Abstract
Employing a pseudo-orthonormal coordinate-free approach, the solutions to the Klein–Gordon and Dirac equations for particles in Melvin spacetime are derived in terms of Heun’s biconfluent functions.
Highlights
The study of relativistic particles in static magnetic fields has a long history and is still attracting considerable attention, especially for cases where someone deals with curved manifolds
When dealing with slowly rotating neutron stars which have been termed as magnetars [2], it has been assumed that their huge magnetic induction in the core and crust, B ∼ 1014–1015 (G), is affecting the spacetime geometry
Another way is to assume that magnetized metrics, as the one belonging to the Melvin class [4, 5], may be reliable candidates for describing these highly compact astrophysical objects with a dominant axial magnetic field [6]
Summary
The study of relativistic particles in static magnetic fields has a long history and is still attracting considerable attention, especially for cases where someone deals with curved manifolds. Within a coordinate-dependent formulation, switching between canonical and pseudo-orthonormal basis, the abovementioned authors are integrating the system of four coupled first-order differential equations, in the first approximation, neglecting the terms in higher orders of the polar radial coordinate ρ. Their solutions are expressed in terms of generalized Laguerre polynomials, to the case of the Dirac equation in cylindrical coordinates on a flat manifold [7]. The Heun functions, either general or confluent, are main targets of recent investigations and have been obtained for massless particles evolving in a Universe described by the metric function written as a nonlinear mixture of Schwarzschild, Melvine, and Bertotti-Robinson solutions [11]
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