Geodesic flows on a black-hole background

TL;DR

This paper explores a novel approach to geodesic flows in Schwarzschild spacetime, linking classical and quantum concepts, and investigates how matter density and wave functions behave near black hole horizons, with implications for quantum gravity.

Contribution

It introduces a new framework for geodesic flows based on noncommutative geometry, analyzing density and wave function dynamics in black-hole backgrounds, including horizon crossing and quantum effects.

Findings

Density bumps merge into a single bump upon collision.

Opposite phase wave function bumps merge into a dipole.

Horizon modes and atomic states persist inside the black hole, influenced by quantum gravity.

Abstract

A recent notion of geodesic flows which comes out of noncommutative geometry but which is also novel in the classical case is studied in detail for a Schwarzschild spacetime. In this framework, the geodesic velocity field is an independent concept which then defines the flow of a density $\rho$ on spacetime or possibly that of an amplitude wave function $\psi$ with $\rho = |\psi|^2$. The proper time flow parameter $s$ is generated collectively by the flow of matter. We show carefully how the $\rho$ evolution can be justified as modelling a large number of geodesics interpolated as a local density. Using Kruskal-Szekeres coordinates, we show that there are no issues crossing the horizon. A novel feature is that whereas two colliding Gaussian bumps in density $\rho$ merge into a single bump, two colliding wave function $\psi$ bumps of opposite phase merge into a dipole with a different density $|\psi|^2$ profile, providing a potential test of our wave-function hypothesis. We also revisit the Klein-Gordon flow or pseudo-quantum mechanics around a black-hole and find that previously found black-hole atom states and modes generated at the horizon when an area of disturbance approaches it are also present inside the black-hole in a reflected fashion. We argue that the behaviour of the horizon modes across the horizon as well as discretisation of the atomic spectrum depend on quantum gravity corrections at the horizon.