The research, carried out by Jia-Zhou Liu, Shan-Ping Wu, Shao-Wen Wei, and Yu-Xiao Liu of Lanzhou University, tackles a subtle but fundamental question in the covariant phase space formalism, the modern mathematical framework in which conserved quantities such as energy and entropy arise as surface terms tied to diffeomorphism symmetries, the gauge redundancies of general relativity. Within this framework, the Iyer-Wald construction derives the entropy of a stationary black hole by evaluating a local functional built from the Lagrangian’s response to curvature, integrated over a cross section of the horizon. Crucially, the derivation assumes that all fields involved, including any matter fields that couple directly to curvature, can be smoothly extended onto the so-called regular bifurcation surface, the two-dimensional slice where the horizon generator vanishes. The Lanzhou team asked what happens when that smoothness assumption breaks down, as it does for some vector and tensor fields whose components become singular in horizon-adapted coordinates.

When a matter field cannot be smoothly extended to the bifurcation surface, the standard argument that forces certain horizon contributions to vanish no longer goes through. In particular, two types of terms can survive in the horizon surface charge variation, the quantity from which the entropy is extracted. The authors organize the full entropy entering the thermodynamic first law into a clean three-part decomposition: the entropy equals the standard Wald entropy, plus a correction coming from the non-Wald part of the entropy-sector Noether charge, plus a further integrable remainder coming from the presymplectic potential, the one-form that encodes the symplectic structure of the field theory on its space of solutions. This decomposition, which they label schematically as the entropy being the Wald term plus two additional pieces, provides a direct computational criterion for deciding when the familiar Wald density is sufficient and when the full horizon calculation is unavoidable.

Importantly, the authors distinguish their contribution from the well-known Jacobson-Kang-Myers ambiguity of the covariant phase space, in which one may shift the Lagrangian and the presymplectic potential by exact forms, reshuffling local expressions without changing the physics. Instead of exploiting that freedom, the team fixes a single representative, obtained by directly varying the displayed Lagrangian, and works consistently within it. After separating out ordinary matter and gauge work terms, such as electromagnetic or vector kinetic contributions that carry independent charges or fixed background variations, everything that remains in the entropy sector is subject to the new decomposition. The two corrective pieces are then evaluated explicitly for three representative families of static, spherically symmetric black holes, each illustrating a different physical possibility.

The first example involves gravity coupled to a background Kalb-Ramond antisymmetric two-form field, a nonminimal coupling inspired by string theory and by models of spontaneous Lorentz symmetry breaking. Here the coupling does modify the Wald entropy itself, rescaling it in terms of the tensor field’s vacuum expectation value, so the naive area law is already altered. Yet when the team carried out the full horizon analysis, both of the new correction terms vanished exactly. The Kalb-Ramond background is regular enough at the bifurcation surface that the standard Iyer-Wald reduction applies, and the entropy entering the first law is precisely the Wald entropy. This case demonstrates that nonminimal coupling by itself does not guarantee corrections; it is the regularity of the matter fields at the horizon that matters.

The second and perhaps most striking example is bumblebee gravity, a vector-tensor theory in which a fixed background vector field acquires a vacuum expectation value that spontaneously breaks Lorentz symmetry in the gravitational sector. The team examined two branches of exact black hole solutions, distinguished by whether the bumblebee vacuum expectation value is spacelike or lightlike. On the spacelike branch, the Noether charge correction vanishes, but a nonzero remainder survives from the presymplectic potential, and integrating that remainder on the space of solutions adds exactly half of the coupling-dependent contribution back into the entropy, yielding a first law entropy larger than the Wald value. On the lightlike branch, where the radial vector component diverges at the horizon in the static frame, both corrections are nonzero and, remarkably, they cancel each other exactly, so that the final entropy reduces to the plain area law even though neither the Wald term nor the individual corrections would suggest it. The cancellation is a genuine surprise, showing that pieces of the entropy can hide in different sectors of the phase space and conspire in unexpected ways.

The third class of examples pushes the analysis into higher-curvature territory: extended Gauss-Bonnet gravity formulated in Weyl geometry, where the connection is torsionless but not metric-compatible, with the failure of metric compatibility encoded in a Weyl vector field. In five spacetime dimensions, the team studied two solution branches, one in which the two coupling constants coincide and one in which they differ. Here both corrective mechanisms are active simultaneously: the curvature dependence of the tensor that multiplies the Riemann tensor in the Lagrangian becomes entangled with the Weyl vector, and the non-Wald Noether charge contribution as well as the presymplectic remainder both enter the answer. Assembling the full entropy from the three pieces, the authors verified that the enlarged thermodynamic first law closes properly, with the energy obtained from the asymptotic surface charge and an additional conjugate potential appearing for the solution’s hair parameter, which can be promoted to a genuine thermodynamic variable.

Beyond its technical elegance, the work carries a practical message for the black hole thermodynamics community. In recent years, debates have swirled around the correct entropy assignments for black holes in Horndeski theories, generalized Proca models, and other nonminimally coupled frameworks, with different calculations sometimes producing conflicting answers. The Lanzhou decomposition offers a common language for resolving such disputes: by checking whether the Noether charge correction and the presymplectic remainder vanish, vanish partially, or cancel, one can systematically determine the entropy that the first law demands. The authors also emphasize that their criterion separates regular tensor backgrounds, singular vector backgrounds, and higher-curvature Weyl-vector examples within one unified covariant phase space treatment, something no single earlier prescription accomplished.

The study leaves open a number of tantalizing directions. Rotating black holes, whose horizons involve an additional angular momentum term and a second Killing field, were deliberately excluded to keep the analysis tractable, and extending the decomposition to spinning and genuinely dynamical horizons is a natural next step. The authors also note that their analysis could be combined with the recently developed extended Iyer-Wald formalism and universal thermodynamic-volume constructions, which aim to give geometric meaning to the full suite of black hole thermodynamic variables. If the extra entropy terms identified here survive in broader settings, they could subtly alter phase diagrams, thermodynamic stability analyses, and even microscopic state counts of black holes in the modified gravity theories that increasingly populate the cosmological literature.

For now, the lesson is a humbling one. The Wald entropy, elegant as it is, rests on regularity assumptions that real solutions of realistic theories can violate. When a vector field refuses to behave smoothly at the horizon’s deepest point, finite pieces of entropy can leak out of the local Wald density and reappear elsewhere in the covariant phase space, sometimes enhancing the entropy, sometimes hiding inside a Noether charge, and sometimes canceling themselves away entirely. Black holes, it seems, still guard their secrets about the quantum nature of spacetime, and even our most trusted formulas must be checked against the full architecture of the theory in which they live.

Subject of Research: Corrections to black hole entropy beyond the Wald term in gravity theories with nonminimal matter-curvature couplings

Article Title: Black hole entropy beyond the wald term in nonminimally coupled gravity: a covariant phase space decomposition

Article References: Liu, J.-Z., Wu, S.-P., Wei, S.-W., & Liu, Y.-X. (2026). Black hole entropy beyond the wald term in nonminimally coupled gravity: a covariant phase space decomposition. The European Physical Journal C, 86(9), Article 1026. https://doi.org/10.1140/epjc/s10052-026-16290-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16290-y

Keywords: black hole entropy, Wald entropy, covariant phase space, nonminimal coupling, bumblebee gravity, Kalb-Ramond field, Gauss-Bonnet gravity, Weyl geometry, first law of thermodynamics, Noether charge, Lorentz symmetry breaking, black hole thermodynamics

General Parallel Relativity is not a new theory of gravity in the sense of predicting different experiments. It is a dynamically equivalent reformulation of Einstein’s general relativity, one in which the fundamental variable is not the metric alone but a parallel frame, a field of orthonormal directions threaded through spacetime. Because the parallel frame can be fixed once and for all, it supplies what the paper calls explicit boundary reference data: a concrete, physical specification of what the coordinates and the local laboratories are doing at the edge of any finite region. That seemingly technical bookkeeping device turns out to be the key to defining energy and entropy without ambiguity, because in general relativity the value of a conserved charge always depends on how the boundary of the region is anchored.

The technical heart of the work lies in the Hamiltonian or covariant phase space formalism. In this framework, every continuous symmetry of the action, such as a diffeomorphism that drags the fields along a vector field, generates a conserved current, and the integral of that current over a closed surface gives a charge. For the charge to be well defined and integrable, meaning that its value does not depend on the path taken through the space of field configurations, the boundary terms of the action must be chosen carefully. Koivisto derives the Noether currents and the associated Noether identities for the parallel formulations of gravity in full generality, covering metric-affine parallelism as well as absolute teleparallelism, and shows how the symplectic potential, the object that encodes the boundary response of the theory, decomposes into pieces associated with the frame, the connection and the matter fields.

The first major payoff is a quasi-local energy-momentum. Diffeomorphisms along the canonical frame directions generate surface charges that measure the energy and momentum contained within any closed two-surface, and crucially these charges include a genuine matter surface term. In older approaches, from Einstein’s infamous pseudotensors through to modern quasi-local constructions, the gravitational contribution to energy has always been reference-dependent and often ambiguous. In modern Hamiltonian language, as the paper notes, those pseudotensors can be understood as quasi-local expressions tied to particular boundary terms, reference choices and displacement vectors. The parallel formulation makes the reference choice explicit and physical: it is the parallel frame itself, fixed by the bulk fields and held fixed at the boundary as part of the Dirichlet data.

To test the construction, Koivisto applies it to two exact solutions: the Reissner-Nordström black hole, the charged solution of Einstein-Maxwell theory, and a dilatonic charged black hole of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger class. In both cases the matter contribution, computed from the conserved energy currents of the electromagnetic and scalar fields, combines with the Misner-Sharp mass, the standard spherically symmetric gravitational energy, to yield precisely the Hamiltonian energy appropriate to the electromagnetic boundary condition being imposed. When the electric potential rather than the charge is held fixed at the boundary, the resulting energy is the Legendre-transformed quantity familiar from ordinary electrostatics, sometimes called the electric enthalpy. The agreement between the Hamiltonian construction and the thermodynamic expectation is not assumed but derived, which is the strongest form of consistency check available in this subject.

The second payoff concerns black hole entropy. At a smooth horizon, the only local symmetry generator that survives on the codimension-two corner is a boost in the two-plane normal to the horizon surface. The corresponding Hamiltonian surface charge is evaluated on a Killing vector that generates the horizon’s time translation, and its value reproduces the Bekenstein-Hawking entropy. What makes this derivation distinctive is that the result satisfies the first law of black hole thermodynamics, the relation between changes in energy, entropy, angular momentum and electric charge, independently and in exact agreement with a completely separate calculation based on the Euclidean boundary action, the Gibbons-Hawking approach in which entropy is read off from the periodicity of imaginary time in the Euclidean section. Two independent routes arriving at the same thermodynamic relation is strong evidence that the surface charge is genuinely the entropy and not an artefact of one particular formalism.

The paper also situates the result within the broader landscape of teleparallel and gauge formulations of gravity. Earlier teleparallel treatments of black hole entropy obtained the entropy either from a bare diffeomorphism Noether charge combined with an assumed first law, or by adapting the Iyer-Wald construction to symmetric teleparallelism, or from the boundary term required for differentiability of the canonical generator in the Poincaré gauge theory sector. Where direct comparison is possible, the black hole energy expressions of those approaches agree with the new ones only asymptotically, as the radius tends to infinity, and differ at finite radius. The new construction has the advantage of being fully covariant and of treating the energy and entropy charges within a single unified framework, with the same boundary data controlling both.

Several subtle points deserve emphasis. The canonical frame that anchors the charges is determined by the bulk fields, but its pullback to the boundary surface is treated as fixed Dirichlet data, and the corner displacement is held fixed when the charge is varied. This care avoids a known pitfall: a parameter-dependent boundary frame can lead to apparent violations of the first law. The normalisation of the time leg of the frame is fixed by the asymptotics, and any other global choice would simply renormalise the conventional charges. The analysis also assumes a non-extremal horizon, since the smooth Euclidean normal disk used in the regularity argument degenerates at extremality, though the extremal geometry is recovered as a limiting case. Intriguingly, in pure absolute parallelism a certain multiplier charge is absent, but quantum corrections may generate curvature-dependent terms carrying a momentum of the same type, hinting at a dynamical, effective-gravity origin of horizon entropy that could connect this classical result to induced-gravity accounts of why black holes are thermal objects.

The broader significance is conceptual as much as computational. Physicists have long known that energy in general relativity is subtle, because the equivalence principle forbids any local, coordinate-independent definition of gravitational energy density. The covariant phase space programme, developed by Wald and many others and recently extended to include edge modes and corner symmetries, reframes the question: conserved quantities are not stored in points of spacetime but live on boundaries, and their values are fixed by the boundary conditions. General Parallel Relativity takes this philosophy to its logical conclusion by building the boundary reference data directly into the fundamental field. Energy-momentum and entropy then appear as two faces of the same structure, two kinds of Hamiltonian surface charge distinguished only by which symmetry of the parallel frame generates them: translations along the frame directions for energy-momentum, and normal-plane boosts for entropy.

Whether these tools will change anything about the black holes that astronomers actually observe is an open question, since the charges evaluated on stationary solutions reproduce familiar results. But for the community working on quantum gravity, holography and the still-unsolved problem of what spacetime is made of, having a single, covariant, internally consistent ledger in which a black hole’s energy and its entropy are entered as charges of the same formal kind is a genuinely useful advance. It suggests that the long-standing tension between gravitational energy and gravitational entropy may be, at least in part, an artefact of insisting on the metric as the only fundamental variable, and that the humble parallel frame, long dismissed as mere notation, carries real physical information about the boundaries of the universe’s accounting system.

Subject of Research: Hamiltonian surface charges for energy-momentum and black hole entropy in General Parallel Relativity

Article Title: Hamiltonian surface charges in general parallel relativity

Article References: Koivisto, T. S. (2026). Hamiltonian surface charges in general parallel relativity. General Relativity and Gravitation, 58(9), Article 103. https://doi.org/10.1007/s10714-026-03602-6

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03602-6

Keywords: General Parallel Relativity, Hamiltonian surface charges, black hole entropy, black hole thermodynamics, teleparallel gravity, quasi-local energy, covariant phase space, Reissner-Nordström black hole, dilatonic black hole, general relativity, horizon boosts, boundary terms