Back in 1973, Jacob Bekenstein proposed that a black hole has entropy proportional to the area of its horizon. Stephen Hawking later confirmed this with his famous radiation, but the microscopic origin of Bekenstein-Hawking entropy remains mysterious. Two recent independent studies now shed light on the problem by treating the black hole’s interior not as a singularity, but as a dynamic de Sitter region with its own thermodynamics. The question is how to reconcile these pictures and what they tell us about the fundamental nature of gravitational entropy.
The authors perform a theoretical synthesis, comparing two frameworks. The first is a family of regular Schwarzschild solutions with a de Sitter core—so-called OCK geometries—where the interior structure is set by an integer parameter n. The second is the local thermodynamics of a de Sitter vacuum (Volovik’s approach), in which the sign of the Hubble parameter H plays a pivotal role. The inner horizon entropy is computed via the Wald Noether charge formalism, while the total entropy of the de Sitter region is obtained by integrating the local density $s_{vac}$ = 3H/4 over the Hubble volume. The entropy balance is then tracked along a classical collapse sequence driven by a decreasing n.
The central result: the sign of gravitational entropy in a de Sitter region is strictly tied to the sign of the Hubble parameter—expansion (H > 0) yields positive entropy, while contraction (H < 0) gives negative entropy. The total entropy of a homogeneous region can be written as $S_{dS}$ = sgn(H) A/4. In the OCK scenario (n > 2), the core expands and the inner Cauchy horizon carries an extra entropy $S_{inne}$r = πh_c^2. During collapse, as n decreases, the inner horizon shrinks and this positive entropy is gradually released. For an illustrative case with n = 3, the inner horizon radius is 0.7676 times the outer horizon’s radius, and its entropy amounts to 59% of A/4. In Volovik’s scenario, a black hole–gravastar harbors a contracting de Sitter core whose total entropy equals −A/4, exactly canceling the entropy of the outer horizon, so the object’s total entropy is zero. Both scenarios satisfy the generalized second law. In the first, excess entropy is carried off by radiation; in the second, entropy rises because the negative contribution disappears. The classical collapse in OCK terminates with a “Minkowski violation” at n = 0—a quantum threshold.
The synthesis of the two approaches reveals that the Hubble parameter H acts as a thermodynamic order parameter, separating two phases of the de Sitter vacuum—expansion and contraction. Gravitational collapse acquires an arrow of time. Transitions between states are exponentially suppressed by entropy differences, so the system naturally relaxes to a Schwarzschild attractor. The integer nature of n in the OCK solutions points to a quantized entropy spectrum $S_{reg}$(n) = A/4 + πh_c^2(n), and the number of allowed core configurations could provide a logarithmic measure of microstates, S = ln Ω(M).
If an instanton connecting the expanding and contracting de Sitter phases is found, the Phase transition would turn into a genuine quantum phase transition. In that picture, the zero-entropy gravastar would be an intermediate state on the way to a classical black hole. Observational signatures are of special interest: a stepwise release of entropy could produce a sequence of gravitational wave bursts or echoes during black hole mergers. Future research should also incorporate rotation, higher dimensions, and modified theories of gravity.
These findings touch the foundations of quantum gravity, the holographic principle, and the thermodynamics of spacetime, and may spur astrophysical searches for regular black holes and gravastars.
The immediate task is to construct an exact dynamical solution that interpolates between an expanding and a contracting core, and to compute the probability of quantum tunneling through the entropy barrier. It is also essential to rigorously prove the quantization of entropy and to calculate the number of microstates Ω(M) from first principles.
This work directly addresses the microscopic nature of Bekenstein-Hawking entropy by linking it to internal degrees of freedom that circumvent the singularity. It also raises the question of how a quantum transition between de Sitter vacuum phases could occur dynamically, and clarifies the status of gravastars in a complete theory of quantum gravity.
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What the work does
The paper compares two formalisms for the internal structure of black holes: OCK geometry with a de Sitter core and Volovik thermodynamics. The entropy of the inner horizon and the total entropy of the de Sitter region are calculated, showing the connection between the sign of entropy and the sign of the Hubble parameter. The entropy balance during classical collapse is analyzed.
Where its strength is
The Hubble parameter is established as a thermodynamic order parameter that separates phases of the de Sitter vacuum, and it is shown that classical collapse has an arrow of time set by the entropy balance.
Where it could go next
Worth trying
Why this may matter
The proposed synthesis could serve as a basis for a classical understanding of black hole entropy and its connection to cosmology, which is potentially important for quantum gravity.
Nearby works in our archive
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For an astrophysical black hole of 10 M☉, the semiclassical description of collapse breaks down at n ~ 0.01—well before the classical Minkowski violation (n=0). In this regime, the inner horizon’s temperature surpasses the Planck temperature, and the geometry turns into a quantum foam.