The constraint behind the clock

In ordinary mechanics the Hamiltonian generates time evolution and time is a parameter you are handed. In General Relativity neither is true: the theory is invariant under relabelling the time coordinate, and that invariance turns the Hamiltonian into a constraint. For the full theory this is the notorious “problem of time.” For a homogeneous universe it is not notorious at all - it is a short, completely explicit calculation, and doing it once fixes the vocabulary for everything that follows. So let us do it.

The standard ADM metric for the Hamiltonian description of General Relativity is

Here $N(t)$ is called the Lapse function, and $N^i$ is the Shift vector. The spatial metric is determined by $h_{ij}$. This is essentially the four-dimensional metric broken down to a three-dimensional metric description. In principle all three of these objects are viable degrees of freedom.

The present "minisuperspace" set up fixes $N^i$ to zero by fiat, and enforces the FLRW metric by fixing

Hence the metric under consideration becomes

and the action $S=\int \sqrt{-g}\left(\frac{1}{2}R-\frac{1}{2}{(\partial \varphi )}^2-V(\varphi )\right)$ in reduced Planck units ($M_P=1$). Homogeneity collapses the field theory to a mechanics problem; per unit comoving volume, dropping a boundary term,

Two things are worth pausing on. First, the lapse $N$ appears without a time derivative. It is not a coordinate that evolves, but a multiplier we get to choose. Second, the scale-factor kinetic term carries a minus sign: the metric on configuration space (the DeWitt metric) is Lorentzian, with $a$ the timelike direction. That indefinite signature is the fingerprint of gravity hiding inside a mechanics problem, and it is why the quantum version is a wave equation, not a diffusion.

The conjugate momenta follow from (4):

The last is not a definition of a velocity - it is a primary constraint¹¹ 1 Here and in what follows, we are using Dirac’s language of constrained Hamiltonian systems. So weakly equals represented by $\approx$ here, means equals up to a constraint. It’s a bookkeeping device., ${\pi }_N\approx 0$: the lapse has no dynamics. Legendre-transforming (and keeping ${\pi }_N$ around, as Dirac instructs),

The entire Hamiltonian is the lapse times $\mathscr{H}$. That is the first sign that something is unusual: in a normal theory $H$ is a fixed function on phase space; here it is proportional to a quantity we are free to scale.

A constraint is only consistent if it is preserved in time. Demanding ${\dot{\pi }}_N=\{{\pi }_N,H\}\approx 0$ and using $\{{\pi }_N,N\}=-1$ gives

so the primary constraint drags a secondary constraint into existence: $\mathscr{H}\approx 0$. No further constraints appear (the chain closes), and $\mathscr{H}$ is first class - it Poisson-commutes with itself and with ${\pi }_N$. Now substitute the momenta back into $\mathscr{H}=0$:

i.e.

That is to say, the Friedmann equation is the Hamiltonian constraint. It is not an equation of motion the system obeys as it evolves; it is a condition that carves out the physical surface $\mathscr{H}=0$ in phase space, and the dynamics never leaves it²² 2 For completeness, the genuine equations of motion generated by $\mathscr{H}$ are $\dot{\varphi }={\pi }_{\varphi }/a^3$ and ${\dot{\pi }}_{\varphi }=-a^3{V}^{\prime }$, which combine to $\ddot{\varphi }+3H\dot{\varphi }+V^{\prime }=0$ - the scalar field equation we know from the study of inflation..

Now, on the constraint surface, the total Hamiltonian vanishes,

and $\mathscr{H}$, the thing that remains, does not evolve the system forward in time. Being first class, it generates gauge transformations. What looks like “evolution” is the flow of the constraint, and the arbitrary function multiplying it is the lapse $N$. Different choices of $N(t)$ are different gauge fixings of one and the same physical history. In other words:

there is no preferred time; picking a clock means fixing the Lapse function.

This is the concrete, minisuperspace face of the problem of time, and is the raison d’être for these notes. $N=1$ is cosmic time; $N=a$ is conformal time; $N=1/H$ is the e-fold clock $N_e=\log a$ in which the inflaton dynamics becomes autonomous and its velocity is bounded, .

Each is a legitimate section of the same gauge orbit, and this is the whole moral of our series, each makes some features manifest and hides others. Just like choosing a gauge in electromagnetism. In part 2, we work through a menu of such choices.

The unreduced phase space $(a,\varphi ,{\pi }_a,{\pi }_{\varphi })$ carries the canonical symplectic form $\omega =d{\pi }_a\wedge da+d{\pi }_{\varphi }\wedge d\varphi$. Restricted to $\mathscr{H}=0$, this is the Gibbons–Hawking–Stewart structure, and serves as a natural measure on classical universes [3, 4]. The system also has a dynamical similarity: rescaling the spatial volume (and the momenta to match) maps solutions to solutions and leaves the field dynamics untouched. This is equivalent to the statement, in canonical dress, that the overall scale of $V$ is classically invisible. Sloan observed that quotienting the symplectic system by this similarity yields a contact system [5, 6]: odd-dimensional, with a genuine dissipative flow whose friction is precisely the non-conservation of the reduced Liouville measure.

Which variable you eliminate in the quotient is itself a choice. Sloan removes the volume and keeps a Hubble-like coordinate $h$, evolving in cosmic time, with contact form $\eta =-dh+\dot{\varphi }d\varphi$ and contact Hamiltonian $H^c=-\frac{3}{8}{h}^2+\frac{1}{2}{\dot{\varphi }}^2+V$. The presentation this series uses instead eliminates $H$ through the Friedmann constraint and keeps $N_e=\log a$ as the time. In this context we have the “time-derivative”

Hence this gauge choice lands on the reduced flow for $(\varphi ,{\varphi }^{\prime })$ with the state-dependent friction $3(1-{\varphi }^{\prime 2}/6)$. These are two sections of the same reduction - plausibly related by a contactomorphism and a reparametrization, a point worth settling but not needed here. The contact reduction itself is Sloan’s; what is new downstream is the e-fold presentation and the use it is put to (the Itzhaki–Kovetz fixed point and its Arnold-singularity structure).

The lesson to carry forward is a hierarchy. Underneath everything sits a constrained symplectic system with no preferred time. Choosing a clock (Part 2) gauge-fixes the lapse. Eliminating $H$ then reduces that system to a lower-dimensional contact description (Part 3), where a surprising amount of rigid geometry becomes visible. And when that reduced description appears to break at a zero-energy vacuum, where the field kinates and the contact picture degenerates. We discuss this in detail (Part 4), where we find that nothing has actually gone wrong with the physics: we have merely run a gauge-and-reduction choice past its domain of validity. The cure is to un-reduce, to give $H$ back and return to the constrained symplectic system we started from here. The series is, in the end, a round trip through this diagram, and this post is the fixed point it departs from and returns to.