Parallel Frames Give Gravity a New Ledger for Black Hole Energy and Entropy

A single mathematical idea has quietly reshaped how physicists think about two of the deepest quantities in Einstein’s theory of gravity: the energy contained in a finite region of spacetime and the entropy of a black hole horizon. In a new study published in General Relativity and Gravitation, Tomi S. Koivisto of the University of Tartu and the National Institute of Chemical Physics and Biophysics in Estonia shows that, in a formulation of general relativity known as General Parallel Relativity, both of these quantities emerge naturally as Hamiltonian surface charges, the conserved quantities that live on the boundaries of a physical system. The result ties together decades of work on gravitational energy, black hole thermodynamics and the modern covariant phase space programme, and it does so with a level of internal consistency that has often eluded earlier attempts.

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: Hamiltonian surface charges in general parallel relativity. (n.d.). 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