Cosmological Solutions (FLRW)

Applying the Einstein field equations to the universe as a whole requires the opposite idealization from Schwarzschild: instead of an isolated mass in empty space, a smooth distribution of matter filling everything. The cosmological principle — that on large scales the universe is spatially homogeneous (no preferred place) and isotropic (no preferred direction) — fixes the geometry almost completely, yielding the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. Its dynamics, the Friedmann equations, are the foundation of modern cosmology: expansion, the Big Bang, cosmic redshift, and late-time acceleration.

We use $c=1$, keep $G$ explicit, mostly-plus signature.

The FLRW metric

Homogeneity and isotropy of space force the metric into the form

$$\boxed{d{s}^2=-d{t}^2+a(t{)}^2[\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\Omega }^2],}$$

where $t$ is cosmic time (proper time of observers at rest in the matter — the "comoving" frame), $a(t)$ is the dimensionless scale factor, and $k$ is the constant spatial curvature, normalizable to

$$k=\{\begin{array}{ll}+1& \text{closed (3-sphere, finite)}\\ 0& \text{flat (Euclidean, infinite)}\\ -1& \text{open (hyperbolic, infinite).}\end{array}$$

The spatial slices are the three constant-curvature geometries; $a(t)$ scales physical distances between comoving points. Only the ratio $a(t)/a(t_0)$ is physical; one conventionally sets $a(t_0)=1$ today.

Kinematics: comoving distance, redshift, the Hubble law

Galaxies at rest in the cosmic fluid have fixed comoving coordinates; their physical separations grow as $a(t)$. Two purely kinematic consequences follow before any dynamics:

• Cosmological redshift. Light emitted at time $t_e$ with wavelength ${\lambda }_e$ and observed today has ${\lambda }_0/{\lambda }_e=a(t_0)/a(t_e)$. Defining redshift $1+z=a(t_0)/a(t_e)$, expansion stretches wavelengths: distant (earlier) sources are redshifted. This is not a Doppler shift through space but a stretching of space.
• Hubble's law. The recession velocity of a nearby galaxy is $v=H_{0}d$, with the Hubble parameter $H\equiv \dot{a}/a$ (and $H_0$ its present value). Hubble's 1929 observation of $v\propto d$ was the first evidence that the universe expands — predicted by Friedmann (1922) and Lemaître (1927) from these equations.

Dynamics: the Friedmann equations

Feeding the FLRW metric and a perfect-fluid source $T^{\mu }{}_{\nu }=\mathrm{diag}(-\rho ,p,p,p)$ into the field equations (with a cosmological constant $\Lambda$) collapses ten equations to two:

$$\boxed{H^2=(\frac{\dot{a}}{a}{)}^2=\frac{8\pi G}{3}\rho -\frac{k}{a^2}+\frac{\Lambda }{3},}$$

$$\boxed{\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho +3p)+\frac{\Lambda }{3}.}$$

The first (the Friedmann equation) is the $00$-component — an energy constraint relating expansion rate to content and curvature. The second (the acceleration equation) is the spatial part and shows what drives acceleration or deceleration. Their consistency is equivalent to the fluid continuity equation (from ${\nabla }_{\mu }{T}^{\mu \nu }=0$):

$$\dot{\rho }+3H(\rho +p)=0.$$

Pressure gravitates — and can accelerate

The acceleration equation carries the deepest lesson: it is $\rho +3p$, not $\rho$, that sources gravitational attraction (the strong energy condition). Ordinary matter and radiation ($\rho +3p>0$) decelerate the expansion. But any component with

$$\rho +3p<0⟺p<-\frac{1}{3}\rho$$

produces repulsion — accelerated expansion. A cosmological constant ($p=-\rho$) does exactly this, which is why $\Lambda$, moved to the right-hand side as dark energy, explains the observed cosmic acceleration.

The cosmic inventory and its evolution

Solving continuity for each component with equation of state $p=w\rho$ gives $\rho \propto a^{-3(1+w)}$, so different contents dilute at different rates:

Because the exponents differ, the universe passes through eras dominated successively by radiation (early, $\rho \propto a^{-4}$ redshifts fastest), then matter, then — once everything else has diluted — the constant dark-energy density. Writing the present fractions as density parameters ${\Omega }_i={\rho }_i/{\rho }_{\text{crit}}$ (with ${\rho }_{\text{crit}}=3H_0^2/8\pi G$), the Friedmann equation becomes the flatness constraint ${\sum }_i{\Omega }_i+{\Omega }_k=1$.

$\Lambda$CDM: the standard model of cosmology

The concordance model fits all major datasets — the cosmic microwave background (Planck), Type Ia supernovae, baryon acoustic oscillations — with a flat ($k=0$) FLRW universe containing roughly

$${\Omega }_{\Lambda }\approx 0.69,{\Omega }_{\text{matter}}\approx 0.31(\text{of which most is cold dark matter}),{\Omega }_{\text{radiation}}\approx 10^{-4}.$$

This is $\Lambda$CDM ($\Lambda$ + Cold Dark Matter). It implies an expansion that decelerated during the matter era and began accelerating around $z\sim 0.7$ as $\Lambda$ took over — the acceleration discovered via supernovae in 1998 (2011 Nobel Prize). Extrapolated backward, $a\to 0$ at finite time: the Big Bang.

The Big Bang and its puzzles

Running the flat, matter/radiation FLRW solution backward, $a(t)\to 0$ produces a genuine curvature singularity — $\rho \to \infty$, and the singularity theorems confirm (given the SEC) that it is unavoidable, not an artifact of exact symmetry. The hot early phase makes firm, confirmed predictions: Big Bang nucleosynthesis (the primordial light-element abundances) and the cosmic microwave background (the relic radiation from recombination, $z\approx 1100$, a near-perfect $2.7$ K blackbody).

Two features the bare model does not explain — the horizon problem (why causally disconnected regions share the same temperature) and the flatness problem (why $\Omega$ is so close to $1$) — motivate cosmic inflation: an early epoch of SEC-violating accelerated expansion (a slowly rolling scalar field, $p\approx -\rho$) that stretches a tiny causal patch to encompass the observable universe and drives $\Omega \to 1$, while seeding structure from quantum fluctuations. The initial singularity itself lies beyond classical GR and is a target for quantum gravity; the problem of time and the beginning are discussed philosophically under time and cosmology.

Summary

• The cosmological principle (homogeneity + isotropy) forces the FLRW metric, with scale factor $a(t)$ and spatial curvature $k\in \{-1,0,+1\}$.
• Expansion gives cosmological redshift $1+z=a_0/a_e$ and Hubble's law $v=H_{0}d$, $H=\dot{a}/a$.
• The Friedmann equations relate $H$ and $\ddot{a}$ to content; acceleration requires $\rho +3p<0$ (SEC violation), supplied by $\Lambda$/dark energy.
• Components dilute as $\rho \propto a^{-3(1+w)}$, giving radiation → matter → $\Lambda$ eras; $\Lambda$CDM (${\Omega }_{\Lambda }\approx 0.69$, ${\Omega }_m\approx 0.31$, flat) is the standard model.
• Backward extrapolation gives the Big Bang singularity, with BBN and the CMB as confirmed relics; the horizon and flatness problems motivate inflation.

Next

The Kerr solution returns to local, strong-field geometry: the rotating black hole, frame dragging, and the ergosphere. Black-hole thermodynamics and gravitational waves are planned in blackholes/ and waves/.