'' + (\alpha')^2 - \alpha'\beta' + \frac{2\alpha'}{r} \right] = 0$$
$$R_{rr} = -\alpha'' - (\alpha')^2 + \alpha'\beta' + \frac{2\beta'}{r} = 0$$
Adding these equations (after multiplying Rtt by e-2(α-β)) gives the remarkably simple result:
$$\frac{2(\alpha' + \beta')}{r} = 0 \quad \Longrightarrow \quad \alpha(r) + \beta(r) = \text{const}$$
The constant can be set to zero by rescaling the time coordinate. Thus α = -β. Now the (θ,θ) component gives:
$$R_{\theta\theta} = e^{-2\beta}\left(r(\beta' - \alpha') - 1\right) + 1 = 0$$
$$\Longrightarrow \quad e^{-2\beta}\left(1 + 2r\alpha'\right) = 1$$
$$\Longrightarrow \quad \frac{d}{dr}\left(r \, e^{2\alpha}\right) = 1$$
This integrates immediately to:
$$e^{2\alpha} = 1 - \frac{C}{r}$$
where C is an integration constant. Matching to the Newtonian limit (where gtt ≈ -(1 + 2Φ/c²) with Φ = -GM/r) identifies C = 2GM/c² ≡ rs, the Schwarzschild radius.
The Metric Components
The Schwarzschild Metric
$$ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 \, dt^2 + \frac{dr^2}{1 - \frac{2GM}{c^2 r}} + r^2 \left(d\theta^2 + \sin^2\theta \, d\phi^2\right)$$
In geometrized units (G = c = 1), the individual metric components are:
$$g_{tt} = -\left(1 - \frac{2M}{r}\right)$$
Gravitational time dilation
$$g_{rr} = \left(1 - \frac{2M}{r}\right)^{-1}$$
Radial distance stretching
$$g_{\theta\theta} = r^2$$
Angular metric (standard sphere)
$$g_{\phi\phi} = r^2 \sin^2\theta$$
Angular metric (standard sphere)
Birkhoff's Theorem
Theorem (Birkhoff, 1923): Any spherically symmetric vacuum solution of Einstein's equations is necessarily static and is given by the Schwarzschild metric.
This is the GR analog of Newton's shell theorem. The profound consequence is that a spherically symmetric pulsating star has no gravitational radiation — the external spacetime remains Schwarzschild regardless of internal dynamics, as long as spherical symmetry is maintained.
Proof sketch: Start with the most general spherically symmetric metric (not assuming staticity): allow α = α(t,r) and β = β(t,r), and include a possible cross term gtr. The (t,r) component of the vacuum equations gives ∂β/∂t = 0, meaning β is time-independent. The remaining equations then force α to be time-independent as well (up to a function of t alone that can be absorbed by reparameterizing the time coordinate). The gtr cross term can be removed by a coordinate transformation. Thus static is not an assumption — it is a consequence.
Coordinate vs. True Singularities
The metric components diverge at two radii: r = 2M (the Schwarzschild radius) and r = 0. These have fundamentally different physical character.
Coordinate Singularity at r = 2M
At r = 2M, gtt → 0 and grr → ∞, but this is an artifact of the coordinate system, not a physical singularity. A freely falling observer crosses r = 2M in finite proper time and experiences nothing locally dramatic. The singularity can be removed by choosing better coordinates (Eddington-Finkelstein or Kruskal-Szekeres, discussed on Page 3).
True Singularity at r = 0
To determine whether a singularity is physical (coordinate-independent), we examine curvature invariants — scalar quantities formed from the Riemann tensor that are the same in all coordinate systems.
Kretschner Scalar Divergence
The Kretschner scalar (the simplest non-trivial curvature invariant) is:
$$K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} = \frac{48 M^2}{r^6}$$
This is a scalar, so it has the same value in every coordinate system. At r = 2M, K = 48M² / (2M)⁶ = 3/(4M⁴), which is perfectly finite. This confirms that r = 2M is merely a coordinate singularity. However, as r → 0:
$$K = \frac{48 M^2}{r^6} \xrightarrow{r \to 0} \infty$$
The divergence of K at r = 0 proves that the curvature itself becomes infinite — this is a genuine physical singularity that cannot be removed by any coordinate transformation. Tidal forces become infinite, and geodesics cannot be extended through this point.
Other curvature invariants confirm this:
$$R = 0 \quad (\text{vacuum}), \qquad R_{\mu\nu}R^{\mu\nu} = 0 \quad (\text{vacuum}), \qquad C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} = \frac{48M^2}{r^6}$$
In vacuum, the Kretschner scalar equals the square of the Weyl tensor since Ricci vanishes.
Physical Schwarzschild Radii
Sun
≈ 3 km
(well inside the Sun)
Earth
≈ 9 mm
(size of a marble)
Sagittarius A*
≈ 12 million km
(0.08 AU)