Gravity from Pre-geometry: Emergent Spacetime

• Gravity from pre-geometry is a framework where classical spacetime and gravitational dynamics emerge from non-metric microphysics.
• It employs mechanisms like spontaneous symmetry breaking and combinatorial critical phenomena to derive the Einstein–Hilbert action from gauge-theoretic foundations.
• The approach offers practical insights into matter coupling, UV completion, and solutions to the cosmological constant problem via emergent geometric structures.

Gravity from pre-geometry refers to the class of theoretical frameworks in which spacetime geometry, the gravitational field, and associated dynamical equations arise as emergent macroscopic phenomena from fundamentally non-geometric, pre-metric, combinatorial, or gauge-theoretic microphysics. Such frameworks are constructed without presupposing a spacetime metric or connection; rather, these classical concepts appear only after appropriate collective or symmetry-breaking effects take place. Recent research demonstrates several realizations of this paradigm, highlighting a unifying narrative that gravitational dynamics—such as those encapsulated by the Einstein–Hilbert action—can be derived from spontaneous symmetry breaking of larger gauge symmetries, combinatorial critical phenomena, noncommutative geometry, or relational/axiomatic principles (Addazi et al., 2024, Addazi, 22 Dec 2025, Yang, 2010, Trugenberger, 2016, Koivisto et al., 2019, Addazi et al., 2 May 2025, Addazi et al., 18 Feb 2026).

1. Structural Foundations of Pre-geometric Gravity

Pre-geometric gravity typically starts from a smooth four-manifold $M$ that is not endowed a priori with a metric, relativity structure, or even coordinates. Instead, the fundamental variables are:

• Connections $A_\mu^{IJ}$ for a higher symmetry gauge group $G$, typically $SO(1,4)$ (de Sitter) or $SO(3,2)$ (anti-de Sitter), with $I,J = 0,...,4$;
• An internal vector “Higgs field” $\Phi^I$ or $\phi^A$, transforming under $G$;
• Optionally, further combinatorial, spin network, or algebraic structures.

The pre-geometric action is manifestly diffeomorphism-invariant by construction using only the Levi-Civita density $\epsilon^{\mu\nu\rho\sigma}$ and internal group invariants, for instance: $A_\mu^{IJ}$0 with $A_\mu^{IJ}$1 the curvature of $A_\mu^{IJ}$2 and $A_\mu^{IJ}$3 a potential enforcing spontaneous symmetry breaking (SSB) (Addazi et al., 2024, Addazi, 22 Dec 2025, Addazi et al., 2 May 2025, Addazi et al., 18 Feb 2026).

Other approaches emphasize pre-metric combinatorics (as in random regular graphs modeling connectivity and Ricci curvature, or the combinatorics of spin networks), or algebraic quantum structures such as noncommutative matrix models (Trugenberger, 2016, Yang, 2010, Steinacker, 2010, Vaid, 2017).

2. Spontaneous Symmetry Breaking and Emergence of Geometry

Gravitational dynamics and a metric structure emerge when the internal Higgs field acquires a nontrivial vacuum expectation value (VEV), typically in one internal direction: $A_\mu^{IJ}$4. This breaks $A_\mu^{IJ}$5, where $A_\mu^{IJ}$6 is the Lorentz group. The gauge fields $A_\mu^{IJ}$7 (for $A_\mu^{IJ}$8) become identified, up to normalization, with the vierbein $A_\mu^{IJ}$9, while $G$0 ($G$1) become the spin connection $G$2 (Addazi et al., 2024): $G$3 The emergent spacetime metric is then $G$4.

In the combinatorial/criticality approach, geometric structure arises through a quantum phase transition. The ordered phase of random-graph models displays locally lattice-like connectivity, allowing the identification of a Riemannian metric and continuum limit (Trugenberger, 2016).

Other frameworks, such as the teleparallel or “premetric” program, relate torsion or excitation forms in the absence of a metric; only after imposing local Lorentz invariance does one recover the equivalence to the Einstein–Hilbert theory (Itin et al., 2016, Koivisto et al., 2019).

3. Reconstruction of Gravitational Dynamics

Upon SSB, the pre-geometric action reduces to an effective action for the emergent vierbein and spin connection, yielding the Einstein–Hilbert and cosmological constant terms, along with topological invariants such as Gauss–Bonnet: $G$5 Here $G$6 and $G$7 are determined by the parameters of the SSB sector: e.g., $G$8, $G$9 (Addazi et al., 2024, Addazi, 22 Dec 2025). The sign of $SO(1,4)$0 reflects whether the underlying gauge group is de Sitter or anti-de Sitter.

Variants such as the MacDowell–Mansouri and Wilczek constructions yield, after symmetry breaking, either

• The Einstein–Hilbert term plus cosmological constant and Gauss–Bonnet density (MacDowell–Mansouri), or
• A pure Einstein–Hilbert term with $SO(1,4)$1 (Wilczek) (Addazi, 22 Dec 2025, Addazi et al., 18 Feb 2026).

At the canonical level, the Hamiltonian analysis confirms that the number of physical degrees of freedom matches General Relativity: two graviton polarizations plus, in general, a massive scalar (from the Higgs sector), which decouples in the deep IR (Addazi et al., 2 May 2025, Addazi, 22 Dec 2025).

4. Coupling to Matter and Emergent Geometric Structures

Post-SSB, all standard matter couplings arise via minimal coupling to the emergent spin connection and vierbein. The full dictionary between pre-geometric gauge data and emergent geometric objects is established:

• $SO(1,4)$2 (vierbein)
• $SO(1,4)$3 (spin connection)
• $SO(1,4)$4 (metric) Matter fields couple to $SO(1,4)$5 via covariant derivatives pre-SSB and recover the standard Dirac, Yang-Mills, and scalar actions in the broken phase (Addazi et al., 2024, Addazi, 22 Dec 2025, Addazi et al., 2 May 2025).

Torsion and Riemann-Cartan curvature emerge naturally: $SO(1,4)$6 This provides a geometric interpretation for all gauge dynamics, including possible couplings to topological or higher-curvature invariants.

5. Quantization, Hamiltonian Structure, and UV Completion

The Hamiltonian analysis of pre-geometric gravity reveals a rich structure of constraints. In the unbroken phase, the only (first-class) constraints are the SO(1,4) Gauss constraints, and the theory is topological—there is neither a local metric nor Hamiltonian constraint generating diffeomorphisms in a metric sense (Addazi et al., 2 May 2025). After SSB, the emergent variables and Dirac brackets reproduce precisely the Arnowitt-Deser-Misner (ADM) formulation and constraint algebra of General Relativity: $SO(1,4)$7 Here, $SO(1,4)$8 and $SO(1,4)$9 are lapse and shift, $SO(3,2)$0 and $SO(3,2)$1 the Hamiltonian and momentum constraints.

Quantization can proceed via implementation of quantum constraints (Wheeler–DeWitt equation) in the emergent geometric sector: $SO(3,2)$2 This provides a conceptual bridge to Loop Quantum Gravity and other canonical quantization schemes (Addazi, 22 Dec 2025, Addazi et al., 2 May 2025). In the UV (pre-geometric) phase, the theory is strictly a Yang-Mills-type gauge theory, hence power-counting renormalizable in the absence of matter, suggesting prospects for UV completion.

The underlying action admits formulation as a constrained BF theory for the full gauge group, with SSB reducing to Palatini gravity plus cosmological constant in the IR (Addazi et al., 2 May 2025).

6. Cosmological Constant, Topological Sectors, and Vacuum Selection

A notable development in pre-geometric gravity is the proposed solution to the cosmological constant problem. The emergent low-energy action generically contains a Gauss–Bonnet topological term with a quantized coupling: $SO(3,2)$3 where $SO(3,2)$4 is the Euler characteristic, and $SO(3,2)$5, matching the de Sitter entropy (Addazi et al., 18 Feb 2026).

The gravitational $SO(3,2)$6-angle $SO(3,2)$7 is periodic, so the cosmological constant is quantized into discrete sectors: $SO(3,2)$8 A periodic Higgs potential for the pre-geometric Higgs field produces a discrete set of vacua labeled by $SO(3,2)$9, and SSB dynamics deterministically selects the sector matching the observed vacuum energy. Transition rates between sectors are exponentially suppressed as $I,J = 0,...,4$0, stabilizing the observed value of $I,J = 0,...,4$1 against quantum fluctuations and radiative corrections (Addazi et al., 18 Feb 2026).

7. Broader Landscape: Alternative Pre-geometric Approaches

While the above describes the gauge-theoretic and symmetry-breaking pathway, other paradigms of pre-geometric gravity include:

Approach	Fundamental Variables	Mechanism of Emergence
Combinatorial	Bits $I,J = 0,...,4$2 encoding random graph connectivity	Quantum phase transition in curvature/cycle order parameter; continuum limit yields $I,J = 0,...,4$3 and $I,J = 0,...,4$4 (Trugenberger, 2016)
Noncommutative	Matrices $I,J = 0,...,4$5 with noncommutative relations	Semi-classical limit of the matrix model yields emergent metric and gravity (Yang, 2010, Steinacker, 2010)
Axiomatic/Purified	Torsion-free connection, conservation laws	Constitutive relation and local linearization yield Coincident General Relativity; metric as Stueckelberg field, graviton as Goldstone boson (Koivisto et al., 2019, Itin et al., 2016)
Loop/Spin network	SU(2) connections and spin networks	Quantum-discrete area eigenvalues aggregate into smooth geometry; effective action reproduces Nambu-Goto and Einstein–Hilbert terms (Vaid, 2017)
Shape dynamics	Diffeomorphism and conformal variables	Relational evolution generates effective metric as an abstraction from matter fluctuations (Koslowski, 2015)

Each of these approaches offers a distinct perspective on the emergence of geometry and gravitational dynamics from fundamentally pre-geometric—combinatorial, algebraic, or relational—structures.

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Gravity from pre-geometry therefore offers a comprehensive framework unifying the emergence of classical spacetime, gravitational dynamics, matter interactions, and even vacuum selection as macroscopic consequences of deeper non-metric, gauge-theoretic or combinatoric quantum structures. The paradigmatic models realize Einstein–Cartan gravity, the Planck scale, cosmological constant, and the full suite of gravitational phenomena as emergent, rather than fundamental, features of Nature (Addazi et al., 2024, Addazi, 22 Dec 2025, Addazi et al., 2 May 2025, Addazi et al., 18 Feb 2026, Yang, 2010, Trugenberger, 2016, Koivisto et al., 2019).