Pre-Geometric Gravity

• Pre-geometric gravity is a theoretical framework where spacetime and gravitational interactions emerge from metric-free, gauge-theoretic, combinatorial, or algebraic structures.
• Models employ mechanisms like spontaneous symmetry breaking of gauge connections and scalar multiplets to dynamically generate effective gravitational actions resembling Einstein’s theory.
• These approaches enhance ultraviolet behavior and resolve classical singularities by embedding gravity within renormalizable Yang-Mills-type actions and non-singular cosmological solutions.

Pre-geometric gravity refers to theoretical frameworks in which spacetime geometry and gravitational interactions emerge from a more fundamental, metric-free structure. Instead of postulating a spacetime manifold equipped with a Lorentzian metric, these approaches typically begin from gauge-theoretic, combinatorial, or algebraic structures that acquire geometric interpretation dynamically, often through mechanisms such as spontaneous symmetry breaking or critical phase transitions. The resulting effective theories recover General Relativity or modified gravity regimes, providing new avenues for UV completion, singularity resolution, and unification with quantum field theory.

1. Historical Precedents and Conceptual Foundations

Pre-geometric ideas trace philosophical and physical lineage back to ancient Greek physics, where thinkers like Plutarch and Posidonius anticipated core notions later formalized in geometric gravity. Plutarch’s explanation for lunar motion invokes an impetus-based mechanism analogous to momentum ($p=mv$), with orbital motion preventing radial decay (“The Moon gets the guarantee of not falling down just from its motion and from the dash associated with its revolution...”). Posidonius described cosmic cohesion as a binding force, with cycles of contraction and expansion in the vacuum (“Matter possesses an intrinsic cohesive force $C$ preventing fragmentation by the surrounding void.”) (0709.1277). These qualitative seeds matured into Newtonian and Einsteinian gravity, with pre-geometric gravity now representing their abstraction to metric-free, algebraic, or gauge-theoretic regimes.

2. Gauge-Theoretic Realizations and Mechanism of Emergence

Modern pre-geometric gravity models commonly use principal bundles with gauge group $G=SO(1,4)$ (de Sitter), $SO(3,2)$ (anti-de Sitter), or $SO(4)$, without imposing a prior metric on the base manifold (Addazi et al., 3 Sep 2024, Addazi, 22 Dec 2025, Wetterich, 2021). The fundamental fields include a gauge connection $A^{AB}$ (antisymmetric in $A,B=0,\dots,4$) and a Higgs-like multiplet $\Phi^A$.

The theory is defined by a Yang-Mills-type action, such as: $$S_0 = -\frac{1}{4g^2} \int F^{AB} \wedge \star F_{AB}$$ with $F^{AB}=dA^{AB}+A^{A}{}_C\wedge A^{CB}$ the field strength. No metric nor tetrad is assumed at this stage. Crucially, a SO(1,4)-invariant potential $V(\Phi)$ enforces spontaneous symmetry breaking: $$V(\Phi)=\lambda \left( \eta_{AB} \Phi^A \Phi^B \mp v^2 \right)^2$$ By minimization, $\langle \Phi^A \rangle = v \delta^A_4$, breaking the gauge symmetry to $SO(1,3)$. The connection decomposes as $A^{I4} \equiv m e^I$ (vierbein) and $A^{IJ} \equiv \omega^{IJ}$ (spin connection).

In unitary gauge, the emergent metric: $g_{\mu\nu} = \eta_{IJ} e^I_\mu e^J_\nu$ where $e^I_\mu = \frac{1}{m v} D_\mu \Phi^I$. Substituting these identifications into the broken phase action yields the Einstein–Hilbert or Einstein–Cartan Lagrangian with cosmological constant: $$S_{\rm eff} = \frac{1}{16\pi G} \int \left[ R^{IJ}(\omega)\wedge e^K \wedge e^L - 2\Lambda\,e^I\wedge e^J\wedge e^K\wedge e^L \right] \epsilon_{IJKL}$$ with emergent parameters: $$\frac{1}{16\pi G} \sim \frac{v^2}{g^2}, \qquad \Lambda = \pm g^2 v^2$$ where the $\pm$ sign distinguishes between de Sitter and anti-de Sitter cases (Addazi et al., 3 Sep 2024, Addazi, 22 Dec 2025, Meluccio, 5 May 2025).

3. Hamiltonian Structure and Constraint Analysis

The canonical analysis starts from a 3+1 decomposition of the gauge connection and multiplet, yielding primary and secondary constraints via Dirac’s algorithm (Addazi et al., 2 May 2025, Addazi, 22 Dec 2025):

Regime	Variables (total)	Constraints	Physical DoF
Pre-geometric UV	90	10 first-class, 44 second-class	3
Geometric (IR)	20	ADM+Lorentz	2–3

In the UV, the system is described by $(A_i^{AB}, \Phi^A)$ and their conjugate momenta, with 3 degrees of freedom corresponding to a massless graviton and a heavy scalar. The unbroken phase’s constraint structure allows for perturbative renormalizability due to the underlying gauge symmetry. After SSB, the IR Hamiltonian becomes that of ADM general relativity, and the remaining degrees of freedom match the standard metric graviton content. Integrating out the scalar recovers the ADM Hamiltonian, establishing compatibility with canonical quantum gravity programs and the Wheeler-DeWitt equation (Addazi, 22 Dec 2025, Addazi et al., 2 May 2025).

4. Cosmological Solutions and Singularity Resolution

Exact solutions to the pre-geometric field equations can be constructed. In the unbroken SO(1,4)-invariant phase, cosmological ansätze of the Yang-Mills connection give rise to pre-geometric analogs of de Sitter universes (Meluccio, 5 May 2025). Notably, the underlying fields $(E_s(t), E_t(t), \Omega(t))$ remain regular for all $t$ and there is no Big Bang singularity. The classical FRW past-incompleteness is resolved via a smooth transition from the geometric (SSB) phase to the high-energy pre-geometric phase, capped by SO(1,4) gauge symmetry. This mechanism provides a plausible pathway for singularity resolution and potential initial conditions for cosmological inflation.

5. Quantum Field Theory Embedding and UV Completion

Pre-geometric gravity models often exhibit improved ultraviolet behavior. In unbroken phase ($SO(1,4)$-invariant), the pure Yang-Mills action plus scalar multiplet is power-counting renormalizable, with all couplings dimensionless prior to SSB (Addazi et al., 3 Sep 2024, Addazi, 22 Dec 2025). The emergent gravity is an IR phenomenon, permitting asymptotic safety or renormalization group completions beyond the Planck scale. Euclidean versions (with $SO(4)$ symmetry) avoid ghost and tachyonic poles in the graviton propagator, with a well-defined functional integral and robust analytic continuation to Lorentzian signature (Wetterich, 2021).

6. Alternative Approaches: Axiomatic, Emergent, and Statistical

Premetric or purified gravity axiomatizes gravitational interactions as arising from conserved currents and pure-gauge connections (Koivisto et al., 2019). The metric emerges as a Stueckelberg field associated with broken translational symmetry, and the graviton appears as the Goldstone boson. Constitutive laws with 14 free parameters reduce uniquely to Coincident General Relativity under diffeomorphism invariance.

From a statistical physics standpoint, group field theory (GFT) provides a microscopic, combinatorial model of pre-geometric spacetime, with continuum geometry emerging at a phase transition characterized by critical exponents in the partition function (Sindoni, 2011). The effective gravitational action arises from coarse-graining, and subleading corrections yield higher-curvature operators. Thermodynamic interpretation of spacetime—entropy, internal energy, and area law—naturally follows from the statistical framework.

7. Principle of Dynamical Emergence and Equivalence

In certain formulations, notably those employing D’Alembert’s principle, gravity emerges not as a primary assumption of curvature, but from the balance of real inertial and gravitational forces (Haro et al., 15 Oct 2025). The equivalence principle is realized dynamically: free fall corresponds to force compensation rather than simply vanishing curvature, with the spacetime metric appearing as the unique solution enforcing local force balance and Lorentz invariance.

Concluding Remarks

Pre-geometric gravity encompasses a broad spectrum of gauge-theoretic, combinatorial, and axiomatic models in which spacetime geometry, gravitational dynamics, and even matter couplings arise as emergent, secondary phenomena. Whether via spontaneous symmetry breaking of larger gauge symmetries, combinatorial critical points, or conservation principles, these approaches provide UV-complete, singularity-free, and often thermodynamically interpretable extensions and generalizations of Einstein gravity. The connection to canonical programs, loop quantum gravity, and statistical approaches suggests a unifying conceptual architecture for quantum spacetime and gravity’s true microscopic origin (Addazi et al., 3 Sep 2024, Meluccio, 5 May 2025, Addazi, 22 Dec 2025, Koivisto et al., 2019, Wetterich, 2021, Sindoni, 2011, Haro et al., 15 Oct 2025).