Fresh look at the effects of gravitational tidal forces on a freely-falling quantum particle

TL;DR

This paper investigates how tidal gravitational forces influence a quantum particle in free fall, deriving a relativistic Schrödinger equation and analyzing the effects through different boundary conditions, with implications for the equivalence principle.

Contribution

It derives a relativistic Schrödinger equation from the Klein-Gordon equation to analyze tidal effects on a quantum particle in free fall, comparing relativistic and tidal-force terms.

Findings

Tidal forces induce harmonic and inverted harmonic oscillator potentials.

Energy spectra are quantized and depend on boundary conditions.

Tidal effects are mass-independent and relate to the equivalence principle.

Abstract

We take a closer and new look at the effects of tidal forces on the free fall of a quantum particle inside a spherically symmetric gravitational field. We derive the corresponding Schr\"odinger equation for the particle by starting from the fully relativistic Klein-Gordon equation in order (i) to briefly discuss the issue of the equivalence principle and (ii) to be able to compare the relativistic terms in the equation to the tidal-force terms. To the second order of the nonrelativistic approximation, the resulting Schr\"odinger equation is that of a simple harmonic oscillator in the horizontal direction and that of an inverted harmonic oscillator in the vertical direction. Two methods are used for solving the equation in the vertical direction. The first method is based on a fixed boundary condition, and yields a discrete-energy spectrum with a wavefunction that is asymptotic to that of a particle in a linear gravitational field. The second method is based on time-varying boundary conditions and yields a quantized-energy spectrum that is decaying in time. Moving on to a freely-falling reference frame, we derive the corresponding time-dependent energy spectrum. The effects of tidal forces yield an expectation value for the Hamiltonian and a relative change in time of a wavepacket's width that are mass-independent. The equivalence principle, which we understand here as the empirical equivalence between gravitation and inertia, is discussed based on these various results. For completeness, we briefly discuss the consequences expected to be obtained for a Bose-Einstein condensate or a superfluid in free fall using the nonlinear Gross-Pitaevskii equation.