---
title: Einsteinian Cubic Gravity Overview
url: https://www.emergentmind.com/topics/einsteinian-cubic-gravity
type: topic
---

# Einsteinian Cubic Gravity Overview

Einsteinian cubic gravity (ECG) is a higher-curvature extension of general relativity that incorporates a unique cubic-in-curvature invariant in such a way that the linearized spectrum on maximally symmetric backgrounds contains only the massless, transverse graviton. In contrast to Lovelock or other quasi-topological theories, ECG remains nontrivial and non-topological in four dimensions, and exhibits novel dynamics, black-hole physics, and holographic features. This article surveys the structure, solutions, thermodynamics, holography, stability properties, and cosmological dynamics of ECG, with particular emphasis on four-dimensional static and rotating black holes, their observational signatures, and the crucial subtlety of ghost freedom.

## 1. Gravitational Action and Field Equations

The ECG action in four spacetime dimensions is given by
\[
S = \frac{1}{16\pi G}\int d^4x \sqrt{-g} \left[ R - 2\Lambda + \lambda \, \mathcal{P} \right],
\]
where $R$ is the Ricci scalar, $\Lambda$ the cosmological constant (often set to zero in asymptotically flat contexts), and $\lambda$ a coupling constant with dimension (length)$^2$ or (mass)$^{-2}$. The key cubic curvature invariant is
\[
\mathcal{P} = 12\,R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}
+ R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}
-12\,R_{abcd}R^{ac}R^{bd}
+8\,R_{a}{}^{b}R_{b}{}^{c}R_{c}{}^{a}.
\]
The equations of motion are fourth-order PDEs in the metric but, for spherically symmetric ansätze, reduce to a single ordinary differential equation (ODE) for the metric function $f(r)$:
\[
-(f-1)r - \lambda G^2\left[ 4f'^3 + 12\frac{f'^2}{r} -24f(f-1)\frac{f'}{r^2} -12f f''\big(f' - \frac{2(f-1)}{r}\big)\right] = 2GM,
\]
where $(\,)'$ denotes $d/dr$ and $M$ is the ADM mass parameter [1610.08019]. The uniqueness and physicality of ECG stem from the requirement that, upon linearization on maximally symmetric backgrounds, only the Einstein graviton propagates—a constraint that tightly fixes the relative coefficients in $\mathcal{P}$ [1607.06463].

## 2. Static, Spherically Symmetric and Rotating Black Holes

For the static spherically symmetric line element,
\[
ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_{(2)}^2,
\]
ECG admits one-parameter families of asymptotically flat black-hole solutions. The equation for $f(r)$ reduces to Schwarzschild for $\lambda \to 0$. For small $\lambda$,
\[
f(r) = 1 - \frac{2GM}{r} - \lambda G^2 \left( \frac{108(2GM)^2}{r^6} - \frac{92(2GM)^3}{r^7} \right) + \mathcal{O}(\lambda^2, r^{-8}).
\]
Near the horizon $r_+$, $f(r_+)=0$, with surface gravity $\kappa=f'(r_+)/2$ and Hawking temperature $T=\kappa/(2\pi)$ [1610.08019]. The Wald entropy is
\[
S = \frac{\pi r_+^2}{G}\left[1 - 48\lambda G^2 \frac{\kappa^2}{r_+^2} \left( \frac{2}{\kappa r_+} + 1 \right) \right].
\]
In charged cases with Maxwell coupling, the system admits two coexisting black-hole branches above a critical coupling, violating uniqueness yet retaining pathologies such as the disappearance of inner (Cauchy) horizons found in Reissner–Nordström [2002.04071].

The extension to slowly rotating black holes, $ds^2=-f(r)\,dt^2+dr^2/f(r)+2a r^2 p(r)\sin^2\theta dt d\phi+\cdots$, leads to coupled ODEs for $f(r)$ and $p(r)$ solvable numerically. ECG corrections shift the horizon angular velocity, ISCO radius, photon sphere, and shadow size, but the deviations become significant only for compact objects with $M\sim\lambda^{1/4}$ [2004.09598].

## 3. Thermodynamic and Phase Structure

ECG black holes exhibit thermodynamic properties that deviate from the Schwarzschild paradigm. For asymptotically AdS or flat cases, the Hawking temperature and entropy can be computed analytically as functions of $r_+$ and $\lambda$; the Abbott–Deser mass matches the integration parameter. The first law, $dM=TdS$, is satisfied for all values of $r_+$ and $\lambda$ [1610.08019, 2111.02663]. Notably, ECG solutions feature two branches for $T<T_\mathrm{max}$: a small $r_+$ branch with positive specific heat (thermodynamically stable) and a large $r_+$ branch with negative specific heat (unstable), merging at $T_\mathrm{max}\sim \lambda^{-1/4}$, signaling a phase transition [1610.08019]. In AdS, the equation of state is quadratic in $T$ and exhibits van der Waals–type criticality when $\lambda<0$, with the critical ratio $P_c v_c/T_c=3/8$ matching the universal value from classical thermodynamics [1610.06675].

Higher-dimensional extensions require two independent metric functions and admit “super-entropic” black holes that violate the reverse-isoperimetric inequality [1610.06675]. The inclusion of gauge fields yields novel features such as the existence of both small and large black holes with the same $(M,Q)$, and the complete absence of inner horizons even in over-extremal regimes [2002.04071].

## 4. Holographic Aspects and Conformal Field Theory Duals

In the AdS context, four-dimensional ECG provides a tractable holographic model for asymptotically locally AdS spacetimes, yielding explicit non-hairy AdS$_4$ black holes. The relationship between bulk and boundary data is as follows [1802.00018]:

- The graviton two-point function charge is $C_T = \frac{3L^2}{\pi G}(1-3\mu f_\infty^2)$, with $f_\infty$ determined by $1-f_\infty+\mu f_\infty^3=0$ (here, $\mu$ is the dimensionless ECG coupling).
- The universal entanglement entropy charge $a^* = \frac{\pi L^2}{4G}(1+3\mu f_\infty^2)$ controls EE across disk regions.
- Thermal entropy and Rényi entropies of the dual CFT$_3$ can be computed using exact or numerically precise bulk black holes.
- The ratio $\eta/s$ is a non-analytic function of the ECG coupling and is strictly bounded below by $1/4\pi$, with positive-energy black holes forbidding any violation of the KSS bound, regardless of the coupling strength.
- The generalized Gibbons–Hawking–York boundary term enters the Euclidean on-shell action weighted by $a^*$, providing a systematic prescription for holographic renormalization and for extracting dual CFT data [1802.00018].

## 5. Stability and Pathology: Ghosts, Laplacian Instabilities, and the EFT Regime

While ECG is constructed to propagate only the massless graviton at the linearized level on maximally symmetric backgrounds, ghost and Laplacian instabilities can arise for non-perturbative values of the coupling. For spherically symmetric black holes with order-unity coupling, odd-parity metric perturbations around the ECG black hole exhibit three dynamical modes (rather than the single pure-spin-2 degree of freedom of GR). One mode always behaves as a ghost (sign mistuned kinetic term), and another acquires a negative sound speed squared for high-$\ell$ angular modes, $c^2_\Omega=-1/2$, resulting in Laplacian instability on arbitrarily short timescales [2305.07217].

These pathologies only disappear when restricting ECG to the strictly perturbative regime (EFT validity), with $\lambda {\cal P}\ll R$, i.e., the cubic term is always subleading compared to the Ricci scalar. Thus, there are no genuinely stable, static, spherically symmetric black-hole solutions in ECG with unsuppressed higher-order curvature terms [2305.07217].

## 6. Observational and Phenomenological Signatures

ECG predicts modest but potentially observable deviations from general relativity in strong-field regions (black holes, compact objects), but is essentially indistinguishable from GR in solar-system or weak-field regimes. Key predictions include:

- Slight increases in the black-hole shadow angular diameter, with $r_\mathrm{ph}$ (photon sphere) and $r_\mathrm{ISCO}$ both shifted slightly outward at linear order in $\lambda$ [1801.03223, 2401.16071].
- Enhanced periastron precession (compared to GR) for slightly positive cubic couplings, providing a direct but currently loose bound from S2 star orbits: $\lambda \lesssim 5 \times 10^{29} M_\odot^4$ [2401.16071].
- Binary inspiral (“zoom–whirl”) waveforms in ECG show suppressed amplitudes and shifted phasing; future LISA or TianQin observations could constrain $\lambda/ (GM)^4$ at the percent level.
- Gravitational lensing by ECG black holes results in milliarcsecond-level shifts in image positions for supermassive black holes, which are accessible to VLTI/GRAVITY and future EHT-like telescopes [1810.10657].
- Shadow measurements at $\lesssim1\%$ precision would constrain the cubic coupling to $|\lambda|\lesssim O(0.1) M_\text{pl}^5/r_S^4$ [2502.01747].

New classes of horizonless compact objects (“frozen gravitational stars,” FGSs) have been numerically constructed; these are naked singularities cloaked by a critical "frozen" surface at the Schwarzschild radius and are observationally indistinguishable from extremal black holes [2410.04575].

## 7. Cosmological Solutions and the Role of the Cubic Term

In cosmology, the cubic curvature term in ECG (and extended "cosmological Einsteinian cubic gravity," CECG) has a pronounced effect on early-universe dynamics. The phase-space analysis of the cosmological Friedmann equations—modified by the cubic term—reveals a universal “inflationary big-bang” attractor (for high curvature, $H^4 \gg 1/(16\beta)$), corresponding to non-standard matter-driven inflation that is the generic past behavior for all positive $\beta$ [2003.10516]. However, the cubic modification alone cannot yield late-time acceleration without a cosmological constant term; the late-time attractor is standard de Sitter only if $\Lambda>0$. The CECG model thus provides a purely geometric mechanism for primordial inflation but requires vacuum energy for late-time accelerated expansion.

---

**References**

1. [1610.08019], [1610.06675], [1607.06463] — foundational construction, static black holes, thermodynamics, and uniqueness.
2. [2305.07217] — instability and ghost analysis beyond EFT regime.
3. [2002.04071], [2111.02663], [1910.10721], [2004.09598] — rotating, charged black holes and black strings.
4. [1802.00018] — holographic dictionary and thermodynamics; [1801.03223], [2502.01747], [1810.10657], [2401.16071], [2410.04575] — observational phenomenology, lensing, strong-field tests, and exotic compact objects.
5. [1707.06308], [2003.10516] — cosmological solutions, critical points, and asymptotic analysis.

Source: https://www.emergentmind.com/topics/einsteinian-cubic-gravity