Extending the Lanczos-potential definition of gravitational entropy for perturbed FLRW space-times beyond the radiation-era, spatially-flat, linear tensor sector

Mena and Tod (Class. Quantum Grav. 24 (2007) 1733) implemented Penrose's suggestion that gravitational entropy should be built from potentials for the Weyl tensor: for purely gravitational (tensor) perturbations of a spatially-flat Friedmann–Lemaître–Robertson–Walker (FLRW) space-time with radiation equation of state they constructed a Lanczos potential, defined a hypersurface-based complex structure, and proposed a gravitational entropy $S_g$ whose one proven good property is that it is finite at the initial singularity exactly when the initial Weyl tensor is finite (i.e. when the singularity is isotropic, in the conformal-extendibility sense). The construction is established only in this restricted setting, and $S_g$ is explicitly time-dependent and non-monotone. The open problem is to extend — or rigorously delimit — this potential-based entropy: to general equations of state, non-zero spatial curvature, scalar and vector perturbations, and nonlinear gravitational clumping; to determine whether the finiteness criterion survives such extensions; to identify a canonical or physically preferred complex structure; to relate $S_g$ to black-hole entropy and other entropy proposals; and to determine whether any such construction can satisfy a second-law-type monotonicity. A 2021 minicourse by Tod confirms the extension is still unclear, and the adjacent Bel–Robinson-based proposal of Clifton, Ellis and Tavakol and its successors pursue a different functional that does not answer this question.

Background

General relativity offers no generally agreed local notion of the entropy stored in the free gravitational field. Penrose's Weyl Curvature Hypothesis (1979) conjectures that the initial singularity of the universe was special — the Weyl tensor (the conformal-curvature part of the Riemann tensor, which encodes the tidal, non-local gravitational degrees of freedom and vanishes in any perfectly homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) space-time) should be finite or vanishing there — and Penrose further suggested that a measure of gravitational entropy should be built from potentials for the Weyl tensor, on the model of the scalar product on the Hilbert space of one-particle states of linearized gravity in flat space. In flat-space linearized theory the space of solutions of the spin-2 wave equation carries a conserved symplectic form $\,\Omega$ (a closed, time-independent bilinear form on pairs of solutions) and a complex structure $J$ (an operator with $J^{2}=-1$, defined by the splitting of solutions into positive- and negative-frequency parts); the inner product $\langle\Phi,\Phi\rangle=\Omega(\Phi,J\Phi)$ is the squared particle number of the coherent quantum state corresponding to the classical solution, and Penrose proposed this norm as a guide to gravitational entropy.

Mena and Tod took this suggestion as far as it has ever been taken in a cosmological setting. A Lanczos potential $L_{abc}$ is a third-rank potential for the Weyl tensor (the analogue of the electromagnetic vector potential for the field strength), locally guaranteed to exist by a general spinor-potential theorem; a second potential, or superpotential, $\phi_{ab}$ with $L_{abc}=\nabla_{[a}\phi_{b]c}$ exists in vacuum space-times but encounters a Ricci-curvature obstruction in non-vacuum ones. For purely gravitational (tensor, i.e. gravitational-wave-type) perturbations of a spatially-flat FLRW background — perturbations with vanishing trace, divergence and fluid-parallel components, so that the perturbed Ricci tensor vanishes and, since the background Weyl tensor vanishes, the first-order Weyl tensor is gauge-invariant — Mena and Tod derived wave equations for the invariant parts of the Lanczos potential, solved them by Fourier transform in terms of Bessel functions for the radiation equation of state ($p=\rho/3$), and, following a prescription of Ashtekar and Magnon for Klein–Gordon fields, defined a complex structure $J$ on each constant-time hypersurface via the positive spatial Laplacian $\Theta=-D_{a}D^{a}$. The gravitational entropy $S_g$ at a chosen time $t_{0}$ is then the squared norm of the perturbation in the associated inner product. Its one proven good property: evaluated at the initial singularity, $S_g$ is finite exactly for perturbations whose initial Weyl tensor is finite, finiteness being understood in the conformal-extendibility sense — the metric (background plus perturbation) can be rescaled so as to extend smoothly through the singularity, i.e. the initial singularity is isotropic. Growing modes (finite initial Weyl tensor) give finite initial $S_g$; decaying modes give divergent initial $S_g$.

The construction is, however, established only in this restricted setting — spatially flat, radiation equation of state, linear theory, tensor modes only — and it is explicitly time-dependent and non-monotone: the symplectic form is not conserved in time, and the mode coefficients entering $S_g$ are non-monotone functions of time, so $S_g$ decreases from its initial value even for purely growing modes. The authors state that the definition has no obvious monotonicity property and suggest that monotone behaviour may require the nonlinear dynamics in which genuine gravitational clumping occurs. They also state that the correct, canonical definition of the complex structure $J$ in curved space — theirs is one natural choice among many, tied to a choice of hypersurface — is problematic, and that it is unclear how any such measure relates to black-hole entropy. For context, the main competing program defines gravitational entropy from the Bel–Robinson tensor (the unique divergence-free rank-four tensor quadratic in the Weyl tensor, interpretable as the super-energy-momentum of the free gravitational field) rather than from potentials: this is of the 'wrong differential order' in Penrose's sense — algebraic in the Weyl tensor rather than built from its potentials — but it is the line along which most later progress on monotone entropy growth during structure formation has occurred.

Problem Statement

The problem is to extend — or rigorously delimit — the potential-based gravitational entropy of Mena and Tod beyond its established setting of linear tensor perturbations of a spatially-flat, radiation-era FLRW space-time. The common objective uniting the following requests is a well-founded, Penrose-type gravitational entropy built from potentials for the Weyl tensor that is valid for realistic perturbed cosmologies and passes the two physical tests Penrose's program demands: the correct initial-singularity behaviour and second-law-type growth during structure formation.

1. Construct the hypersurface-based Lanczos-potential/superpotential entropy $S_g$ (or a justified modification of it) for at least one of: (a) a general perfect-fluid equation of state $p=(\gamma-1)\rho$ with $\gamma\neq4/3$; (b) non-zero spatial curvature $k\neq0$; (c) scalar (density) or vector perturbations, for which the perturbed Ricci tensor no longer vanishes and the curvature obstruction to the superpotential must be confronted; or (d) a controlled nonlinear regime in which genuine gravitational clumping occurs.

2. Determine whether the characteristic finiteness criterion survives the extension: is $S_g$ finite at the initial singularity exactly when the initial Weyl tensor is finite in the conformal-extendibility (isotropic-singularity) sense?

3. Identify canonical or physically preferred choices of the complex structure $J$ in the extended settings, or establish that no such choice exists and that the hypersurface dependence is irreducible.

4. Clarify the relation, if any, between $S_g$ and other gravitational-entropy notions — black-hole (Bekenstein–Hawking) entropy, curvature-invariant-based measures, density-contrast measures, and Bel–Robinson-based proposals — or prove that no consistent relation exists.

5. Determine whether a modified or nonlinear potential-based construction can satisfy a second-law-type monotonicity appropriate for gravitational clustering, or prove that no potential-based entropy with property (2) can be monotone.

A complete answer addresses facets 1, 2 and 5 — it establishes, with construction and proof, a potential-based gravitational entropy passing the finiteness and monotonicity tests, thereby implementing Penrose's program, or proves that no such extension exists — and answers facets 3 and 4, positively or negatively. For facet 1 it suffices to treat at least one of the cases (a)–(d), and a functional counts as a justified modification of $S_g$ only if it remains built from potentials for the Weyl tensor and is assessed against the same finiteness and monotonicity tests. Partial answers resolving individual facets for stated model classes are meaningful intermediate results; the facets are scientifically linked components of this one objective, not independent problems.

References

1. F. C. Mena and P. Tod, 'Lanczos potentials and a definition of gravitational entropy for perturbed Friedman–Lemaitre–Robertson–Walker spacetimes', Class. Quantum Grav. 24 (2007) 1733–1745. DOI: 10.1088/0264-9381/24/7/004, arXiv:gr-qc/0702057, https://arxiv.org/abs/gr-qc/0702057
2. P. Tod, 'Conformal Methods in General Relativity with application to Conformal Cyclic Cosmology: A minicourse given at the IXth IMLG Warsaw 2018', arXiv:2102.02701 (2021), https://arxiv.org/abs/2102.02701
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