Lorentz-Invariant Geometric Reduction

• Lorentz-Invariant Geometric Reduction is a framework in differential geometry and field theory that reduces system degrees of freedom while maintaining full Lorentz symmetry.
• It employs methods such as frame bundle reductions, Higgs mechanisms, and coordinate reparametrizations to constrain geometric and dynamical structures.
• These techniques lead to effective lower-dimensional dynamics, ghost-free Lagrangians, and insights into momentum space geometry and relative locality.

Lorentz-invariant geometric reduction encompasses a suite of techniques and structures—primarily within differential geometry, field theory, and variational calculus—that reduce or constrain the geometric or dynamical content of a physical theory while manifestly preserving Lorentz symmetry. These reductions operate at multiple levels: from the global topology of fiber bundles to symmetry breaking mechanisms in field theory, coordinate and variational reparametrizations, and the geometric formulation of ghost-free Lorentz-invariant Lagrangians. The central theme is to identify or engineer substructures or effective dynamics that retain the full Lorentz group as a symmetry, even as the system is “reduced” in dimension, rank, or functional freedom.

1. Frame Bundle Reductions and Lorentzian Structures

A principal manifestation of Lorentz-invariant geometric reduction is the reduction of the structure group of the frame bundle of a smooth manifold from GL$(n,\mathbb{R})$ to Lorentzian or related subgroups. Given an $n$-dimensional smooth manifold $M$, its frame bundle $F(M)$ is a principal GL$(n,\mathbb{R})$-bundle whose fibers consist of ordered bases of $T_pM$. A Lorentzian metric $g$ of signature $(1,3)$ on $M$ selects an O$(1,3)$-structure: the associated orthonormal frame bundle $F_O(M)$ is a principal O$(1,3)$-subbundle whose frames are $g_p$-orthonormal. This O$(1,3)$ reduction both defines the metric and determines the geometric arena for Lorentzian geometry.

The Levi-Civita connection is uniquely characterized as the symmetric connection preserving the O$(1,3)$ reduction, given by Cartan’s structure equations with torsion-free and metric-compatibility constraints. Bundle reductions to alternative subgroups, such as the conformal Weyl group ($\mathbb{R}_+ \times$ O$(1,3))$, the teleparallel case (trivial subgroup), or the unimodular group SL$(n,\mathbb{R})$, produce other geometric frameworks such as Weyl geometry, teleparallel gravity (Weitzenböck connection), or unimodular structures. The 3+1 “time gauge” used in canonical gravity arises as a further O$(3)$-reduction, with significant implications for canonical variables and quantization approaches (2002.01410).

2. Lorentz-Invariant Dimensional Reduction via Higgs Mechanisms

A field-theoretic approach to Lorentz-invariant geometric reduction leverages spontaneous symmetry breaking of higher-dimensional local Lorentz groups via a generalized Higgs mechanism. In the construction of (Das et al., 2018), the gravitational sector is formulated in a first-order gauge-theory language, coupling gravity to a scalar (Higgs) field in the vector representation of SO$(D-1,1)$. The scalar potential forces the Higgs field to acquire a vacuum expectation value aligned with a specific direction in the internal Lorentzian space, breaking SO$(D-1,1) \rightarrow$ SO$(D-2,1)$. As a consequence, the components of the spin connection mixing the “reduced” direction become massive and decouple at low energies, and the dynamics reduces to D–1 dimensions.

This mechanism preserves manifest Lorentz invariance in both the original and the reduced theories and provides a geometric alternative to compactification or brane-world reductions. Iterative applications or modifications with non-minimal couplings generalize the scheme to a variety of higher-derivative and $f(R)$-type gravities. The approach, however, does not directly resolve cosmological constant or moduli stabilization problems inherent to conventional compactification techniques (Das et al., 2018).

3. Coordinate and Functional Reparametrization: Geometric Constraint Recapture

Lorentz-invariant geometric reduction also refers to analytic techniques by which Lorentz symmetry-breaking terms or geometric constraints are recast into Lorentz-invariant formulations via field redefinitions or coordinate “stretching.” In two-field scalar models with explicit Lorentz-violating terms, as studied in (Bandeira et al., 8 Oct 2025), geometric constraints that arise in models of domain walls in constrained media are recast by suitably redefining the spatial coordinate for one of the fields. Static, first-order (BPS) equations with Lorentz-breaking couplings can be rendered equivalent to Lorentz-invariant equations in a redefined “stretched” coordinate $\epsilon(x) = \int^x [1 + b g(\phi_s(y))]\,dy$, where $b$ and $g$ parameterize the deformation and $\phi_s$ is a known solution.

By judicious choice of model functions, Lorentz-invariant geometric reduction exactly reproduces the (geometrically constrained) kink solutions of the corresponding Lorentz-invariant models. More generally, the framework enables the systematic construction of new classes of exact solutions—including asymmetric and negative-energy-density defects—that retain first-order reduction by coordinate transformation. The Lorentz-invariant nature is thus restored for static configurations, with implications for the structure and stability of topological defects (Bandeira et al., 8 Oct 2025).

4. Lorentz-Invariant Geometric Reduction in Variational Problems

An explicit example of Lorentz-invariant geometric reduction in variational analysis is given by the sharp Sobolev inequality on the circle, as established in (Le, 2023). The functional

$$\mathcal{F}[v] = \int_{\mathbb{S}^1}\big[4(v')^2 - v^2\big]\,d\theta$$

subject to the constraint $\int_{\mathbb{S}^1} v^{-2}\,d\theta = 2\pi$, is shown to be Lorentz-invariant: both the functional and the constraint are preserved under the action of Lorentz boosts on the circle parameter. The extremizers correspond to parametrizations of spacelike plane sections of the lightcone in $(1+2)$-dimensional Minkowski space.

The proof employs an iterative geometric reduction alternately applying symmetric-decreasing rearrangement and Lorentz transformation (“boosts”) to monotonically converge to the unique minimizing function. This method is extensible to higher-dimensional spheres, yielding sharp Sobolev and Yamabe inequalities via Lorentz-invariant geometric reductions. The geometric underpinning is the embedding of functional minimizers as codimension-1 spacelike hypersurfaces in the null cone, where the Lorentz group acts transitively on constancy classes of the relevant invariants (Le, 2023).

5. Lorentz-Invariant Geometric Reduction in Field Theories: Ghost-Free Lagrangians

A geometric reduction formalism for Lorentz-invariant, ghost-free Lagrangians is established in (Li, 2015), unifying all known healthy (second-order) Lorentz-invariant field theories via the total antisymmetrization of index contractions and the use of differential form language. For any set of fields—scalars, vectors, $p$-forms, or spin-2—the general Lagrangian is constructed by contracting higher derivative terms in two totally antisymmetric index chains using generalized Kronecker deltas. This design ensures the Euler–Lagrange equations are at most second order, evading Ostrogradsky instabilities.

The essential geometric feature is that every dangerous second derivative is packaged as an exact form, and variational derivatives yield only total derivatives or (by $d^2=0$) vanish, as in Lovelock gravity. A novel “duality,” generalizing Hodge duality, arises from the possible exchange of these chains, mirrored in wedge product expressions of double forms or the exchange of Levi-Civita symbols. This method provides a single geometric principle—“the boundary of a boundary vanishes” ($d^2=0$)—as the foundation for the construction and reduction of all compatible Lagrangians, including Galileons, $p$-form Galileons, Proca-Galileons, Lovelock gravity, de Rham-Gabadadze-Tolley (dRGT) mass terms, and Horndeski theories (Li, 2015).

6. Lorentz-Invariant Geometrization of Deformed Momentum Space

In momentum space geometry and relative locality, Lorentz-invariant geometric reductions classify all possible nonlinear, Lorentz-preserving deformations of the momentum-space metric and composition law. As demonstrated in (Astuti et al., 2015), the most general analytic expansion to $O(p^2)$ for the momentum-space metric and addition law admits only a finite-dimensional two-parameter space of deformations modulo diffeomorphisms and scaling. Associativity of the addition law is generically lost; any genuinely Lorentz-invariant deformation of momentum composition is necessarily non-associative and introduces curvature in momentum space, as characterized by the associator and its connection curvature. Nonmetricity further emerges when the deformed metric cannot be preserved under parallel transport in the presence of these deformations, rendering a geometric reduction of phase-space locality that strictly retains Lorentz covariance. This structure is pivotal in formulations that exploit relative locality and in models attempting to encode Planck-scale deformations of kinematics (Astuti et al., 2015).

7. Broader Geometric and Physical Context

The unifying motif of Lorentz-invariant geometric reduction is the systematic identification of geometric, algebraic, or analytic procedures that reduce the degrees of freedom or symmetries of physical models while strictly maintaining Lorentz invariance. This includes the construction of subbundles, decomposition of connections, ghost-free Lagrangians via index antisymmetrization, variational minimization under Lorentz-invariant constraints, and the identification of effective lower-dimensional actions after symmetry breaking. The methodology articulates the geometric and symmetry principles underlying metric compatibility, conformal/teleparallel/unimodular formulations, momentum-space geometry, and field-theoretic model building, and provides a blueprint for generalizations to higher spins, mixed symmetry fields, and non-Lorentz-invariant phases.

A plausible implication is that advances in Lorentz-invariant geometric reduction frameworks will further systematize the search for consistent, predictive, and renormalizable models of gravity and matter—especially in regimes (e.g., quantum gravity, higher-spin field theory, or extended objects) where dimensionality, symmetry, and geometric structure are dynamically emergent or constrained.