Newtonian gravity and Coulomb’s law rely on the concept of point-like entities, finite mass or charge confined to zero volume, which implies infinite fields at their locations. While these theories form the backbone of classical physics and remain accurate for everyday scales, they clash with quantum mechanics. Heisenberg’s uncertainty principle prohibits particles from having precise positions and momenta; instead, they exist as probability distributions in phase space.

The point-particle assumption pervades classical theories, from electromagnetism to general relativity (GR), but fails in extreme regimes. For example, electrons bound to atomic nuclei or neutrons compressed into stars defy classical descriptions. The collapse of such systems into black holes, often modeled as singularities, highlights the inadequacy of point-like idealizations.

This paper builds on efforts to resolve these limitations. Previous work eliminated the Coulomb singularity of the Dirac electron by coupling its wave function to its own electric field, yielding a distributed charge density and correct magnetic properties without invoking advanced quantum field theory. For GR, we argue that the classical energy-momentum tensor, built from singular point particles, must be reimagined quantum-mechanically. Rather than ad hoc fixes, we propose replacing each point particle with a wave-like field (e.g., a Klein-Gordon or Yukawa-type solution), ensuring consistency with quantum uncertainty.

Our approach differs from past attempts to address singularities. Some studies explored self-gravitating quantum systems, finding nonsingular spacetimes; others modified gravity by introducing auxiliary fields. A separate line of research investigated exotic compact objects like boson stars. Crucially, these works often imposed quantum or classical corrections externally. In contrast, we emphasize that particles are fundamentally extended wave packets, not mathematical points, and their quantum nature should directly shape spacetime.

Unlike models of "regular" black holes that smooth singularities by altering matter fields, our framework treats quantum extension as intrinsic. By embedding wave-like particles into GR’s foundations, we unify spacetime dynamics with quantum mechanics, bridging a gap that has persisted since the birth of modern physics.

Based on: Remove point-mass concept-remove singularities from GR

Yousef Sobouti and Haidar Sheikhahmadi