Cosmology in General Relativity

General relativity's boldest application is to the entire universe. If mass and energy curve spacetime, then the total content of the cosmos must shape the geometry of everything at once. Apply the field equations to a universe filled smoothly with matter and you do not get a static box — you get a spacetime that expands or contracts. Einstein found this so unpalatable that in 1917 he added a term to hold the universe still; when Hubble showed in 1929 that the galaxies really are flying apart, Einstein called that fudge his "greatest blunder." The expansion was general relativity's prediction all along.

This page applies the machinery of curved spacetime to the whole universe: the FRW metric that describes an expanding cosmos, the scale factor and Hubble's law that quantify the stretch, and the Friedmann equations that govern the expansion's fate. It is the same metric-and-geodesics story as before, now written across the sky.

The FRW metric: a universe with one clock

On the largest scales the universe looks the same everywhere (homogeneous) and the same in every direction (isotropic) — the cosmological principle. The most general spacetime with that symmetry is the Friedmann–Robertson–Walker (FRW) metric:

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

Two objects carry all the physics. The scale factor a(t) is the "size of the universe" as a function of cosmic time — double it and every distance between galaxies doubles. The constant k fixes the spatial curvature: k > 0 a closed 3-sphere, k = 0 flat, k < 0 open and saddle-like. Galaxies sit at fixed comoving coordinates (r,\theta,\phi); it is a(t) that grows, carrying them apart like raisins in rising dough.

Hubble's law and cosmological redshift

Because every distance scales with a(t), the rate at which two galaxies separate is proportional to how far apart they already are. Define the Hubble parameter H = \dot a / a; then the recession velocity of a galaxy at distance d is

Slide H_0 below to see the Hubble diagram — recession velocity against distance — tilt. Its slope is the expansion rate, and measuring that slope (with supernovae as standard candles) is how we clock the universe. The reciprocal 1/H_0 \approx 14 billion years is a rough measure of its age.

The Friedmann equations: the fate of the cosmos

Feed the FRW metric into Einstein's field equations and the ten equations collapse, by symmetry, to just two — the Friedmann equations — governing a(t):

\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3}, \qquad \frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}.

Read them as an energy balance for the universe. The first says the expansion rate H^2 is driven by the density \rho, tempered by the curvature k, and pushed by the cosmological constant \Lambda (dark energy). The second is cosmic F = ma: ordinary matter and radiation (\rho + 3p/c^2 > 0) decelerate the expansion, pulling inward — but a positive \Lambda pushes outward and can make the expansion accelerate.

This is not academic. In 1998, distant supernovae revealed that the expansion is speeding up — the \Lambda term dominates, and dark energy makes up about 68\% of the universe. The balance of terms also fixes the critical density \rho_c = 3H_0^2/(8\pi G) that separates a universe that recollapses from one that expands forever; our universe sits, to remarkable precision, at exactly this critical, spatially flat value (k \approx 0). Run the equations backward and a \to 0: a hot, dense beginning — the Big Bang, some 13.8 billion years ago.

This is the single most important subtlety in cosmology, and the answer overturns the naive picture. Distant galaxies are not moving through space like debris from an explosion; they sit essentially still at fixed comoving coordinates, and it is space itself that expands between us. The cosmological redshift is therefore not a Doppler shift from motion — it is light being stretched as the space it travels through grows. And the payoff: because the recession is the stretching of space rather than motion in space, it is not bounded by c. Galaxies beyond the Hubble distance (where v = H_0 d exceeds c) recede faster than light, and we still see many of them, because the special-relativistic speed limit governs motion through space at a point, not the expansion of space globally. There is no contradiction — only a reminder that general relativity is a theory of geometry, and geometry can do things a flat-space intuition forbids.

The name is a trap. Three corrections. First, the Big Bang did not happen somewhere; it happened everywhere at once, because the whole of space was compressed — there was no surrounding void for it to explode into. Every point you can see today was, at a \to 0, arbitrarily close to every other. Second, the galaxies are not fragments hurtling away from a central blast; the redshift is the stretching of space (above), not shrapnel velocity. Third, "the size of the universe" is a(t), a relative scale — if the universe is spatially infinite it was always infinite, merely denser. The FRW metric describes a uniform stretching of everything, with no centre, no edge, and no outside. The Big Bang is better pictured as the whole of space, everywhere, once being unimaginably hot and dense — and then expanding.