Cubic Order Multi-Gravity Potential

• Cubic order multi-gravity potential is a framework that introduces cubic interactions through discrete lattice models and curvature invariants to extend classical gravity theories.
• It modifies standard gravitational predictions by incorporating nonlinear corrections that influence black hole metrics, thermodynamics, and cosmological evolution.
• The theory addresses stability and quantum consistency by carefully designing cubic terms to cancel ghost modes and yield testable signatures in strong-field regimes.

A cubic order multi-gravity potential refers to the mathematical structure, physical implications, and observable signatures of gravitational theories whose action or field equations contain cubic interactions—either in the Riemann tensor (and contractions thereof) or, in lattice or multi-field analogues, in the interactions between multiple gravitational degrees of freedom. These higher-order terms are crucial for formulating discrete models of gravity, constructing effective field theories for strong-field or quantum gravity dynamics, and developing modified gravity frameworks with distinct propagator structure, stability properties, black hole solutions, and cosmological behavior.

1. Lattice Model Realizations and the Cubic Structure

Discrete approaches to gravity provide an explicit formulation of multi-gravity potentials by promoting the spacetime metric to a set of symmetric, non-degenerate $n\times n$ matrices $g_i$ assigned to each site $i$ of an $n$-dimensional hyper-cubic lattice. The key dynamical ingredient is a local action constructed from nearest-neighbor link interactions—i.e., the action is a sum over links $\langle ij\rangle$, involving combinations of $g_i$ and $g_j$ such as their arithmetic average $g_{(ij)} = \frac{1}{2}(g_i + g_j)$ and the matrix difference $\Delta g_{(ij)}$.

The resulting action, suppressing dimensional prefactors, is of the form

$$S = -\sum_{\langle ij\rangle} \sqrt{\det g_{(ij)}}\left[\det(I + g_i^{-1} \Delta g_{(ij)})^{2 - n/2} - 1\right]$$

where $I$ is the identity and $s$ is the lattice spacing. This construction yields a "cubic" (hypercubic) multi-gravity structure, as each site's degrees of freedom (in the field space of positive-definite symmetric matrices) interact locally in a manner analogous to higher-order potentials.

Expanding this action for small fluctuations $g_i = I + h_i$ ($|h_i|\ll 1$), one finds that to quadratic order the action reduces to a discretization of the linearized Einstein-Hilbert action in harmonic gauge, but systematically, higher-order (cubic and beyond) terms contribute nontrivial multi-gravity interaction potentials at next-to-leading orders in the lattice spacing: $$S \sim \sum_{\langle ij \rangle} \left(\text{quadratic in } h_i-h_j + \text{cubic corrections}\right) + O(h^4)$$ These terms directly encode nonlinearly interacting gravitational "fields" situated on a cubic network, representing a prototypical cubic order multi-gravity potential (Tate et al., 2012).

2. Covariant Theories: Curvature Cubic Invariants

In continuum gravitational theories, cubic order contributions usually refer to additions of curvature-cubed invariants to the action. The most general such terms in $D$ dimensions consist of independent contractions: $$\mathcal{P} = 12 R_{\mu}{}^{\rho}{}_{\nu}{}^{\sigma} R_{\rho}{}^{\gamma}{}_{\sigma}{}^{\delta} R_{\gamma}{}^{\mu}{}_{\delta}{}^{\nu} + R_{\mu\nu}{}^{\rho\sigma} R_{\rho\sigma}{}^{\gamma\delta} R_{\gamma\delta}{}^{\mu\nu} - 12 R_{\mu\nu\rho\lambda} R^{\mu\rho} R^{\nu\lambda} + 8 R_\mu{}^\nu R_\nu{}^\rho R_\rho{}^\mu$$ as in Einsteinian cubic gravity or alternative, parameterized sets of basis invariants as in generalized Lovelock or quasi-topological gravities (Bueno et al., 2016, Hennigar et al., 2017, Erices et al., 2019).

Uniquely in four dimensions, the theory labeled Einsteinian cubic gravity (ECG) is neither trivial nor topological, and propagates only a massless, transverse graviton around maximally symmetric backgrounds. This is achieved by selecting special combinations of cubic invariants so that all other possible ghost or scalar degrees of freedom decouple—a requirement not generally satisfied by generic cubic theories (such as Gauss-Bonnet or Lovelock variants, which are topological or vanish in $D=4$). The construction uses algebraic procedures involving auxiliary Riemann tensors and derivatives of the action with respect to the auxiliary parameters; these methods allow for a systematic classification of viable cubic order multi-gravity potentials with "healthy" spectrum.

3. Black Hole Phenomenology and Thermodynamics

In ECG and closely related cubic gravity theories, neutral and charged, static and rotating black hole solutions have been constructed and fully characterized in $D=4$. The metric function $f(r)$ satisfies a nonlinear second-order ODE, determined exclusively by a single function: $$f(r):\,\text{nonlinear ODE}(f, f', f'', r; \lambda)$$ where $\lambda$ is the cubic coupling, and the invariant $\mathcal{P}$ modifies the horizon, photon sphere, and asymptotics compared to Schwarzschild or Reissner-Nordström cases (Bueno et al., 2016, Sánchez et al., 3 Feb 2025). Explicitly, cubic terms shift the horizon radius:

• $\lambda > 0$ reduces the black hole horizon and photon sphere sizes relative to GR,
• $\lambda < 0$ increases them.

Thermodynamic quantities are modified analytically:

• Hawking temperature and Wald entropy are computed as $T(\lambda, r_h)$ and $S(\lambda, r_h)$, where $r_h$ is the horizon radius.
• The first law $dM = T dS$ holds exactly, verified independently in the presence of cubic corrections.

Importantly, for certain parameter regimes, black holes exhibit positive specific heat (thermodynamic stability), a feature absent in Schwarzschild solutions.

For rotating solutions, on-shell amplitude techniques show that the ECG corrections to the classical potential are expressed as differential operators acting on the non-rotating solution, generalizing the Newman–Janis algorithm to higher-derivative gravity (Burger et al., 2019). Consequently, the black hole shadow, quasi-normal modes, gravitational waveforms from extreme mass ratio inspirals, light bending, and orbital precession all acquire $\lambda$-dependent corrections, opening up prospects for direct observational tests (Li et al., 29 Jan 2024, Sánchez et al., 3 Feb 2025).

4. Cosmology and Dark Sector Phenomenology

Cubic order multi-gravity potentials constructed from $P$-type invariants or $f(P)$-function modifications have direct cosmological consequences (Erices et al., 2019):

• In a Friedmann–Lemaître–Robertson–Walker background, the cubic terms yield contributions to both the background and perturbation equations, remaining of second order in derivatives.
• At early times, these terms can drive inflationary (de Sitter) expansion with an effective cosmological constant determined solely by the cubic terms—even in the absence of a bare $\Lambda$.
• At late times, the new terms form an effective dark-energy sector; depending on the parameters (or the function $f(P)$), the resulting equation-of-state parameter $w_{DE}$ can be quintessencelike, phantomlike, or exhibit phantom-divide crossing without relying on an explicit cosmological constant.

This flexibility allows the models to accommodate a wide variety of cosmological histories, with the cubic terms producing testable modifications to structure formation, expansion history, and cosmic acceleration.

5. Modified Dynamics, Stability, and Quantum Implications

In metric-affine gravity, cubic order invariants built from curvature, torsion, and nonmetricity serve a critical role in stabilizing additional vector and axial modes that typically render quadratic theories unstable due to Ostrogradsky ghosts. The kinetic matrix for these modes, with coefficients tunable via the many available cubic invariants, can be made positive-definite, eliminating gradient and ghost instabilities (Bahamonde et al., 20 Nov 2024).

Moreover, the inclusion of cubic invariants enables the existence of Reissner–Nordström–like black holes with dynamical torsion and nonmetricity "charges". Unlike electromagnetic charge, these arise from the extra gravitational degrees of freedom and can be massive—a property that avoids several no-go theorems constraining massless higher-spin interactions in quantum gravity regimes. This suggests that cubic multi-gravity potentials can provide a physically viable arena for extended and higher-spin gravity models.

6. Observational and Theoretical Implications

Cubic order multi-gravity potentials have significant implications for both fundamental theory and observations:

• In compact variable-mass systems and binary mergers, worldline effective field theory approaches demonstrate that Riemann–cubed terms modify the conservative two-body binding potential at 2PN order, introducing corrections as $1/r^6$, $1/r^7$, and via acceleration and velocity-dependent operators (Kulkarni et al., 2 Oct 2024). These corrections may imprint detectable phase shifts in gravitational wave signals.
• In cosmological settings, the effective dark energy sector driven by cubic terms can produce late-time acceleration without fine-tuned cosmological constants.
• For astrophysical black holes, cubic corrections affect the photon sphere and thus the shadow, yielding possible constraints on the coupling $\lambda$ from EHT-like observations or accurate measurements of S2 star precession around SgrA*.

From a quantum and statistical perspective, mean-field and random matrix analyses of lattice models indicate that the cubic order potentials carry subtle pathologies—such as unbounded potentials—that are also present in continuum Einstein–Hilbert gravity, highlighting challenges for the nonperturbative quantization of gravity (Tate et al., 2012).

7. Summary Table: Cubic Order Multi-Gravity Potentials—Key Elements

Aspect	Representative Formula/Feature	Primary Reference(s)
Lattice realization	$$S = -\sum_{\langle ij\rangle} \sqrt{\det g_{(ij)}}\,\cdots$$	(Tate et al., 2012)
Covariant cubic invariant	$\mathcal{P} =$ specified cubic contractions of Riemann	(Bueno et al., 2016, Hennigar et al., 2017)
Black hole metric correction	$f(r)$ solves: $f''$–type 2nd order ODE with cubic terms	(Bueno et al., 2016, Sánchez et al., 3 Feb 2025)
Rotating solutions	$V(r) = -G m_1 m_2 \cos(a\cdot \nabla)[1/r - C/r^6]$ (rotation operator)	(Burger et al., 2019)
Cosmology/dark energy	$S = \int \sqrt{-g}[R/2\kappa + f(P)]$ ; $P$ built from Riemann contractions	(Erices et al., 2019)
Stability (MAG)	Cubic invariants cancel vector/axial instabilities in kinetic matrix	(Bahamonde et al., 20 Nov 2024)
2-body potential in WEFT	$V_\alpha \sim G^2 m_1 m_2/r^6 \times$ (velocity/acceleration terms)	(Kulkarni et al., 2 Oct 2024)

The cubic order multi-gravity potential, as realized in current research, emerges as a critical structure: discrete lattice (hypercubic) models, covariant curvature-invariant theories, and effective field-theoretic frameworks all converge on the importance of cubic terms for extending the landscape of modified gravity, controlling additional degrees of freedom, ensuring stability, and generating testable signatures across strong and weak gravity regimes.