Bounds on the Lyapunov Exponent of Circular Null Orbits in $n$-Dimensional Black-Hole Spacetimes

Abstract: Unstable circular null orbits provide a geometric bridge between black-hole optics, photon rings, strong gravitational lensing, and the eikonal sector of quasinormal ringing. In four-dimensional Einstein gravity, the instability rate of such orbits, measured by the Lyapunov exponent $λ$, obeys model-independent upper bounds when the matter sector satisfies standard energy conditions. We extend this analysis to static, spherically symmetric, asymptotically flat black holes in arbitrary dimensional Einstein gravity, allowing for an anisotropic matter distribution. We show that, under the tangential null energy condition, the Lyapunov exponent admits a dimension-dependent upper bound in terms of the generalized surface gravity $κ(r)$ and the metric function $μ(r)$, both evaluated at the photon sphere $r =r_γ$. For $n$- dimensional black hole spacetime, the bound is $λ\leq\sqrt{n-3}\, {κ<em>γ}/{\sqrt{μ</em>γ}}$. Related bounds are derived in terms of the critical impact parameter (shadow radius), the orbital frequency, a local acceleration scale, and eikonal quasinormal-mode damping. All principal inequalities reduce to the known four-dimensional results for $n=4$. Also, the bounds involving the critical impact parameter and orbital frequency are saturated by the Schwarzschild-Tangherlini vacuum solution. The results provide a compact set of consistency conditions linking the dimensionality of spacetime to the instability of photon trapping in Einstein gravity.

• The paper derives the bound λ ≤ √(n−3) κγ/√μγ by showing that radial-pressure terms cancel from the photon-sphere derivative, leaving the tangential null energy condition as the key assumption.
• The result yields equivalent constraints on shadow size, orbital frequency, local acceleration, Unruh temperature, and eikonal quasinormal-mode damping, with the normalized instability satisfying λRsh ≤ √(n−3).
• The bounds recover the four-dimensional results and are exactly saturated by Schwarzschild–Tangherlini black holes, while extensions to rotating or higher-curvature spacetimes remain open.

This paper extends the model-independent upper bounds on the Lyapunov exponent $\lambda$ of unstable circular null orbits from four-dimensional to $n$-dimensional static, spherically symmetric, asymptotically flat black holes in Einstein gravity (2608.18436). The matter sector is modeled as an anisotropic fluid with energy density $\rho$, radial pressure $p_r$, and tangential pressure $p_t$ in each of the $(n-2)$ angular directions. The central result is that the tangential null energy condition (NEC) at the photon sphere suffices to establish a dimension-dependent bound, with all principal inequalities reducing to the four-dimensional results of Gallo and Mäadler when $n=4$ and being saturated by the Schwarzschild-Tangherlini vacuum solution.

Geometric setup

The spacetime is described by

$$ds^2 = -e^{-2\delta(r)}\mu(r)\,dt^2 + \frac{dr^2}{\mu(r)} + r^2 d\Omega_{n-2}^2,$$

with an event horizon at $\mu(r_H)=0$ and asymptotic flatness conditions on $\mu$ and $n$0. The field equations give $n$1 and $n$2. The matter is required to satisfy the dominant energy condition, and horizon regularity fixes $n$3. The metric function relates to the generalized Misner-Sharp mass via $n$4.

The circular null orbit at $n$5 satisfies the geometric condition $n$6, where $n$7. Equivalently, the photon-sphere characteristic function

$n$8

vanishes at $n$9. Since $\rho$0 for a regular horizon while $\rho$1, continuity guarantees existence of at least one photon sphere. The paper adopts the dimension-dependent photon-sphere radius bounds of Song, Fu, and Cen (Cen et al., 4 Jan 2026), noting that the lower bound requires an additional monotonicity assumption.

Lyapunov exponent and the key identity

Following Cardoso et al. (0812.1806), the coordinate-time Lyapunov exponent is $\rho$2, which can be expressed through the characteristic function as

$\rho$3

in agreement with the higher-dimensional result of Gallo and Villanueva (Gallo et al., 2015). Bounding $\rho$4 therefore reduces to controlling $\rho$5.

The central technical observation is that differentiating $\rho$6 and using stress-energy conservation evaluated at the photon sphere causes all radial-pressure terms to cancel exactly, yielding

$\rho$7

Consequently, only the local tangential NEC, $\rho$8, is needed; no sign assumption on $\rho$9 is required. This gives $p_r$0 and hence

$p_r$1

Principal bounds and their forms

Introducing the generalized surface gravity $p_r$2 following Abreu and Visser (Abreu et al., 2010)—the redshifted proper acceleration of a static observer—the paper derives the structural identity $p_r$3, so that at the photon sphere $p_r$4. The principal bound becomes

$p_r$5

Under the radial NEC one further has $p_r$6, giving $p_r$7. The bound admits several equivalent formulations, summarized below:

Form	Inequality	Assumptions
Surface gravity	$p_r$8	Tangential NEC at $p_r$9
Shadow radius	$p_t$0	Tangential NEC
Orbital frequency	$p_t$1	Tangential NEC
Local acceleration	$p_t$2	Radial NEC
Unruh temperature	$p_t$3	Radial NEC
Critical exponent	$p_t$4	Tangential NEC

Here $p_t$5 is the critical impact parameter (the shadow radius $p_t$6 for an observer at infinity), $p_t$7, and $p_t$8 is the proper acceleration of a static observer at the photon sphere. The dimensionless combination $p_t$9 measures radial instability per optical timescale, so the inequality caps this ratio by $(n-2)$0: increasing the spacetime dimension weakens the normalized bound.

Eikonal quasinormal modes

Under the standard eikonal geodesic/QNM correspondence, $(n-2)$1, the shadow bound translates into constraints on ringdown observables:

$(n-2)$2

The authors are explicit about the scope of this translation: the geodesic/QNM correspondence holds for test fields in the eikonal limit but can fail for gravitational perturbations in higher-curvature theories, as demonstrated by Konoplya and Stuchlík for Einstein-Lovelock black holes (Konoplya et al., 2017). The geodesic bounds themselves are independent of this correspondence; it enters only when converting them into QNM constraints.

Consistency checks and sharpness

Setting $(n-2)$3 recovers exactly the Gallo-Mädler inequalities $(n-2)$4, $(n-2)$5, and $(n-2)$6. For the Schwarzschild-Tangherlini solution, the identity for $(n-2)$7 saturates its NEC bound, and one obtains $(n-2)$8, matching the exact result of Cardoso et al. The vacuum solution therefore saturates the shadow, frequency, and critical-exponent bounds, establishing that the factor $(n-2)$9 is sharp rather than an artifact of loose estimates. The earlier bound of Bianchi, Grillo, and Morales (Bianchi et al., 2020), derived for charged black holes with $n=4$0, is recovered as a special case; the present derivation requires neither a specific solution nor that restriction.

Under the stronger assumptions of Song, Fu, and Cen (weak/dominant-type inequalities plus non-positive trace), one obtains $n=4$1, $n=4$2, and hence the radius-based corollary $n=4$3, and—with the additional monotonicity assumption—a horizon-scale bound $n=4$4. These are explicitly flagged as less universal than the principal result because they depend on global matter assumptions beyond the local tangential NEC.

Limitations and open questions

Several qualifications are stated plainly in the paper. First, the entire analysis is restricted to static, spherically symmetric geometries in pure Einstein gravity; whether analogous dimension-dependent bounds hold in Einstein-Gauss-Bonnet or Lovelock theories remains open, since the modified field equations alter the relation between matter variables, metric functions, and the characteristic function $n=4$5. Second, extension to rotating higher-dimensional black holes is substantially harder because null trapping is described by a photon region rather than a single photon sphere, and no universal instability bound is known there. Third, the QNM corollaries inherit the limitations of the eikonal geodesic/QNM correspondence, which fails for gravitational perturbations in higher-curvature theories. Finally, the observational relevance is limited: current photon-ring and gravitational-wave data concern black holes consistent with four-dimensional general relativity, so the higher-dimensional bounds serve primarily as theoretical consistency relations rather than directly testable predictions.

Conclusion

The paper establishes that for any static, spherically symmetric, asymptotically flat $n=4$6-dimensional Einstein black hole with anisotropic matter, the instability rate of circular null orbits obeys $n=4$7, together with equivalent bounds in terms of the shadow radius, orbital frequency, local acceleration, Unruh temperature, and eikonal QNM damping. The derivation hinges on an exact cancellation of radial-pressure terms in $n=4$8, leaving only the tangential NEC as the essential physical input. All bounds reduce correctly to the four-dimensional case and are saturated by Schwarzschild-Tangherlini, confirming that the $n=4$9 factor captures the genuine dimensional dependence of maximal photon-orbit instability within this class of spacetimes.