The work, carried out by L. C. Garcia de Andrade of the Institute for Cosmology and Natural Philosophy, builds on the teleparallel equivalent of General Relativity, known as TEGR. Teleparallel gravity is mathematically equivalent to Einstein’s theory in its predictions, but it reformulates gravity in terms of torsion rather than curvature. Where general relativity describes gravity as the bending of spacetime, teleparallelism describes the same physics as a twisting of the internal frames that define direction and orientation at every point. Because the two pictures agree on observable outcomes, teleparallelism offers a complementary mathematical toolkit—one that makes certain questions about torsion, which are invisible in the standard formulation, suddenly tractable.

The heart of the new analysis lies in a class of topological defects called global textures. These are remnants of the early universe, formed when a quantum field that once filled all of space settled into its present state in a non-uniform way. Textures are thought to have seeded clumps of matter in galaxies and the voids between them, and they carry a characteristic energy scale set by the symmetry-breaking field. Crucially, the author treats textures not just as gravitational sources but as seeds of dynamical dark energy, drawing on earlier ideas that torsion could mimic the repulsive gravity associated with the universe’s accelerated expansion—a connection first explored by Gasperini in Einstein–Cartan cosmology and extended by Wanas to dark energy scenarios.

To extract a numerical constraint, the study matches an interior solution—a global texture obeying Einstein’s field equations—to an exterior flat spacetime, using the Israel–Lichnerowicz matching conditions generalized to Einstein–Cartan geometry. These conditions demand that the metric be continuous across the boundary and that the jump in its first derivatives be balanced by torsion. Working in null coordinates, the so-called light-cone approximation where the coordinate u equals t minus r, the author simplifies the matching equations dramatically. The texture metric used is the Notzold solution, characterized by a small parameter epsilon equal to eight pi G eta squared, where eta is the symmetry-breaking scale, bounded above by roughly ten to the eighteen GeV.

The payoff is striking. Applying the matching conditions to the angular part of the metric, and adopting the tachyonic approximation in which the radial coordinate exceeds the time coordinate, the torsion trace component T-squared-sub-1-2 comes out bounded below ten to the minus forty-four GeV. That is thirteen orders of magnitude tighter than the ten to the minus thirty-one GeV limit that Kostelecky and collaborators obtained in 2008 using dual maser experiments to search for torsion-induced Lorentz violation. The key to the estimate is identifying the texture correlation length with the inverse of the Hubble constant, roughly ten to the minus forty-two GeV in natural units, which converts the geometric matching condition into a cosmological bound. A second, independent component of the torsion trace, denoted K, is found to be approximately half of the first, confirming the internal consistency of the framework.

The analysis then turns to cosmic strings, the line-like cousins of textures. In teleparallel gravity, the connection splits into a Riemannian part and a contortion tensor, and with the teleparallel spin connection set to zero, the geodesic equation can be written entirely in terms of contortion. For a cosmic string metric with tension G mu, the contortion component K-sub-3-1-3 reduces to a simple expression involving the string tension and the radial coordinate. Plugging in the observational bound on string tension, eight G mu less than or equal to eight times ten to the minus seven, and taking a characteristic distance of one parsec, the author recovers a torsion trace bound of ten to the minus twenty-four per second—remarkably, the same order of magnitude that Nitsch derived in 1979 using metric-affine gravity and the physics of spinning particles on Earth.

Perhaps the most visually evocative result concerns what happens to particles near a torsion-carrying string. Solving the geodesic equations shows that a test particle moving in the plane orthogonal to the string follows a planar spiral, with angular velocity equal to one minus eight G mu divided by the radius. The author interprets this two-dimensional spiral as the projection of a three-dimensional Cartan spiral staircase, a structure previously explored by Alexander, Jenks, and Magueijo in the context of parity-violating cosmology. Because the radial acceleration is positive when the integration constant is positive, the teleparallel string is repulsive rather than attractive—reinforcing the picture of torsion-laden defects as candidates for the dark energy that drives cosmic acceleration. In this scenario, particles do not fall into the defect; they are flung outward while winding around it like threads on a screw.

The final strand of the argument connects torsion to gravitational waves, bringing the analysis into contact with real detector data. Gravitational waves propagating through teleparallel spacetime induce torsion waves, and the relation between the wave amplitude and the torsion component takes a simple form: the strain is proportional to the torsion divided by the gravitational-wave frequency. For cosmological gravitational waves at frequencies around ten to the minus seven hertz, a torsion of ten to the minus twenty-four per second implies a strain of order ten to the minus seventeen, comfortably within the sensitivity band of modern interferometers. For black-hole merger signals at higher frequencies near ten to the minus five hertz, the predicted torsion-induced strain drops to about ten to the minus twenty-three, close to the ten to the minus twenty-one strains actually recorded by the LIGO–Virgo collaboration. Using Rumpf’s older terrestrial bound of ten to the minus fifteen per second instead would amplify the predicted strain to ten to the minus eight, illustrating how dramatically torsion can enhance gravitational-wave amplitudes and how future detectors might discriminate between torsion scenarios.

Beyond the numbers, the study sketches a broader vision in which topological defects serve as laboratories for physics that no particle accelerator can reach. The author notes that the torsion–Hubble connection hinted at in the texture bound may even bear on the Hubble tension, the persistent discrepancy between different measurements of the universe’s expansion rate, following recent proposals that torsion condensates could alleviate the conflict. Future work, the paper suggests, will extend the framework to Einstein–Cartan–Kalb–Ramond gravity, where a fully antisymmetric three-index field introduces axion-like torsion, and to hybrid defects such as domain walls threaded by cosmic strings, including strings threading teleparallel wormholes.

For now, the headline result stands: by treating cosmic textures as gravitational matching problems in null coordinates, teleparallel gravity delivers the most stringent constraint yet on a component of Cartan torsion, pushing it below ten to the minus forty-four GeV. Whether nature actually exploits torsion at any measurable level remains open, but the study demonstrates that the universe’s largest structures—defects stretched across cosmic horizons—can probe its subtlest geometric twists with a precision that rivals, and in this case surpasses, the finest instruments ever built. If torsion exists, the spiral staircases winding through spacetime just got a great deal harder to hide.

Subject of Research: Constraints on Cartan spacetime torsion from teleparallel topological defects and gravitational waves

Article Title: Torsion constraints driven by TEGR topological defects in null coordinates, GWs and matching conditions

Article References: de Andrade, L. C. G. (2026). Torsion constraints driven by TEGR topological defects in null coordinates, GWs and matching conditions. The European Physical Journal C, 86(9), Article 1038. https://doi.org/10.1140/epjc/s10052-026-15957-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-15957-w

Keywords: torsion, teleparallel gravity, TEGR, cosmic textures, cosmic strings, topological defects, dark energy, gravitational waves, LIGO, Einstein-Cartan theory, matching conditions, contortion

A new theoretical study by Salvatore Capozziello and Giuseppe Meluccio, published in The European Physical Journal C, argues that the answer lies beneath geometry itself. The authors show that all three members of the geometric trinity, along with a broad family of extended gravitational theories, can emerge from a single pre-geometric framework in which spacetime has no metric at all, no notion of distance, angle or even causality. In this picture, curvature, torsion and non-metricity are not fundamental features of the fabric of spacetime. They are emergent properties, arising when a deeper gauge symmetry is spontaneously broken at what is expected to be an energy scale near the Planck length.

The starting point is a formulation of gravity that resembles the Yang-Mills gauge theories underpinning the Standard Model of particle physics. In ordinary metric-affine theories of gravity, the gravitational field is described by two independent sets of variables: the spacetime metric, or equivalently the tetrad fields, and the affine connection that tells observers how to transport vectors from point to point. Pre-geometric gravity, by contrast, begins on a four-dimensional differentiable manifold stripped of any metric structure. The fundamental degrees of freedom are the components of a gauge potential associated with the (anti-)de Sitter group, and the dynamics are governed by an additional Higgs-like field, in direct analogy with the Higgs mechanism that gives elementary particles their masses.

The crucial event in this framework is spontaneous symmetry breaking. A real Higgs-like field acquires a nonzero vacuum expectation value along a direction in an internal five-dimensional space, and this process reduces the fundamental gauge symmetry of spacetime from the de Sitter group SO(1,4) or the anti-de Sitter group SO(2,3) down to the Lorentz group SO(1,3). The components of the gauge field that contain the broken internal direction become the tetrads from which the spacetime metric is constructed, while the surviving components become the spin connection of the Lorentz group. In earlier work, notably a proposal by Frank Wilczek dating back to 1998, it was shown that when the fluctuations of the Higgs-like field are frozen out at low energies, this mechanism yields precisely the Einstein-Hilbert Lagrangian of general relativity, complete with a cosmological constant term and correctly identified Planck mass.

What remained unexplored, Capozziello and Meluccio point out, was the fate of the affine connection in this emergent phase, and whether more general gravitational actions could be reconstructed. Their paper addresses both gaps. First, the authors derive the gauge-fixing conditions on the pre-geometric gauge potential that generate the different affine connections used across the metric-affine landscape. The Levi-Civita connection of general relativity, the unique torsionless and metric-compatible connection on a pseudo-Riemannian manifold, emerges from a specific combination of the pre-geometric field components and their first derivatives. The Weitzenböck connection of TEGR, which is flat and metric-compatible, arises from the strikingly simple condition of setting the pre-geometric spin connection components to zero. The coincident gauge of STEGR, in which the affine connection vanishes altogether, likewise follows from a corresponding pre-geometric constraint.

Second, the authors construct pre-geometric versions of the actions of the geometric trinity. The Einstein-Hilbert action based on the Ricci scalar R can be obtained from the Wilczek Lagrangian supplemented by an additional term proportional to a pre-geometric volume-like quantity, with a suitable choice of coupling constant cancelling the emergent cosmological constant. The TEGR action, built on the torsion scalar T, emerges from a pre-geometric Lagrangian assembled from three carefully defined scalars that, after symmetry breaking, reproduce the quadratic combinations of the torsion tensor appearing in the torsion scalar. Notably, the antisymmetrisation of spacetime indices in these constructions ensures that no second-order time derivatives of the Higgs-like field appear, keeping the theory free of the Ostrogradsky instabilities that plague many higher-derivative theories.

The non-metricity case demanded the most ingenuity. Unlike curvature and torsion, non-metricity is not defined through the spin connection of the Lorentz group, which is antisymmetric in its internal indices, but through that of the general linear group, which admits a symmetric part. This means the pre-geometric gauge potential cannot be that of the (anti-)de Sitter group at all; it must be generalised to a broader symmetry, which the authors argue should be the five-dimensional general linear group GL(5). A further subtlety arises from group theory: the 25 generators of GL(5) do not decompose neatly into the 16 generators of GL(4) plus translations alone, but also include dual translation generators and a dilation generator. The authors show that a specific gauge-fixing condition, setting certain components of the gauge potential to zero, eliminates the unwanted degrees of freedom and allows the exact recovery of the STEGR Lagrangian based on the non-metricity scalar Q, with an emergent Planck mass identified in terms of the pre-geometric coupling constants and the vacuum expectation value.

Perhaps the most far-reaching result is that the construction generalises well beyond the trinity. The authors provide pre-geometric Lagrangian densities whose spontaneous symmetry breaking yields f(R), f(T) and f(Q) theories, as well as combined f(R,T,Q) theories in which f is a sufficiently smooth function of all three geometric invariants. Every pre-geometric action discussed is gauge invariant, generally covariant and background independent. In this sense, the metric-affine framework and the pre-geometric framework are shown to be fully compatible, with pre-geometry supplying a single unifying physical source for all the geometric notions that previously appeared as independent starting points.

The implications for quantum gravity could be significant. Pre-geometric theories are formulated in a Yang-Mills style on a manifold without a metric, exactly at the energy scale where general relativity is expected to break down, and the gauge choices of the emergent geometric theories are revealed to be gauge-fixing conditions on a single gauge potential rather than constraints on two independent variables. The authors argue this makes pre-geometric theories, in a sense, more natural than their metric-affine counterparts. There are also potential phenomenological signatures: pre-geometric theories like the Wilczek model possess three degrees of freedom rather than the two of a massless graviton, with the additional scalar degree of freedom excitable only in an ultra-high-energy regime near the Planck scale. Deviations from general relativity associated with this field could in principle be falsifiable, and the authors suggest the extra dynamics might even cure the ultraviolet divergences of geometric origin that afflict the emergent theories.

The study also connects to a broader classification scheme, dubbed the eightfold way of gravity, in which metric-affine theories are organised by the number of nonzero geometric invariants they involve. Special relativity corresponds to none, the geometric trinity to exactly one, theories such as Einstein-Cartan gravity to two, and the most general metric-affine theories to all three. Viewed through this configuration space, the new results indicate that the five-dimensional general linear group could serve as the unifying symmetry group of all metric-affine theories when these are understood as emergent from pre-geometric gravity. The authors note that a full Hamiltonian analysis of these theories, an exhaustive catalogue of symmetry-breaking patterns for the non-metricity case, and possible higher-dimensional Kaluza-Klein-style extensions remain open directions for future work, with several of these topics to be treated in a forthcoming paper.

Subject of Research: The emergence of the geometric trinity of gravity from pre-geometric gauge theory via spontaneous symmetry breaking

Article Title: The pre-geometric origin of geometric trinity of gravity

Article References: Capozziello, S., & Meluccio, G. (2026). The pre-geometric origin of geometric trinity of gravity. The European Physical Journal C, 86(9), Article 1073. https://doi.org/10.1140/epjc/s10052-026-16359-8

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16359-8

Keywords: general relativity, geometric trinity, pre-geometric gravity, spontaneous symmetry breaking, torsion, non-metricity, teleparallel gravity, Yang-Mills gauge theory, affine connection, quantum gravity, metric-affine gravity, Higgs mechanism

To understand what is at stake, it helps to recall what torsion is and why anyone cares. In Einstein’s general relativity, spacetime geometry is described entirely by curvature, and the connection that tells parallel vectors how to move is symmetric. Einstein–Cartan theory relaxes that assumption, allowing spacetime to carry torsion, a twisting of the geometry sourced by the intrinsic spin of matter. Because ordinary matter carries very little net spin density, torsion is expected to be fantastically weak, and no experiment has yet measured its coupling to fermions directly. Any theoretical scheme that promises a concrete numerical bound therefore attracts attention, which is precisely why the original paper, authored by L. C. Garcia de Andrade and published in July 2026, seemed exciting.

The original argument ran roughly as follows. Consider a pair of domain walls, thin sheets of concentrated energy, in a spacetime whose geometry is described by a tetrad, the set of local orthonormal frames that generalizes the metric. In the teleparallel formulation, the spin connection is set to zero, so the torsion two-forms are simply the exterior derivatives of the tetrads. Garcia de Andrade computed these torsion components, derived equations for a wall-separation function h(t,z), solved them, and then studied Dirac fermions living on the walls. The effective interaction entering the Dirac equation combined the electromagnetic coupling eB with the torsion coupling g_T multiplied by an axial torsion field S. Setting that combination to zero in a thin-wall limit, the paper extracted g_T as a ratio of the magnetic field to the torsion field, arriving at a bound near ten to the minus four.

Castro’s comment dismantles this chain at its first link. Recomputing the torsion two-forms directly from the tetrad ansatz with vanishing spin connection, he obtains expressions that differ from the published ones by factors of two and by misplaced coefficients. For example, where the original paper reports a coefficient of minus four pi sigma times z h times the time derivative of h in one torsion component, the direct calculation gives minus eight pi sigma times the same combination. Since every downstream equation for the wall dynamics is built on these two-forms, the discrepancies are not cosmetic. They propagate into the differential equations governing h(t,z) and ultimately into the claimed wave equation and its effective propagation velocity, which had been interpreted as a signature of gravitational radiation from the walls.

A second, independent problem involves a subtle but crucial distinction between coordinate components and tetrad components. The original paper writes the torsion components in a form that requires reading the coefficients in the tetrad basis rather than directly in the coordinate basis of time and spatial differentials. When Castro performs the conversion carefully, the equation for the wall-separation function integrates to a logarithmic expression, h squared equals z squared plus an inverse constant times the logarithm of a linear function of z squared and t. This is not the polynomial solution printed in the original paper. Even taking the original’s own scalar equations at face value, Castro shows that their exact integration yields a logarithm, not the cubic polynomial reported, unless an unstated perturbative expansion is smuggled in. The small-h approximation invoked in the original work, he demonstrates, does not rescue the polynomial form either; it merely imposes extra consistency conditions on integration constants.

The consequences ripple outward. The wave equation for the wall separation, and with it the interpretation of an effective velocity and the possibility of gravitational waves emitted by the coalescing walls, does not follow from the corrected calculation. Castro is careful about the limits of his claim: he does not prove that no gravitational radiation or propagating torsional mode can arise in some corrected model. He shows only that the specific derivation in the original paper is invalid as written. He also notes that even within the original’s own reduced flatness constraint, the solution h squared equals two At plus B means the dramatic coalescence of the walls at time zero occurs only for a special choice of the integration constant B, not as a forced consequence of the equations.

The fermionic sector fares no better in Castro’s assessment. Solving the zero-mode Dirac equations with the effective combination A equal to eB plus g_T times S, he finds Gaussian solutions whose signs differ from those published. For positive A, one chiral component is Gaussian-damped and normalizable while the other grows without bound; for negative A, the roles reverse. This is the familiar chiral selection pattern of localized zero modes, and it matters enormously for the condensate. Castro further shows that the scalar Dirac bilinear used to construct the condensate was computed incorrectly: with the beta matrix as defined in the original paper, the bilinear mixes left- and right-handed components and, in any normalizable chiral sector, actually vanishes, even though the positive probability density remains localized. The original analysis, he argues, conflated a scalar condensate with a positive density, two mathematically distinct objects.

Then comes the dimensional problem, which may be the most damning point of all. The original bound rested on identifying the combination eB plus g_T S with a wall thickness or separation, a quantity with dimensions of length. But in natural units a magnetic field has dimensions of mass squared, while an axial torsion vector carries dimensions of mass, so the sum cannot equal a length without additional dimensional parameters. In Castro’s corrected reading, the combination A instead fixes a localization length, roughly one over the square root of the absolute value of A. A thin, tightly localized fermion mode corresponds to a large A, not to A equal to zero. Imposing the cancellation condition, as the original paper did, actually makes the localization length diverge, the opposite of a thin wall. Estimating g_T therefore requires specifying an independent physical scale, and the magnetic field and torsion field alone cannot determine the coupling.

The numbers illustrate how far the original estimate strays. Using the original paper’s own magnetic field of ten to the minus nine gauss and torsion scale of ten to the minus thirty gigaelectronvolts, Castro finds that the cancellation condition yields g_T of order minus six gigaelectronvolts, not ten to the minus four. With a localization length of one centimeter, the corrected formula gives a coupling of order hundreds of gigaelectronvolts; with one meter, the result stays near the cancellation value. The magnetic length associated with the field is about eight centimeters. No unique number emerges without an independently specified scale, which is exactly the kind of hidden assumption that can make a striking bound evaporate under scrutiny.

Castro closes with a measured summary of eight points and a constructive caveat: none of this rules out the possibility that a corrected model of magnetized, torsion-carrying double walls could support localized fermion modes or eventually yield genuine phenomenological constraints on axial torsion. But the specific conclusions and the numerical bound in the original paper, he concludes, require a full rederivation from the ground up. For a field where torsion remains one of gravity’s best-hidden secrets, the episode is a reminder that the path to measuring spacetime’s twist runs through careful bookkeeping as much as through bold ideas, and that in theoretical physics, the factors of two are never just details.

Subject of Research: Corrections to the derivation of axial torsion coupling bounds from fermion condensates in Einstein–Cartan double domain walls

Article Title: Comment on “Einstein–Cartan fermion condensates trapped in double walls induce axial torsion coupling bounds [Eur. Phys. J. C (2026) 86: 785]”

Article References: Castro, L. B. (2026). Comment on “Einstein–Cartan fermion condensates trapped in double walls induce axial torsion coupling bounds [Eur. Phys. J. C (2026) 86: 785]”. The European Physical Journal C, 86(10), Article 1148. https://doi.org/10.1140/epjc/s10052-026-16388-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16388-3

Keywords: Einstein–Cartan gravity, spacetime torsion, teleparallel gravity, fermion condensates, domain walls, Dirac zero modes, axial torsion coupling, gravitational waves, tetrad formalism, dimensional analysis, theoretical particle physics, general relativity

New general relativity belongs to the broader family of teleparallel theories of gravity. Instead of describing gravity through the curvature of a Levi-Civita connection, as Einstein did, teleparallel theories use a flat connection with torsion, built from an object called the tetrad field. The original formulation dates back to the work of Hayashi and Shirafuji in 1979, who proposed a general quadratic Lagrangian in the three basic torsion invariants. That construction introduces three free dimensionless parameters, conventionally labelled with the coefficients associated with the axial, vector and symmetric traceless pieces of the torsion tensor. Special choices of these parameters recover the teleparallel equivalent of general relativity, often abbreviated TEGR, which is dynamically identical to Einstein’s theory despite its geometrically distinct formulation.

The difficulty is that most choices of the three parameters do not yield a healthy theory. A physically admissible subset of the parameter space must simultaneously satisfy three demanding conditions: the theory must be free of ghost instabilities, meaning that its propagating degrees of freedom must not carry negative kinetic energy; it must genuinely propagate a massless spin-2 mode, the gravitational wave carrier expected of any relativistic theory of gravity; and it must reproduce Newtonian gravity in the weak-field, slow-motion limit, so that ordinary attraction is recovered. Earlier work, including studies of gravitational waves and stability in NGR by Golovnev, Semenova and Vandeev and by Bahamonde and collaborators, has mapped out which corners of parameter space survive these tests. Beltrán Jiménez and Dialektopoulos had already identified non-linear obstructions for consistent versions of the theory, foreshadowing the kind of problem the new analysis uncovers.

López, Coley and Yildirim asked a deceptively simple question: assuming that a static, spherically symmetric configuration in NGR possesses a local black-hole horizon, what do the field equations say about the theory’s free parameters? A local black-hole horizon, in the sense used by Ashtekar, Krishnan, Hayashi and other researchers in black-hole mechanics, is a quasi-locally defined null surface that traps light, and its existence can be imposed without any detailed knowledge of the interior region. The authors considered configurations in vacuum, in the presence of a perfect fluid, and in the presence of an electromagnetic field, covering Reissner-Nordström-like charged holes as well as matter-filled geometries. They worked within a static, spherically symmetric, perturbative framework, expanding the field equations to first order in a small parameter near the horizon.

The result is a systematic exclusion argument rather than the discovery of a new solution. Whenever the existence of a horizon is imposed, the field equations, together with the matter conservation equation, drive the NGR parameters toward the very values that correspond to known pathological models: theories that contain ghost instabilities, theories that fail to propagate the spin-2 graviton, or theories that lack a consistent Newtonian limit. Conversely, in any corner of parameter space that passes the ghost-freedom, spin-2 propagation and Newtonian limit tests, the required black-hole configurations either do not exist or reduce to the TEGR case. In other words, the class of acceptable theories and the class of acceptable black holes do not overlap except in the trivial intersection with general relativity’s teleparallel twin.

Crucially, the authors emphasize that the obstruction is not geometric. They showed that the remaining admissible geometries are perfectly regular at the horizon: the torsion scalar invariants that define the NGR Lagrangian remain finite there, and the spacetime admits a consistent black-hole interpretation. This point matters because it separates two very different kinds of failure. If the invariants diverged at the horizon, the Lagrangian density of the teleparallel theory would itself be undefined there, and the horizon and its interior would be excluded from the manifold, a phenomenon the same group recently documented in the Schwarzschild geometry of TEGR, where regular and singular subclasses emerge according to the behavior of the Lorentz sector. Here, by contrast, the geometry survives; it is the underlying theory that breaks down. The pathology is theoretical, not a breakdown of spacetime at the horizon.

The perturbative machinery behind this conclusion is considerable. The authors constructed general static, spherically symmetric teleparallel geometries with free radial functions, substituted them into the NGR field equations, and solved the resulting system perturbatively near the horizon. Appendix material catalogues the auxiliary functions and the exact expressions for the coefficients entering the matter and geometrical sectors, including cases with electric charge, where combinations of the horizon radius and the charge appear in the parameter constraints. The upshot of the algebra is that the consistency conditions at the horizon act as a filter: only a measure-zero slice of parameter space allows the equations to close, and that slice coincides with the pathological regions or with TEGR itself.

Why should anyone outside mathematical relativity care? The answer lies in the current experimental landscape. Gravitational-wave astronomy and black-hole imaging have made it possible to test gravity in the strong-field regime for the first time, and modified gravity theories are frequently invoked as explanations for dark energy, dark matter, or the earliest moments of the cosmos. But a modification of gravity that cannot support the very objects we observe—stellar-mass black holes detected through LIGO and Virgo, and the supermassive shadows imaged by the Event Horizon Telescope—faces an immediate observational problem. If NGR admits no physically meaningful non-vacuum black holes, then any attempt to use it to model astrophysical compact objects is on unstable ground, and the theory’s viability must be sought in other arenas, such as cosmology, if at all.

The authors are careful to state the scope of their result. The conclusion holds within the static, spherically symmetric, perturbative framework they examined, and their definition of a black-hole horizon is deliberately minimal, remaining valid regardless of the detailed nature of the interior, including proposals for so-called singularity-free black holes. Rotating black holes, dynamical collapse, and fully non-linear solutions remain open questions, and extending the analysis to those regimes is a natural next step. Nevertheless, the pattern established here echoes earlier findings in teleparallel torsion theories, where black-hole existence has repeatedly proven to be a severe consistency filter. For new general relativity, the message of this work is stark and elegant at once: demand a healthy graviton, a Newtonian limit, and a black hole horizon, and the theory hands you back nothing new—only the geometries it inherits from Einstein.

Subject of Research: Black hole solutions in new general relativity, a torsion-based modification of general relativity

Article Title: On non-vacuum black holes in new general relativity

Article References: López, D. F., Coley, A. A., & Yildirim, B. (2026). On non-vacuum black holes in new general relativity. General Relativity and Gravitation, 58(9), Article 101. https://doi.org/10.1007/s10714-026-03605-3

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03605-3

Keywords: new general relativity, teleparallel gravity, black holes, torsion invariants, event horizons, ghost instabilities, Newtonian limit, spin-2 graviton, modified gravity, general relativity, spherically symmetric spacetimes, non-vacuum solutions

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

The foundation of the new analysis lies in teleparallel gravity, an alternative formulation of Einstein’s theory in which the central dynamical object is the tetrad, a field that attaches a local frame to every point in spacetime. In this picture, the torsion scalar T replaces the Ricci scalar R of general relativity, and the simplest version of the theory, known as the teleparallel equivalent of general relativity, reproduces all the standard predictions of Einstein’s framework. The advantage of the teleparallel route is technical: its field equations remain second order, avoiding the mathematical pathologies that plague many higher-derivative extensions. Extending this theory by adding an arbitrary function of the torsion scalar yields f(T) gravity, which has been widely studied as a way to explain the accelerated expansion of the late universe without invoking a cosmological constant or dark energy.

But f(T) gravity alone, the authors argue, cannot capture all of the higher-order torsional corrections that matter in the extreme gravity regimes of the very early universe. To fill this gap, theorists introduced the teleparallel analogue of the Gauss-Bonnet invariant, denoted T_G, a four-dimensional topological quantity built from combinations of the contortion tensor that acquires dynamical significance when coupled to nontrivial functions. The resulting theory, f(T, T_G) gravity, keeps the field equations at second order while enriching the gravitational dynamics with genuinely new terms that become important precisely where classical descriptions break down, namely near the hypothetical moment of the Big Bang singularity.

The central achievement of the new paper is an exact analytical reconstruction. Rather than solving the modified Friedmann equations numerically, the team began with six carefully chosen bouncing scale factors, the mathematical functions that describe how the size of the universe evolves in time. These included the symmetric bounce, in which contraction mirrors expansion around the bounce point; the superbounce, in which a sharply contracting cosmos reverses without a singularity; the oscillatory bounce, which produces repeated cycles of contraction and expansion; the matter bounce, rooted in loop quantum cosmology and capable of generating a scale-invariant spectrum of primordial perturbations; the Little Rip class of phantom-like evolutions; and the ekpyrotic bounce, in which slow contraction driven by a steep negative potential smooths the cosmos before a controlled transition to expansion.

For each scenario, the researchers expressed the scale factor as a function of the torsion scalar and examined three distinct classes of gravitational Lagrangian. The first, an additive separable form written as g(T) plus h(T_G), lets the torsion and Gauss-Bonnet sectors contribute independently and splits the Friedmann equation into two manageable ordinary differential equations. The second, a multiplicative coupling of the form T_G multiplied by g(T), captures direct interactions between the two geometric invariants. The third, written as T plus T_G g(T), preserves the standard teleparallel theory at low energies while allowing higher-order corrections to dominate near the bounce. In several cases the reconstructed solutions involve sophisticated special functions, including the Gauss hypergeometric function, the error function, and the incomplete Gamma function, reflecting the rich structure hidden inside these torsion-based models.

The analysis also tracked the key physical quantities through the bounce. Plotting the scale factor, the Hubble parameter, the energy density, and the effective pressure for every model, the authors verified that each scenario remains finite at the transition: the Hubble parameter passes smoothly from negative values during contraction, through zero at the bounce, to positive values during expansion, while the energy density and pressure stay bounded. This regular behavior stands in sharp contrast to standard cosmology, where curvature invariants and densities diverge at the initial singularity, signaling the breakdown of general relativity itself at the moment when the theory is needed most.

A crucial test of physical viability comes from the vacuum condition, the requirement that the reconstructed gravitational Lagrangian reduce correctly in the absence of matter, so that flat Minkowski spacetime is recovered when the torsional invariants vanish. The team found that this criterion discriminates sharply between the model classes. The multiplicative coupling and its minimally extended variant generally satisfy the constraint for the symmetric bounce and the superbounce, while the separable additive model fails, producing a trivial, physically unacceptable Lagrangian. For the matter bounce, only the additive model survives the vacuum test, while the coupled forms require setting the matter density parameter to zero, a condition that undermines a scenario fundamentally driven by dust-like matter. For the ekpyrotic case, the vacuum condition reduces one model to a purely torsion-dominated contribution, revealing a tension between mathematical consistency and the matter content the scenario requires.

The broader significance of the work lies in its unifying scope. Previous studies of bouncing cosmologies in modified gravity, including landmark analyses of superbounce and loop-quantum ekpyrosis in curvature-based F(R), F(G), and F(T) theories, typically treated single backgrounds with numerical or approximate reconstruction methods. By contrast, the new analysis derives exact reconstruction schemes for six distinct cosmological histories within one consistent teleparallel Gauss-Bonnet framework, allowing a direct, side-by-side comparison of their gravitational structures. The authors emphasize that the resulting models can describe a smooth transition from contraction to expansion without exotic matter violating energy conditions, positioning f(T, T_G) gravity as a serious competitor to inflation as a description of the earliest moments of cosmic history.

Of course, mathematical elegance is not the same as observational truth. The authors note that the reconstructed Lagrangians are reliable primarily near the bounce and must ultimately be tested against data. A natural next step, they suggest, is to compare the predicted Hubble expansion profiles with observations from cosmic chronometers, type Ia supernovae, and baryon acoustic oscillation measurements, which would prune the wide family of solutions down to a smaller set of observationally viable models. Extensions of the analysis should also address the stability of the solutions under perturbations and the shape of the primordial perturbation spectrum they generate. If those tests succeed, the picture they paint is striking: a universe that never began with a singularity, but with a quiet, torsion-driven rebound from a previous contracting epoch, described exactly by equations that Einstein’s own theory could never provide.

Subject of Research: Exact cosmological bouncing solutions in modified teleparallel f(T, T_G) gravity

Article Title: Exact cosmological bouncing solutions in (f(T,T_G)) gravity

Article References: Sultan, A. M., Sabir, A. A., & Said, J. L. (2026). Exact cosmological bouncing solutions in $$f(T,T_G)$$ gravity. The European Physical Journal C, 86(9), Article 1061. https://doi.org/10.1140/epjc/s10052-026-16154-5

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16154-5

Keywords: f(T, T_G) gravity, teleparallel gravity, bouncing cosmology, initial singularity, Gauss-Bonnet term, torsion scalar, matter bounce, ekpyrotic universe, oscillatory bounce, superbounce, modified gravity, early universe