Geometric criticality and regular quasibound spectra of axial Proca fields in Schwarzschild spacetime
Abstract
We investigate how a local geometric critical point of the odd-parity Proca potential relates to the global quasibound spectrum in Schwarzschild spacetime. After normalization by μ2, the axial potential Vax=(1−2M/r)[ℓ(ℓ+1)/r2+μ2] has the same radial form as the timelike-geodesic effective potential. Its outer minimum and inner maximum merge at r=6M, the Schwarzschild innermost stable circular orbit, when (Mμc)2=ℓ(ℓ+1)/12. With δ=1−(Mμ)2/(Mμc)2, we derive exact fold scalings: stationary-point separation ∝δ1/2, barrier--well depth ∝δ3/2, and curvature magnitudes ∝δ1/2. Continued-fraction calculations for the first three axial quasibound branches with ℓ=1,2,3, cross-checked for selected modes by independent Riccati shooting, show regular frequency continuation across the threshold and approximately linear near-critical behavior. At fixed radius the radial operator is affine in δ, supporting a conditional simple-root analytic interpretation. The main result is the explicit distinction between geometric potential criticality and spectral degeneracy for the branches examined.
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BibTeX
@misc{ribeiro2026geometric,
title = {{Geometric criticality and regular quasibound spectra of axial Proca fields in Schwarzschild spacetime}},
author = {Antonio Sousa Ribeiro and Francisco Artur Pinheiro Alves Junior},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34528},
eprint = {2610.34528},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34528},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric criticality and regular quasibound spectra of axial Proca fields in Schwarzschild spacetime
AU - Antonio Sousa Ribeiro
AU - Francisco Artur Pinheiro Alves Junior
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34528
UR - https://arxiv.org/abs/2610.34528
AB - We investigate how a local geometric critical point of the odd-parity Proca potential relates to the global quasibound spectrum in Schwarzschild spacetime. After normalization by μ^2, the axial potential V_ax=(1−2M/r)[ℓ(ℓ+1)/r^2+μ^2] has the same radial form as the timelike-geodesic effective potential. Its outer minimum and inner maximum merge at r=6M, the Schwarzschild innermost stable circular orbit, when (Mμ_c)^2=ℓ(ℓ+1)/12. With δ=1−(Mμ)^2/(Mμ_c)^2, we derive exact fold scalings: stationary-point separation ∝δ^1/2, barrier--well depth ∝δ^3/2, and curvature magnitudes ∝δ^1/2. Continued-fraction calculations for the first three axial quasibound branches with ℓ=1,2,3, cross-checked for selected modes by independent Riccati shooting, show regular frequency continuation across the threshold and approximately linear near-critical behavior. At fixed radius the radial operator is affine in δ, supporting a conditional simple-root analytic interpretation. The main result is the explicit distinction between geometric potential criticality and spectral degeneracy for the branches examined.
ER -