A Holographic, Hydrodynamic Model of a Schwarzschild Black Hole

TL;DR

This paper introduces a holographic, hydrodynamic model of Schwarzschild black hole interiors, depicting them as a degenerate fluid and deriving novel equations of state and quantum properties through a Schr"odinger-like framework.

Contribution

It presents a new holographic hydrodynamic model of black hole interiors, linking interior fluid dynamics with exterior radiation via a Schr"odinger-like equation and novel equations of state.

Findings

Derived a holographic pressure-mass density relation: P=ρ/9

Estimated total particle count as ~2.8 times horizon quantum areas

Found Fermi energy exceeds Hawking thermal energy, indicating degeneracy

Abstract

Schwarzschild (non-rotating and chargeless) black holes are classically understood to be voids of extreme gravitation. In this study, we propose a holographic model for their interiors, envisioning them instead as a hydrodynamic medium. Motivated by the neutrino composition in Hawking radiation (81%), we model the interior as a degenerate fluid, mirrored by the horizon via AdS/CFT duality. A Schwarzschild metric revised with a signum function as the power of the ratio $r_S/r$ distinguishes interior linear-well dynamics from exterior Schwarzschild geometry, rimming the horizon with singularity-like gravitational attraction. A Hamiltonian analysis of the total action leads to formulating a Schr\"odinger-like equation, which offers an alternative representation as the contracted Einstein field equations under a holographic-hydrodynamic framework. This eventually yields an equation of state between holographic pressure and black hole mass density: $P=\rho/9$. Ideal gas analysis reveals a total particle count of $\sim2.8$ times the number of horizon quantum areas, with the Fermi energy far exceeding the Hawking thermal energy, ensuring degeneracy. As our discussion, we explore the mass shell free-fall model of a BH with holographic pressure, and dissect the spherical wave solutions to the Schr\"odinger-like equation describing confined interior fields and freely propagating exterior quanta (i.e., Hawking radiation).