The Critical Point
The curve breaks. The manifold cracks. And in the fracture, all of thermodynamics is revealed.
The Divergence
There is a point in the state space of every fluid where the distinction between liquid and gas vanishes. Not gradually — it simply ceases to exist. The critical point. At this point, the isothermal compressibility diverges. The heat capacity blows up. Correlation lengths stretch to infinity. And in geometrothermodynamics, the scalar curvature of the thermodynamic metric diverges to negative infinity.
This is not coincidence. The divergence of $R$ at the critical point is the geometric signature of a phase transition of the second kind. The manifold itself cannot bear the weight of the critical fluctuations, and its curvature screams.
A Van der Waals Fluid
Consider the van der Waals equation of state:
$$\left(P + \frac{a}{v^2}\right)(v - b) = T$$
where $v = V/N$ is the molar volume, $a$ controls attraction, and $b$ the excluded volume. The critical point occurs at:
$$v_c = 3b, \quad T_c = \frac{8a}{27b}, \quad P_c = \frac{a}{27b^2}$$
Near this point, one introduces reduced variables $\tilde{P} = P/P_c$, $\tilde{v} = v/v_c$, $\tilde{T} = T/T_c$ and finds that the thermodynamic metric components diverge. The scalar curvature $R$ behaves as:
$$R \sim \frac{1}{(\tilde{T} - 1)^\gamma}$$
where $\gamma$ is the critical exponent. For mean-field theory (van der Waals), $\gamma = 1$, but experimentally $\gamma \approx 1.24$ for the 3D Ising universality class. The geometry sees the critical point in both cases — but the divergence rate carries the universality class within it.
The curvature scalar is a universal probe. It does not care about microscopic details; it sees only the large-scale structure of fluctuations, encoded in the exponents.
What the Divergence Means
When $R \to -\infty$ at the critical point, what are we saying? We are saying that the space of equilibrium states becomes infinitely curved — infinitely responsive — to perturbations. A tiny change in temperature or pressure sends you careening across the manifold. The distance between states near the critical point is enormous in geometric terms, even though the states are microscopically similar.
This is the geometric origin of critical opalescence. The fluid becomes opaque because the length scale of density fluctuations diverges, and the divergence is the divergence of curvature. Light scatters because the manifold is folding in on itself.
Below, Above, and At
Below the critical temperature, the fluid separates into liquid and gas phases. In the state space, this manifests as a region where the isothermal compressibility is negative — mechanically unstable. The geometry resolves this by the Maxwell construction, which replaces the unphysical region with a horizontal line in the $P$-$V$ diagram. In the geometric language, the equilibrium manifold has a fold, and the physical states live on the stable branches.
At the critical point, the two branches merge. The fold pinches off. The manifold is smooth, but its curvature is not. The singularity is intrinsic — not an artifact of coordinates, but a true geometric feature, like the singularity at the event horizon of a black hole (or rather, like the smoothness of the horizon in the right coordinates).
Aftermath
After the critical point, there is no liquid, no gas, only a supercritical fluid. The manifold is smooth and curved, and the critical point becomes a scar on its surface — a memory of the phase transition, visible forever in the curvature.