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Black Holes and the Classic Tests

Solve Einstein's equations for a star, meet the event horizon, and put general relativity on trial against Mercury, starlight, and the ticking of clocks.

The Schwarzschild solution

Within months of Einstein's 1915 equations, Karl Schwarzschild found the first exact solution — the geometry of empty space outside any static, spherical mass M (a star, a planet, the Sun). In the usual coordinates the Schwarzschild metric reads:

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

The Schwarzschild metric: the gravitational field of the Sun, Earth, or a black hole, all in one line. Far away (r >> r_s) it smoothly becomes flat Minkowski space.

Everything is controlled by one length, the Schwarzschild radius:

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

For the Sun r_s is about 3 km; for the Earth about 9 mm. Both sit far inside the body, so no horizon forms — the vacuum solution only applies outside.

Black holes and the event horizon

The metric does something dramatic at r=r_s: the time coefficient vanishes and the radial one blows up. If a mass is compressed entirely within its own Schwarzschild radius, that surface becomes an event horizon — a one-way membrane. Inside it, every future-pointing light cone tips inward toward r=0; not even light can climb back out. The object is a black hole.

Classical general relativity says a black hole is utterly black. That statement is not the last word: Hawking showed that quantum effects near the horizon make a black hole glow ever so faintly (Hawking radiation) and slowly evaporate — a first hint of the quantum-gravity frontier this track's Vol-II sibling on Cosmology & Beyond takes up.

Test 1: the perihelion precession of Mercury

A Newtonian planet traces a perfectly closed ellipse, returning to the same perihelion forever. The Schwarzschild geodesic adds a small correction that makes the ellipse slowly rotate — the orbit becomes a rosette that never quite closes. Per orbit, the perihelion advances by approximately:

\Delta\phi \approx \frac{6\pi G M}{c^2\,a\,(1 - e^2)}

The relativistic perihelion advance per orbit, for semi-major axis a and eccentricity e. For Mercury it sums to the famous 43 arcseconds per century.

An orbit in the Schwarzschild geometry precesses instead of closing: the ellipse's long axis slowly rotates, tracing a rosette. Increase the relativistic correction and the precession grows. (Switch to light-bending mode to preview the next test.)

This was general relativity's first triumph. Mercury's orbit was already known to precess by an unexplained 43″ per century after every other effect (the pull of the other planets, the Sun's oblateness) was subtracted. Einstein's formula produced that number with no free parameters — a genuine postdiction that reportedly gave him heart palpitations.

Test 2: the bending of starlight

Because light follows null geodesics, it too is deflected as it grazes a mass. For a ray with impact parameter b passing a mass M, general relativity predicts a total deflection angle of:

Starlight grazing the Sun is bent, so the apparent position of a background star shifts outward. Measuring that shift during a total eclipse was the 1919 test that made Einstein world-famous.

\delta\phi = \frac{4GM}{c^2\,b} = \frac{2\,r_s}{b}

The GR light-deflection angle — exactly twice what a naive equivalence-principle argument (time curvature alone) gives, because space curvature contributes equally for light.

Let us actually put numbers into the formula and predict the deflection of starlight grazing the edge of the Sun — the very quantity Eddington measured.

  1. Collect the constants: G = 6.67\times10^{-11}\,\mathrm{N\,m^2/kg^2}, the Sun's mass M_\odot = 1.99\times10^{30}\,\mathrm{kg}, c = 3.00\times10^{8}\,\mathrm{m/s}, and take the impact parameter as the Sun's radius, b = R_\odot = 6.96\times10^{8}\,\mathrm{m}.
  2. First get the Sun's Schwarzschild radius: r_s = 2GM_\odot/c^2 \approx 2.95\times10^3\,\mathrm{m} — the Sun's whole mass 'measured as a length,' about 3 km.
  3. Now the deflection: \delta\phi = 2r_s/R_\odot = 2(2.95\times10^3)/(6.96\times10^8) \approx 8.5\times10^{-6} radians.
  4. Convert to arcseconds (1\,\mathrm{rad}=206265''): 8.5\times10^{-6}\times206265 \approx 1.75''. This matches Eddington's measurement — and is exactly double the $0.87''$ that the equivalence principle alone would give.

Test 3 and beyond: redshift, waves, and the frontier

The third classic test is gravitational redshift: a clock deeper in a gravitational well runs slower, so light climbing out of the well loses frequency and reddens. For a weak field over a height h (the Pound-Rebka experiment, 1959), and exactly in the Schwarzschild geometry, respectively:

\frac{\Delta\nu}{\nu} \approx \frac{gh}{c^2}, \qquad \frac{\nu_\infty}{\nu} = \sqrt{1 - \frac{r_s}{r}}

Gravitational redshift. It is why GPS satellites, whose clocks run faster high up, must be corrected by about 38 microseconds per day to keep navigation accurate.

General relativity has since passed every test thrown at it. In 2015 LIGO directly detected gravitational waves — ripples in spacetime from merging black holes, arriving exactly as Einstein's equations predict a century earlier — and in 2019 the Event Horizon Telescope imaged the shadow of a black hole. The theory is arguably the best-tested in physics.