Sloan (2021) reformulated the cosmological sector of general relativity (FLRW and Bianchi models) as a frictional system governed by a Herglotz-type, action-dependent variational principle on an odd-dimensional contact manifold that makes no reference to the unobservable scale factor, and whose symplectification reproduces the Einstein-Hilbert action. The paper explicitly flagged five open directions: extension to full inhomogeneous general relativity, quantization, a rigorous thermodynamic and arrow-of-time account, measure-theoretic completeness, and the operational meaning of scale-freedom in inhomogeneous settings. Subsequent work has resolved the first of these: a scale-free, action-dependent (contact) formulation exactly equivalent to the Einstein-Hilbert dynamics now exists for full classical general relativity (Sloan 2025; Bell and Sloan 2026a-c). What remains open is the physical completion of the program, in two linked layers: (i) a quantum theory of the contact formulation, built on a phase space without symplectic structure and without a conserved Hamiltonian, and its relation to Wheeler-DeWitt quantization and Loop Quantum Cosmology, which promote the very volume operator this formulation discards; and (ii) rigorous statistical-mechanical foundations: a definition of entropy and typicality on contact phase space, a precise status for the Past Hypothesis, and a complete contact-measure prescription for probabilistic comparisons of cosmological solutions. Both layers remain open: quantization of contact systems is explicitly described as unresolved in work through 2026, and the existing arrow-of-time accounts, while conceptually developed, defer rigorous typicality and measure results.

Background

Cosmological modelling in general relativity is conventionally performed on symmetry-reduced 'minisuperspace' models: the FLRW spacetimes (Friedmann-Lemaître-Robertson-Walker; homogeneous and isotropic universes) and the Bianchi models (homogeneous but anisotropic universes, classified by the Lie algebra of their spatial Killing vector fields). The geometry is encoded in a global scale factor aa (or spatial volume v=a3v = a^{3}) together with matter degrees of freedom, and the dynamics follow from the Einstein-Hilbert action (the standard action of general relativity, built from the spacetime Ricci curvature scalar). The overall scale, however, is not observable in isolation: physical measurements depend only on ratios of scale, on the Hubble parameter H=a˙/aH = \dot a / a (the expansion rate), and on matter variables. The same physical history is represented by infinitely many choices of overall scale, or of the fiducial cell used to regulate non-compact spatial slices. This redundancy has concrete consequences: measures on the space of cosmological solutions that integrate over the scale direction are formally infinite and require ad hoc cut-offs; matter equations exhibit the familiar 'Hubble friction' damping, which appears as an external feature of the conservative (Hamiltonian) variational structure; and the scale factor is precisely the variable whose dynamics becomes singular at the big bang.

Contact geometry provides the odd-dimensional analogue of symplectic geometry, the natural setting of Hamiltonian mechanics. A contact manifold carries a one-form whose powers define a volume, and contact Hamiltonian mechanics extends symplectic Hamiltonian mechanics to dissipative (frictional) systems. The Herglotz variational principle realizes this: one extremizes a Lagrangian that depends on the action itself (an 'action-dependent' Lagrangian), and the resulting Euler-Lagrange-type equations are generically non-conservative. Contact systems possess a canonical invariant measure, the contact analogue of the Liouville measure (Bravetti and Tapias 2015), and the framework was systematized by Bravetti, Cruz and Tapias (2017).

Sloan (2021), the source of this problem, showed that the cosmological sector of general relativity admits such a description: a Herglotz-type action for FLRW and Bianchi class A models that contains no scale factor. Its extremization reproduces the usual dynamics of all physical observables (matter variables and the Hubble parameter, which acts as a frictional drag variable), and its symplectification — reinstating the scale direction to recover an even-dimensional symplectic system — reproduces the Einstein-Hilbert action for these models. The reduced description requires one fewer initial datum, exhibits measure focusing (the contact measure contracts along the flow, giving a directionality of time without presupposing a thermodynamic arrow), and ties into the earlier result of Koslowski, Mercati and Sloan that relational, scale-free variables admit a unique deterministic classical evolution through the big bang singularity. The paper closed by flagging five unresolved directions: (1) extension beyond homogeneous minisuperspace to full, inhomogeneous general relativity; (2) quantization of the frictional/contact formulation, possibly via open-system (e.g. Lindblad-type) techniques, and its comparison with Wheeler-DeWitt quantization (the canonical quantization of minisuperspace, yielding the Wheeler-DeWitt equation) and Loop Quantum Cosmology (LQC, the symmetry-reduced application of loop quantum gravity), both of which promote a volume operator; (3) a rigorous thermodynamic and statistical-mechanical account — entropy, the Past Hypothesis (the standard posit that the universe began in an unusually low-entropy state, grounding the thermodynamic arrow of time), and the status of cosmological initial conditions; (4) a complete measure prescription for probabilistic comparisons of cosmological solutions; and (5) the operational meaning of eliminating scale in inhomogeneous settings, where local rods (such as the historic metre des archives) do define distances.

The subsequent classical development has been substantial. 'Dynamical similarity' — a transformation rescaling the unit of Hamilton's principal function, under which solutions of a given system are identified — was identified as the structural symmetry at play (Gryb and Sloan 2021), and Bravetti, Jackman and Sloan (2023) showed that quotienting a broad class of Hamiltonian systems by such scaling symmetries generically yields an equivalent contact Hamiltonian system, 'Poincaré's dream' of a scale-invariant description. Sloan (2023) extended the Herglotz action to homogeneous cosmologies generally. In parallel, an action-dependent (k-contact and multicontact) formalism for dissipative field theories was developed, including a variational derivation of an action-dependent Einstein-Hilbert Lagrangian (Gaset and Mas 2023; de León et al 2025) — an adjacent modification of general relativity rather than a reproduction of it. Sloan (2025) then constructed what the source had asked for: a Herglotz Lagrangian for full general relativity in which the spacetime metric is decomposed into a fixed-determinant part γ\gamma and a conformal factor, the latter being the redundant scale degree of freedom; the construction introduces action densities sas_a, makes no reference to the conformal mode, and the resulting equations of motion were shown to be exactly those derived from the Einstein-Hilbert action (with worked inhomogeneous examples such as the Schwarzschild interior and Lemaître-Tolman-Bondi dust). Bell and Sloan (2026a,b) supplied the general multisymplectic (De Donder-Weyl) framework, extending the symmetry reduction to classical field theories including singular, gauge-symmetric Lagrangians, and Bell and Sloan (2026c) completed the first-order Palatini formulation of general relativity as a non-conservative, action-dependent field theory, identifying the conformal mode as the generator of a dynamical similarity and verifying that the linearized reduced theory propagates the expected two gravitational degrees of freedom.

What this program has not produced is the physical layer beyond classical dynamics. A quantum theory based on the contact formulation must confront structural obstacles: an odd-dimensional phase space has no symplectic form, hence no Poisson algebra to represent as operator commutators; and the contact Hamiltonian is not conserved along the flow, so Schrödinger-type unitary evolution is unavailable, pushing one toward open-system or non-unitary quantizations whose probability interpretation is delicate. On the statistical-mechanical side, no rigorous entropy definition on contact phase space exists, and the measure prescription (choice of comparison surfaces, non-compact directions, behaviour of cut-offs under the focusing evolution) is incomplete; the recent arrow-of-time account of Gryb and Friederich (2025), built on Janus points (states of a time-symmetric system from which entropy-like or complexity measures increase in both temporal directions) and attractors, explicitly defers rigorous typicality results.

Problem Statement

The contact (Herglotz) formulation of cosmology is now established classically, from homogeneous minisuperspace through full general relativity. The unresolved problem is the physical completion of this formulation, consisting of two linked components that together determine whether the scale-free, frictional description is a complete foundation for cosmology or only a classical reformulation:

(1) Quantum theory. Construct a consistent quantum theory whose classical limit reproduces the contact/Herglotz dynamics of cosmological systems — i.e. a quantum description built on the reduced, scale-free degrees of freedom (matter variables and momenta, shape/anisotropy variables, and the Hubble parameter together with the contact structure), rather than on the volume operator that Wheeler-DeWitt quantization and Loop Quantum Cosmology promote. This requires establishing which quantization route is viable for an odd-dimensional, non-conservative system — canonical quantization of a symplectification (which reinstates the scale direction, so serving as the standard-theory comparison), a direct contact quantization, or open-system/non-unitary evolution such as Lindblad-type dynamics — including the probability interpretation of the resulting evolution, and it requires determining the theory's behaviour at the classical big-bang singularity that the contact formulation traverses deterministically, together with a like-for-like comparison of its predictions against Wheeler-DeWitt and Loop Quantum Cosmology treatments of the same models.

(2) Statistical-mechanical and measure foundations. Formulate the rigorous probabilistic content of the contact framework: a well-defined notion of entropy and typicality on contact phase space; the precise status of the Past Hypothesis within the framework (in particular whether the attractor/Janus-point account grounded in measure focusing can be made rigorous); and a complete contact-measure prescription for probabilistic comparisons of cosmological solutions — the choice of comparison surfaces (for example, of constant Hubble parameter), the treatment of residual non-compact directions, and the behaviour of regularization cut-offs under time evolution, i.e. whether they commute with the focusing dynamics.

The two components are the surviving remainder of the five open directions flagged in the source: the extension to inhomogeneous general relativity has since been resolved classically, and the operational meaning of scale-freedom in inhomogeneous settings is absorbed into that construction and its reading of the conformal mode as non-observable (Gryb and Sloan 2021; Bell and Sloan 2026c), leaving quantization and the statistical-mechanical and measure foundations as the open core. They share one objective: to establish what the frictional, scale-reduced ontology physically implies once the classical equivalence with Einstein-Hilbert dynamics is secured. A complete answer addresses both components, a component counting as resolved only when all of its listed sub-asks are addressed; the components may be advanced separately, but each is evaluated against the same classical contact formulation of general relativity.

References

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  2. Alessandro Bravetti, Hans Cruz, Diego Tapias, 'Contact Hamiltonian Mechanics', Annals of Physics 376, 17-39 (2017). DOI 10.1016/j.aop.2016.11.003, arXiv:1604.08266, https://arxiv.org/abs/1604.08266
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