Pre-Geometric Gravity (PGG)

• Pre-Geometric Gravity (PGG) is a theoretical framework where spacetime geometry and gravitational dynamics emerge from gauge-theoretic foundations and spontaneous symmetry breaking.
• It employs a fundamental gauge connection and Higgs-like fields to derive familiar structures of General Relativity through mechanisms like symmetry breaking from SO(1,4) or SO(3,2) groups.
• The framework provides insights into quantum gravity, the cosmological constant problem, and the resolution of singularities by dynamically generating mass scales and effective gravitational actions.

Pre-Geometric Gravity (PGG) designates a class of theories in which spacetime geometry and gravitational dynamics arise as emergent phenomena from a more fundamental gauge-theoretic framework. These approaches treat connections and “Higgs-like” fields, rather than the metric or tetrad, as fundamental, with the familiar structure of General Relativity (GR) and its extensions emerging via spontaneous symmetry breaking (SSB) from a larger internal gauge group—typically SO(1,4) (de Sitter) or SO(3,2) (anti-de Sitter). The Einstein-Hilbert action, cosmological constant, and even the Planck scale are rendered as derived quantities from the symmetry-breaking process. PGG frameworks offer a potential bridge between GR, canonical quantum gravity, and particle physics, while providing a novel route for quantum gravitational UV completions and novel cosmological mechanisms.

1. Fundamental Structure and Gauge-Theoretic Foundations

Pre-geometric gravity is formulated on a differentiable 4-manifold with no a priori metric structure. The fundamental fields are:

• A gauge connection $A_\mu^{AB}(x)$ valued in SO(1,4) or SO(3,2) ($A,B=0,\ldots,4$), and
• A scalar “Higgs-like” field $\phi^A(x)$ transforming in the fundamental representation.

The dynamics are encoded in gauge-invariant Lagrangians constructed from the field strength

$$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$

and the covariant derivative

$$\nabla_\mu\phi^A = \partial_\mu\phi^A + A_{B\mu}^{\;\;A}\phi^B.$$

Action densities are built solely from these objects and the Levi-Civita symbols $\epsilon^{\mu\nu\rho\sigma}$ and $\epsilon_{ABCDE}$ since no metric is available prior to SSB. Two archetypal Lagrangians are prominent:

• The Wilczek action

$$\mathcal{L}_{\rm W} = k_{\rm W}\,\epsilon_{ABCDE}\,\epsilon^{\mu\nu\rho\sigma}\, F_{\mu\nu}^{AB}\,\nabla_{\rho}\phi^C\,\nabla_{\sigma}\phi^D\,\phi^E,$$

• The MacDowell–Mansouri action

$$\mathcal{L}_{\rm MM} = k_{\rm MM}\,\epsilon_{ABCDE}\,\epsilon^{\mu\nu\rho\sigma}\, F_{\mu\nu}^{AB}\,F_{\rho\sigma}^{CD}\,\phi^E,$$

with distinct physical consequences for emergent dynamics (Addazi et al., 2024, Addazi, 22 Dec 2025).

A generic action then reads: $$S = \int d^4x\,(\mathcal{L}_{\rm MM/W} + \mathcal{L}_{\rm SB})$$ where $A,B=0,\ldots,4$0 is an SSB-inducing potential.

2. Mechanism of Spontaneous Symmetry Breaking and Emergence of Geometry

PGG models implement SSB through a “Mexican-hat” or cosine-type potential favoring a nonzero vacuum expectation value (VEV) for $A,B=0,\ldots,4$1: $A,B=0,\ldots,4$2 This breaks SO(1,4) or SO(3,2) to SO(1,3). Accordingly, one splits the connection: $A,B=0,\ldots,4$3 where $A,B=0,\ldots,4$4 is the emergent vierbein and $A,B=0,\ldots,4$5 a mass parameter.

The metric emerges as: $A,B=0,\ldots,4$6

The emergent action (after SSB) takes the Einstein–Cartan or Einstein–Hilbert form, e.g.,

$A,B=0,\ldots,4$7

where $A,B=0,\ldots,4$8 is the Gauss–Bonnet density. The Planck mass and the cosmological constant are given by see-saw-like relations, e.g.,

$A,B=0,\ldots,4$9

These illustrate that both fundamental mass scales emerge dynamically (Addazi et al., 2024, Addazi, 22 Dec 2025).

3. Field Equations and Gravity Sector in Broken and Unbroken Phases

Variation of the action yields:

• Metric (vierbein) variation gives Einstein’s equations with cosmological constant:

$\phi^A(x)$0

• Spin connection variation yields the (algebraic) Cartan equation for torsion:

$\phi^A(x)$1

with canonical spin current $\phi^A(x)$2.

In the unbroken phase, a unified gauge-theoretic field equation couples all pre-geometric fields and interpolates between high-energy UV dynamics and low-energy GR: $\phi^A(x)$3 with the combined “pre–EC tensor” capturing unified gravitational sources (Meluccio, 5 May 2025, Addazi, 22 Dec 2025).

4. Hamiltonian Formulation, Constraint Structure, and Degrees of Freedom

The Hamiltonian analysis is grounded in Dirac’s algorithm for constrained systems:

• Canonical pairs: $\phi^A(x)$4, $\phi^A(x)$5
• Constraints classified as:
  • 10 first-class (generating SO(1,4) or SO(3,2) gauge transformations)
  • 44 second-class (resulting from the field definitions and primary constraints)

Degrees of freedom count in the UV unbroken phase: $\phi^A(x)$6 interpreted as a massless spin-2 graviton (2 d.o.f.) plus a scalar mode from $\phi^A(x)$7 (Addazi et al., 2 May 2025, Addazi, 22 Dec 2025).

Upon symmetry breaking and integrating out the scalar, the canonical GR ADM formalism is recovered, with the Hamiltonian constraint structure exactly matching General Relativity (Addazi, 22 Dec 2025, Addazi et al., 2 May 2025).

5. Cosmological Solutions and Singularity Resolution

An explicit homogeneous and isotropic ansatz in the pre-geometric phase yields a regular, non-singular solution: $\phi^A(x)$8 that after SSB maps to the de Sitter metric: $\phi^A(x)$9 with all geometric and gauge fields smooth as $$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$0 (Meluccio, 5 May 2025). There is no curvature singularity in the fundamental variables, and the “Big Bang” is dynamically resolved as the SSB transition from the pre-geometric to the geometric phase.

6. Quantum Aspects, Wheeler–DeWitt Equation, and UV Completions

Quantization of PGG is addressed via:

• A pre-geometric Wheeler–DeWitt equation:

$$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$1

with $$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$2 a wave-functional on the space of gauge and Higgs fields. In the low-energy, symmetry-broken sector, this reduces to the standard Wheeler–DeWitt equation for the 3-metric (Addazi et al., 2 May 2025, Addazi, 22 Dec 2025).

• UV completion pathways include:
  • Topological BF-formulation and “simplicity” constraints, with classical equations imposing the emergence of Einstein–Cartan gravity as the SSB phase (Addazi et al., 2 May 2025).
  • Power-counting renormalizability of the unbroken gauge theory, suggesting connections to asymptotic safety or other functional RG fixed points (Addazi et al., 2024).

PGG also encompasses group field theory (GFT) approaches, in which continuum spacetime and the gravitational Hamiltonian emerge from the critical behavior of pre-geometric quantum field theories over group manifolds (Sindoni, 2011).

7. Cosmological Constant, Holography, and Dynamical Dark Energy

PGG provides new mechanisms for the cosmological constant (CC) problem:

• The Gauss–Bonnet coupling after SSB scales as the de Sitter entropy,

$$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$3

• The CC is quantized into discrete topological sectors with vacuum selection governed by the Higgs barrier, suppressing quantum transitions between vacua by $$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$4, and giving a dynamical explanation for the CC’s smallness (Addazi et al., 18 Feb 2026, Addazi et al., 28 Apr 2026).
• The dynamical sector incorporates additional pseudo-Nambu–Goldstone bosons (“hairons” or gravi-axions) whose mass scale $$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$5 and whose dynamics potentially account for cosmic acceleration. Quadratic (f(Lovelock)) extensions of the original action yield a phenomenology closely mimicking $$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$6CDM, yet with testable deviations in gravitational slip, GW propagation, and cosmological time variation of $$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$7 (Addazi et al., 8 May 2026, Addazi et al., 28 Apr 2026).

8. Matter Couplings, Emergent Principles, and Extensions

PGG provides a natural “dictionary” mapping pre-geometric building blocks onto standard matter couplings:

• Scalar, spinor, gauge kinetic terms, and matter couplings arise via SSB from gauge-invariant densities built from $$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$8, mapping onto conventional minimally-coupled QFTs in curved spacetime (Addazi et al., 2024).
• The emergence of full diffeomorphism invariance and the equivalence principle follows automatically: the residual SO(1,3) invariance ensures local Lorentz covariance, while active coordinate transformations become solution-generating symmetries (Addazi et al., 2024, Meluccio, 5 May 2025).
• PGG archetypes generalize further to include propagating torsion (Einstein–Cartan theory), extended quadratic invariants, and potential unification with GUT groups via larger gauge structures (e.g., SO(1,13)), and naturally support “hyperunification” scenarios (Chkareuli, 2017, Addazi, 22 Dec 2025).

9. Summary Table: Core Structural Features

Feature	Emergent After SSB	UV Pre-geometric Phase
Metric/vierbein	$$F_{\mu\nu}^{AB} = 2\,\partial_{[\mu}A_{\nu]}^{AB} + 2\,A_{C[\mu}^{\;\;A}A_{\nu]}^{CB}$$9	Field is undefined
Spin connection	$$\nabla_\mu\phi^A = \partial_\mu\phi^A + A_{B\mu}^{\;\;A}\phi^B.$$0	Part of full connection
Planck mass and cosmological constant	$$\nabla_\mu\phi^A = \partial_\mu\phi^A + A_{B\mu}^{\;\;A}\phi^B.$$1 (see-saw)	Not fundamental
Degrees of freedom	2 (graviton) + 1 (scalar, heavy)	3 (full gauge–Higgs system)
Diffeomorphism invariance	Dynamically restored	Built in via Levi–Civita construction
CC problem	Topologically protected	Quantized by θ-angle, high-entropy barrier
Quantum gravity scheme	Canonical ADM/Wheeler–DeWitt	Wavefunctional on $$\nabla_\mu\phi^A = \partial_\mu\phi^A + A_{B\mu}^{\;\;A}\phi^B.$$2

PGG thus realizes GR and its principles as emergent, dynamical consequences of gauge symmetry breaking, and delivers a UV-complete, background-independent program with rich phenomenological and cosmological implications (Addazi, 22 Dec 2025, Addazi et al., 2024, Meluccio, 5 May 2025, Addazi et al., 2 May 2025, Addazi et al., 18 Feb 2026, Addazi et al., 28 Apr 2026, Addazi et al., 8 May 2026, Chkareuli, 2017, Sindoni, 2011).