# Photon motion in Kerr-de Sitter spacetimes

## TL;DR

This paper analyzes photon trajectories in Kerr-de Sitter spacetimes, revealing how impact parameters influence possible photon motions and classifying spacetime types based on effective potential behaviors.

## Contribution

It introduces a comprehensive analysis of non-equatorial photon motion in Kerr-de Sitter spacetimes, including impact parameter constraints and a new classification scheme.

## Key findings

- Constraints on impact parameters depend on spacetime parameters.
- Vortical motion at constant radius is excluded for negative impact parameters.
- Spacetime classification based on effective potential behavior.

## Abstract

We study general non-equatorial motion of photons in the Kerr-de Sitter black hole and naked singularity spacetimes. The motion is governed by the impact parameter $X$ related to axial symmetry of the spacetime, and impact parameter $q$ related to its hidden symmetry. Appropriate 'effective potentials' governing the latitudinal and radial motion are introduced and their behaviour is examined by 'Chinese boxes' technique giving regions allowed for the motion in terms of the impact parameters. Limits on the impact parameter $X$ and $q$ are established in dependence on the spacetime parameters $M, \Lambda, a$. The motion can be of orbital type (crossing the equatorial plane, $q>0$) and vortical type (tied above or bellow the equatorial plane, $q<0$). It is shown that for negative values of $q$ the reality conditions imposed on the latitudinal motion yield stronger constraints on the parameter $X$ than that following from the reality condition of motion in the radial direction, which, e. g., exclude possibility of existence of vortical motion of constant radius. Several other consequences are determined and a classification of Kerr-de Sitter spacetimes reflecting the behaviour of the effective potentials is given.