The auxiliary-metric formulation of Born-Infeld New Massive Gravity

Abstract: Born-Infeld New Massive Gravity (BINMG) completes New Massive Gravity to all orders in curvature through the determinant of the metric shifted by the Einstein tensor. We recast it with an independent auxiliary metric $q_{μν}$, whose algebraic equation of motion $q_{μν}=g_{μν}+\fracσ{m^2}G_{μν}(g)$ recovers the determinant action exactly on the regular branch and resums the infinite curvature series into a single relation. In the densitized variable $P^{μν}=\sqrt{-q}\,q^{μν}$ the three-dimensional action is polynomial, with all derivative dependence carried by the coupling $P^{μν}G_{μν}(g)$. The formulation makes known properties follow with substantially less algebra: the unique vacuum follows in one line, and the quadratic action yields a single Pauli-Fierz massive spin-2 field with the Fierz-Pauli tuning generated rather than imposed. On locally AdS backgrounds the conserved charges, BTZ mass and angular momentum, central charge, and entropy reduce to the Einstein results times a common factor. The formulation also isolates the nonlinear degree-of-freedom problem in the right variables, leaving the full Dirac count to separate work.

• The paper presents an auxiliary-metric reformulation of BINMG that simplifies the determinant structure while preserving ghost-free massive spin-2 propagation.
• It derives a polynomial action from the non-polynomial Born-Infeld form, enabling explicit evaluation of the unique maximally symmetric vacuum and linearized spectrum.
• The approach facilitates computation of conserved charges and BTZ thermodynamics by algebraically rescaling Einstein quantities without cumbersome curvature expansions.

Auxiliary-Metric Reformulation of Born-Infeld New Massive Gravity: Summary and Analysis

Overview and Motivation

Born-Infeld New Massive Gravity (BINMG) provides a resummation of three-dimensional New Massive Gravity (NMG) to all orders in curvature via a determinant structure involving the Einstein tensor. BINMG is distinguished from NMG by its non-polynomial determinant action, which encodes an infinite tower of curvature corrections yielding a unique maximally symmetric vacuum and maintaining ghost-free massive spin-2 propagation. Traditional approaches to BINMG suffer from operational complexity due to the determinant's nonlinearity.

This paper presents a structurally streamlined auxiliary-metric formulation of BINMG, introducing an independent auxiliary metric $q_{\mu\nu}$ with an algebraic equation of motion that recovers the original determinant action exactly. The reformulation brings computational transparency to vacuum structure, linearized spectrum, conserved charges, and positions the nonlinear Hamiltonian analysis in a natural mathematical setting.

New Massive Gravity and BINMG Structure

NMG augments three-dimensional Einstein gravity with a quadratic curvature invariant that precisely yields the Pauli-Fierz theory for a massive spin-2 field, eliminating pathological scalar modes via trace tuning. BINMG elevates this structure by adopting a Born-Infeld determinant involving the shifted metric $g_{\mu\nu} + m^2 G_{\mu\nu}$, which at low curvature recovers the NMG quadratic order but resums the entire infinite curvature series at higher orders. This structure has non-trivial implications in AdS/CFT correspondence and boundary terms in higher dimensions.

Auxiliary-Metric Reformulation

The auxiliary-metric action introduces $q_{\mu\nu}$ as an independent metric with an algebraic equation $q_{\mu\nu} = g_{\mu\nu} + m^2 G_{\mu\nu}(g)$, linearizing the Born-Infeld determinant and encapsulating the infinite series within a single relation. The action, reformulated in densitized variables $P^{\mu\nu} = (\det{q}) q^{\mu\nu}$, becomes polynomial and localizes all derivative dependence in the coupling $P^{\mu\nu} G_{\mu\nu}(g)$. This enables the compact derivation of field equations, explicit computation of vacua, and simplified linearized analysis.

Before eliminating $q_{\mu\nu}$, the theory is second order in both metrics; after substitution, the original fourth-order BINMG structure is recovered. The auxiliary approach also reveals a structural resemblance to bimetric theories, though the auxiliary metric lacks an independent kinetic term.

Vacuum Structure and Linearized Spectrum

A key result is the manifestation of BINMG's unique maximally symmetric vacuum, contrasted with the dual vacua of NMG. The auxiliary-metric equation simplifies to $q_{\mu\nu} = a g_{\mu\nu}$ with the proportionality fixed by the metric field equation. The singular endpoint—where BINMG transitions to nondynamical, massless gravity—is not imposed but emerges from the algebraic structure.

The quadratic action derived from the auxiliary formulation yields exactly one Pauli-Fierz massive spin-2 field with Fierz-Pauli mass tuning as an output of the Born-Infeld potential rather than an input assumption. The mass formula $M_{BINMG} = - \sigma m^2 a = -\sigma m^2 + \Lambda$ ensures normalizable propagator and no ghosts for the relevant sign choices, establishing unitarity of the linearized spectrum directly.

Conserved Charges and BTZ Thermodynamics

On locally AdS backgrounds, including the BTZ black hole, the auxiliary metric reduces algebraically to $q_{\mu\nu} = a g_{\mu\nu}$, enabling immediate evaluation of physical observables. Mass, angular momentum, central charge, and entropy are simple rescalings of corresponding Einstein quantities, via $g_{\mu\nu} + m^2 G_{\mu\nu}$0, avoiding curvature expansions and determinant variations. This rescaling extends to holographic quantities and boundary CFT observables, offering a pragmatic algebraic shortcut for the evaluation of AdS correlators and entropic constructions.

The sign of the overall factor $g_{\mu\nu} + m^2 G_{\mu\nu}$1 is dictated by the bulk mass unitarity, with implications for boundary charge positivity and AdS/CFT duality structure.

Nonlinear Degree-of-Freedom Count

The nonlinear Hamiltonian analysis, central to establishing the absence of the Boulware-Deser ghost, is facilitated by the auxiliary-metric formulation. The polynomial action, after ADM decomposition, isolates scalar constraints whose second-class nature determines the degree-of-freedom reduction, matching the manifestly healthy linearized spectrum. While a complete Dirac constraint classification is reserved for future work, the auxiliary structure positions all relevant variables in a tractable algebraic framework, emphasizing the roles of densitized variables and constraint propagation.

Theoretical and Practical Implications

The auxiliary-metric formulation of BINMG clarifies key theoretical structures: vacuum uniqueness, mass tuning, and resummation of infinite curvature corrections. It also provides substantial practical computational advantages for conserved charges, thermodynamics, and boundary observables in 3D gravity. The bimetric resemblance suggests avenues for comparative studies with bigravity models and ghost-free extensions.

Future developments will focus on the nonlinear Hamiltonian constraint analysis in the auxiliary setting, with potential implications for consistent higher-dimensional generalizations, holographic studies, and effective field theory constructions in lower-dimensional gravity.

Conclusion

The auxiliary-metric reformulation of Born-Infeld New Massive Gravity delivers a compact algebraic framework that retains complete fidelity with the determinant action, while markedly simplifying operational aspects. It makes the vacuum structure, linearized spectrum, and conserved charges explicit, and positions the nonlinear degree-of-freedom problem in an analytically tractable setting. This structure affords both a firmer theoretical foundation and enhanced practical utility for computations in three-dimensional massive gravity (2606.20247).