Abstract

We propose that black holes can be modeled as information-theoretic stability optimizers: physical systems that maximize entropy while minimizing divergence between interior and exterior information distributions. Using the definition of stability reward as a function of coupling between systems (see Information-Theoretic Stability as Reward Function), we reformulate the laws of black-hole thermodynamics as expressions of informational coupling across the horizon. The event horizon is interpreted as a stability boundary where entropy flux across the surface balances the internal rate of informational compression. This perspective unifies the thermodynamic, holographic, and information-geometric descriptions of black holes and suggests a general stability principle underlying gravitational dynamics.

1. Introduction

The thermodynamics of black holes reveals a deep correspondence between gravitational dynamics and information theory (Bekenstein 1973; Hawking 1975; Bousso 2002). The entropy–area law, the temperature of Hawking radiation, and the holographic principle all imply that black holes organize information in an optimal fashion. Here we interpret these properties through the lens of information-theoretic stability (see Information-Theoretic Stability as Reward Function): stability as a reward function of coupling between systems. The exterior field adapts to the black hole’s geometric constraints; the black hole’s interior adapts to infalling matter. Each side’s stability is a function of its coupling to the other across the horizon.

2. Information-Theoretic Stability

2.1 Definition

Following Information-Theoretic Stability as Reward Function, for any system $A$ adapting to a reference $B$, the stability reward

$$R_s(t) = -\frac{1}{\delta t}\,D_{\mathrm{KL}}(p_{t+\delta}||p_t),$$

is a function of $A$’s coupling to $B$, not an intrinsic property of $A$. At equilibrium $R_s(t)\to0$: the system has fully adapted to its reference and entropy production is balanced by entropy dissipation.

2.2 Extension to Continuous Fields

Let the microstate field of spacetime be represented by a probability density $p(x^\mu,t)$ defined on a 4-volume $\mathcal{V}$ with metric $g_{\mu\nu}$. Then

$$R_s = -\int_{\mathcal{V}} p(x^\mu,t)\, \partial_t \log\!\frac{p(x^\mu,t)}{p(x^\mu,t-\delta)}\, \sqrt{-g}\,d^4x$$

defines the continuous stability functional. Local conservation of probability implies a continuity equation

$$\nabla_\mu J^\mu = 0,\qquad J^\mu = p\,u^\mu,$$

where $u^\mu$ is the four-velocity of informational flow.

3. The Event Horizon as Stability Boundary

3.1 Entropy Flux Balance

Consider a stationary black hole with horizon area $A$. Let $H_{\text{int}}$ and $H_{\text{ext}}$ denote the Shannon entropies of internal and external fields. At the horizon $\mathcal{H}$,

$$\partial_t H_{\text{int}} + \partial_t H_{\text{ext}} = 0,$$

so that entropy flux into the black hole equals entropy flux out via Hawking radiation (Hawking 1975). The horizon thus enforces a global stability constraint:

$$R_s^{(\text{int})} + R_s^{(\text{ext})} = 0.$$

This expresses the first law of black-hole mechanics as informational equilibrium.

3.2 Divergence Minimization

Let $p_{\text{int}}$ and $p_{\text{ext}}$ denote the respective state distributions on either side of $\mathcal{H}$. The horizon minimizes joint divergence

$$D_{\mathrm{joint}} = D_{\mathrm{KL}}(p_{\text{int}}||p_{\text{ext}}) + D_{\mathrm{KL}}(p_{\text{ext}}||p_{\text{int}}),$$

subject to fixed total entropy $S_{\text{tot}}$. At equilibrium, $\nabla D_{\mathrm{joint}}=0$; informational exchange through the horizon reaches stationary balance.

4. Black-Hole Thermodynamics as Stability Optimization

4.1 Entropy–Area Relation

The Bekenstein–Hawking entropy

$$S_{BH} = \frac{k_B c^3 A}{4G\hbar}$$

is interpreted as the maximal entropy compatible with global informational stability. Variation of the horizon area yields

$$\delta S_{BH} = \frac{c^3 k_B}{4G\hbar}\,\delta A = \frac{\delta Q}{T_H},$$

where $T_H$ is Hawking temperature. This expresses thermodynamic work $\delta Q$ as informational flux maintaining stability across the boundary.

4.2 Second Law as Global Stability Increase

The generalized second law (Bekenstein 1973) states $\partial_t(S_{BH}+S_{\text{ext}})\ge0$. In stability terms,

$$\partial_t R_s^{(\text{tot})} \ge 0,$$

so global informational stability cannot decrease. Black holes therefore function as maximum-stability reservoirs for information flows in the universe.

5. Information Preservation and Hawking Radiation

Hawking radiation provides the feedback mechanism restoring global stability when perturbations disturb equilibrium. Each emitted quantum carries away precisely the information necessary to maintain $R_s^{(\text{int})}+R_s^{(\text{ext})}=0$. Apparent information loss arises because external observers measure only partial distributions $p_{\text{ext}}$; in the full joint manifold, divergence is conserved.

6. Holography and Stability Mapping

The holographic principle (’t Hooft 1993; Susskind 1995) asserts that the informational content of a volume is encoded on its boundary. In stability terms, the boundary mapping $\Phi:\mathcal{V}\to\partial\mathcal{V}$ preserves the stability functional:

$$R_s[\mathcal{V}] = R_s[\partial\mathcal{V}],$$

analogous to a holographic duality where bulk and boundary share identical divergence dynamics. This recasts AdS/CFT correspondence (Maldacena 1998) as a geometric equality of stability optimization across dimensions.

7. Cosmological Implications

At cosmological scale, black holes act as attractors for informational instability: they absorb high-divergence regions and return the universe toward maximal entropy subject to global stability conservation. The universe thereby evolves toward configurations that minimize global divergence gradients—a universal tendency consistent with Jaynesian entropy maximization (Jaynes 1957) and thermodynamic self-organization (Crooks 1999; Lloyd 2000).

8. Conclusion

Black holes exemplify the principle of information-theoretic stability optimization. They maintain minimal divergence between interior and exterior informational states, converting local instability into global equilibrium through entropy exchange at the horizon. This view unifies gravitational thermodynamics, quantum information, and statistical inference within a single variational framework, suggesting that the dynamics of spacetime itself instantiate the general stability principle of information.

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