Geometry, Topology, and Confined Spectra Around Anisotropic Cosmic Strings

Abstract

We study scalar, Dirac, and electromagnetic test fields on a two-parameter family of static spacetimes, labeled by an angular winding number $k$ and an anisotropy parameter $α$ that shortens one direction of the transverse metric relative to the other. Only the members with $k=1$ and $α<1$ are conical, as for an ordinary straight cosmic string. The remaining members are nonconical and, except in the flat cases, curved and nonvacuum. A common spin-weighted connection Laplacian organizes the local equations, while the curvature coupling, global field conditions, and constraints are fixed separately for each physical field. Explicit frame transitions remove spurious winding dependence in the flat limit. For the nonconical members of the family, isothermal coordinates yield an exact factorization of the indicial equation and determine its monodromy: the indicial exponents are independent of the anisotropy parameter, although the finite-domain eigenvalues are not. We derive variational bounds and a weak-anisotropy selection rule for the scalar spectrum. The electromagnetic equations are obtained from the four-potential in Lorenz gauge, with gauge-invariant transverse-electric (TE) and transverse-magnetic (TM) reductions and an annular transverse-electromagnetic (TEM) mode. Finite-element computations on an annulus, validated against benchmarks, are consistent with the bounds, confirm the selection rule, give MIT bag Dirac and conducting-wall Maxwell spectra, and reproduce the invariance of the transverse-electromagnetic capacity under $k\mapsto2-k$, an exact consequence of a radial inversion of the Dirichlet energy.