Black Hole Subsystem Dynamics

• Black Hole Subsystem (BHSub) is a distinct, self-segregated group of black holes that form a dynamical core within larger stellar or quantum systems.
• They govern energy regulation in star clusters through mass segregation and binary interactions, influencing core expansion and cluster evolution.
• BHSubs provide insights into quantum information flow and the black hole information paradox via scaling laws, observable diagnostics, and dynamical feedback mechanisms.

A Black Hole Subsystem (BHSub) is a physically or mathematically distinguished subset of degrees of freedom within a larger gravitational or astrophysical system, manifesting as a spatial, dynamical, or quantum-theoretical “subsystem” dominated by black holes or associated with black hole microstructure. BHSubs play pivotal roles in diverse contexts ranging from the collisional dynamics in globular clusters and nuclear star clusters, through quantum modeling of black hole information and entropy, to observational signatures and theoretical probes of both astrophysical and microscopic black holes.

1. Dynamical Evolution and Structure in Stellar Systems

In globular clusters and nuclear star clusters, a BHSub typically refers to a self-segregated population of stellar-mass black holes forming a compact subcluster at the system’s center. Two-component dynamical models, in which the heavy component (black holes) and the light component (normal stars) are initially distributed similarly, reveal rapid mass segregation: the more massive black holes lose kinetic energy through two-body relaxation and sink toward the center (Breen et al., 2013, Breen et al., 2013). The resulting subsystem evolves with a compact Lagrangian radius $r_{h,2}$ and a half-mass radius distinct from that of the light stars, as described by

$$(r_{h,2})^{5/2} / (r_h)^{5/2} \sim (M_2^{3/2} / M^{3/2}) (m_2/m) \Big(\frac{\ln \Lambda_2}{\ln \Lambda}\Big)$$

where $M_2$, $m_2$ are the total and individual masses of the BHs, and $M$, $m$ pertain to the cluster as a whole.

After initial concentration, the BHSub undergoes "core collapse" and enters a prolonged regime where its dynamics and mass loss are tightly regulated by interactions with the host stellar background, not its local dynamical timescale (Breen et al., 2013). Its energy-generation capacity, primarily via three-body interactions and binary formation, matches the demands of the global cluster, in accord with Hénon’s Principle.

2. Energy Balance and Regulation: Hénon’s Principle

Hénon’s Principle is central to BHSub evolution in collisional stellar systems. Energy generated in the BHSub through dynamical binary processes must equilibrate with the energy flux required for the overall system’s expansion. The dimensionless expansion rate $\zeta$ ties the core energy flux to the cluster’s half-mass relaxation time $t_{rh}$:

$\dot{E} \simeq \frac{|E|}{t_{rh}\zeta}$

In two-component systems, although the BHSub has a much shorter $t_{rh,2}$ than the global $t_{rh}$, its energy generation rate is slaved to the latter (Breen et al., 2013). This self-regulation implies that BH mass loss rates—principally through the ejection of BHs in dynamically energetic encounters—are almost insensitive to the precise mass ratios ($m_2/m_1$, $M_2/M_1$), provided they are sufficiently large ($m_2/m_1 \gtrsim 10, M_2/M_1 \sim 10^{-2}$). The predicted near-universal loss rate is

$\dot{M}_2 \simeq - (\beta\zeta M)/(\alpha t_{rh}),$

with typical values $\alpha \approx 0.15$, $\beta \approx 2.2$, and $\zeta$ of order $0.09$–$0.12$ for Plummer models.

This near-independence from subsystem details encodes a self-consistent feedback: the cluster’s need for energy (driven by bulk relaxation) dictates the rate and longevity of the central BHSub, which in turn governs phenomena such as core expansion, delay of core collapse, or the emergence of so-called dark star clusters heavily dominated by black holes.

3. Formation, Observational Diagnostics, and Scaling Relations

Advanced numerical and machine learning studies of simulated star cluster populations have yielded robust scaling laws and methods to infer the presence and characteristics of a BHSub using observable cluster properties (Sedda et al., 2018, Askar et al., 2018, Askar et al., 2018). The size of a BHSub in a globular cluster is operationally defined as the radius enclosing equal mass in black holes and stars ($R_\mathrm{BHS}$), comparable to the "influence radius" for supermassive black holes. Empirical “fundamental plane” relationships relate key intrinsic properties:

Property	Scaling Law (Illustrative)	Typical Value
Density	$\log\rho_\mathrm{BHS} = A\log[L_V/r_{hl}^2] + B$	$10-10^5$ $M_\odot\,\mathrm{pc}^{-3}$
Mass	$M_\mathrm{BHS} = \alpha\,R_\mathrm{BHS} + \beta$	$10^2-10^3\,M_\odot$
Number	$\log N = G\log M + H$	$10$–$10^3$
Mean Mass	$\log m = I\log R + J$	$14$–$22$ $M_\odot$

When applied to Milky Way globular clusters, these methods suggest that many clusters retain hundreds of BHs in relatively low-density BHSubs, with dynamical and core-structural fingerprints visible in their photometric and kinematic data. Moreover, similar scaling applies to diagnosing intermediate-mass black holes versus stellar-mass BHSubs by exploiting surface brightness and velocity dispersion profiles.

4. BHSubs in Young Clusters and Collective Accretion

In young massive clusters, rapid mass segregation following supernova-driven gas loss may result in the prompt formation of a dense BHSub hosting hundreds to thousands of black holes (Kaaz et al., 2020). After the dissipation of stellar feedback ($\gtrsim 50$ Myr), clusters may re-accrete gas, leading to highly enhanced accretion rates onto the BHSub:

$$\dot{M}_\mathrm{BH,ss} = N^\alpha\left(\frac{\rho_c}{\rho_\infty}\right)\dot{M}_\mathrm{BH,\infty},$$

with $N$ the number of BHs and $\alpha$ encoding collective enhancement (potentially approaching unity in efficiently cooled gas). The predicted accretion luminosity,

$L_\mathrm{total}\sim10^{40}\ \mathrm{erg\ s}^{-1},$

can, in principle, enable detection of luminous, unresolved X-ray sources, but observations (e.g., in the Antennae Galaxies) have not yet found such signals, imposing constraints on the BH retention fraction or the efficacy of feedback in these environments.

5. Quantum Subsystems and the Fundamental Structure of Quantum Gravity

Beyond astrophysics, the concept of a BHSub is central to formal discussions of quantum information and the black hole information paradox (Giddings, 2021). In this context, a BHSub is a factor of the Hilbert space—$$\mathcal{H} = \mathcal{H}_\mathrm{BH} \otimes \mathcal{H}_\mathrm{env}$$—whose dynamical interplay with the environment encodes unitarity, information retrieval, and (potentially) nonlocal interactions.

The "black hole theorem" posits that, under three assumptions (existence as a quantum subsystem, identical exterior evolution for different black hole states, and complete evaporation/removal), the evolution is many-to-one and so violates unitarity. Resolving this demands either relaxing the notion of strict subsystem independence or introducing interactions (parametrized corrections to the Hamiltonian, e.g.,

$\Delta H = \int dV\, H^{\mu\nu}(x)\,T_{\mu\nu}(x)$

with $T_{\mu\nu}$ the stress tensor and $H^{\mu\nu}(x)$ state-dependent operators supported near the horizon) that allow the black hole's internal state to affect the exterior. This has direct implications for observable signatures in electromagnetic and gravitational wave channels and establishes a deep connection between subsystem structure and the nature of quantum gravity.

6. Subsystem Entropy, the Page Curve, and Information Paradox

BHSubs are also fundamental in discussions of subsystem entropy and information retention, especially in the context of black hole evaporation. The Page curve describes the average entropy $S(\rho_A)$ of a quantum subsystem $A$ (e.g., the black hole or the Hawking radiation) as a function of its fractional size:

$$\mathbb{E}[ S(\rho_A) ] \approx \begin{cases} \log d_A - \frac{d_A}{2 d_B}, & d_A \leq d_B \ \log d_B - \frac{d_B}{2 d_A}, & d_B < d_A \end{cases}$$

where $d_A$ and $d_B$ are dimensions of the subsystem and its complement (Dahlsten, 29 May 2025). This curve rises to a maximum at the “Page time” (when $d_A \approx d_B$) and then decreases, reflecting the flow of information from the black hole to its outgoing radiation—a central feature of unitary models of black hole evaporation. The mathematical underpinning comes from random matrix theory and the invariance of Haar measure, while the physical significance is that typical subsystems of a larger pure system are nearly maximally entangled until the halfway point.

7. Subsystems, Complexity, and Quantum Measurement in Holography

In the AdS/CFT context, the response of subsystem complexity and entanglement to measurement (projection) sheds light on the interplay between geometric and quantum subsystem structure (Jian et al., 2023). Projection on a complementary region inserts an end-of-the-world (EOW) brane in the dual geometry, fundamentally altering the entanglement wedge and causing abrupt “phase transitions” in complexity as measured by the volume-within-RT-surface (Ryu–Takayanagi) construction:

$C_A = \max \frac{V}{G_N L}$

where changes in the dominant RT surface translate into discrete jumps in complexity of the unmeasured subsystem. For setups where the subsystem of interest contains or is complementary to a black hole, these geometrical transitions directly encode new structure and phase behavior in the complexity and entanglement patterns.

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Through these multiple facets—dynamical, observational, computational, and quantum-information-theoretic—the BHSub paradigm illuminates how black holes, as subsystems within larger systems, mediate energy, encode and transfer information, and serve as fundamental probes of gravitational dynamics and quantum structure.