https://doi.org/10.1140/epjc/s10052-026-15283-1
Regular Article - Theoretical Physics
On hyperboloidal foliations in the study of black hole quasinormal modes
1
School of Intelligent Manufacturing, Zhejiang Guangsha Vocational and Technical University of Construction, 322100, Jinhua, Zhejiang, China
2
Postdoctoral Department, Turon University, 180100, Karshi, Uzbekistan
3
Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista, 12516-410, Guaratinguetá, SP, Brazil
4
Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, 225009, Yangzhou, China
5
Escola de Engenharia de Lorena, Universidade de São Paulo, 12602-810, Lorena, SP, Brazil
6
Natural Sciences Department, Metropolitan State University, 55106, Saint Paul, MN, USA
7
Le Moyne College, 13214, Syracuse, NY, USA
8
Mathematics and Statistics Department, Metropolitan State University, 55106, Saint Paul, MN, USA
a
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Received:
8
October
2025
Accepted:
31
December
2025
Published online:
22
January
2026
Abstract
In this work, we demonstrate that the hyperboloidal foliation technique, applied to the study of black hole quasinormal modes, where the spatial boundary is shifted from spacelike infinity to the future event horizon and null infinity, is effectively equivalent to the continued fraction approach, in which the asymptotic wave function typically diverges at both ends of spatial infinity. Specifically, a given hyperboloidal slicing, corresponding to a particular choice of coordinates, always uniquely determines a scheme for extracting the asymptotic form of the wave function at the spatial boundary. Owing to the mathematical equivalence, it follows that the efficiency and precision observed using the hyperboloidal approach should be attributed, not to avoiding the pathological behavior at the spatial boundaries, but primarily to other factors, such as the use of Chebyshev grids.
© The Author(s) 2026
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