Surface Tension and Negative Pressure Interior of a Non-Singular `Black Hole'

Abstract: The constant density interior Schwarzschild solution for a static, spherically symmetric collapsed star has a divergent pressure when its radius $R\le\frac{9}{8}R_s=\frac{9}{4}GM$. We show that this divergence is integrable, and induces a non-isotropic transverse stress with a finite redshifted surface tension on a spherical surface of radius $R_0=3R\sqrt{1-\frac{8}{9}\frac{R}{R_s}}$. For $r < R_0$ the interior Schwarzschild solution exhibits negative pressure. When $R=R_s$, the surface is localized at the Schwarzschild radius itself, $R_0=R_s$, and the solution has constant negative pressure $p =-\bar\rho$ everywhere in the interior $r<R_s$, thereby describing a gravitational condensate star, a fully collapsed non-singular state already inherent in and predicted by classical General Relativity. The redshifted surface tension of the condensate star surface is given by $\tau_s=\Delta\kappa/8\pi G$, where $\Delta\kappa=\kappa_+-\kappa_-=2\kappa_+=1/R_s$ is the difference of equal and opposite surface gravities between the exterior and interior Schwarzschild solutions. The First Law, $dM=dE_v+\tau_s dA$ is recognized as a purely mechanical classical relation at zero temperature and zero entropy, describing the volume energy and surface energy change respectively. Since there is no event horizon, the Schwarzschild time t of such a non-singular gravitational condensate star is a global time, fully consistent with unitary time evolution in quantum theory. The $p=-\bar\rho$ interior acts as a defocusing lens for light passing through the condensate, leading to imaging characteristics distinguishable from a classical black hole. A further observational test of gravitational condensate stars with a physical surface vs. black holes is the discrete surface modes of oscillation which should be detectable by their gravitational wave signatures.

Overview of Surface Tension and Negative Pressure Interior of a Non-Singular 'Black Hole'

The paper by Pawel O. Mazur and Emil Mottola proposes a novel interpretation of the constant density interior Schwarzschild solution for a static, spherically symmetric collapsed star circumventing the conventional view of black hole singularity formation. The authors explore the mathematical intricacies of general relativity to explore the physical implications of a configuration where the pressure diverges in the interior solution. This divergence has traditionally been disregarded, yet Mazur and Mottola argue that it leads to the formation of non-isotropic transverse stress that includes a finite surface tension.

Key Findings:

1. Integrable Pressure Divergence: The authors reveal that the divergence in pressure, typically expected to approach infinity within the star, is integrable. This integrability results in a non-isotropic pressure, where transverse stress differs significantly from radial pressure.
2. Non-singular Configuration: By allowing the pressure to be negative in certain regions, the paper challenges traditional singularity theorems based on the strong energy condition. The condition $\rho + \sum_{i=1}^3 p_i \ge 0$, significant in proving singularity formation, is circumvented, predicting a negative pressure interior that acts as a defocusing lens.
3. Surface Tension: The mathematical treatment leads to the derivation of a finite surface tension at the interface of two solutions. This surface tension is proposed as the signature of a gravitational condensate star, offering a coherent stationary spacetime without event horizons.
4. Mechanics of Collapse: Extending classical general relativity mechanics, the authors recognize a purely mechanical classical relationship governing energy changes. This involves volume and surface contributions, illuminating a path divergent from the thermodynamic view initiated by black hole mechanics.
5. Physical Characteristics: The unique characteristics of gravitational condensate stars are highlighted, suggesting that these entities might allow light to pass through the interior rather than being absorbed. This defocusing effect introduces distinctive imaging properties unlike classical black holes.

Implications:

The proposed gravitational condensate star offers a compelling alternative to the singularity model in gravitational collapse. It challenges the inevitability of singularities by considering sufficient pressures that create defocusing lenses, effectively reshaping our understanding of spacetime under extreme gravity conditions. This interpretation can pave the way for future developments in astrophysics and quantum gravity, where standard expectations rooted in singularities might be replaced by entities maintaining coherence and causal structure in quantum theory.

The model suggests advances in astrophysical observations, proposing testable differences between gravastars and black holes based on lensing effects and gravitational wave signatures emanating from surface tensions and dynamics.

In conclusion, this research opens the avenue to speculate on the next developments in AI and computational physics simulations, where such theoretical frameworks can be rigorously tested. As computing power escalates, reaching the precision necessary to differentiate gravitational wave signals and optical images characteristic of these non-singular configurations, it will provide further evidence to challenge or corroborate this intriguing theory.