A 2024 thesis by Chakraborty systematized the state of classical gravitational entropy (GE) and concluded that every existing proposal has a restricted domain: purely geometric Weyl-scalar recipes are coordinate-independent but thermodynamically disconnected and require ad hoc modifications for axisymmetric or rotating spacetimes, while the thermodynamically grounded Clifton–Ellis–Tavakol (CET) construction, based on the Bel–Robinson tensor, is algebraically unique only for Petrov types D and N, depends on the observer, and lacks a first-principles local gravitational temperature. Later 2025–2026 literature characterizes the non-unique CET decomposition in Petrov type I, derives a geometric free-field temperature within locally rotationally symmetric (Petrov D) geometries, and introduces new alternative (horizon- or charge-based) entropy proposals, but no construction or no-go result settles the existence of a universal, thermodynamically grounded gravitational entropy with the correct Bekenstein–Hawking limit, nor the principled relation between bulk fluid and horizon temperature in dynamical cosmologies that currently makes generalized-second-law (GSLT) tests ambiguous. The problem remains open.

Background

In general relativity, entropy is unambiguously associated only with horizons (the Bekenstein–Hawking entropy of black holes, SBH=A/4S_{\rm BH} = A/4 in Planck units, where AA is the horizon area), while matter fields carry their own thermodynamic entropy. Penrose's Weyl curvature hypothesis (WCH) additionally asserts that the free gravitational field should carry an entropy that vanishes at the initial cosmological singularity and grows as structure forms, the Weyl tensor being the part of the spacetime curvature that encodes the tidal, gravitational degrees of freedom not directly tied to local matter. Giving this idea a precise definition is the gravitational entropy (GE) problem.

Two families of classical GE proposals dominate the literature. In the geometric family, Rudjord, Grøn and Hervik (2008) and Romero, Thomas and Pérez (2012) define a GE from a vector field Ψ=P e^r\Psi = P\,\hat e_r built from a scalar PP constructed out of the Weyl tensor, with entropy density s=ks∣∇⋅Ψ∣s = k_s\lvert\nabla\cdot\Psi\rvert fixed to reproduce the Bekenstein–Hawking value on a black-hole horizon. These recipes are coordinate- and frame-independent and apply to any metric, but they are purely geometric constructs with no link to relativistic thermodynamics, and in axisymmetric or rotating spacetimes they must be modified by hand — replacing PP by the Weyl invariant CabcdCabcdC_{abcd}C^{abcd} or adding angular components to Ψ\Psi — to remain regular, with no general derivation of these choices. In the thermodynamic family, Clifton, Ellis and Tavakol (CET, 2013) define the effective energy-momentum tensor of the free gravitational field as a rank-2 square root of the Bel–Robinson tensor (the rank-4 tensor, quadratic in the Weyl tensor, that is uniquely divergence-free and is interpreted as the super-energy-momentum of free gravity), and then use a gravitational temperature TgravT_{\rm grav} to integrate an entropy production rate. This construction is grounded in relativistic thermodynamics and reproduces the Bekenstein–Hawking entropy when integrated over a Schwarzschild interior, but it is observer-dependent, and the required algebraic square-root decomposition of the Bel–Robinson tensor is unique only for Petrov types D and N. The Petrov classification is the algebraic classification of the Weyl tensor by its eigenvalue structure: type D is 'Coulomb-like' (Schwarzschild, Kerr), type N is wavelike (radiative fields), while type I is the generic case, with types II and III intermediate; generic spacetimes, including most inhomogeneous cosmologies, are of type I.

A separate, entangled issue is the gravitational temperature. The CET scheme requires a local TgravT_{\rm grav} that is chosen, not derived: it can vanish (e.g. for the Ellis wormhole) or be ill-suited away from horizons, and reproducing known semiclassical horizon temperatures works only in special limits. On the cosmological side, the generalized second law of thermodynamics (GSLT — the requirement that matter plus horizon entropy be non-decreasing) is tested on dynamical horizons using the Kodama–Hayward temperature (the temperature associated with the Kodama vector, a preferred time direction in spherically symmetric dynamical spacetimes generalizing the Killing observer). Chaplygin-gas cosmologies, a family of unified dark-matter/dark-energy fluid models that includes the variable modified Chaplygin gas (VMCG) and its 'new' variant (NVMCG), provide concrete test cases. Chakraborty's 2024 thesis showed that for the same VMCG model, using a bulk thermodynamically derived temperature T(z)T(z) versus the Kodama–Hayward horizon temperature reverses the GSLT conclusion, and that the NVMCG violates the GSLT on apparent and event horizons in parts of its parameter space — leaving the physical interpretation of such violations and the very choice of temperature for GSLT tests unresolved.

Problem Statement

Determine whether a thermodynamically grounded, universal gravitational entropy for classical general relativity exists, together with its local gravitational temperature. Either answer form resolves the objective: a construction satisfying conditions (1)–(4) below, or a no-go result establishing that no such construction exists. A complete answer must address, as one linked objective: (1) a definition of gravitational entropy that is well defined, unique under explicitly stated assumptions (unique up to explicitly stated observer or gauge choices, for each Petrov type including the generic Petrov type I, and recovering the conditional uniqueness of the CET Bel–Robinson square-root decomposition in the type-D and type-N cases), and regular (a finite entropy density free of the uncontrolled divergences — e.g. at symmetry axes or ring singularities — that force ad hoc modifications) for generic spacetimes — the generic Petrov type I and the other algebraic types beyond Petrov types D and N (types II and III), as well as the axisymmetric and rotating metrics (such as Kerr) and NUT-type geometries (metrics carrying a gravitomagnetic monopole, such as Taub–NUT) whose gravitational entropy previously required ad hoc modifications — not just on a case-by-case basis with ad hoc modifications of the auxiliary vector field Ψ\Psi or scalar PP; (2) reduction to the Bekenstein–Hawking entropy for black holes; (3) behavior consistent with the second law of thermodynamics, including monotonic growth with structure formation in cosmology, in keeping with Penrose's Weyl curvature hypothesis; and (4) a first-principles derivation of the local gravitational temperature entering the thermodynamic construction, valid beyond locally rotationally symmetric (spacetimes whose geometry is invariant under spatial rotations about a preferred observer) / Petrov type D geometries, reproducing the Hawking, Unruh, and horizon (Kodama–Hayward) temperatures in their appropriate limits, and non-degenerate in spacetimes where the GE should be computable — e.g. traversable wormholes such as the Ellis wormhole, where the CET prescription gives Tgrav=0T_{\rm grav}=0 and makes the computation of GE impossible. Conditions (1)–(4) are jointly required: a construction must satisfy all four together — with (4) supplying the temperature entering the construction of (1) — and a no-go result must establish that they cannot be satisfied jointly; an answer addressing only a proper subset does not resolve the objective.

The related but distinct question of how the bulk fluid temperature relates to the dynamical-horizon (Kodama–Hayward) temperature in near-equilibrium cosmologies — which currently makes GSLT tests for Chaplygin-type models such as the NVMCG ambiguous — is a separate open question raised in the thesis epilogue; it is recorded here as context, not as a required deliverable of this objective.

References

  1. Samarjit Chakraborty, 'On the thermodynamic aspects of gravity', PhD thesis (2024), arXiv:2401.10975, DOI 10.48550/arXiv.2401.10975, https://arxiv.org/abs/2401.10975
  2. Timothy Clifton, George F. R. Ellis, Reza Tavakol, 'A gravitational entropy proposal', Class. Quantum Grav. 30 (2013) 125009, arXiv:1303.5612, DOI 10.1088/0264-9381/30/12/125009, https://arxiv.org/abs/1303.5612
  3. Øystein Rudjord, Øyvind Grøn, Sigbjørn Hervik, 'The Weyl curvature conjecture and black hole entropy', Phys. Scr. 77 (2008) 055901, DOI 10.1088/0031-8949/77/05/055901, https://doi.org/10.1088/0031-8949/77/05/055901
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