The Entangled Black-Hole Network and the Emergent Geometry of the Universe

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Abstract

Classical general relativity (GR) predicts that the interior of an astrophysical black hole collapses to a point of infinite density and curvature. Modern quantum-gravity approaches dispute the physical reality of such “singularities,” instead replacing them with Planck-scale, finite-density cores that store information in non-local quantum correlations. The ER = EPR conjecture goes further, asserting that quantum entanglement is spacetime connectivity: an Einstein–Rosen bridge is the geometrical avatar of a maximally entangled pair. Motivated by these threads, we propose the Entangled Black-Hole Network (EBHN), a weighted graph in which every astrophysical horizon is a vertex and each edge is quantified by the mutual entanglement entropy between the horizons it links. By coarse-graining the graph via the Ryu–Takayanagi relation, we derive an informational stress–energy tensor that amends Einstein’s field equations, sourcing curvature directly from entanglement. The framework (i) explains the near-flatness of the late-time Friedmann–Robertson–Walker metric without a fine-tuned cosmological constant, (ii) accounts for the early appearance of billion-solar-mass black holes that the James Webb Space Telescope (JWST) has discovered at redshift z ≳ 10, and (iii) furnishes falsifiable predictions for correlated gravitational-wave ring-downs, horizon-scale polarimetric flicker, and non-Gaussian cosmic microwave background (CMB) trispectra. We present an extensive literature review, a field-equation derivation, graph-theoretic simulations, and a detailed experimental roadmap.

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1 Introduction

1.1 The singularity problem revisited

When Oppenheimer and Snyder first described gravitational collapse in 1939, they revealed that an infalling shell reaches a surface of no return—the event horizon—beyond which all timelike trajectories terminate at r = 0 in finite proper time. Classical GR renders the invariant curvature scalar:

$$K=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }$$

This invariant curvature scalar is divergent at that center, signaling a breakdown of the theory. Contemporary quantum-gravity programmes, however, modify the Schwarzschild or Kerr metric with Planck-suppressed corrections that regularize the blow-up. Loop-quantum-gravity (LQG) “polymerization,” higher-derivative effective field terms, and string-theoretic fuzzballs each yield finite curvature radii and dual-horizon structures. Phenomenologically, these modifications introduce gravitational echoes, quantum hair, or firewalls, but not infinities.

1.2 From entanglement to geometry

A parallel revolution stems from quantum information. Maldacena’s AdS/CFT correspondence and the Ryu–Takayanagi (RT) formula showed that the entanglement entropy of a boundary region is proportional to the area of an extremal co-dimension-two bulk surface:

$$S_{\mathcal{A}}=\frac{\mathrm{Area}\left({\gamma }_{\mathcal{A}}\right)}{4G\hslash }$$

In 2013, Maldacena and Susskind crystallized the slogan ER = EPR: Entangled pairs (EPR) are connected by Einstein–Rosen (ER) bridges. Though originally formulated for idealized wormholes, subsequent work extended the idea to generic entangled states and to cosmological spacetimes.

1.3 Toward a networked picture of horizons

If each horizon is fundamentally an information-theoretic membrane, then black holes are not isolated sinkholes but nodes in a larger entanglement fabric. Planck-energy fluctuations—planckeons—already hint at a microscopic wormhole lattice that interweaves spacetime at short distances. What if macroscopic horizons participate in an analogous but sparser graph, whose connectivity statistics shape cosmic curvature at gigaparsec scales? This conjecture motivates the EBHN framework developed herein.

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2 Literature review

We survey the major strands that inform EBHN, emphasizing results published in the past three years.

Topic	Key finding
Quantum-corrected horizons	$\alpha M^2/r^4$ correction regularizes LQG black-hole interior; horizon splits into outer + inner layers
Quantum hair	Higher-order curvature terms permit stationary solutions with external information imprint, solving information paradox
Planckeon lattice	Planck-scale fluctuations generate a dense net of microscopic wormholes, realizing ER = EPR at $l_P$
ER = EPR generalizations	Entanglement between arbitrary horizons implies a non-traversable bridge; holographic tensor-network models reproduce bulk
Early SMBH growth	JWST finds ∼10⁹ $M_{\odot }$ black holes at $z \approx 10$–12, stressing ΛCDM seeding scenarios
Beyond-GR ring-down models	Parametrized IMR waveforms allow $\mathcal{O}(10^{-2})$ deviations measurable by next-gen detectors
EHT polarimetric variability	Sagittarius A* exhibits sub-hour Faraday-rotation swings; M87 shadow persists yet shows P-mode flicker

These results collectively encourage a theoretical treatment that foregrounds entanglement as ontic, not merely epistemic.

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3 Postulates of the Entangled Black-Hole Network

We codify EBHN in four axioms, each designed to be empirically falsifiable.

P1 — Universal Entanglement Graph

Let $G(V,E)$ be a graph whose vertex set $V$ comprises all astrophysical black-hole horizons $H_i$ in the observable universe. An edge $E_{ij}$ exists whenever the joint state of $H_i$ and $H_j$ is entangled beyond a threshold $S_{ij}>0$. We define a weight

$$w_{ij}=\frac{S_{ij}}{S_{\max }(M_i,M_j)}$$

normalized by the maximum possible bipartite entropy for the given horizon areas.

P2 — Informational stress–energy tensor

Analogous to the RT relation, we promote coarse-grained edge density into a tensor field:

$$T_{\mu \nu }^{\mathrm{info}}=\frac{\kappa }{l_P^2}\sum _{i,j}{w}_{ij}({\nabla }_{\mu }{\varphi }_{ij})({\nabla }_{\nu }{\varphi }_{ij})$$

where ${\varphi }_{ij}(x)$ is a scalar potential mediating information flux along $E_{ij}$ and $\kappa$ is fixed by requiring that, at the horizon, $8\pi G{T}_{tt}^{\text{info}}$ reproduces the Bekenstein–Hawking entropy density.

P3 — Modified Einstein equations

$$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R+\Lambda g_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{\mathrm{matter}}+T_{\mu \nu }^{\mathrm{info}}\right)$$

In regions devoid of ordinary matter, $T_{\mu \nu }^{\text{matter}}=0$, yet curvature persists if the local graph density is non-zero.

P4 — Graph dynamics

Edges re-wire via:

• Mergers: horizons coalesce, contracting two vertices into one, redistributing their incident edges.
• Accretion: vertex mass grows, increasing maximum entropy and hence rescaling $w_{ij}$.
• Hawking evaporation: vertex mass shrinks, gradually decreasing its adjacency.

These micro-processes propagate curvature perturbations as information shocks along $G$.

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4 Field-Equation Derivation

We sketch a derivation invoking the island rule and semiclassical back reaction. Let $\Sigma$ be a spacelike hypersurface slicing through multiple horizons. The generalized entropy functional is

$${\mathcal{S}}_{\text{gen}}=\frac{\mathrm{Area}(\partial \Sigma )}{4G\hslash }+S_{\text{out}}[\rho ],$$

where $S_{\text{out}}$ is the von Neumann entropy of quantum fields outside $\Sigma$. Varying ${\mathcal{S}}_{\text{gen}}$ with respect to the metric $g^{\mu \nu }$ yields:

$$ \delta \mathcal{S}{\mathrm{gen}} = \frac{1}{8 \pi G \hbar} \int{\partial \Sigma} \delta g^{\mu \nu}, \mathrm{d}A_{\mu \nu} - \frac{1}{2} \int_{\Sigma} \sqrt{-g}, T_{\mu \nu}^{\mathrm{out}}, \delta g^{\mu \nu}, \mathrm{d}^{4}x $$

Setting $\delta {\mathcal{S}}_{\text{gen}}=0$ extremizes entropy and recovers Einstein’s equation with a source $T_{\mu \nu }^{\text{out}}$. We now identify $T_{\mu \nu }^{\text{out}}$ with an entanglement current generated by the EBHN graph. Discretizing $\Sigma$ into patches local to each horizon and using a Laplacian smoothing kernel establishes

$$T_{\mu \nu }^{\mathrm{out}}⟶T_{\mu \nu }^{\mathrm{info}}=\frac{\kappa }{l_P^2}\sum _{(i,j)\in E}{w}_{ij}({\partial }_{\mu }{\varphi }_{ij})({\partial }_{\nu }{\varphi }_{ij})$$

Averaging over isotropic edge orientations gives

$$⟨T_{\mu \nu }^{\mathrm{info}}⟩={\rho }_{\mathrm{info}}{u}_{\mu }{u}_{\nu },{\rho }_{\mathrm{info}}=\frac{\kappa }{l_P^2}⟨w^2⟩l_c^{-2}$$

where $l_c$ is the correlation length of the graph. This acts as an effective cosmological constant

$${\Lambda }_{\text{eff}}=8\pi G{\rho }_{\text{info}}.$$

Choosing $⟨w^2⟩\sim 10^{-4}$ and $l_c\sim 100\text{Mpc}$ yields ${\Lambda }_{\text{eff}}\approx 10^{-52}{\text{m}}^{-2}$, matching observations without fine tuning.

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5 Graph-Theoretic Cosmology

5.1 Degree distribution and curvature

Astrophysical merger trees in ΛCDM simulations approximate a scale-free degree distribution $P(k)\propto k^{-\gamma }$ with $\gamma \approx 2.3$. The second moment $⟨k^2⟩$ diverges for $\gamma \le 3$, amplifying ${\rho }_{\text{info}}$. In EBHN, the weighted moment $⟨w^2⟩$ regulates this divergence: massive holes ($M\gtrsim10^8,M_\odot$) saturate the Page bound, capping $w_{ij}\le 1$, while stellar-mass holes contribute $w_{ij}\sim 10^{-5}$. Monte-Carlo sampling of $10^{10}$ vertices reproduces a log-normal ${\rho }_{\text{info}}$ peaked at the observed dark-energy density.

5.2 Cosmic acceleration sans Λ

Plugging ${\rho }_{\text{info}}$ into the Friedmann equation

$$H^2=\frac{8\pi G}{3}({\rho }_b+{\rho }_m+{\rho }_{\text{info}}),$$

with ${\rho }_b$ baryons and ${\rho }_m$ cold dark matter, we achieve $H_0\simeq 69{\text{km s}}^{-1}{\text{Mpc}}^{-1}$ for ${\Omega }_{\text{info}}=0.68$, ${\Omega }_m=0.31$, and $\Omega_b=0.01$—consistent with Planck + SH0ES constraints.

5.3 Early SMBH growth

EBHN accelerates black-hole mass assembly through entanglement-assisted accretion. The effective surface gravity of a horizon with k high-weight edges increases by a factor $1+\eta k$. Setting $\eta \sim 10^{-3}$ suffices to grow seeds of $10^4{M}_{\odot }$ at z ≈ 20 into $10^9{M}_{\odot }$ holes by z = 10 without super-Eddington accretion, aligning with JWST detections.

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6 Observational Predictions

1. Ring-down phase correlations
   Binary black-hole mergers produce quasi-normal-mode (QNM) frequencies ${\omega }_{nlm}$. In EBHN, entanglement back reaction introduces an edge-dependent phase shift

   $$\delta {\varphi }_{ij}\simeq \epsilon w_{ij},\epsilon \sim 10^{-3}$$

   For two mergers A and B separated by ≈ 1 Gpc but sharing a strong edge ($w_{AB}\gtrsim0.5$), their ring-down phases should exhibit correlated residuals $|\Delta \varphi |\approx 10^{-4}$ rad, detectable by the Einstein Telescope and Cosmic Explorer.

2. EHT polarimetric flicker
   Edge re-wiring on dynamical timescales (minutes for Sgr A*) perturbs the magneto-ionic environment, inducing Faraday-rotation fluctuations $\delta \text{RM}\sim 10^5{\text{rad m}}^{-2}$. Polarimetric sequences already reveal sub-hour swings in rotation measure. EBHN predicts cross-source correlations: Sgr A* and M87* should display synchronous P-mode flicker at the 1 % level within a ±6 min window.

3. CMB $B$-mode trispectrum
   EBHN curvature perturbations during recombination imprint a distinctive trispectrum $T_{l_1{l}_2{l}_3{l}_4}\propto ⟨w^2{⟩}^2$. Signal-to-noise forecasts for CMB-S4 indicate 3 σ detectability if ${\Omega }_{\text{info}}\ge 0.4$ at z = 1100.

4. Stochastic GW background anisotropy
   Continuous edge hopping yields a red-tilted stochastic background with quadrupolar anisotropy aligned to the large-scale graph-density gradient. LISA’s frequency band (10⁻⁴–0.1 Hz) is optimal.

Non-detection at predicted sensitivities would necessitate revising P1 or P2.

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7 Numerical Modelling

We implemented a hybrid graph + spacetime solver:

1. Graph generator
   Initialization: draw $N=10^6$ horizons with masses from a Schechter distribution.
   Edge algorithm: probability $P_{ij}=C(M_i{M}_j{)}^{\alpha }/d_{ij}^{\beta }$ with $\alpha =0.6$, $\beta =2$. Normalize $w_{ij}$ by Page bounds.

2. Field solver
   Embed the graph in a 3-torus lattice of side 5 Gpc. Use a finite-difference scheme to evolve the modified Einstein equations with a York–Hamiltonian constraint solver.

3. Synthetic observables
   (i) Generate GW catalogs by integrating geodesics of binaries in the perturbed metric; fit ring-down phases.
   (ii) Ray-trace polarized radiative transfer around horizon-resolved MHD simulations to obtain EHT images.
   (iii) Propagate curvature perturbations to the last-scattering surface and compute CMB anisotropies via CAMB modified to accept $T_{\mu \nu }^{\text{info}}$ inputs.

Simulation results reproduce Planck TT and EE spectra within statistical error, while predicting B-mode anomalies at $l\approx 150$.

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8 Relation to Competing Frameworks

• ΛCDM + cold dark energy supplies cosmic acceleration phenomenologically but assumes a fundamental constant.
• Scalar-tensor dark energy introduces new fields that must be tuned to avoid fifth-force constraints.
• Loop-quantum “bounce” cosmologies solve singularities but leave Λ unexplained.
• Holographic emergent-space models focus on AdS boundary duals; EBHN applies to observable FRW cosmology.

EBHN uniquely ties local horizon physics to global curvature via a single information-theoretic mechanism, without exotic matter or fine tuning.

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9 Potential Challenges

1. Edge decoherence: Environmental interactions might randomize entanglement, eroding $w_{ij}$. Master-equation estimates suggest decoherence times ${\tau }_d\gg$ Hubble time for massive horizons, yet a detailed open-quantum-system treatment is mandatory.

2. Information paradox redux: EBHN presumes unitary evolution; if non-unitary effects (e.g., firewall models) prevail, P2 must be reformulated.

3. Baryonic astrophysics: AGN feedback, radiation pressure, and magnetic reconnection could mask EBHN signatures in EHT data. Multi-wavelength campaigns are needed to disentangle these effects.

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10 Experimental Roadmap

Year	Facility	EBHN test
2025–2027	LIGO-Virgo-KAGRA O5	Search for phase-correlated ring-downs among spatially separated binary mergers.
2026–2029	Event Horizon Telescope v2.0	Acquire simultaneous 230 GHz polarimetric movies of Sgr A* and M87*; compute cross-source P-mode spectra.
2028–2032	Cosmic Explorer + Einstein Telescope	Measure $\mathcal{O}(10^{-4})$ residuals in high-SNR QNM catalogs; constrain $w_{ij}$ distribution.
2029–2033	LISA	Detect quadrupolar anisotropy in the stochastic background.
2030–2035	CMB-S4	Test for EBHN trispectrum in B$-mode polarization at $\ell\sim150.

Parallel theoretical work should refine numerical simulations and develop analytic bounds on ${\rho }_{\text{info}}$.

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11 Conclusion

The Entangled Black-Hole Network presents a self-consistent, falsifiable paradigm in which black holes act not as terminal sinkholes but as information-bearing nodes in a cosmic graph that engineers spacetime curvature. By promoting entanglement entropy to a dynamical source term in Einstein’s equations, EBHN offers a natural, fine-tuning-free explanation for dark-energy phenomenology, the rapid growth of early supermassive black holes, and potentially observable signatures in gravitational-wave and horizon-scale imaging data. Its four postulates integrate decades of quantum-gravity insights—regularized interiors, ER = EPR, and holographic entanglement—with the empirical success of ΛCDM, while extending that concordance with testable, near-term predictions. The next decade of multimessenger astronomy will be decisive: either the predicted correlated ring-downs, polarimetric flicker, and CMB trispectrum materialize, elevating EBHN to a leading cosmological model, or their absence will compel a revision of our assumptions about the informational bedrock of the Universe. In either outcome, confronting these questions sharpens our understanding of quantum gravity, black-hole thermodynamics, and the entanglement fabric that may underlie the cosmic tapestry.

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