Pre-Geometric Gravity: Emergent Spacetime

• Pre-geometric gravity is a theoretical framework where spacetime emerges from a non-metric gauge field and a Higgs-like field after spontaneous symmetry breaking.
• The approach uses a metricless starting point with de Sitter or anti-de Sitter gauge symmetries, leading to emergent tetrads, spin connection, and Einstein-Cartan dynamics.
• It bridges quantum gravity and cosmology by deriving general relativity, dark energy dynamics, and quantized geometrical features from deeper pre-geometric structures.

Pre-geometric gravity denotes a class of theories in which spacetime geometry is not fundamental but emergent. In the gauge-theoretic versions developed most explicitly in recent work, the starting point is a four-dimensional manifold without a prior metric, equipped instead with an internal de Sitter or anti-de Sitter gauge symmetry, a gauge connection $A_\mu^{AB}$, its curvature $F_{\mu\nu}^{AB}$, and a Higgs-like internal vector $\phi^A$. When the larger gauge symmetry is spontaneously broken to the Lorentz group $SO(1,3)$, part of the connection is reinterpreted as the tetrad and part as the spin connection, so that the metric, the Einstein-Hilbert or Einstein-Cartan action, the Planck scale, and the cosmological constant appear as broken-phase quantities rather than microscopic inputs (Addazi et al., 2024). In this sense, pre-geometric gravity is a research program in which geometry is a low-energy ordered phase of a more primitive gauge, topological, or constitutive structure (Addazi, 22 Dec 2025).

1. Metricless starting point and fundamental variables

The central technical feature of pre-geometric gravity is that the unbroken theory is formulated without an inverse metric. In the gauge-Higgs constructions based on $SO(1,4)$, $SO(2,3)$, or $SO(3,2)$, the only intrinsically available object for contracting spacetime indices is the Levi-Civita symbol $\epsilon^{\mu\nu\rho\sigma}$, treated as a tensor density. General covariance is therefore maintained by building scalar densities directly from $\epsilon^{\mu\nu\rho\sigma}$, the internal Levi-Civita symbol $\epsilon_{ABCDE}$, the curvature $F_{\mu\nu}^{AB}$0, and the Higgs-like field $F_{\mu\nu}^{AB}$1, rather than from $F_{\mu\nu}^{AB}$2, which does not yet exist. The basic covariant derivative is

$F_{\mu\nu}^{AB}$3

and the field strength is

$F_{\mu\nu}^{AB}$4

This is the sense in which the underlying phase is genuinely non-metric (Addazi, 22 Dec 2025).

A broader pre-geometric literature uses different microscopic variables while preserving the same logical order. In Euclidean pregeometry, for example, an $F_{\mu\nu}^{AB}$5 Yang-Mills gauge field $F_{\mu\nu}^{AB}$6 and a vector field $F_{\mu\nu}^{AB}$7 are taken as fundamental, with the metric defined as the composite bilinear $F_{\mu\nu}^{AB}$8. The microscopic action is built from two-derivative terms and admits a bounded Euclidean functional integral for $F_{\mu\nu}^{AB}$9 and $\phi^A$0, while general relativity appears only in the long-distance effective action (Wetterich, 2021). In the premetric or “purified gravity” program, the starting axioms are conservation of energy-momentum and the equivalence of gravitation and inertia; there the metric emerges as a Stueckelberg field and the graviton is interpreted as a Goldstone boson of broken symmetry (Koivisto et al., 2019). These variants share the claim that geometry is derived rather than assumed, but they differ substantially in microscopic ontology and in what is treated as fundamental.

2. Spontaneous symmetry breaking and the emergence of geometry

The standard gauge-Higgs mechanism of pre-geometric gravity is a spontaneous symmetry breaking

$\phi^A$1

with some papers using $\phi^A$2 for the anti-de Sitter case. The Higgs-like internal vector acquires a vacuum expectation value written as $\phi^A$3 or $\phi^A$4, depending on index convention. In the broken phase, the connection decomposes as

$\phi^A$5

so that the “translational” components become tetrads and the remaining components become the Lorentz spin connection. The metric is then reconstructed in the usual way,

$\phi^A$6

and with it the inverse metric and the volume element $\phi^A$7 (Addazi et al., 2024).

This reorganization is the defining emergence claim of the subject. Before symmetry breaking there is only a gauge field and a Higgs-like field on a manifold with differential structure; after symmetry breaking there are tetrads, curvature, torsion, and a metric spacetime. Recent formulations therefore describe the unbroken phase as a state with “no geometry,” while the broken phase generates simultaneously the metric, the Einstein-Hilbert action, the Planck scale, the cosmological constant, de Sitter entropy, and horizon degrees of freedom (Addazi et al., 28 Apr 2026).

The same mechanism is used to argue that the two classical tenets of general relativity are not microscopic postulates. In one line of argument, general covariance is already present at the density level in the pre-geometric theory, while diffeomorphism invariance in the usual gravitational sense becomes meaningful only after the metric emerges. Likewise, the equivalence principle is tied to the residual Lorentz symmetry of the broken phase, since the unbroken subgroup is precisely $\phi^A$8 (Addazi et al., 2024). A plausible implication is that in these models local Lorentz symmetry is more primitive than metric geometry, and metric geometry is a derived description adapted to the broken vacuum.

3. Broken-phase actions, Einstein-Cartan dynamics, and the Geometric Trinity

Two pre-geometric Lagrangians dominate the recent literature.

Model	Pre-geometric Lagrangian	Broken-phase output
MacDowell-Mansouri	$\phi^A$9	Einstein-Hilbert term, cosmological constant term, Gauss-Bonnet term
Wilczek	$SO(1,3)$0	Einstein-Hilbert term plus cosmological constant
Trinity extensions	$SO(1,3)$1, $SO(1,3)$2, $SO(1,3)$3 in suitable pre-geometric form	GR, TEGR, and STEGR after symmetry breaking and gauge choice

After spontaneous symmetry breaking, the MacDowell-Mansouri theory yields

$SO(1,3)$4

while the Wilczek theory yields

$SO(1,3)$5

In the MacDowell-Mansouri case, the broken-phase action contains the Gauss-Bonnet density $SO(1,3)$6; in the Wilczek case it does not. This is the origin of the repeated claim that the Einstein-Hilbert action is not fundamental but a low-energy effective action of a broken gauge phase, with the smallness of $SO(1,3)$7 controlled by a see-saw relation involving a large symmetry-breaking scale $SO(1,3)$8 (Addazi, 22 Dec 2025).

The field-equation analysis sharpens the relation to first-order gravity. In pre-geometric Einstein-Cartan theory, the broken-phase equations reproduce the Einstein equations and the Cartan torsion equations. For the Wilczek model, the gauge-field equation splits into vacuum Einstein and Cartan equations. For the MacDowell-Mansouri model, the Einstein equation is again recovered, while the spin-connection equation is modified linearly by curvature because of the Gauss-Bonnet contribution. The underlying Einstein-Cartan structure is explicit: torsion is algebraic in standard Einstein-Cartan theory and does not propagate (Meluccio, 5 May 2025).

The same pre-geometric logic has been extended beyond curvature-based gravity. A recent formulation derives the entire Geometric Trinity of Gravity from pre-geometry: general relativity from the Wilczek action, the teleparallel equivalent of general relativity from a torsion-scalar combination built from pre-geometric second derivatives of $SO(1,3)$9, and the symmetric teleparallel equivalent from a pre-geometric construction whose broken-phase limit reproduces the non-metricity scalar $SO(1,4)$0. In that analysis, GR and TEGR fit naturally with $SO(1,4)$1 or $SO(1,4)$2, whereas the STEGR construction requires enlarging the gauge group to $SO(1,4)$3 so that non-metricity can be represented (Capozziello et al., 16 Jun 2026).

4. Canonical structure, degrees of freedom, and quantization

The Hamiltonian analysis of pre-geometric gravity is carried out most fully for the Wilczek-type theory supplemented by a symmetry-breaking potential. The canonical momenta are computed directly from the pre-geometric fields $SO(1,4)$4, and the Hamiltonian density is polynomial: $SO(1,4)$5 In the spontaneously broken phase, the canonical structure reproduces the ADM or Einstein-Cartan Hamiltonian if and only if the time gauge is imposed. In this sense, the infrared theory is not merely Lagrangian-equivalent to gravity; its canonical structure matches the familiar one after the relevant gauge choice (Addazi et al., 2 May 2025).

In the ultraviolet unbroken phase, the theory is degenerate and must be analyzed with Dirac’s constrained Hamiltonian formalism. One recent count finds $SO(1,4)$6 dynamical phase-space variables, $SO(1,4)$7 gauge-fixing conditions, $SO(1,4)$8 first-class constraints, and $SO(1,4)$9 second-class constraints, giving

$SO(2,3)$0

These three physical degrees of freedom are interpreted as a massless graviton with $SO(2,3)$1 polarizations plus one massive scalar, identified with the remaining physical component $SO(2,3)$2 after the would-be Goldstone modes are absorbed in unitary gauge. Integrating out the massive scalar in the broken phase yields the Arnowitt-Deser-Misner Hamiltonian of general relativity (Addazi, 22 Dec 2025).

The same work connects pre-geometric gravity to canonical quantum gravity in two ways. First, it proposes a pre-geometric Wheeler-DeWitt equation

$SO(2,3)$3

where $SO(2,3)$4 is a quantum state of the gauge field and the Higgs-like field, rather than of the metric alone. Second, it gives an extended BF formulation in which the generalized simplicity constraint

$SO(2,3)$5

yields the pre-geometric action from BF data. This provides a direct bridge to constrained-BF and Loop Quantum Gravity-style formulations, while preserving the interpretation that geometry appears only after symmetry breaking (Addazi et al., 2 May 2025).

5. Cosmology, cosmological constant, entropy, and dark energy

Pre-geometric gravity has been used both for early-universe and late-universe questions. In the Einstein-Cartan extension with matter, the broken-phase equations reduce exactly to the standard flat-universe Friedmann equations with cosmological constant. In vacuum, a symmetry-reduced ansatz for the unbroken theory admits an exact solution that maps, after symmetry breaking, to a flat FLRW metric with

$SO(2,3)$6

that is, to a spatially flat de Sitter universe. The unbroken theory is then interpreted as a non-singular pre-geometric phase, while the Big Bang is reinterpreted as the time when the classical metric description begins to apply (Meluccio, 5 May 2025).

A second research line uses pre-geometric gravity to address the cosmological constant problem. In the MacDowell-Mansouri and Wilczek models, the same symmetry-breaking parameter $SO(2,3)$7 that generates the Planck scale suppresses $SO(2,3)$8 as $SO(2,3)$9 or $SO(3,2)$0. This see-saw relation has been combined with de Sitter entropy

$SO(3,2)$1

leading to the claim that

$SO(3,2)$2

In that formulation, the Higgs-like order parameter is treated as an “information field,” and vacuum stability is attributed not to radiative cancellations but to an entropic barrier. A transition from the observed high-entropy de Sitter vacuum to a UV state with $SO(3,2)$3 is assigned an amplitude

$SO(3,2)$4

The same work introduces hairons, described as pseudo-Nambu-Goldstone bosons associated with pre-geometric Wilson lines and orbifold or gravitational instantons, with mass $SO(3,2)$5 and the correspondence

$SO(3,2)$6

They are interpreted as horizon “hair” or “qubits” that encode the informational structure of the de Sitter vacuum (Addazi et al., 28 Apr 2026).

A closely related proposal identifies the Gauss-Bonnet coupling as a gravitational $SO(3,2)$7-angle. In the MacDowell-Mansouri broken phase one has

$SO(3,2)$8

so that $SO(3,2)$9. Because

$\epsilon^{\mu\nu\rho\sigma}$0

the Gauss-Bonnet term contributes only through topology in four dimensions, and the associated $\epsilon^{\mu\nu\rho\sigma}$1-angle is periodic. This leads to a discrete tower of topological sectors labeled by an integer $\epsilon^{\mu\nu\rho\sigma}$2, with the cosmological constant quantized by the sector label and tunneling between neighboring sectors suppressed as

$\epsilon^{\mu\nu\rho\sigma}$3

In that picture, the observed small cosmological constant corresponds to a very large topological sector stabilized by an enormous barrier in the Higgs potential (Addazi et al., 18 Feb 2026).

Late-time cosmic acceleration has also been modeled by adding the simplest quadratic extension of MacDowell-Mansouri pre-geometric gravity that preserves the topological pre-volume form symmetry. After symmetry breaking, the theory becomes $\epsilon^{\mu\nu\rho\sigma}$4 gravity, dual to a Galileon-like Horndeski scalar-tensor theory. The gravitational Gauss-Bonnet $\epsilon^{\mu\nu\rho\sigma}$5-angle becomes dynamical and is reinterpreted as a gravi-axion, whose ultra-light effective mass sets the dark-energy scale. In the phenomenological analysis reported there, the model fits DESI BAO+FS data with

$\epsilon^{\mu\nu\rho\sigma}$6

gives $\epsilon^{\mu\nu\rho\sigma}$7 and $\epsilon^{\mu\nu\rho\sigma}$8, and satisfies the tensor-speed bound $\epsilon^{\mu\nu\rho\sigma}$9 (Addazi et al., 8 May 2026).

6. Scope of the term, alternative programs, and conceptual constraints

The label pre-geometric gravity is used with different strengths in the literature, and several papers explicitly distinguish a strong and a weak sense. In one weak-field reconstruction, a conformastatic metric is derived from a relativistic formulation of D’Alembert’s principle and a dynamical interpretation of the equivalence principle, without invoking Einstein’s field equations; the same work states that this is “pre-geometric” only in the limited sense that geometry is reconstructed from force balance, proper time, and Lorentz invariance, not from non-spatiotemporal microphysics (Haro et al., 15 Oct 2025). Related geometrizations of Newtonian dynamics construct effective metrics for conservative-force motion and recover the Schwarzschild metric in the static spherical gravitational case, again without deriving gravity from a deeper non-geometric substrate (Friedman et al., 2019). Other authors use a tetrad-first language in which matter distribution is encoded geometrically in parameterized absolute parallelism, or a dynamical-3-space ontology in which the metric is induced and secondary; both are emergent relative to metric general relativity, but neither matches the gauge-Higgs notion of a metricless Yang-Mills-like pre-phase (Wanas et al., 2014, Cahill, 2011).

This plurality of meanings is one source of controversy. A common misconception is that any derivation of a metric without Einstein’s equations is automatically “pre-geometric” in the strong quantum-gravity sense. The literature does not support that identification. Several papers instead reserve the stronger terminology for models in which the unbroken phase has no metric, no tetrad, and sometimes no standard notion of curvature or causality, with geometry appearing only after a phase transition (Addazi et al., 2024).

A second source of caution comes from obstruction theory. Notes on the tetradic Einstein-Hilbert-Palatini action show that in many non-Lorentzian “concrete geometries” the cosmological term plays no role for $\epsilon^{\mu\nu\rho\sigma}$0, and that for $\epsilon^{\mu\nu\rho\sigma}$1 the full EHP theory is trivial. The same obstruction logic extends to abstract algebra-valued, graded, and supergeometric settings. This does not refute gauge-Higgs pre-geometric gravity, which is built differently, but it does indicate that gravity is not generically realized in arbitrary generalized geometries. A plausible implication is that viable pre-geometric completions are highly constrained rather than automatic (Martins et al., 2019).

Overall, pre-geometric gravity is best understood as a family resemblance term centered on one precise claim: the metric description of spacetime is emergent. In its strongest current form, that claim is implemented by a metricless gauge theory with spontaneous symmetry breaking, from which general relativity, Einstein-Cartan theory, teleparallel variants, de Sitter entropy, and cosmological sectors arise as broken-phase phenomena. In weaker usages, the term marks reconstructions of geometry from force laws, constitutive relations, or more primitive geometric data. The contemporary research program is therefore unified less by a single formalism than by a common reversal of logical order: gravity is not the starting point but the infrared manifestation of deeper, pre-geometric structure.