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 $ρ$ on spacetime or possibly that of an amplitude wave function $ψ$ with $ρ= |ψ|<sup>2$. The proper time flow parameter $s$ is generated collectively by the flow of matter. We show carefully how the $ρ$ 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 $ρ$ merge into a single bump, two colliding wave function $ψ$ bumps of opposite phase merge into a dipole with a different density $|ψ|<sup>2$ 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.

• The paper develops a geodesic-flow framework in which an independently evolved velocity field drives density and amplitude transport, and validates it against particle ensembles with mass conservation errors below 0.002%.
• The authors use regular Kruskal–Szekeres coordinates to model flows crossing the horizon, finding smooth transport, phase-sensitive wave-packet collisions, and horizon behavior that depends on numerical resolution and short-distance cutoffs.
• The paper extends Klein–Gordon flows into the black-hole interior, identifying horizon modes and cutoff-dependent discrete spectra while emphasizing that their physical interpretation requires quantum-gravity regularization and remains unconfirmed.

Kumar and Majid study a notion of geodesic flow originating in noncommutative geometry, applied here to a purely classical Schwarzschild spacetime (2603.03222). The framework departs from standard treatments of geodesics: rather than deriving velocities from particle positions, it treats the geodesic velocity field $X$ as an independent dynamical object obeying its own equation, which then drives the evolution of densities $\rho$ and, optionally, amplitudes $\psi$ with $\rho = |\psi|^2$. The paper develops this formalism on a black-hole background using Kruskal–Szekeres coordinates, validates it against ensembles of actual geodesics, and extends earlier work on Klein-Gordon flows to the interior of the horizon.

The geodesic velocity field as primary object

The central equations are the geodesic velocity equation

$\dot X + \nabla_X X = 0,$

the density flow equation $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$, and the amplitude flow equation $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$, all written with convective derivatives along $X$. Two structural facts organize the theory: along the flow, $|X|^2$ is conserved (so unit-speed fields remain unit speed), and the divergence evolves according to an equation involving $(\nabla_\mu X^\nu)(\nabla_\nu X^\mu)$ and $\rho$0, connecting the formalism to the Ricci tensor's role in free fall. The authors emphasize that the density equation determines $\rho$1 from $\rho$2 but not conversely — the usual ordering of position-then-velocity is inverted.

A practical question is how to choose the initial field $\rho$3, since any unit-speed time-oriented vector field is admissible. The paper's proposal is to require $\rho$4 in the interior of the region of interest, supplemented by boundary data encoding prescribed flux. The authors also discuss the relation to relativistic fluid mechanics: identifying $\rho$5 with the fluid four-velocity $\rho$6 and $\rho$7 with number density $\rho$8 yields a special case where $\rho$9 does not evolve but $\psi$0 obeys a nonlinear evolution driven by pressure gradients; this does not contradict fluid mechanics because no external evolution parameter $\psi$1 appears there. The parameter $\psi$2 itself is interpreted as a collective proper time generated by the matter flow, and justifying this interpretation is one of the paper's stated goals.

Validation against statistical geodesic bunches

To establish that the smooth flow faithfully represents many individual geodesics, the authors sprinkle $\psi$3 dust particles from a Gaussian distribution centered at $\psi$4 in the exterior region ($\psi$5), assign each particle the initial velocity $\psi$6 at its location, evolve them along their own geodesics, and reconstruct interpolated profiles $\psi$7 and a crude tangent-interpolated $\psi$8 via Gaussian kernels. Comparing these against the PDE solutions of the velocity and density flow equations shows a fair match, with differences growing in $\psi$9 and attributed to shape change of the peak that the Gaussian interpolation cannot capture by construction. Mass conservation holds to better than 0.002% up to $\rho = |\psi|^2$0. This supports the interpretation of $\rho = |\psi|^2$1 as a collective proper time and constitutes the proof of concept for the whole framework.

The construction of admissible initial fields is treated carefully. Imposing both Dirichlet conditions on two boundary edges generically produces kink artifacts at the corner, so only one Dirichlet condition is used, with Neumann on another edge. Three families are exhibited: covariantly constant fields along a line of constant $\rho = |\psi|^2$2, the normalized Killing field $\rho = |\psi|^2$3, and the symmetric field $\rho = |\psi|^2$4 in both components, the last being nonsingular at $\rho = |\psi|^2$5 and extendable across regions.

Flows through the horizon and colliding wave packets

Because Kruskal–Szekeres coordinates are regular at the horizon, the formalism can be tested where Schwarzschild coordinates fail. A Gaussian density bump initialized in region IV crosses the horizons smoothly, passing into region I and then region II, with total mass conserved to 0.007%. A close-up approach to the singularity at the corner $\rho = |\psi|^2$6 shows the bump flattening and probability mass leaving the computational domain along the hyperbola $\rho = |\psi|^2$7.

The amplitude-level dynamics produces the paper's most distinctive claim: two colliding Gaussian bumps in $\rho = |\psi|^2$8 merge into a single bump, whereas two $\rho = |\psi|^2$9-bumps of opposite phase collide into a dipole whose $\dot X + \nabla_X X = 0,$0 profile differs qualitatively. The authors argue this provides a potential empirical test of the hypothesis that densities admit underlying spacetime wave functions. They are explicit about the caveat: wave functions over all of spacetime evolving in a collective proper time $\dot X + \nabla_X X = 0,$1 do not correspond to ordinary quantum mechanics, and until the interpretation of $\dot X + \nabla_X X = 0,$2 and of spacetime wave functions is settled, the dipole-versus-bump distinction cannot be promoted to a physical prediction.

The collision computation also exposes a structural limitation of the classical-$\dot X + \nabla_X X = 0,$3 formalism. When two flows share a single velocity field, once two wave functions approximately coincide they must thereafter evolve together; genuine crossing would require either a quantum or tensor-product treatment of $\dot X + \nabla_X X = 0,$4 itself, which the authors defer. Numerically, the two-bump velocity solution develops ripples that grow toward the horizon; the authors demonstrate these are discretization artifacts (they shrink at finer resolution for small $\dot X + \nabla_X X = 0,$5) and apply Gaussian smoothing, while conceding that even after smoothing the residual of the velocity equation remains non-negligible at late times and cannot be reduced by refinement alone.

Klein-Gordon flow and pseudo-quantum mechanics

The second half of the paper turns to quantum geodesic flows on the algebra $\dot X + \nabla_X X = 0,$6 of differential operators, whose Schrödinger picture is the Klein-Gordon flow $\dot X + \nabla_X X = 0,$7. The Heisenberg-picture momentum equations justify $\dot X + \nabla_X X = 0,$8 as proper time at operator level. For static backgrounds, fixing energy with respect to the Killing field $\dot X + \nabla_X X = 0,$9 reduces the flow to "pseudo-quantum mechanics" — Schrödinger-like evolution in $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$0 that is not ordinary quantum mechanics since no single observer's time equals $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$1.

In Kruskal–Szekeres coordinates, fixed-energy modes take the form $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$2 with $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$3, reducing the radial problem to a Schrödinger-like equation in $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$4 with measure $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$5. Reproducing the phenomenon found previously in Schwarzschild coordinates, a dissipating Gaussian approaching the horizon generates "horizon modes" from around $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$6: oscillations of arbitrarily small wavelength bunched at $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$7 that absorb the original packet entirely by $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$8. Wave-function entropy increases throughout, while $\dot\rho + X(\rho) + \rho\,\mathrm{div}(X) = 0$9 moves away from the horizon due to the modes. The same behavior occurs inside the horizon for an interior Gaussian, with mirror-image modes generated on the inner side. The authors state plainly that the apparent penetration of horizon modes across the horizon may be a numerical artifact: the discretization cuts off the infinitely fast oscillations near $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$0, and reliable evolution requires keeping the domain boundary at least roughly $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$1 the resolution away from the horizon. Their conclusion for the continuum theory is that fixed-energy solutions are not differentiable at the horizon, and that resolving the horizon itself requires quantum-gravity-scale regularization — discreteness or noncommutativity — which would cut off the highest frequencies and leave information in a thin "skin" at the horizon.

Stationary modes of the Klein-Gordon operator separate into spherical harmonics times radial functions satisfying an equation analogous to the hydrogen atom, with critical energy $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$2 separating bounded from oscillatory families. Outside the horizon the spectrum is continuous, attributed to the absence of a boundary condition at the horizon itself. Inside, imposing a Neumann condition at a cutoff $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$3 near the horizon and regularity at $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$4 yields a discrete spectrum indexed by the number $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$5 of bulk zero-crossings; the eigenvalues depend only gradually on the cutoff down to the resolution scale. The implication drawn is direct: quantum-gravity corrections modeled as a finite resolution at the horizon determine the internal atomic spectrum, which is otherwise invisible from outside.

Limitations and open questions

The paper is candid about several restrictions. All numerical work is restricted to the radial sector ($$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$6 only); extending to angular variables would be needed, for example, to exhibit a gravitational Bohm–Aharonov effect of a wave function flowing around and interfering with itself. The interpolation of tangent vectors into $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$7 is admittedly crude, since vectors at different points should be parallel transported before averaging. The multivaluedness of $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$8 when trajectories cross remains unresolved within the classical-vector-field formalism. The horizon-mode penetration and the discrete interior spectrum both hinge on cutoff assumptions whose physical content — whether Planck-scale discreteness or noncommutative geometry actually supplies them — is asserted as plausible but not established. Whether the dipole signature of opposite-phase collisions survives once the interpretation of $$\dot\psi + X(\psi) + \tfrac12 \psi\,\mathrm{div}(X) = 0$$9 is clarified is left open.

Conclusion

The paper establishes, via direct comparison with ensembles of geodesics, that the noncommutatively-motivated geodesic flow formalism is a consistent classical tool in which the velocity field is primary and densities or amplitudes are secondary. On the Schwarzschild background it demonstrates horizon-crossing without coordinate pathology, identifies a phase-dependent collision signature distinguishing amplitude flows from density flows, confirms generally covariantly the horizon-mode phenomenon and hydrogen-like stationary states in Kruskal–Szekeres coordinates, and extends both to the black-hole interior, where a cutoff-dependent discrete spectrum emerges. The recurring theme is that behavior at the horizon — mode structure, differentiability, spectral quantization — is controlled by whatever short-distance regularization quantum gravity provides, a dependence the paper makes explicit rather than assuming away.