Gravitational eigenstates in weak gravity: I. Dipole decay rates of charged particles

Abstract

The experimental demonstration that neutrons can reside in gravitational quantum stationary states formed in the gravitational field of the Earth indicates a need to examine in more detail the general theoretical properties of gravitational eigenstates. Despite the almost universal study of quantum theory applied to atomic and molecular states very little work has been done to investigate the properties of the hypothetical stationary states that should exist in similar types of gravitational central potential wells, particularly those with large quantum numbers. In this first of a series of papers, we attempt to address this shortfall by developing analytic, non-integral expressions for the electromagnetic dipole state-tostate transition rates of charged particles for any given initial and final gravitational quantum states. The expressions are non-relativistic and hence valid provided the eigenstate wavefunctions do not extend significantly into regions of strong gravity. The formulae may be used to obtain tractable approximations to the transition rates that can be used to give general trends associated with certain types of transitions. Surprisingly, we find that some of the high angular momentum eigenstates have extremely long lifetimes and a resulting stability that belies the multitude of channels available for state decay.

Gravitational Quantisation and Dark Matter

“…Very little work has been done however to investigate the potential existence or theoretical properties of gravitational eigenstates and their consequences for astrophysics (Ernest 2009a(Ernest , 2009b. The simplest gravitational eigenstate system would feasibly consist of two neutral, spin-zero elementary-type particles in a bound state.…”

“…The analysis for this has been done elsewhere and we give a brief summary here. (For more detail see Ernest, 2009a.) The gravitational form of Schrodinger's equation for a two particle system is written as…”

“…This hybrid 1/rlogarithmic potential differs from the equivalent point-mass, 1/r point potential substantially at low r although in principle the characteristics of the high-l, n quantum states of 1/r potentials that make them suitable as fundamental components of dark matter eigenspectral arrays will have equivalent states in the eigenstate ensembles of the real logarithmic halo potentials. In the central point-mass potential case it was possible (Ernest 2009a(Ernest , 2009b) to obtain approximations for the energies and wavefunction forms of the eigenstates which can be used to obtain quantitative values for interaction rates etc. It is clearly important to be able to develop approximations for similar states in the real logarithmic potentials of the galactic halo case which we now do below.…”

“…A short account of the possible qualitative model of the formation of dark halos was given by Ernest (2006) and we summarise that approach here. There is still considerable work to be done in developing and testing the validity of the approach but the qualitative scenario does provide a general basis for how formation would need to proceed.…”

“…(Mass captured in the halos is sequentially removed from the more rapidly expanding, necessarily under-density, midhalo regions as expansion proceeds.) Interestingly however, from Carr's black hole number density-mass spectrum equation, it can be calculated that the value th r takes for the separation of such adjacent halos at the formation time translates, after universal expansion, into the order of magnitude for the number density of present day galaxies (Ernest, 2006).…”