The expanding sphere and the equation for a(t)

To keep infinity under control, start with something finite and take a limit at the end:

• At some initial time $t_i$, take a sphere of radius $R_{\max,i}$, filled uniformly with matter of density $\rho_i$. A sphere is the simplest shape that keeps the model the same in every direction.
• Give it a Hubble flow: every particle moves away from the centre with velocity proportional to its distance.

• Treat the matter as "dust": particles with no pressure, moving slowly compared with light. This describes the real universe from about 50,000 years after the Big Bang (when matter took over from radiation) until a few billion years ago (when dark energy took over).

Divide the sphere into thin spherical shells. Label each by its initial radius $r_i$, and let $r(r_i, t)$ be its radius at time $t$. All velocities and forces point radially, so each shell stays a shell.

By the shell theorem, a shell feels only the mass inside it, and as long as shells don't cross, that mass never changes:

Initially every shell moves outward faster than the ones inside it, so none are about to cross. Guth's argument: the equations, valid up to any first crossing, would have to predict that crossing — and they never do. So there is none.

Newton's second law for the shell is then

Now the key step. Measure each shell's radius relative to where it started, $u(r_i, t) \equiv r(r_i, t)/r_i$. Dividing the equation by $r_i$ and writing $r = u\,r_i$:

The $r_i^3$ from the mass has cancelled against the $r_i^3$ from rescaling the force. Neither the equation nor its starting values depend on $r_i$ any more, so every shell follows the same function of time. That function is the scale factor, $u = a(t)$, and

Guth emphasises that this works because gravity is exactly inverse-square: with a force law of most other forms, the cancellation would fail and the expansion would not stay uniform. (A force growing in proportion to distance would also preserve uniformity — that is how a cosmological constant behaves, a later topic.) The radius $R_{\max,i}$ never appeared, so letting the sphere become infinite changes nothing.

The mass inside a shell is fixed while its volume grows as $a^3$, so the density falls as

Putting this back into the rescaled equation gives the equation of motion of the universe (for matter only):

The physically meaningful combination is $\ddot a/a$: the deceleration rate depends only on the density at that moment. (In Guth's convention, set up formally in Lecture 7, $a$ carries units of metres per notch, and these units cancel only in $\ddot a/a$.)

The equation $\ddot a = -\tfrac{4\pi}{3}G\rho_i/a^2$ can be integrated once. Multiply by $\dot a$ and notice that the result is a total time derivative:

Multiplied by $m\,r_i^2$, $E$ is the kinetic plus potential energy of a test particle of mass $m$ on the outer edge of the sphere. (For a particle inside the sphere this is not its true potential energy, because the outer shells also contribute; Guth notes the interpretation is cleanest at the edge.) So, as Lecture 7 develops, the universe behaves like a ball thrown upward:

• $E > 0$: the expansion slows but never stops.
• $E < 0$: the expansion stops at a maximum size and the universe recollapses.
• $E = 0$: the borderline case, just barely expanding forever.

The next lecture works out these three behaviours in time and names them closed, open and flat; what those names mean for the geometry of space comes later in the course.

• A uniformly expanding sphere of dust, followed shell by shell with Newton's gravity, stays uniform: $r = a(t)\,r_i$.
• This works only because gravity is exactly inverse-square; the result does not depend on the sphere's size.
• Matter density falls as $\rho \propto 1/a^3$, and the scale factor obeys $\ddot a = -\tfrac{4\pi}{3}G\rho\,a$.
• $E = \tfrac12\dot a^2 - \tfrac{4\pi}{3}G\rho_i/a$ is conserved; its sign decides between eternal expansion and recollapse.