Minimal Massive Gravity (MMG)

Updated 27 July 2025

Minimal Massive Gravity (MMG) is a three-dimensional massive gravity model that propagates a single massive spin-2 field and resolves the bulk/boundary unitarity clash through a unique J-tensor modification.
It features consistent matter couplings and a rich vacuum structure including maximally symmetric, warped, and Lifshitz spacetimes, which are crucial for understanding black hole entropy and cosmological bounces.
MMG’s distinctive field equations and symmetry structure enable unified bulk and boundary unitarity, establishing a precise AdS/CFT correspondence with computable dual central charges.

Minimal Massive Gravity (MMG) is a three-dimensional model of massive gravity that propagates a single massive spin-2 degree of freedom and is specifically constructed to resolve the “bulk/boundary unitarity clash” present in earlier models such as Topologically Massive Gravity (TMG). MMG achieves this by introducing a crucial curvature-squared modification—the so-called J-tensor—resulting in improved boundary properties without sacrificing the desirable bulk features of TMG. The theory admits consistent, nontrivial matter couplings, a rich vacuum structure (including warped geometries and Kaluza–Klein vacua), and smooth cosmological bounces, while providing a unique extension among 3D gravity models with a single propagating massive graviton.

1. Field Equations and Theoretical Structure

The defining field equation of MMG in metric variables is

$\gamma\,E_{\mu\nu} \equiv \bar\Lambda_0\, g_{\mu\nu} + \bar\sigma\, G_{\mu\nu} + \frac{1}{\mu}\, C_{\mu\nu} + \frac{\gamma}{\mu^2}\, J_{\mu\nu} = 0$

where

$G_{\mu\nu}$ is the Einstein tensor,
$C_{\mu\nu}$ is the Cotton tensor,
$J_{\mu\nu}$ is a particular curvature–squared tensor defined by

$J^{\mu\nu} = \frac{1}{2\,\det g}\, \epsilon^{\mu\rho\sigma}\, \epsilon^{\nu\tau\eta}\, S_{\rho\tau} S_{\sigma\eta}$

with $S_{\mu\nu}$ being the Schouten tensor.

$\bar\sigma, \bar\Lambda_0, \gamma, \mu$ are theory parameters.

Unlike standard curvature invariants, $J_{\mu\nu}$ does not obey a Bianchi identity; however, its divergence vanishes for solutions of the full MMG field equations (on-shell). This is central for the theory’s third-way consistency, distinguishing MMG from models derived strictly from metric actions.

Parameter space is richer than in TMG due to an additional parameter $\alpha$ present in the formulation using dreibein, Lorentz connection, and auxiliary fields. After appropriate field redefinitions, all parameter choices can be mapped into a single connected domain where bulk and boundary unitarity are simultaneously satisfied (Arvanitakis et al., 2014).

2. Matter Coupling and Conservation Laws

A naive minimal coupling of matter fails in MMG due to the non-Bianchi property of $J_{\mu\nu}$ . Consistency instead requires a modified matter source tensor $\mathcal{T}_{\mu\nu}$ that is quadratic in the ordinary stress tensor:

$E_{\mu\nu} = \mathcal{T}_{\mu\nu}$

with

$\mathcal{T}_{\mu\nu} = T_{\mu\nu} + (\text{terms proportional to } 1/\mu) [\epsilon^{\mu\rho\sigma} D_\rho T_{\sigma\nu}] + (\text{quadratic corrections in } T_{\mu\nu})$

The explicit construction—demonstrated in both auxiliary field and improved dark energy parameterizations—ensures compatibility with matter conservation ( $D^\mu T_{\mu\nu} = 0$ ) (Arvanitakis et al., 2014, Cebeci, 2020). For matter Lagrangians that are functionals only of the metric (connection-independent), the quadratic source reduces to known expressions, but for connection-dependent matter (e.g., spinors) further structure appears, with effective gravitational couplings dependent on matter bilinears (Cebeci, 2020).

3. Vacuum, Cosmological, and Exact Solutions

MMG admits a range of vacuum and exact solutions exceeding that of TMG:

Maximally symmetric vacua: Solutions $G_{\mu\nu} = -\Lambda g_{\mu\nu}$ with $\Lambda$ found by solving a quadratic equation:

$\gamma\,\bar\Lambda_0 + \bar\sigma\,\Lambda + \frac{\Lambda^2}{4\mu^2} = 0$

yielding up to two distinct (A)dS vacua except at the "merger point," where they coalesce (Arvanitakis et al., 2014, Arvanitakis, 2015).

Warped and Kaluza–Klein spacetimes: MMG supports warped (A)dS solutions, including non-Einstein black holes at the merger point and near-horizon geometries factorizing as (A)dS $_2 \times S^1$ (Arvanitakis et al., 2014, Charyyev et al., 2017).
Homogeneous and Lifshitz solutions: The solution space encompasses the full catalog of algebraic Type-O, Type-N, and Type-D solutions of TMG, with further genuinely MMG solutions such as stationary Lifshitz spacetimes with dynamical exponent $z=-1$ and anisotropic Lifshitz metrics (Charyyev et al., 2017, Altas et al., 2015).
Domain walls and black holes: With correct choice of matter and parameter regimes, MMG allows domain wall cosmologies interpolating between distinct vacua only if the graviton is tachyonic; otherwise, such walls are forbidden in the unitarity region. Non-BTZ AdS black holes satisfying Brown–Henneaux boundary conditions exist inside the physically allowed region (Arvanitakis, 2015).

4. Linearized Spectrum and Chiral/Logarithmic Behavior

The linearized spectrum includes:

A single massive spin-2 excitation with modified mass

$M' = \frac{1}{2} \left( \sigma - \frac{\gamma}{2\mu^2} \right) + \ldots$

obeying the BF bound in AdS; left- and right-moving massless modes correspond to the Brown–Henneaux boundary degrees of freedom (Tekin, 2014, Alishahiha et al., 2014).

Chiral point: The graviton mass can be tuned to zero, leading to chiral gravity where one Virasoro central charge vanishes, and the gravitational excitations become purely right- or left-moving. At these critical points, logarithmic solutions with weakened boundary falloffs appear, directly linked to logarithmic CFT duals and the presence of logarithmic partners for the stress tensor (Giribet et al., 2014, Alishahiha et al., 2014).
Log Gravity and degeneracy: At the chiral point and in the decoupling limit of the Cotton tensor, MMG supports a large class of non-Einstein, non-stationary metrics (including BTZ deformations and time-dependent solutions) characterized by finite conserved charges even when asymptotics are weakened (Giribet et al., 2014).

5. Conservation Laws, Black Hole Entropy, and Symmetry Structure

Conserved global charges (mass, angular momentum) in MMG are constructed using background Killing vectors by extending the Abbott–Deser–Tekin formalism. Despite the absence of a Bianchi identity off-shell (since MMG's field equations do not come from a metric-only action), on-shell conservation holds, enabling computation of physically meaningful charges for black hole and warped spacetimes (Tekin, 2014, Yekta, 2015).

Black hole entropy: The entropy of MMG black holes, such as BTZ, is computed using an extension of the Tachikawa/Wald approach. The entropy acquires three separate contributions: the standard Bekenstein–Hawking term (proportional to the outer horizon), a correction from the gravitational Chern–Simons term (involving the inner horizon), and a novel term due to MMG (also proportional to the outer horizon). The total entropy agrees precisely with the microscopic Cardy formula using the central charges of the dual CFT (Setare et al., 2015).
Symmetry algebra: The asymptotic symmetry group comprises two Virasoro algebras (Brown–Henneaux), with central charges explicitly computable in terms of MMG parameters. The bulk and boundary conditions enforcing unitarity select a single, physically viable branch of parameter space (Arvanitakis et al., 2014).
Gauge structure and charges: MMG exhibits three gauge symmetries: local Lorentz, translation (diffeomorphism), and an “m”-type symmetry associated with the auxiliary field. In certain parameter regimes, their algebras close on an appropriately extended field space. The construction and integrability of conserved charges require a careful redefinition of field variations (combining Lie, Kosmann, and auxiliary transformations) to ensure compatibility with Wald’s method and to yield charges that correctly account for all symmetries, including those from the auxiliary sector (Liu et al., 23 Jul 2025).

6. Holography, Unitarity, and Uniqueness

A central virtue of MMG is the explicit resolution of the bulk/boundary unitarity problem: both the massive graviton in the AdS $_3$ bulk and the dual boundary CFT can be rendered unitary for the same parameter choices, a property singular among known 3D massive gravity models (Arvanitakis et al., 2014, Altas et al., 2015). The AdS/CFT dictionary applies, with central charges

$c_{\pm} = \frac{3\ell}{2 G_3}\left( \sigma + \frac{\gamma}{2\mu^2} \pm \frac{1}{\mu\ell} \right)$

inheriting dependence on all deformation parameters. The model’s uniqueness is formalized by demonstrating that no further nontrivial consistent deformations (at cubic or quartic order in curvature) exist that preserve both the degree of freedom counting and the third-way consistency (Altas et al., 2015).

Extensions of MMG, such as Generalized MMG (GMMG) and MMG $_2$ , further enrich the solution space and boundary properties, but MMG remains unique as the solitary quadratic extension supporting a single massive graviton with resolved bulk–boundary consistency.