Year: 2026
Type: article
Abstract: We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold M as part of the construction of quantum geodesics on the algebra $${\mathcal {D}}(M)$$ D ( M ) of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first-order differential calculus on this algebra in the presence of curvature. The corresponding Schrödinger picture has wave functions on spacetime and proper time evolution by the Klein–Gordon operator, with stationary modes being solutions of the Klein–Gordon equation. As an application, we describe gravatom solutions of the Klein–Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon.
Source: Letters in Mathematical Physics
Authors: Edwin Beggs, Shahn Majid
Institutions: Swansea University, Queen Mary University of London
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Funder: Leverhulme Trust
Award: G6131219727
OpenAlex ID: W7123336272