What happens to Petrov classification on horizons of axisymmetric dirty black holes

TL;DR

This paper investigates how the Petrov classification of axisymmetric dirty black holes changes near horizons, revealing differences between on-horizon, off-horizon, and boosted frames, and classifies the possible relations among these types.

Contribution

It introduces the concept of boosted Petrov type and analyzes the relations between on-horizon, off-horizon, and boosted Petrov types for axisymmetric black holes.

Findings

Petrov types can differ between frames near horizons.

Only types D and O can be the same in all three characteristics.

The analysis includes ultra-extremal horizons and their algebraic structures.

Abstract

We consider axisymmetric stationary dirty black holes with regular non-extremal or extremal horizons, and compute their on-horizon Petrov types. The Petrov type (PT) in the frame of the observer crossing the horizon can be different from that formally obtained in the usual (but singular in the horizon limit) frame of an observer on a circular orbit. We call this entity the boosted Petrov type (BPT), as the corresponding frame is obtained by a singular boost from the regular one. The PT off-horizon can be more general than PT on-horizon and that can be more general than the BPT on horizon. This is valid for all regular metrics, irrespective of the extremality of the horizon. We analyze and classify the possible relations between the three characteristics and discuss the nature and features of the underlying singular boost. The three Petrov types can be the same only for space-times of PT D and O off-horizon. The mutual alignment of principal null directions and the generator in the vicinity of the horizon is studied in detail. As an example, we also analyze a special class of metrics with utra-extremal horizons (for which the regularity conditions look different from the general case) and compare their off-horizon and on-horizon algebraic structure in both frames.