Fermion self-trapping in the optical geometry of Einstein-Dirac solitons

Abstract

We analyze gravitationally localized states of multiple fermions with high angular momenta, in the formalism introduced by Finster, Smoller, and Yau [Phys Rev. D 59, 104020 (1999)]. We show that the resulting solitonlike wave functions can be naturally interpreted in terms of a form of self-trapping, where the fermions become localized on shells the locations of which correspond to those of "bulges" in the optical geometry created by their own energy density.

Nonlinear effects in the excited states of many-fermion Einstein-Dirac solitons

“…Each curve represents a continuous family of solutions parametrized by z, with the fermion energy increasing for each subsequent excited state. At low redshift, the curves approximate the expected nonrelativistic relationship ω ∝ z 1=4 [15], before the relativistic transition occurs at z ∼ 1. This causes the onset of damped oscillatory behavior, with each curve oscillating around the appropriate infinite-redshift "power-law" solution [16].…”

“…The mass-radius relations exhibit spiraling behavior, in common with models of astrophysical phenomena, such as white dwarfs and neutron stars, and there exists a maximum mass analogous to, e.g., the Chandrasekhar limit. At low redshift, these curves approximate the nonrelativistic relation m ∝ R −1=3 [15], before spiraling towards their respective infinite-redshift solutions.…”

“…First, note the behavior of the ground state curve: as N f is increased, the amplitude of the oscillations becomes larger, and the first few minima develop into sharp points. We have previously interpreted this behavior by way of a fermion self-trapping effect (see [15] for details), where these sharp points correspond to the sudden appearance of new trapping regions in the solutions corresponding to those redshift values.…”

“…In comparison with the two-fermion case, these oscillations are of much larger amplitude, so much so that the first minimum in the fermion fields is very close to zero. We have previously demonstrated [15] that these oscillations can be interpreted in terms of a fermion self-trapping effect, with the positions of the fermion field peaks corresponding to the locations of stable null circular geodesics (photon spheres) in the soliton spacetime. The fermion field minima occur at the locations of the accompanying unstable photon spheres, from which the fermions are effectively repelled.…”

“…Comparison between the fermionic and bosonic cases is presented in [12,13], while axisymmetric solutions corresponding to single fermion states have recently been found [14]. Ground state solutions with large numbers of fermions have also been analyzed by the current authors [15] and their structure interpreted in the form of a fermion self-trapping effect. In this paper, we shall consider the behavior of the corresponding excited states.…”