Nested Apparent Horizons

• Nested apparent horizons are multiple, mutually embedded marginally trapped surfaces observed in dynamic spacetimes like black hole mergers and inhomogeneous cosmologies.
• Their identification employs precise horizon conditions, bifurcation analysis, and computational techniques including spectral methods and shooting approaches.
• Studying these structures reveals insights into gravitational dynamics, stability criteria, and the potential quantum discretization of horizon areas.

Nested apparent horizons are configurations in which multiple apparent horizons or marginally trapped surfaces of differing character are simultaneously present and embedded within each other in a connected spacetime region. Such nested structures arise in a broad array of physical scenarios—including dynamical black hole mergers, scalar-tensor gravitational collapse, strong Hawking backreaction regimes, and inhomogeneous cosmological spacetimes. Their paper provides incisive tests of the dynamical nature of horizon formation, stability criteria, and quantum aspects of horizon area discreteness.

1. Definitions, Characterization, and Horizon Types

In general relativity and its extensions, an apparent horizon (AH) is a closed surface on a spacelike hyperslice where the expansion of outgoing null geodesics, $\theta_{(\ell)}$, vanishes: $\theta_{(\ell)} = 0.$ A marginally outer trapped surface (MOTS) is any closed surface where the outgoing expansion vanishes and the ingoing expansion is nonpositive. In dynamical or inhomogeneous backgrounds, multiple such surfaces can arise and coexist in nested fashion.

Several precise horizon-like structures are relevant in this context:

• Apparent Horizon (AH): The outermost boundary of the trapped region, defined by $\nabla_\mu k^\mu = 0$ for the outgoing null normal %%%%2%%%%.
• Marginally outer trapped surfaces (MOTSs): Generic surfaces meeting $\theta_{(\ell)}=0$, possibly not globally outermost.
• Absolute Apparent Horizon (AAH) (in Szekeres models): The locus where the "most radial" null direction is turned toward decreasing areal radius, given by a turning condition involving $R_{,z}$ and the metric dipole.
• Light Collapse Region (LCR): The spacetime region where all outgoing null geodesics are forced to decreasing $R$, bounded by the AAH.

Nested apparent horizons correspond to spatial configurations such as: $$\mathcal{S}_\mathrm{in} \subset \mathcal{H}_\mathrm{out}$$ where $\mathcal{S}_\mathrm{in}$ is a (possibly "superextremal") MOTS and $\mathcal{H}_\mathrm{out}$ the enclosing apparent horizon.

2. Formation Mechanisms in Dynamical and Inhomogeneous Spacetimes

Nested apparent horizons are generically produced via dynamical processes that admit multiple simultaneously trapped regions with distinct causal or geometric character.

2.1. Rotating Black Hole Binaries and "Overspun" MOTSs

In numerically evolved binaries of nearly extremal Kerr black holes, superposed Kerr–Schild (SKS) initial data are constructed by prescribing a conformal 3-metric,

$g_{ij} = \psi^4 \,\tilde{g}_{ij}^{\rm SKS},$

with excision surfaces $\mathcal{S}_\mathrm{in}$ tuned (via a spin-parameter $\Omega_r$) so that

$$\zeta_{\rm in} = \frac{8 \pi S_{\rm in}}{A_{\rm in}} > 1.$$

A true apparent horizon $\mathcal{H}_{\rm out}$ always forms around, with

$$\zeta_{\rm out} = \frac{8 \pi S_{\rm out}}{A_{\rm out}} < 1.$$

This demonstrates existence of a superextremal inner MOTS, always shielded by a subextremal apparent horizon (Lovelace et al., 2014).

2.2. Binary Black Hole Mergers: MOTS Nesting and Annihilation

During the head-on merger of two nonspinning black holes, the sequence is:

• Pre-merger: two disjoint, stable individual MOTSs ($\Sigma_1$, $\Sigma_2$).
• At the merger: prompt formation and bifurcation of a common outer MOTS ($\Sigma_\mathrm{outer}$, spacelike and stable) and a distorted inner MOTS ($\Sigma_\mathrm{inner}$).
• Post-merger: the individual horizons persist for finite time and are eventually annihilated with partners inside $\Sigma_\mathrm{outer}$, which survives as the ultimate boundary.
• Throughout, additional short-lived, topologically intricate ("exotic") MOTSs appear, always nested within $\Sigma_\mathrm{outer}$ (Pook-Kolb et al., 2021).

2.3. Nested Horizons in Brans–Dicke and Cosmological Spacetimes

In spherically symmetric inhomogeneous Brans–Dicke solutions (Vitagliano et al., 2013, Faraoni et al., 2012), the condition for an apparent horizon is

$g^{ab} \partial_a R \partial_b R = 0,$

with $R$ the areal radius depending on evolving background. These systems can admit multiple real roots, interpreted as:

• Inner black hole horizon, shielding the central singularity.
• Outer (possibly cosmological) horizon, set by the cosmic expansion.

These horizons can merge (annihilate) or be born in pairs at critical epochs, producing nested horizon configurations.

2.4. Quantum–Backreaction–Induced Nested Horizons

In the presence of sufficiently intense Hawking backreaction, the semiclassical stress-energy of outgoing quanta can distort the background geometry enough to dynamically generate additional apparent horizons outside the original black hole. Employing an outgoing Vaidya geometry,

$$ds^2 = -\left(1 - \frac{2M(u)}{r}\right) du^2 - 2 du \, dr + r^2 d\Omega^2 \,,$$

a mass profile $m(r,u)$ with sufficient gradient can produce multiple solutions to

$f(r,u) \equiv r - 2m(r,u) = 0,$

yielding inner ($r_\mathrm{AH}^{(1)}$) and outer ($r_\mathrm{AH}^{(2)}$) apparent horizons (Silverman, 4 Nov 2025).

3. Extremality, Quantitative Measures, and Stability

Quantitative assessment of nested apparent horizons employs both geometric inequalities and operator-theoretic stability analysis.

3.1. Quasilocal Spin–Area Inequality

For any closed 2-surface $\mathcal{H}$,

$$A = \oint_{\mathcal H} dA \,, \qquad S = \frac{1}{8\pi} \oint_{\mathcal H} \omega_A \phi^A\, dA$$

where $\omega_A$ is the normal-bundle connection, and $\phi^A$ a rotation vector (approximate Killing). Extremality is defined via

$\zeta \equiv \frac{8\pi S}{A} \le 1,$

with $\zeta=1$ only for extremal Kerr.

3.2. Booth–Fairhurst Extremality Parameter

The Booth–Fairhurst extremality is

$$e = \frac{1}{4\pi} \oint_{\mathcal H} \omega_A \omega^A \, dA$$

which depends on boost gauge. A unique, gauge-invariant lower bound $e_0$ can be defined by fixing $D^A \omega_A = 0$, leading to

$$4\pi e_0 = -\oint \Omega D^{-2} \Omega \, dA, \quad e \geq e_0$$

with $e_0^{\rm Kerr}(1) = (4+\pi)/8 \approx 0.893$ in the extremal Kerr limit.

Nested configurations always exhibit $e_{0,\rm in}>1$ for superextremal inner MOTS and $e_{0,\rm out}<1$ for their enclosing apparent horizons (Lovelace et al., 2014).

3.3. Stability Operator Analysis

The MOTS stability operator governs the dynamical status of a horizon: $$L[\phi] = -\Delta_\Sigma \phi + 2 \omega^a D_a \phi + (1/2 \mathcal{R} - 2 |\sigma_+|^2 - \text{div}_\Sigma \omega - \omega_a \omega^a) \phi$$ Surfaces with positive principal eigenvalue $\lambda_0>0$ are strictly stable, forming smooth, spacelike worldtubes and true horizon barriers. Instability ($\lambda_0<0$ or higher negative eigenvalues) signals merger or annihilation events and underlies the short‑lived nature of exotic nested MOTSs in merger simulations (Pook-Kolb et al., 2021).

4. Dynamical Evolution and Horizon Bifurcation

The general mechanism for the appearance (birth) or disappearance (merger) of nested apparent horizons is mathematically controlled by the bifurcation properties of the trapping equation. In spherically symmetric models with a mass profile $m(r)$, the horizon condition $f(r) = r - 2m(r) = 0$ yields a pair of nested horizons when

$\frac{\partial m}{\partial r}|_{r_*} = \frac{1}{2}$

as a critical slope. The process parallels saddle–node bifurcations, with roots appearing or annihilating in pairs as the background evolves or as the stress-energy content varies (Silverman, 4 Nov 2025).

In Brans–Dicke and cosmological cases, analogous parametric or discriminant conditions demarcate periods of nested horizons: $$\Delta = A^{2(\alpha+1)} + 4H^2\mathcal{C}A^{2(\alpha-1)(\alpha+1)/\alpha}$$ First positive value of $\Delta$ marks the appearance of a nested pair; vanishing of the discriminant signals merger and disappearance (Faraoni et al., 2012, Vitagliano et al., 2013).

5. Quantum Discreteness and Area Quantization in Nested Horizons

A distinct proposal emerging from semiclassical studies posits that the proper separation (area difference) between nested apparent horizons induced by intense Hawking backreaction may be quantized following a Bohr–Sommerfeld–type adiabatic invariant: $\oint \kappa\,dA = 2\pi n \hbar$ where $\kappa$ is the surface gravity and $A$ the area. This implies

$$\Delta A = 4\pi (r_2^2 - r_1^2) = n\epsilon\ell_{\mathrm{P}}^2$$

for integer $n$, Planck length $\ell_P$, and coefficient $\epsilon \simeq 8\pi$.

This structure is reminiscent of the discrete area spectrum in loop quantum gravity, as well as resonance quantization of dynamical trapping regions. The minimal allowed $\Delta r \sim \ell_{\mathrm{P}}$ suggests a geometric remnant or minimal thickness for evaporating black holes (Silverman, 4 Nov 2025).

6. Physical Implications and Interpretation

Nested apparent horizons provide a fine-grained quasi-local structure not captured by event horizons:

• In mergers, they supply a spectrum of intermediate null surfaces, with only strictly stable MOTSs corresponding to the quasi-local black hole boundary; all others mediate transitions or signal nonstationary, nonunique interior structures (Pook-Kolb et al., 2021).
• In cosmological and Brans–Dicke backgrounds, nesting dynamics model how central singularities are alternately exposed or cloaked by black hole and cosmological horizons as background parameters evolve (Vitagliano et al., 2013, Faraoni et al., 2012).
• In quantum-backreaction scenarios, multi-horizon geometry may encode Planck-scale discreteness and influence the end-state of evaporation, pointing to potential observable imprints in high-precision simulations or analog gravity systems (Silverman, 4 Nov 2025).

7. Methodologies and Computational Techniques

Robust identification and analysis of nested apparent horizons require:

• Spectral Einstein Code (SpEC): For accurate dynamical trapping-surface finding in binary black hole evolutions (Lovelace et al., 2014).
• Axisymmetric generalized shooting methods: To locate complex MOTSs, including folded or self-intersecting ones, by recasting the elliptic horizon condition as a set of ordinary differential equations for generating curves $\gamma(s)$ (Pook-Kolb et al., 2021).
• Parametric and discriminant analysis: For identifying bifurcation points and root structure of horizon-locating equations in analytic solutions and semi-analytic models (Vitagliano et al., 2013, Faraoni et al., 2012).
• Gauge-invariant extremality measures: For consistent quasilocal spin and extremality assessment across dynamically evolving surfaces (Lovelace et al., 2014).

These computational and analytical tools are critical for uncovering and rigorously characterizing the rich phenomenology of nested apparent horizons in physically realistic scenarios.