Lyapunov Exponents in Gravitational Systems

• Lyapunov exponents are quantitative measures that define the exponential rate of divergence between nearby trajectories, indicating stability or chaos in dynamical systems.
• In gravitational systems, these exponents are computed via effective potential derivatives, linking orbital instability to the decay rates of quasinormal modes and observable black hole signatures.
• Applications in modified gravity and gravitational decoupling reveal how perturbations to black hole metrics alter Lyapunov profiles, offering insights into gravitational wave ringdowns and shadow phenomenology.

A Lyapunov exponent is a quantitative diagnostic of orbital stability in a dynamical system, measuring the average exponential rate at which nearby trajectories diverge in phase space. In the context of general relativity and black hole physics, Lyapunov exponents play a central role in characterizing the (in)stability of geodesic orbits, the decay of perturbations (quasinormal modes), and the dynamical response of solutions under gravitational decoupling and related extensions.

1. Mathematical Definition and Computation

For a dynamical system described by coordinates $x^i(t)$ following $\dot{x}^i = F^i(x)$, a Lyapunov exponent $\lambda$ is defined as

$$\lambda = \lim_{t\to\infty} \lim_{|\delta x(0)|\to 0} \frac{1}{t} \ln\frac{|\delta x(t)|}{|\delta x(0)|},$$

where $\delta x(t)$ is a deviation vector evolved under the linearized equations. In the context of geodesics on a manifold, this expression quantifies the growth rate of perturbations $\delta x^i$ along a reference trajectory, and a positive $\lambda$ signals local exponential instability.

For equatorial circular geodesics in a static, spherically symmetric spacetime $ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega^2$, the principal Lyapunov exponent for radial perturbations can be directly related to the second derivative of the effective potential $V_{\text{eff}}(r)$. The formula used in gravitational decoupling applications is

$$\lambda^2 = \frac{f(r_c)}{2r_c^2} V''_{\text{eff}}(r_c),$$

where $V''_{\text{eff}}(r_c)$ is evaluated at the radius $r_c$ of the circular orbit under consideration (Paiva et al., 16 Dec 2025).

2. Role in Geodesic Stability

The sign and magnitude of the Lyapunov exponent inform the fate of small radial perturbations to bound orbits:

• $\lambda > 0$: Instability—perturbed orbits exponentially depart from the reference orbit.
• $\lambda = 0$: Marginal stability (e.g., at the ISCO—innermost stable circular orbit).
• $\lambda < 0$: Theoretically possible for some generalized systems, corresponding to local stability.

In black hole spacetimes, the Lyapunov exponent associated with the unstable photon sphere (null geodesics) determines the decay rate of perturbations and sets the imaginary part of the fundamental quasinormal mode frequency:

$$\operatorname{Im}(\omega_{\text{QNM}}) \sim \lambda.$$

This underlies the correspondence between geodesic dynamics and linear response theory for black holes.

3. Applications in Modified Gravity and Gravitational Decoupling

In the gravitational decoupling framework, Lyapunov exponents are used to probe the influence of supplementary stress-energy components (encoded in a deformation parameter $\alpha$ or analogous coupling) on the orbital structure and stability of regular or "hairy" black hole spacetimes. Deviations from the Schwarzschild solution, typically parametrized by a "hair parameter" $\ell$ or via the decoupling parameter, produce substantial shifts in the Lyapunov exponent profiles, potentially leading to observable changes in the stability of both timelike and null orbits (Paiva et al., 16 Dec 2025).

Notably, the analysis in "Gravitational decoupling and regular hairy black holes: Geodesic stability, quasinormal modes, and thermodynamic properties" demonstrates that the instability timescale of circular orbits, governed by Lyapunov exponents, is sensitive to the details of the decoupling sector. This directly connects the geometric properties of modified black hole metrics to measurable astrophysical signatures.

4. Interpretation in Dynamical Systems and General Relativity

In dynamical systems theory, a positive Lyapunov exponent is a classical hallmark of chaos and unpredictability, although in general relativistic applications (geodesic motion in stationary spacetimes) the interpretation is more restricted—significant only for local (not global) stability and typically not implying full chaotic dynamics.

In the general relativity context, the Lyapunov exponent is especially relevant for:

• The eikonal limit of quasinormal modes: the QNM frequencies at high angular momentum are controlled by the Lyapunov exponent of the unstable photon sphere.
• The timescale for relaxation after perturbations (ringdown), important for gravitational wave signals.
• The occurrence and nature of phenomena such as "gravitational lensing echoes" and black hole shadow properties, where the Lyapunov exponent governs the decay or persistence of features associated with light rings.

5. Summary Table: Lyapunov exponent in gravitational decoupling

Context	Significance	Key Reference
Black hole geodesic motion	Quantifies (in)stability of orbits	(Paiva et al., 16 Dec 2025)
Quasinormal mode decay	Imaginary part set by $\lambda$	(Paiva et al., 16 Dec 2025)
Gravitational decoupling modifications	Shifts in $\lambda$ = shifts in observable signatures	(Paiva et al., 16 Dec 2025)

6. Practical Computation and Physical Implications

For any static, spherically symmetric metric resulting from a gravitational decoupling scheme, the Lyapunov exponent can be computed by:

1. Determining the effective potential $V_{\text{eff}}(r)$ for the given metric.
2. Locating the extremum (e.g., circular orbit or photon sphere radius $r_c$).
3. Evaluating $f(r)$ and $V_{\text{eff}}''(r)$ at $r_c$ and inserting into $\lambda^2 = f(r_c)V_{\text{eff}}''(r_c)/2r_c^2$.

A larger value of the decoupling parameter or hair parameter typically alters $f(r)$ and thus $\lambda$, leading to potentially longer or shorter instability timescales. Such effects are not only diagnostic of the underlying gravitational model but can, in principle, be constrained observationally via gravitational wave signals and shadow phenomenology.

7. Research Directions and Significance

Current research explores the systematic use of Lyapunov exponents to:

• Distinguish between general relativistic and modified gravity black holes.
• Quantify the stability and chaos of orbits in gravitational decoupling extensions.
• Connect the spacetime geometry (as modified by additional sources $\Theta_{\mu\nu}$) to measurable timescales for relaxation and to the spectrum of quasinormal modes (Paiva et al., 16 Dec 2025).
• Provide insights into the dynamical response in strong gravity and its implications for gravitational wave astronomy and black hole imaging.

The Lyapunov exponent thus serves as a bridge between the mathematical structure of decoupled gravitational systems and observational signatures accessible in precision astrophysics and gravitational wave detections.