---
title: Pre-Geometric Gravity (PGG)
url: https://www.emergentmind.com/topics/pre-geometric-gravity-pgg
type: topic
---

# Pre-Geometric Gravity (PGG)

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 [2409.02200][2512.20681].

A generic action then reads:
\[
S = \int d^4x\,(\mathcal{L}_{\rm MM/W} + \mathcal{L}_{\rm SB})
\]
where \(\mathcal{L}_{\rm SB}\) 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 \(\phi^A\):
\[
\langle\phi^A\rangle = v\,\delta^A_4
\]
This breaks SO(1,4) or SO(3,2) to SO(1,3). Accordingly, one splits the connection:
\[
A_\mu^{ab} \to \omega_\mu^{ab} \ (\text{spin connection}),\qquad A_\mu^{a4} \to m\,e_\mu^a \ (\text{tetrad}),
\]
where \(e_\mu^a\) is the emergent vierbein and \(m\) a mass parameter.

The metric emerges as:
\[
g_{\mu\nu} = \eta_{ab}\,e_\mu^a\,e_\nu^b.
\]

The emergent action (after SSB) takes the Einstein–Cartan or Einstein–Hilbert form, e.g.,
\[
\mathcal{L}_{\rm eff} = \frac{M_P^2}{2}e\,R - M_P^2\Lambda\,e - 4k_{\rm MM}v\,e\,\mathcal{G},
\]
where \(\mathcal{G}\) is the Gauss–Bonnet density. The Planck mass and the cosmological constant are given by see-saw-like relations, e.g.,
\[
M_P^2=32\,k_{\rm MM}\,v\,m^2, \quad \Lambda = 3m^2.
\]
These illustrate that both fundamental mass scales emerge dynamically [2409.02200][2512.20681].

## 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:
  \[
  G_\nu^\mu + \Lambda \delta_\nu^\mu = M_P^{-2}\,\tau_\nu^\mu
  \]
- **Spin connection variation** yields the (algebraic) Cartan equation for torsion:
  \[
  \tilde T_{\nu\rho}^{\mu} = T_{\nu\rho}^\mu + 2\delta_{[\nu}^\mu T_{\rho]\sigma}^{\sigma} = M_P^{-2}\,\sigma_{\nu\rho}^\mu
  \]
  with canonical spin current \(\sigma_{\nu\rho}^\mu\).

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:
\[
\mathcal{G}_{AB}^\mu = \mathcal{M}_P^{-2}\,\mathcal{T}_{AB}^\mu
\]
with the combined “pre–EC tensor” capturing unified gravitational sources [2505.02925][2512.20681].

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

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

- Canonical pairs: \((A_i^{AB},\Pi^i_{AB})\), \((\phi^A,\Pi_A)\)
- 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:
\[
2N_{\rm dof} = 90 - 20 - 2\times 10 - 44 = 6 \quad (\rightarrow~3~\text{physical d.o.f.})
\]
interpreted as a massless spin-2 graviton (2 d.o.f.) plus a scalar mode from \(\phi^4\) [2505.01272][2512.20681].

Upon symmetry breaking and integrating out the scalar, the canonical GR ADM formalism is recovered, with the Hamiltonian constraint structure exactly matching General Relativity [2512.20681][2505.01272].

## 5. Cosmological Solutions and Singularity Resolution

An explicit homogeneous and isotropic ansatz in the pre-geometric phase yields a regular, non-singular solution:
\[
E_t(t),~E_s(t),~\Omega(t),~\Phi(t)
\]
that after SSB maps to the de Sitter metric:
\[
ds^2=-dt^2+e^{2Ht}d\vec{x}^2, \quad H=\sqrt{\Lambda/3}
\]
with all geometric and gauge fields smooth as \(t\to-\infty\) [2505.02925]. 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:
  \[
  \widehat{\mathcal{H}}(A,\phi,\Pi^i,\Pi_A)\;\Psi[A,\phi]=0
  \]
  with \(\Psi\) 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 [2505.01272][2512.20681].

- UV completion pathways include:
  - Topological BF-formulation and “simplicity” constraints, with classical equations imposing the emergence of Einstein–Cartan gravity as the SSB phase [2505.01272].
  - Power-counting renormalizability of the unbroken gauge theory, suggesting connections to asymptotic safety or other functional RG fixed points [2409.02200].

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 [1105.5687].

## 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,
  \[
  |\alpha_{\rm GB}|\sim \frac{M_P^2}{\Lambda} \sim S_{dS} \sim 10^{120}
  \]
- The CC is quantized into discrete topological sectors with vacuum selection governed by the Higgs barrier, suppressing quantum transitions between vacua by \(e^{-10^{120}}\), and giving a dynamical explanation for the CC’s smallness [2602.16840][2604.26032].

- The dynamical sector incorporates additional pseudo-Nambu–Goldstone bosons (“hairons” or gravi-axions) whose mass scale \(m_\theta\sim H_0\) and whose dynamics potentially account for cosmic acceleration. Quadratic (f(Lovelock)) extensions of the original action yield a phenomenology closely mimicking \(\Lambda\)CDM, yet with testable deviations in gravitational slip, GW propagation, and cosmological time variation of \(G_{\rm eff}\) [2605.07344][2604.26032].

## 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 \(A,\phi\), mapping onto conventional minimally-coupled QFTs in curved spacetime [2409.02200].

- 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 [2409.02200][2505.02925].

- 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 [1703.01492][2512.20681].

## 9. Summary Table: Core Structural Features

| Feature                                | Emergent After SSB                       | UV Pre-geometric Phase                 |
|-----------------------------------------|------------------------------------------|----------------------------------------|
| Metric/vierbein                        | \(e_\mu^a = m^{-1}A_\mu^{a4}\)           | Field is undefined                     |
| Spin connection                        | \(\omega_\mu^{ab} = A_\mu^{ab}\)         | Part of full connection                |
| Planck mass and cosmological constant  | \(M_P^2 \sim v\,m^2\) (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 \((A,\phi)\)         |

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 [2512.20681][2409.02200][2505.02925][2505.01272][2602.16840][2604.26032][2605.07344][1703.01492][1105.5687].

Source: https://www.emergentmind.com/topics/pre-geometric-gravity-pgg