Understand
In this paper, we study the existence of universal horizons in a given static spacetime, and find that the test khronon field can be solved explicitly when its velocity becomes infinitely large, at which point the universal horizon coincides with the sound horizon of the khronon.
- Choosing the timelike coordinate aligned with the khronon, the static metric takes a simple form, from which it can be seen clearly that the metric is free of singularity at the Killing horizon, but becomes singular at the universal horizon.
- Applying such developed formulas to three well-known black hole solutions, the Schwarzschild, Schwarzschild anti-de Sitter, and Reissner-Nordstr\"om, we find that in all these solutions universal horizons exist and are always inside the Killing horizons.
- In particular, in the Eddington-Finkelstein and Painleve-Gullstrand coordinates, in which the metrics are not singular when crossing both of the Killing and universal horizons, the peeling-off behavior of the khronon is found only at the universal horizons, whereby we show that the values of surface gravity of the universal horizons calculated from the peeling-off behavior of the khronon match with those obtained from the covariant definition given recently by Cropp, Liberati, Mohd and Visser.