Tests and Predictions of General Relativity

A theory is only as good as its confrontations with reality. General relativity is the most stringently tested theory of gravity ever written, and it has passed every trial — often predicting effects before anyone could measure them, sometimes to eleven decimal places. This page walks the classic gauntlet: the anomaly in Mercury's orbit that Newton could not explain, the bending of starlight that made Einstein famous overnight, the reddening of light climbing out of gravity, the relativity corrections your phone's GPS depends on, and the gravitational waves that opened a new astronomy in 2015.

Every one of these follows from the Schwarzschild geometry (or its rotating and radiating cousins) and the rule that free things follow geodesics. The theory has no free parameters to tune — given the masses, the predictions are fixed. That is what makes the agreement so persuasive.

Test 1: Mercury's wandering perihelion

By the 1850s astronomers had a problem. Mercury's elliptical orbit does not close on itself: its point of closest approach to the Sun, the perihelion, creeps forward a little each orbit, tracing a slow rosette. Most of that creep — the pull of the other planets — Newtonian mechanics explained. But 43 arcseconds per century stubbornly remained. Astronomers even invented a phantom inner planet, Vulcan, to account for it. There was no Vulcan.

General relativity nailed it with no new bodies at all. The Schwarzschild metric adds a small 1/r^3 correction to the Newtonian orbit, so the ellipse does not quite close and its axis rotates by

\Delta\phi = \frac{6\pi GM}{c^2\,a\,(1 - e^2)}\quad\text{per orbit}.

For Mercury — nearest the Sun, so deepest in its curvature — this comes to exactly 43 arcseconds per century. Einstein wrote that when he saw the numbers match he had "palpitations of the heart." Reveal the figure to watch the ellipse turn (wildly exaggerated).

Test 2: light bending, and the 1919 eclipse

The equivalence principle already told us gravity bends light. General relativity makes it quantitative: a ray grazing a mass M at impact parameter b is deflected by

\alpha = \frac{4GM}{c^2 b},

which is exactly twice the value a naive "Newtonian photon" would give — the factor of two comes from the curvature of space adding equally to the curvature of time. For a ray skimming the Sun's edge, \alpha = 1.75''. In May 1919 Arthur Eddington photographed stars near the eclipsed Sun and found them shifted by just that amount. The headline "Lights All Askew in the Heavens" made Einstein world-famous within a week. The deflection falls off as 1/b — double the impact parameter, halve the bend.

Today this same bending is a workhorse tool: gravitational lensing magnifies distant galaxies, splits quasars into multiple images, and weighs invisible dark matter by how it warps the light passing through it.

Tests 3–5: redshift, GPS, and gravitational waves

Gravitational redshift. Light climbing out of a gravitational well loses energy and reddens by \Delta f/f = \Delta\Phi/c^2. Pound and Rebka measured it in 1959 over a 22.5-m Harvard tower; today optical clocks resolve the shift over a height change of a single centimetre. The Sun's own light is redshifted, and white dwarfs more so.

GPS. Relativity is not a curiosity here — it is engineering. Each satellite clock runs fast by about 45\ \mu\text{s/day} from weaker gravity (general relativity) and slow by about 7\ \mu\text{s/day} from its orbital speed (special relativity), a net gain of 38\ \mu\text{s/day}. Left uncorrected, positions would drift by about 10\ \text{km} per day. The satellites' clocks are deliberately detuned before launch to compensate.

Gravitational waves. Accelerating masses radiate ripples in spacetime itself, travelling at c. On 14 September 2015, LIGO caught the waves from two black holes (about 36 and 29 solar masses) spiralling together and merging a billion light-years away — a signal stretching spacetime by less than one part in 10^{21}, a fraction of a proton's width across a 4-km arm. It won the 2017 Nobel Prize and opened an entirely new way to observe the universe.

It is a lovely bookkeeping story. If you pretend light is a fast Newtonian particle falling in the Sun's gravity, you get a deflection of 2GM/c^2 b — the "half" answer, which Einstein himself first published in 1911 using only the equivalence principle (time curvature). The other half comes from something the equivalence principle alone cannot see: the curvature of space. The full Schwarzschild metric bends both the time part g_{tt} and the space part g_{rr}, and for a fast-moving ray the spatial curvature contributes just as much as the temporal, doubling the answer to 4GM/c^2 b. Slow-moving objects (planets) barely feel the spatial part, so Newton works well for them; light, moving at c, feels both equally. The 1919 measurement therefore tested not just that gravity bends light, but the full, space-and-time structure of general relativity — and the factor of two was the whole point.

A frequent muddle. There are two relativistic clock effects on a GPS satellite, and they have opposite signs. Special relativity (orbital speed \sim 3.9\ \text{km/s}) makes the moving clock run slow by about 7\ \mu\text{s/day}. General relativity (the satellite sits higher in Earth's potential, where gravity is weaker and time runs faster) makes it run fast by about 45\ \mu\text{s/day}. The gravitational effect wins, for a net gain of about 38\ \mu\text{s/day}. Two mistakes to dodge: don't quote only the special-relativistic piece (it has the wrong sign for the net drift), and don't forget that a microsecond of clock error is a third of a kilometre of position error, because signals travel at c. The engineers had to build the correction in — GPS is a working relativity experiment you carry in your pocket.