Bridging time across null horizons

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. This work is motivated by the broader effort to understand the role of time in general relativity and reviews a unified framework for handling null boundaries in analytical and numerical approaches. The insights developed here offer practical tools for numerical relativity, gravitational wave astronomy, and other explorations of the large-scale structure of spacetimes.