Gravity Before Geometry: How Spacetime’s Curvature, Torsion and Non-Metricity May Emerge From a Deeper Symmetry

One of the strangest facts in modern theoretical physics is that Einstein’s general relativity is not unique. Physicists have known for decades that two apparently different theories, built on entirely different geometric ideas, produce exactly the same predictions for gravity. General relativity attributes gravity to the curvature of spacetime. The teleparallel equivalent of general relativity, known as TEGR, attributes it instead to torsion, a kind of twisting of spacetime. A third formulation, the symmetric teleparallel equivalent of general relativity, or STEGR, locates gravity in non-metricity, the failure of parallel transport to preserve the lengths of vectors. Three different geometric properties, three different theories, one identical physics. This trio has come to be known as the geometric trinity of gravity, and the question of why such radically different descriptions should be equivalent has puzzled researchers for years.

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