Geodesic flows on a black-hole background

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.