Abstract & Overview
The issue of defining the volume of black holes has significant implications for quantum gravity. Drawing on concepts from quantum theory and general relativity, several motivations for introducing discreteness in geometry can be proposed. However, to seriously consider any proposal for a discrete geometry, the identification problem and the challenge of defining a distance function within such a geometry must be addressed. This paper proposes the faithful embedding of sets representing spaces in finite geometry—a specific type of discrete geometry characterized by a finite set of points—into Riemannian manifolds as a solution to these problems. Similar to a classical measuring apparatus that interprets and understands quantum results in classical terms, classical geometry serves as a bridge between the discreteness of the physical world and our continuous understanding of the properties of space. In this framework, the volumetric density of information contained within a black hole is established, providing a consistent volume for the Schwarzschild black hole observed by all observers. Furthermore, the study finds that the minimum volume of the Schwarzschild black hole is non-zero. This fact implies that a black hole can only evaporate until its event horizon radius reaches the Planck length, signifying that black hole remnants are stable. Consequently, the total collapse of a black hole is prevented by the finite nature of the geometry describing physical space.