Schwarzschild Solution

Just weeks after Einstein published his field equations in November 1915, Karl Schwarzschild found the first exact solution while serving on the Eastern Front of World War I. His solution describes the spacetime geometry outside any spherically symmetric, non-rotating mass, a star, a planet, or, taken to its logical extreme, a black hole. It remains the most important exact solution in general relativity and the foundation for most experimental tests of the theory.

Deriving the Solution

The most general static, spherically symmetric line element can be written as:

$$ds^2 = -e^{2\alpha(r)} c^2 \, dt^2 + e^{2\beta(r)} dr^2 + r^2 \, d\Omega^2$$

where $d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2$ is the metric on the unit 2-sphere. Substituting into the vacuum Einstein equations $R_{\mu\nu} = 0$ yields two ordinary differential equations for $\alpha(r)$ and $\beta(r)$. The solution, after imposing asymptotic flatness ($g_{\mu\nu} \to \eta_{\mu\nu}$ as $r \to \infty$), gives:

$$e^{2\alpha} = e^{-2\beta} = 1 - \frac{r_s}{r}$$

The parameter $r_s = 2GM/c^2$ is determined by matching to Newtonian gravity at large distances. For the Sun, $r_s \approx 3$ km; for the Earth, $r_s \approx 9$ mm.

Gravitational Redshift

A photon climbing out of a gravitational well loses energy. If a photon is emitted at radius $r_e$ with frequency $\nu_e$ and observed at radius $r_o > r_e$, the observed frequency is:

$$\frac{\nu_o}{\nu_e} = \sqrt{\frac{1 - r_s/r_e}{1 - r_s/r_o}}$$

For a photon emitted at the surface of the Sun and observed on Earth (where $r_s/r_o \approx 0$):

$$\frac{\Delta\nu}{\nu} \approx \frac{GM}{Rc^2} \approx 2.1 \times 10^{-6}$$

This tiny shift was first measured by Pound and Rebka in 1959 using the Mossbauer effect, and is now routinely accounted for in GPS satellite timing. Clocks run slower in stronger gravitational fields, a direct consequence of the Schwarzschild metric.

Orbital Motion and Mercury's Perihelion

The geodesic equation in the Schwarzschild geometry can be reduced, by symmetry, to motion in the equatorial plane ($\theta = \pi/2$). Conservation of energy and angular momentum gives an effective one-dimensional problem:

$$\frac{1}{2}\left(\frac{dr}{d\tau}\right)^2 + V_{\text{eff}}(r) = \frac{1}{2}(E^2 - c^2)$$

with the effective potential:

$$V_{\text{eff}}(r) = -\frac{GMc^2}{r} + \frac{L^2}{2r^2} - \frac{GML^2}{c^2 r^3}$$

The first two terms are the familiar Newtonian potential and centrifugal barrier. The third term is the relativistic correction, it is always attractive and dominates at small $r$, preventing the centrifugal barrier from growing without bound. This has dramatic consequences.

For nearly circular orbits, the relativistic correction causes the orbit to precess. The perihelion advances by:

This was Einstein's first great triumph. When the calculation yielded the correct value for Mercury's anomalous precession, he reportedly felt that "something actually snapped" inside him.

Light Bending

Photons also follow geodesics, but null geodesics ($ds^2 = 0$). In the Schwarzschild geometry, a light ray passing a mass $M$ at closest approach distance $b$ is deflected by an angle:

$$\delta = \frac{4GM}{bc^2}$$

For light grazing the Sun, $\delta = 1.75''$: twice the Newtonian prediction. This was confirmed by Eddington's famous solar eclipse expedition of 1919, which made Einstein a global celebrity. Today, gravitational lensing is a powerful tool in astronomy: it allows us to map the distribution of dark matter, detect exoplanets, and magnify distant galaxies.

The Schwarzschild Radius

The metric has an apparent singularity at $r = r_s = 2GM/c^2$: the coefficient of $dt^2$ vanishes and the coefficient of $dr^2$ diverges. For ordinary stars, $r_s$ lies deep inside the star where the vacuum solution does not apply. But if an object is compressed below its Schwarzschild radius, the singularity is exposed and profound consequences follow.

Coordinate vs. Physical Singularity

Is $r = r_s$ a genuine singularity of spacetime, or merely a breakdown of coordinates? The key diagnostic is to compute a curvature invariant, a scalar quantity that does not depend on the coordinate choice. The Kretschner scalar is:

$$K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} = \frac{48 G^2 M^2}{c^4 r^6}$$

At $r = r_s$, this is finite: $K = \frac{12}{r_s^4}$. At $r = 0$, it diverges. Therefore $r = r_s$ is a coordinate singularity: an artifact of Schwarzschild coordinates, while $r = 0$ is a physical singularity where curvature truly diverges.

A freely falling observer crosses $r = r_s$ in finite proper time and notices nothing special locally. The coordinate singularity can be removed by choosing better coordinates.

Better Coordinate Systems

Eddington-Finkelstein Coordinates

Define an advanced time coordinate $v = t + r + r_s \ln|r/r_s - 1|$ (in units where $c = 1$). The metric becomes:

$$ds^2 = -\left(1 - \frac{r_s}{r}\right) dv^2 + 2 \, dv \, dr + r^2 \, d\Omega^2$$

This is perfectly regular at $r = r_s$. Ingoing light rays have $v = \text{const}$, making it clear that they cross the horizon smoothly. Outgoing light rays at $r < r_s$ are actually dragged inward, nothing can escape.

Kruskal-Szekeres Coordinates

The maximal analytic extension uses Kruskal-Szekeres coordinates $(T, X)$, related to Schwarzschild coordinates by:

$$T^2 - X^2 = \left(1 - \frac{r}{r_s}\right) e^{r/r_s}$$

The entire spacetime is mapped smoothly. The horizon $r = r_s$ becomes two null lines $T = \pm X$, and the singularity $r = 0$ becomes a spacelike curve $T^2 - X^2 = 1$. The Kruskal diagram reveals the full causal structure: there is not only a black hole region (future singularity) but also a "white hole" region (past singularity) and a second asymptotic region, though the physical relevance of these extensions is debated.

Birkhoff's Theorem

A remarkable uniqueness result: any spherically symmetric vacuum solution of Einstein's equations is necessarily static and given by the Schwarzschild metric. This means that a pulsating, spherically symmetric star produces no gravitational waves, the exterior spacetime is always Schwarzschild. It is the gravitational analogue of the shell theorem in Newtonian gravity.

Birkhoff's theorem has a profound consequence: the gravitational field outside a collapsing star depends only on its total mass, not on the details of the collapse, as long as spherical symmetry is maintained.

Key Insights

• The Schwarzschild metric is the unique spherically symmetric vacuum solution, determined entirely by the mass $M$
• It predicts gravitational redshift (confirmed by Pound-Rebka), orbital precession (confirmed by Mercury), and light bending (confirmed by Eddington)
• The Schwarzschild radius $r_s = 2GM/c^2$ marks a coordinate singularity, not a physical one, freely falling observers cross it smoothly
• The true physical singularity at $r = 0$ has divergent curvature and represents a breakdown of classical general relativity
• Kruskal-Szekeres coordinates reveal the full causal structure, including the black hole interior
• Birkhoff's theorem guarantees that any spherically symmetric vacuum spacetime is Schwarzschild

Looking Ahead

The Schwarzschild solution reveals that when mass is compressed within its Schwarzschild radius, an event horizon forms. In the next lesson, we explore black holes in full generality, including rotating (Kerr) black holes, black hole thermodynamics, Hawking radiation, and the information paradox. These objects, once considered mathematical curiosities, are now observed directly through gravitational waves and the Event Horizon Telescope.