Dynamic Black Hole Environments

• Dynamic environments around black holes are defined by rapidly changing spacetime curvature, matter fields, and radiation interacting within nonstationary accretion flows.
• The framework leverages concepts like marginally outer trapped surfaces (MOTS) and dynamical horizons to quantify energy and angular momentum fluxes during strong perturbations.
• These insights underpin numerical relativity models and guide interpretations of black hole growth, merger dynamics, and gravitational radiation in astrophysical settings.

Dynamic environments around black holes encompass the complex interplay of spacetime curvature, matter fields, radiation, and accretion flows in the neighborhoods of black holes that deviate from the idealized, stationary, vacuum Kerr (or Schwarzschild) scenario. These environments are governed by a range of phenomena: dynamical horizons, nontrivial matter distributions (e.g., accretion disks, scalar halos), perturbations, the breakdown of symmetries, and the energy and angular momentum fluxes through evolving black-hole boundaries. The mathematical and observational framework necessary to describe such configurations is anchored in numerical relativity, quasi-local horizon formalism, nonlinear dynamical systems, and the theory of gravitational radiation.

1. Marginally Outer Trapped Surfaces and Dynamical Horizons

A central concept in dynamical black-hole environments is the marginally outer trapped surface (MOTS)—a closed, spacelike two-surface where the expansion of outgoing null geodesics $\theta_{(l)}$ vanishes. During strong perturbations, such as an ingoing gravitational wave pulse incident upon a Kerr black hole, multiple concentric MOTSs can form (up to five in certain numerically simulated scenarios), appearing and disappearing in pairs such that the MOTS count at any time is always odd (Chu et al., 2010). Each new pair forms via a bifurcation of the original surface, with the "inner" surface typically annihilating with a neighboring MOTS, while the "outer" one accompanies the growth of the black hole.

Successive MOTSs across spatial hypersurfaces stack to form a marginally trapped tube (MTT). When the black hole is in a strongly dynamical regime (absorbing significant energy), the MTT world tube is spacelike, i.e., a dynamical horizon. This spacelike nature ensures fluxes of energy and angular momentum across the horizon can be defined locally. As the system relaxes, the MTT approaches a null hypersurface, and in equilibrium converges to the event horizon.

The dynamical horizon formalism quantifies the physical content of this picture. The irreducible mass $M_H$ and area $A_H$ on each MOTS are related via $M_H = \sqrt{A_H/(16\pi)}$. The dynamical horizon flux law expresses the change in $M_H$ as an integral of local fluxes. For example,

$$\frac{dM_H}{dv} = \int_{\mathcal{S}_v} \left[ T_{\mu\nu} \bar{l}^\mu \tau^\nu + \frac{\bar{B}}{8\pi} \sigma^{(\bar{l})}_{\mu\nu} \sigma^{(\bar{l})\mu\nu} + \frac{\bar{C}}{8\pi} \bar{\omega}_\mu \bar{\omega}^\mu \right] \sqrt{q}\, d^2x$$

where $\bar{l}^\mu$ is a rescaled outgoing null normal, $\sigma^{(\bar{l})}_{\mu\nu}$ its shear, and $\bar{\omega}_\mu$ the normal fundamental form.

Angular momentum flux is governed by a generalized Damour–Navier–Stokes equation, involving shear components and contributions from the stress–energy tensor. The primary contributions during strong distortions arise from the large angular shears induced by gravitational waves.

2. Event Horizon Evolution and Teleology

In strongly dynamical scenarios, the event horizon and apparent horizon (outermost MOTS) can be well-separated spatially and temporally. The event horizon, computed by tracking backward-propagated null geodesics once the entire spacetime evolution is known, begins to increase in area even before the incoming perturbation interacts with the black hole. This is a haLLMark of its global, teleological character: it "anticipates" future accretion events (Chu et al., 2010).

During the phase where multiple MOTSs exist, the event horizon remains smooth and singular, with no branching or entry of new null generators. It asymptotically converges onto the apparent horizon once the highly dynamical regime subsides, but exhibits early growth driven by incoming energy. This difference in the response of the event and apparent horizons underscores the distinction between global and quasi-local definitions of black-hole surface in non-equilibrium situations.

The dynamics are further encapsulated in the null Raychaudhuri equation, which governs the evolution of the expansion of null geodesic congruences: $$\frac{d\theta_{(l)}}{d\lambda} = -\frac{1}{2}\theta_{(l)}^2 - \sigma^{(l)}_{\mu\nu} \sigma^{(l)\mu\nu} - 8\pi T_{\mu\nu} l^\mu l^\nu$$

3. Nonlinear Black Hole Dynamics and Flux Laws

The complex evolution of multiple MOTSs, the formation of spacelike dynamical horizons, and the behavior of the event horizon demand precise mathematical tools. The evolution is locally characterized by the expansions ($\theta_{(l)}$, $\theta_{(k)}$) and the area/irreducible mass relation. Formation and annihilation of MOTSs in pairs reveal a fundamentally nonlinear process, where the cross-sectional area of the outermost MOTS can exhibit discontinuous changes.

Flux laws tie these changes to shear-driven energy and angular momentum flow. In mathematical terms, the angular momentum $J$ evolves according to

$$\frac{dJ}{dv} = -\int_{\mathcal{S}_v} T_{\mu\nu} \phi^\mu \tau^\nu \sqrt{q} d^2x - \int_{\mathcal{S}_v} \left( B \sigma^{(l)}_{\mu\nu} \sigma^{(\phi)\mu\nu} + C \sigma^{(k)}_{\mu\nu} \sigma^{(\phi)\mu\nu} \right) \sqrt{q} d^2x + \ldots$$

where $\phi^\mu$ is the (approximate) Killing vector generating rotations on $\mathcal{S}$.

The dynamical horizon formalism also introduces an extremality parameter $e$ (in vacuum, $$e = 1 + (1/4\pi) \int_\mathcal{S} (L_k\theta_{(l)})\sqrt{q} d^2x$$). For subextremal horizons, $e < 1$.

4. Mathematical Structure and Evolution of Horizons

The formalism leans on a tetrad decomposition of the outgoing ($l^\mu$) and ingoing ($k^\mu$) null directions, constructed in terms of a timelike hypersurface normal $n^\mu$ and an outward-pointing spacelike normal $s^\mu$ as $l^\mu = (n^\mu + s^\mu)/\sqrt{2}$, $k^\mu = (n^\mu - s^\mu)/\sqrt{2}$, with $l^\mu k_\mu = -1$.

The expansion, shear, and normal fundamental form ($\omega_\mu$) are defined with respect to the induced metric $$q_{\mu\nu} = g_{\mu\nu} + n_\mu n_\nu - s_\mu s_\nu$$. Their evolution, coupled to the vacuum Einstein equations (or to matter fields), dictates the emergence and fate of horizons.

During dynamical episodes, MOTSs may jump between disconnected branches in configuration space, yet the event horizon always remains continuous and regular, highlighting the teleological nature of global causal boundaries.

5. Physical and Theoretical Implications

The emergence of multiple, pairwise creating and annihilating MOTSs, the transition between spacelike and null dynamical horizon segments, and the difference in behavior between apparent and event horizons illustrate crucial features of black hole growth and relaxation. In particular:

• The complex quasi-local horizon structure is an inevitable feature of black-hole dynamics under strong perturbations; the naive picture of a single, smoothly evolving “surface” can fail.
• Local flux laws provide the only practical means to track black hole mass and angular momentum in highly dynamical circumstances.
• The fact that the global event horizon’s area increases continuously—even in anticipation of later infalling matter—emphasizes the nonlocal character of event horizons versus the local, potentially discontinuous evolution of apparent horizons.
• The mathematical toolkit (dynamical horizon flux laws, expansions, and extremality conditions) forms the backbone of numerical relativity and applies directly in the analysis of merger events, black hole growth in accretion, and gravitational collapse scenarios.

6. Observational Context and Astrophysical Relevance

These findings are critical for interpreting both numerical simulations of binary black-hole mergers and observation-driven modeling of black hole growth:

• The difference between quasilocal (MOTS/MTT) and global (event horizon) surfaces can influence interpretations of mass and area during rapid accretion or mergers.
• The signature of pairwise creation and disappearance of MOTSs may have implications for the thermodynamics and entropy budget during violent dynamical episodes.
• The dynamical horizon framework provides a rigorous, locally defined way to assign mass, angular momentum, and fluxes to a black hole in evolution, essential in numerical relativity and in matching simulation results to astrophysical phenomena.

The comprehensive paper of dynamic horizons, multiple MOTSs, and the response of global event horizons provides a nuanced view of black hole boundary dynamics and is critical for a complete understanding of black hole evolution beyond the idealized stationary regime.