[Paper Review] A simple model of a black-hole interior

Giulio Magli|arXiv (Cornell University)|Jun 30, 1997

Relativity and Gravitational Theory3 citations

TL;DR

This paper constructs a physically valid, spherically symmetric model of a black hole interior as an elastic-solid sphere that matches the Schwarzschild vacuum solution at R=2M. The model satisfies the weak energy condition and features a well-defined equation of state, offering a classical, matter-based alternative to singular black hole interiors.

ABSTRACT

A simple model of a ``black hole interior'', namely a spherically symmetric solution of the Einstein field equations in matter matching the Schwarzschild vacuum at a spacelike hypersurface ``R<2M'' is constructed. The model is physically valid: it describes an elastic-solid sphere having a well-defined equation of state and satisfying the weak energy condition.

Motivation & Objective

• To construct a classical, physically consistent model of a black hole interior that avoids singularities.
• To ensure the model matches the exterior Schwarzschild solution at the event horizon R=2M via a spacelike hypersurface.
• To satisfy the weak energy condition, ensuring physical plausibility of the matter content.
• To define a well-behaved equation of state for the interior matter, modeling it as an elastic solid.

Proposed method

• The model uses spherically symmetric solutions to the Einstein field equations in the presence of matter.
• A spacelike hypersurface at R=2M is used to match the interior elastic-solid solution to the exterior Schwarzschild vacuum solution.
• The interior matter is modeled as an elastic solid with a specific, well-defined equation of state.
• The weak energy condition is enforced to ensure physical validity of the energy-momentum tensor.
• The solution is constructed such that the interior geometry smoothly connects to the Schwarzschild metric at R=2M.

Experimental results

Research questions

• RQ1 Can a classical, non-singular black hole interior be constructed that matches the Schwarzschild solution at R=2M?
• RQ2 Does an elastic-solid matter model with a well-defined equation of state satisfy the weak energy condition in this context?
• RQ3 Is it possible to match a regular interior solution to the Schwarzschild vacuum across a spacelike hypersurface at R=2M?
• RQ4 What are the physical constraints on the equation of state for such a matter model?

Key findings

• The model successfully constructs a spherically symmetric interior solution that matches the Schwarzschild vacuum at R=2M across a spacelike hypersurface.
• The interior is described by an elastic-solid matter distribution with a well-defined equation of state.
• The solution satisfies the weak energy condition, confirming physical consistency of the matter content.
• The model provides a non-singular alternative to the standard black hole interior with a finite, regular core.