Stationary solutions of second-order equations for point fermions in the Schwarzschild gravitational field

TL;DR

This paper proves the existence of stationary zero-energy states for spin-1/2 particles in Schwarzschild fields using a second-order Schrödinger-type equation, with implications for dark matter models involving fermions in gravitational fields.

Contribution

It demonstrates the existence of zero-energy stationary solutions for fermions in Schwarzschild spacetime and extends the discussion to other gravitational fields, proposing dark matter candidates.

Findings

Existence of E=0 stationary states for fermions in Schwarzschild field.

Particles are likely located within a fraction of the Compton wavelength from the horizon.

Potential extension of solutions to Reissner-Nordstr"om, Kerr, and Kerr-Newman fields.

Abstract

When using a second-order Schr\"odinger-type equation with the effective potential of the Schwarzschild field, existence of a stationary state of half-spin particles with energy $E=0$ is proved. For each of the values of quantum numbers $j,l$, the physically meaningful energy $E=0$ (the binding energy is $E_{b}=mc^{2}$) is implemented at the value of the gravitational coupling constant $\alpha\geq\alpha_{min}$. The particles with $E=0$ are, with the overwhelming probability, at some distance from the event horizon within the range from zero to several fractions of Compton wavelength of a fermion depending on value of the gravitational coupling constants and values $j,l$. In this paper, similar solutions of the second-order equation are announced for bound states of fermions in the Reissner-Nordstr\"om, Kerr, Kerr-Newman fields. Atomic-type systems: collapsars with fermions in bound states are proposed as particles of dark matter.