Measuring the Shape of the Hottest Liquid in the Universe
A new paper in Physical Review C introduces a geometric framework for the quark-gluon plasma — and opens a research programme that connects information theory, differential geometry, and nuclear physi
Subject: summary CQ11049
New geometric tool measures how far quark-gluon plasma is from equilibrium
When atomic nuclei collide at nearly the speed of light, they briefly produce a quark-gluon plasma — the hottest liquid in the universe — which cools into an ordinary flowing fluid astonishingly quickly. Physicists have lacked a coordinate-independent way to measure how “far” this plasma is from thermal equilibrium at any given moment. In this paper, I treat the space of possible plasma states as a curved geometric surface and use a metric from the mathematics of statistics to define distances and curvature on that surface. The approach yields three new results. The standard viscosity of the plasma emerges directly from the shape of the surface at equilibrium, without being assumed. At moderate departures from equilibrium, conventional approximations underestimate the true geometric distance by 15 to 35 percent, pinpointing where standard methods start to fail. And the surface has a precisely computable saddle-like curvature (Ricci scalar = −5/21), meaning that different types of internal stress become increasingly distinguishable as the plasma evolves. This geometric framework provides a new diagnostic — effectively a thermometer based on geometry rather than temperature — for the extreme conditions created in heavy-ion experiments at CERN and Brookhaven.
A new paper in Physical Review C introduces a geometric framework for the quark-gluon plasma — and opens a research programme that connects information theory, differential geometry, and nuclear physics
Keywords: quark-gluon plasma, Fisher-Rao metric, information geometry, relativistic kinetic theory, hydrodynamisation, heavy-ion collisions, CERN, RHIC, Riemannian geometry, transport coefficients, Physical Review C
A paper I have been working on for some time has just been accepted for publication in Physical Review C. The title is “Riemannian geometry of the moment manifold in relativistic kinetic theory and the approach to hydrodynamics,” and it does something that, as far as I am aware, has not been done before: it takes the space of possible states of the quark-gluon plasma — the exotic liquid produced in heavy-ion collisions at CERN and Brookhaven — and treats it as a curved geometric surface, computes the distances and curvature on that surface exactly, and shows that the results tell us something concrete about when and how the plasma transitions from chaos to fluid behaviour.
This post explains what the paper does, why it matters, and where the research programme goes from here.
What is the quark-gluon plasma, and why does it matter?
For the first few microseconds after the Big Bang, the universe was too hot for protons and neutrons to exist. Instead, their constituent quarks and gluons moved freely in a dense, strongly interacting soup: the quark-gluon plasma, or QGP. Today, the only way to recreate this state of matter is to smash heavy atomic nuclei together at nearly the speed of light. This is done at two facilities: the Large Hadron Collider at CERN in Geneva, and the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory on Long Island.
When gold or lead nuclei collide at these energies, they produce a fireball that reaches temperatures exceeding four trillion degrees Celsius — roughly 250,000 times the temperature at the centre of the Sun. The fireball expands, cools, and within about 10⁻²³ seconds converts back into ordinary particles that fly outward and hit the detectors. Everything we know about the QGP is reconstructed from those final-state particles.
One of the most striking discoveries of the RHIC and LHC heavy-ion programmes is that the QGP behaves as a nearly perfect liquid. Its viscosity, relative to its entropy density, is the lowest of any known substance — lower than water, lower than liquid helium, lower than anything else ever measured. And it reaches this liquid-like behaviour extraordinarily quickly: within about 1 fm/c (roughly 3 × 10⁻²⁴ seconds) of the collision, the system is already well described by the equations of fluid dynamics.
This raises a sharp theoretical question. How does a system that starts in a violently out-of-equilibrium state — two nuclei shattering into thousands of quarks and gluons — reach thermal equilibrium so fast? And how do we even define “how far from equilibrium” the system is at any given moment?
The problem with existing diagnostics
The standard tools for tracking the approach to equilibrium are numerical ratios: the Knudsen number (comparing the mean free path of particles to the system size) and the inverse Reynolds number (comparing viscous corrections to the equilibrium pressure). When these ratios are small, the system is close to equilibrium. When they are large, it is far away.
These ratios are useful, but they have a fundamental limitation. They are not metrical — they do not define a proper notion of distance on the space of possible plasma states. They cannot tell you the shortest path between two states, or how the space of states is curved, or whether two states that look similar by one ratio are actually far apart in a more fundamental sense. They are thermometers, not maps.
A more sophisticated tool is the Kullback-Leibler divergence, a quantity from information theory that measures how different two probability distributions are. In kinetic theory, where every state of the plasma corresponds to a probability distribution of particle momenta, the KL divergence gives a natural measure of how far a given state is from equilibrium. It is used in many theoretical analyses. But the KL divergence is not symmetric (the “distance” from A to B is not the same as from B to A), it does not satisfy the triangle inequality, and it does not define geodesics (shortest paths) or curvature. It is an asymmetric divergence, not a distance.
The question the paper addresses is: can we do better? Can we define a proper, coordinate-invariant geometric distance on the space of plasma states — one that has geodesics, curvature, and all the structure of Riemannian geometry?
What the paper does
The answer is yes, and the tool that provides it is the Fisher-Rao metric.
The Fisher-Rao metric is a concept from information geometry, a branch of mathematics that studies the geometry of spaces of probability distributions. Given any parametric family of distributions, the Fisher-Rao metric defines a Riemannian metric on the parameter space: it tells you how to measure distances, angles, and curvature among distributions. It is the unique metric (up to a constant) that is invariant under smooth reparameterisations of the parameters — a result known as Čencov’s theorem, proved in the 1970s. The metric has a direct relationship to the KL divergence: at the infinitesimal level, the Fisher-Rao metric is the Hessian (second derivative) of the KL divergence. So it inherits the information-theoretic content of the KL divergence while adding the full structure of Riemannian geometry.
In the paper, I apply this to the DNMR 14-moment truncation of relativistic kinetic theory. The DNMR framework (named after Denicol, Niemi, Molnár, and Rischke, who developed it in a series of papers around 2012) describes the state of a relativistic gas by 14 macroscopic quantities: the energy density, three components of the flow velocity, a bulk viscous pressure, five independent components of the shear stress tensor, and four components of the heat flux. These 14 quantities define a point on a 14-dimensional manifold. By equipping this manifold with the Fisher-Rao metric, I turn it into a Riemannian space with well-defined distances, geodesics, and curvature.
The paper works in a controlled setting: conformal Bjorken flow (a specific symmetry pattern appropriate for the early stages of a heavy-ion collision), with a massless gas, in the relaxation-time approximation for the collision kernel. This is a deliberately simplified setting — not because the physics is trivial, but because it is the setting in which exact results can be obtained and verified against the exact kinetic-theory solution.
Three results
The paper delivers three main results.
First: viscosity from geometry. At thermal equilibrium, the Fisher-Rao metric on the shear-stress direction reduces to a single number: g = 2/15. This is an exact, dimensionless constant that drops out of the momentum-space integrals. When you multiply it by the appropriate thermodynamic prefactor, you recover the standard shear viscosity of the conformal gas: η = 4Pτ_R/5, where P is the pressure and τ_R is the relaxation time. The viscosity is not an input to the geometry; it is an output. The transport coefficient is encoded in the shape of the equilibrium manifold.
Second: the finite-shear distance gap. Away from equilibrium, the manifold is curved, and the straight-line (tangent-space) approximation to the distance — the one implicitly used in linearised hydrodynamics — underestimates the true geodesic distance. The paper computes both exactly and compares them against the exact solution of the Boltzmann equation in the relaxation-time approximation. The gap is 15 to 35 percent at moderate shear. This is a concrete, quantitative statement about where the linearised Navier-Stokes approximation starts to fail: it under-resolves the geometric distance to equilibrium by a specific, computable amount.
Third: the curvature of the shear sector. The five-dimensional submanifold describing the shear stresses has a computable Ricci curvature scalar: R = −5/21. This is an exact number, not an approximation. The negative sign means the manifold is hyperbolic — shaped like a saddle rather than a sphere. In geometric terms, nearby geodesics diverge: two states that start close together on the shear manifold separate as they evolve. The ten sectional curvatures (one for each pair of shear directions) are also computed exactly, and they organise into a hierarchy that reflects the symmetry-breaking pattern from SO(3) (full rotational symmetry) down to SO(2) (the azimuthal symmetry of the Bjorken flow). This hierarchy is a geometric fact about the equilibrium manifold within the truncation.
Why it matters
The QGP is the most extreme liquid state of matter accessible in the laboratory. Understanding how it thermalises is a central goal of the heavy-ion physics programmes at CERN and Brookhaven. Current methods for extracting transport properties from experimental data use computationally expensive hydrodynamic simulations combined with Bayesian statistical inference — an approach developed by Bernhard, the JETSCAPE collaboration, and others, in which one varies the input parameters of the simulation (including the viscosity) and compares against thousands of observables to find the best fit.
This Bayesian approach implicitly varies the 14-moment manifold globally. A metric on that manifold turns such variation into geometric statements about sensitivity and distinguishability: how much does the observable change when you move in this direction on the parameter space? The Fisher-Rao metric is the canonical answer to that question.
The geometric framework also addresses a conceptual puzzle that has driven much of the theoretical work on the QGP over the past decade: the attractor phenomenon. Around 2015, Heller and Spalinski showed that certain solutions of viscous hydrodynamics possess special trajectories — attractors — toward which all initial conditions converge, even when the system is violently far from equilibrium. This was a breakthrough: it explained why hydrodynamics works so early in a heavy-ion collision, before the system has had time to reach full thermal equilibrium in the conventional sense. The attractor picture has since been sharpened through resurgence (relating the divergent gradient expansion to non-perturbative contributions), adiabatic hydrodynamisation (tracking slow versus fast modes), and bootstrap methods.
But a basic geometric question has remained unanswered. The attractor is a curve in the space of states. If you put a metric on that space, you can ask: is the attractor a geodesic — a shortest path? Does it minimise some geometric functional? Or is it geometrically generic, with no special Riemannian characterisation? These questions cannot even be formulated without a metric, and until this paper, no Riemannian metric had been placed on the moment manifold of relativistic kinetic theory.
The Fisher-Rao framework provides the setting in which the question can finally be asked precisely. In this paper, the attractor turns out not to be a geodesic of the Fisher-Rao metric on the Bjorken submanifold. That is a negative result, but a meaningful one: it rules out the simplest natural geometric characterisation and constrains the alternatives. It also produces a concrete observable — the geodesic curvature of the attractor as a curve on the Fisher-Rao manifold — that measures how much the attractor deviates from the shortest geometric path.
What is next
The PRC paper is the foundation of a broader research programme. It establishes the framework, proves the core theorems, and delivers the first exact results — all within a deliberately simplified setting. The programme that follows extends the framework in five directions, each of which removes one of the simplifying assumptions while preserving the geometric structure.
Beyond conformal Bjorken flow. The paper works in a setting with maximal symmetry: conformal (massless) matter in a longitudinally expanding geometry. Real QGP matter has finite quark masses, which break conformal symmetry and introduce bulk viscosity. The moment manifold then acquires new dimensions — the bulk viscous pressure and, for non-zero baryon density, the heat flux — and the shear and bulk sectors couple. The geometry changes: the block-diagonal structure of the equilibrium Fisher metric, which relies on the conformal SO(3) symmetry via Schur’s lemma, breaks once the shear and bulk blocks mix. Computing the curvature of this enlarged, coupled manifold is the most immediate extension. The paper already contains a numerical computation of the two-dimensional Bjorken submanifold curvature as a function of the particle mass; the full six-dimensional thermal-plus-shear manifold remains to be completed. Including transverse expansion — moving from the Bjorken symmetry to the Gubser flow profile — breaks another symmetry and allows transverse dynamics to enter. Each extension changes the geometry in computable ways and tests whether the main features (negative curvature, the 15-to-35-percent distance gap, the viscosity-from-geometry relation) are stable or are specific to the conformal limit.
Beyond the relaxation-time approximation. The relaxation-time approximation replaces the full Boltzmann collision integral with a single exponential relaxation toward equilibrium. It is analytically tractable but physically simplified. The equilibrium geometric quantities (g = 2/15, R = −5/21) are insensitive to the collision kernel within the class of kernels that share the same equilibrium distribution — a result proved in the paper using Schur’s lemma on the SO(3)-irreducible decomposition of the moment basis. But the dynamics on the manifold — the trajectories, the relaxation rates, the attractor — are kernel-dependent. Extending the framework to realistic QCD-inspired collision kernels, or to kernels extracted from lattice QCD, is a natural next step.
Connection to Bayesian inference. The heavy-ion community increasingly uses Bayesian methods to extract transport coefficients from data. The JETSCAPE collaboration, building on the pioneering work of Bernhard and colleagues, runs thousands of hydrodynamic simulations with different input parameters, compares them against experimental observables from RHIC and the LHC, and uses Bayes’ theorem to construct a posterior probability distribution over the space of transport coefficients. This approach has produced the most precise current estimates of the QGP’s shear viscosity, bulk viscosity, and other transport properties.
The Fisher-Rao metric is the natural metric on the space of models used in Bayesian inference: it measures how statistically distinguishable two parameter choices are, given the data. In the heavy-ion context, the simulation inputs parameterise the same moment manifold that the kinetic-theory Fisher-Rao metric lives on. If these two Fisher metrics are related — the kinetic-theory one on the moment space and the Bayesian one on the parameter space — then the geometric framework provides a principled way to identify which directions in parameter space are well-constrained by data and which are degenerate. This could complement the emulator-based Bayesian workflows currently used by the experimental collaborations, by providing geometric diagnostics that do not require running the full simulation chain.
The attractor as a geometric object. The attractor is not a geodesic of the Bjorken-submanifold Fisher-Rao metric. But it may be a geodesic of a different metric, or a curve that minimises a different geometric functional (for example, one that includes a potential term from the collision integral). Characterising the attractor geometrically — or proving that no natural Riemannian characterisation exists — is an open problem that the framework is designed to address.
Higher-order moment truncations. The 14-moment truncation retains moments up to angular momentum ℓ = 2. Including ℓ = 3 and ℓ = 4 moments enlarges the manifold and changes its curvature. The Schur-lemma argument guarantees that the shear-sector curvature is unchanged under orthogonal extensions, but non-orthogonal extensions (which arise when higher-ℓ moments couple to the shear sector at finite deviation from equilibrium) will modify the geometry. Computing these corrections is necessary for quantifying how much of the curvature structure is a genuine feature of relativistic kinetic theory and how much is an artefact of the 14-moment truncation.
Broader applications. The mathematical framework applies to any kinetic system described by a finite set of moments. This includes non-relativistic plasmas (relevant to fusion energy research and magnetohydrodynamics), astrophysical systems (neutron star interiors, where dense nuclear matter at extreme pressures approaches hydrodynamic behaviour; accretion disc dynamics, where angular momentum transport is governed by effective viscosity), and cosmological fluids (the photon-baryon plasma before recombination, where the transition from tight coupling to free-streaming is a kinetic-to-hydrodynamic crossover in reverse). Each of these systems has its own moment manifold, its own Fisher-Rao geometry, and its own set of open questions about thermalisation and transport. The relativistic QGP is the first application because it is the most extreme and the most precisely studied, but the tools are general.
The bigger picture
This paper sits at the intersection of three fields that do not usually talk to each other: nuclear physics, information geometry, and Riemannian geometry. Nuclear physicists have developed extraordinarily sophisticated tools for simulating heavy-ion collisions, but the geometric structure of the state space they navigate has not been explored. Information geometers have studied the Fisher-Rao metric in depth, but almost exclusively on finite-dimensional statistical models far removed from kinetic theory. Riemannian geometers have powerful theorems about curvature and geodesic behaviour, but they have not been applied to the moment manifolds of relativistic kinetic theory.
The paper brings these together in a specific, controlled, exactly solvable setting. It does not solve the thermalisation problem. It does not replace hydrodynamic simulations. What it does is provide a new set of coordinate-invariant, information-theoretically grounded diagnostic tools for a problem that the community has been studying with coordinate-dependent, scale-ratio-based diagnostics for two decades. The 15-to-35-percent distance gap is a concrete example: it is a number that linearised hydrodynamics cannot see, that the KL divergence alone cannot produce, and that emerges naturally from the Riemannian geometry of the moment manifold.
Whether the curvature structure survives beyond the 14-moment truncation, whether the attractor admits a geometric characterisation, and whether the Fisher-Rao metric connects to the Bayesian inference machinery used by experimentalists — these are the open questions that define the programme ahead. The PRC paper is the starting point. The programme it opens — extending the framework to non-conformal matter, realistic collision kernels, higher-order truncations, and connections to Bayesian data analysis — is designed to turn the geometry of the moment manifold from an exact result in a controlled setting into a practical tool for the nuclear physics community. The mathematics is there. The exact results are proved. What remains is to see how far the geometry carries.
I will be writing about each of these extensions as the work progresses.
Craig S. Wright is a researcher (University of Exeter). The paper “Riemannian geometry of the moment manifold in relativistic kinetic theory and the approach to hydrodynamics” is accepted for publication in Physical Review C. Manuscript CQ11049.
Contact: cw881@exeter.ac.uk · ORCID: 0000-0001-9374-0507