Cubic Metric-Affine Gravity (MAG)

• Cubic Metric-Affine Gravity (MAG) is a four-dimensional extension of GR where both the metric and affine connection are dynamic, incorporating torsion and nonmetricity.
• The inclusion of cubic invariants enriches the theory, allowing ghost-free propagation and stable vector, axial, and massive tensor modes through rigorous algebraic constraints.
• Novel solutions, such as modified black holes and gravitational waves with non-Einsteinian polarizations, highlight the potential observational signatures of cubic MAG.

Cubic Metric-Affine Gravity (MAG) is a four-dimensional gauge-theoretic extension of General Relativity wherein both the metric $g_{\mu\nu}$ and independent affine connection $\tilde\Gamma^\lambda{}_{\mu\nu}$ are dynamical, with their associated non-Riemannian features—torsion $$T^\lambda{}_{\mu\nu} = 2\tilde\Gamma^\lambda{}_{[\mu\nu]}$$ and nonmetricity $$Q_{\lambda\mu\nu} = \tilde\nabla_\lambda g_{\mu\nu}$$—entering the action through quadratic and, crucially, cubic invariants constructed from the curvature, torsion, and nonmetricity tensors. The inclusion of cubic terms results in a classical theory with a significantly richer geometric and dynamical structure, including new propagating degrees of freedom and exact solutions absent in the purely quadratic case, and most notably provides a consistent ghost-free framework for the propagation of vector, axial, and massive tensor modes that evade numerous no-go theorems for higher-spin fields (Bahamonde et al., 20 Nov 2024, Bahamonde et al., 5 Nov 2025).

1. Action Principle and Structural Features

The general cubic Metric-Affine Gravity action is given by

$$S = \frac{1}{16\pi} \int d^4x \sqrt{-g} \left\{ -R + \mathcal{L}_{\rm Quad} + \mathcal{L}_{\rm curv\!-\!tors}^{(3)} + \mathcal{L}_{\rm curv\!-\!nonm}^{(3)} + \mathcal{L}_{\rm curv\!-\!tors\!-\!nonm}^{(3)} \right\}$$

where $-R$ is the usual Einstein–Hilbert term, and the Lagrangian densities $\mathcal{L}_{\rm Quad}$, $\mathcal{L}_{\rm curv\!-\!tors}^{(3)}$, $\mathcal{L}_{\rm curv\!-\!nonm}^{(3)}$, and $\mathcal{L}_{\rm curv\!-\!tors\!-\!nonm}^{(3)}$ systematically incorporate all parity-even, metric-compatible quadratic and cubic invariants in the curvature, torsion, and nonmetricity.

Explicitly, the structure up to cubic order is

$$\mathcal{L}_{\rm Quad} = -\,R + \sum_{i=1}^{13} a_i\,\tilde R^2 + \sum_{I=1}^7 m_I^2\,(\text{irreducible }T,Q)^2 + \sum_{I<J} \alpha_{IJ} (T\cdot Q)$$

$$\mathcal{L}_{\rm curv\!-\!tors}^{(3)} = \sum_{p=1}^{46} h_p\,\tilde R\,T^2, \qquad \mathcal{L}_{\rm curv\!-\!nonm}^{(3)} = \sum_{p=47}^{107} h_p\,\tilde R\,Q^2, \qquad \mathcal{L}_{\rm curv\!-\!tors\!-\!nonm}^{(3)} = \sum_{p=108}^{209} h_p\,\tilde R\,T\,Q$$

where $a_i$, $m_I$, $\alpha_{IJ}$, and $h_p$ are free coupling constants detailed in the referenced appendices. These terms generate, respectively, second- and third-order invariants in the field strengths, with the cubic sector collecting all algebraically inequivalent invariants mixing curvature with up to two explicit torsion or nonmetricity tensors.

A distinct feature of cubic MAG is the large parameter space: the cubic sector alone can be parametrized by up to 209 couplings, subject to algebraic conditions required for physical consistency.

2. Field Equations and Irreducible Decomposition

Variation of the action with respect to the independent fields yields three sets of field equations:

• For the metric $g_{\mu\nu}$ (the “metric equation”):

$$\delta S / \delta g_{\mu\nu}=0 \implies G^{\mu\nu} + (\text{quadratic and cubic currents in } T,Q) = 0$$

• For the torsion $T^\lambda{}_{\mu\nu}$:

$$\delta S / \delta T^\lambda{}_{\mu\nu}=0 \implies \sum a_i (\nabla\tilde R)^{\mu\nu}{}_\lambda + \sum m_I^2 T^{\mu\nu}{}_\lambda + \sum h_p (\tilde R T T + \tilde R Q Q) = 0$$

• For the nonmetricity $Q_{\lambda\mu\nu}$:

$$\delta S / \delta Q_{\lambda\mu\nu}=0 \implies \sum a_i (\nabla\tilde R)^{\lambda\mu\nu} + \sum m_I^2 Q^{\lambda\mu\nu} + \sum h_p (\tilde R Q Q + \tilde R T Q) = 0$$

Key analytical steps for solution include separation of the connection into its Levi-Civita and distortion components, grouping terms according to variational derivatives, and exploiting the irreducible decomposition of torsion and nonmetricity. The latter can be summarized as:

• Torsion: vector $T_\mu$, axial $S_\mu$, tensor $t_{\lambda\mu\nu}$
• Nonmetricity: Weyl $W_\mu$, vector $\Lambda_\mu$, axio-tensor $\Omega_{\lambda\mu\nu}$, trace-free $q_{\lambda\mu\nu}$

The explicit constitutive tensors and their index contractions, as functions of the quadratic and cubic couplings, govern the propagation and mixing of these modes.

3. Stability of Vector and Axial Modes

Quadratic MAG is well known for propagating unhealthy vector and axial degrees of freedom—manifesting as either Ostrogradsky ghosts or Hamiltonians unbounded from below—due to mixing terms such as $R\nabla_\mu\Lambda^\mu$ or $(\nabla\Lambda)^2$, and cross-derivative couplings between inequivalent field irreps. The cubic invariants, when suitably tuned, can cancel these problematic interactions.

Precisely, the presence of 209 independent cubic couplings $h_p$ allows imposition of 39 algebraic stability constraints, yielding a 194-dimensional “safe” parameter hypersurface. On this subspace, the kinetic matrix for the set $(S_\mu,T_\mu,W_\mu,\Lambda_\mu)$ is block diagonal: $$\kappa_{XY} = -2 \begin{pmatrix} 2l_6 & 0 & 0 & 0 \ 0 & 2l_1 & l_{30} & l_{43} \ 0 & l_{30} & 2l_{11} & l_{70} \ 0 & l_{43} & l_{70} & 2l_{16} \end{pmatrix}$$ where the $l_i$ are explicit linear combinations of the original couplings.

Ghost-free propagation and the absence of tachyons across all backgrounds require: $$l_6 \le 0 ,\quad l_1 + l_{11} + l_{16} \le 0 ,\quad 4l_{11}l_{16} + 4l_1(l_{11}+l_{16})-(l_{30}^2 + l_{43}^2 + l_{70}^2) \ge 0 ,$$

$$l_{30}l_{43}l_{70} + 4l_1l_{11}l_{16} - l_{16}l_{30}^2 - l_{11}l_{43}^2 - l_1l_{70}^2 \le 0$$

These algebraic conditions guarantee stability under linear perturbations for all irreducible vector and axial modes.

4. Black Hole and Wave Solutions in Cubic MAG

Cubic MAG admits novel exact solutions with no analog in either GR or quadratic MAG. Two notable classes are:

Reissner–Nordström-like Black Holes

The static, spherically symmetric ansatz allows non-vanishing irreducible torsion and nonmetricity components alongside the metric: $$ds^2 = \Psi_1(r) dt^2 - \Psi_2^{-1}(r) dr^2 - r^2 d\Omega^2$$

$$T^t{}_{tr}=t_1(r),\quad T^r{}_{tr}=t_2(r),\quad Q_{ttt}=q_1(r),\ \dots$$

Upon solving the 123 algebraic stability constraints in addition to the field equations, the general solution features four independent charges:

• Mass $m$
• Spin $\kappa_s$
• Dilation $\kappa_d$
• Shear $\kappa_{\rm sh}$

The metric function is modified vs. GR,

$$\Psi(r) = 1 - \frac{2m}{r} + \frac{\kappa_s^2(d_1+4h_{25})/3 + \frac12\kappa_d^2(2a_6-4a_2-8a_{14}-d_1) + \frac1{15}\kappa_{\rm sh}^2\Sigma(h_i,a_i,d_1)}{r^2}$$

All vector/axial modes remain massless, while irreducible tensors $$t_{\lambda\mu\nu},q_{\lambda\mu\nu},\Omega_{\lambda\mu\nu}$$ acquire universal, $r$-independent mass terms proportional to a parameter $w\sim m_t^2 >0$.

Gravitational Wave (pp-wave) Solutions

New families of exact gravitational wave solutions exist in cubic MAG, exhibiting dynamical torsion and nonmetricity content (Bahamonde et al., 5 Nov 2025). The general pp-wave Ansatz is

$ds^2 = 2 du\,dv - dx^2 - dy^2 - H(u,x,y) du^2$

with all torsion and nonmetricity components functions of $(u,x,y)$ and Lie-dragged by the null vector $k^\mu=\partial_v$.

Depending on which post-Riemannian sectors are present:

• In the Riemann–Cartan branch ($Q\equiv0$), a torsion function $t_{22}(u,x,y)$ contributes an explicit term to $H(u,x,y)$,
• In the Weyl–Cartan branch ($W_\mu \neq 0$ but ${\nearrow Q}_{\alpha\mu\nu}=0$), both $t_{22}$ and a harmonic scalar $w(u,x,y)$ can be present,
• In the general metric-affine case, a traceless nonmetricity vector $\Lambda_\mu$ with at most linear $(x,y)$ dependence, alongside $t_{22}$ and $w$, affects the metric function. The solution reads: $$H(u,x,y) = \mathring H(u,x,y) + l_1 t_{22}^2 + l_2 w^2 + l_3 \lambda^2$$ where the coefficients $l_{1,2,3}$ depend on the fundamental couplings.

5. Polarization Structure and Physical Implications

The algebraic classification of pp-wave solutions in MAG, by Type N conditions on the field strengths, parallels the Petrov–Segre scheme in GR but must incorporate irreducible post-Riemannian components. The physical content is manifest in the geodesic deviation equation for test particles, where, in addition to the standard tensor polarizations ($\mathcal A_+,\mathcal A_\times$) of helicity $\pm2$, cubic MAG generically produces a breathing scalar mode ($\mathcal A_\circ$): $$\mathcal A_\circ = -\tfrac14\Phi_{00}, \qquad \Phi_{00} \propto (\partial_{xx}+\partial_{yy})H$$ Whenever post-Riemannian amplitudes $t_{22},w,\lambda\neq0$, the breathing mode is nonvanishing; thus, cubic MAG predicts a genuine helicity-0 polarization in gravitational radiation, a phenomenological signature in principle observable by gravitational wave interferometers sensitive to non-Einsteinian modes (e.g., LIGO/Virgo bounds on $h_b$, $h_L$). Present limits only exclude dominant scalar components, but advanced detectors may probe the subdominant regime ($10^{-23}$–$10^{-24}$) relevant for cubic-MAG waves.

Stability of these solutions is safeguarded by the same algebraic ghost-free constraints originating from the action-level analysis.

6. Massive Tensors, No-Go Theorems, and Theoretical Significance

Cubic MAG circumvents long-standing obstacles for consistently interacting higher-spin fields that afflict quadratic theories. In quadratic MAG, irreducible massless tensor fields associated with torsion and nonmetricity generically lead to strong-coupling, causality, and positivity pathologies, violating constraints such as the Weinberg–Witten theorem and the Velo–Zwanziger condition.

In the cubic framework, the same tensor modes ($t_{\lambda\mu\nu}$, $q_{\lambda\mu\nu}$, $\Omega_{\lambda\mu\nu}$) acquire genuine Proca-type mass terms proportional to the parameter $w > 0$:

• They propagate with healthy, bounded-from-below Hamiltonians,
• Avoid causality (Velo–Zwanziger) issues,
• Evade restrictions forbidding massless higher-spin conserved currents.

Consequently, in stable cubic MAG models, all propagating post-Riemannian degrees of freedom are either

• spin-0 (scalars: $T_\mu$, $W_\mu$, $\Lambda_\mu$),
• spin-1 (vectors: $T_\mu$, $S_\mu$, $W_\mu$, $\Lambda_\mu$),
• or massive spin-2–like fields ($t_{\lambda\mu\nu}$, $q_{\lambda\mu\nu}$, $\Omega_{\lambda\mu\nu}$);

no massless higher-spin degrees of freedom remain, sidestepping the classic no-go theorems and providing a self-consistent, stable low-energy classical theory for metric-affine gravity (Bahamonde et al., 20 Nov 2024, Bahamonde et al., 5 Nov 2025).

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Sector	Degrees of Freedom	Status in Stable Cubic MAG
Vector/axial (Torsion, Nonmetricity)	$T_\mu$, $S_\mu$, $W_\mu$, $\Lambda_\mu$	Ghost-free, massless, block-diagonal kinetic
Irreducible tensors	$t_{\lambda\mu\nu}$, $q_{\lambda\mu\nu}$, $\Omega_{\lambda\mu\nu}$	Massive, positive-definite Hamiltonians
Metric (spin-2, $g_{\mu\nu}$)	Graviton	Einsteinian in quadratic limit

A plausible implication is that observational signatures from non-Einsteinian gravitational waves or black holes carrying post-Riemannian fluxes would serve as a probe of the extended dynamical content of cubic metric-affine gravity.