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Abstract General relativity, as a diffeomorphism-invariant theory, allows the description of physical phenomena in a wide variety of coordinate systems. In the presence of boundaries, such as event horizons and null infinity, time coordinates must be carefully adapted to the global causal structure of spacetime to ensure a computationally efficient description. Horizon-penetrating time is used to describe the dynamics of infalling matter and radiation across the event horizon, while hyperboloidal time is used to study the propagation of radiation toward the idealized observer at null infinity. In this paper, we explore the historical and mathematical connection between horizon-penetrating and hyperboloidal time coordinates, arguing that both classes of coordinates are simply regular choices of time across null horizons. We review the height-function formalism in stationary spacetimes, providing examples that may be useful in computations, such as source-adapted foliations or Fefferman–Graham–Bondi coordinates near null infinity. We discuss bridges connecting the boundaries of spacetime through a time hypersurface across null horizons, including the event horizon, null infinity, and the cosmological horizon.
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This content will become publicly available on January 17, 2026
Hyperboloidal approach to quasinormal modes
Oscillations of black hole spacetimes exhibit divergent behavior near the bifurcation sphere and spatial infinity. In contrast, these oscillations remain regular when evaluated near the event horizon and null infinity. The hyperboloidal approach provides a natural framework to bridge these regions smoothly, resulting in a geometric regularization of time-harmonic oscillations, known as quasinormal modes (QNMs). This review traces the development of the hyperboloidal approach to QNMs in asymptotically flat spacetimes, emphasizing both the physical motivation and recent advancements in the field. By providing a geometric perspective, the hyperboloidal approach offers an elegant framework for understanding black hole oscillations, with implications for improving numerical simulations, stability analysis, and the interpretation of gravitational wave signals.
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- Award ID(s):
- 2309084
- PAR ID:
- 10585625
- Publisher / Repository:
- Frontiers
- Date Published:
- Journal Name:
- Frontiers in Physics
- Volume:
- 12
- ISSN:
- 2296-424X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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