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Particle and light motion in a space-time of a five-dimensional rotating black hole

Valeri Frolov, Dejan Stojkovic

TL;DR

The paper analyzes particle and light trajectories in the $5$-dimensional Myers-Perry black hole, showing that a Killing tensor enables separability of the Hamilton-Jacobi equations and a first-order description of motion. It derives constants of motion ($E$, $\Phi$, $\Psi$, and a second-order $K$) and explicit radial and angular equations, revealing radial/ angular dynamics governed by effective potentials. A key result is the absence of stable circular orbits in equatorial planes, contrasting with four-dimensional Kerr, while identifying two principal null congruences that are geodesic but possess nonzero shear in five dimensions. The work highlights the role of hidden symmetries in higher-dimensional black holes and paves the way for generalizations to arbitrary dimensions.

Abstract

We study motion of particles and light in a space-time of a 5-dimensional rotating black hole. We demonstrate that the Myers-Perry metric describing such a black hole in addition to three Killing vectors possesses also a Killing tensor. As a result, the Hamilton-Jacobi equations of motion allow a separation of variables. Using first integrals we present the equations of motion in the first-order form. We describe different types of motion of particles and light and study some interesting special cases. We proved that there are no stable circular orbits in equatorial planes in the background of this metric.

Particle and light motion in a space-time of a five-dimensional rotating black hole

TL;DR

The paper analyzes particle and light trajectories in the -dimensional Myers-Perry black hole, showing that a Killing tensor enables separability of the Hamilton-Jacobi equations and a first-order description of motion. It derives constants of motion (, , , and a second-order ) and explicit radial and angular equations, revealing radial/ angular dynamics governed by effective potentials. A key result is the absence of stable circular orbits in equatorial planes, contrasting with four-dimensional Kerr, while identifying two principal null congruences that are geodesic but possess nonzero shear in five dimensions. The work highlights the role of hidden symmetries in higher-dimensional black holes and paves the way for generalizations to arbitrary dimensions.

Abstract

We study motion of particles and light in a space-time of a 5-dimensional rotating black hole. We demonstrate that the Myers-Perry metric describing such a black hole in addition to three Killing vectors possesses also a Killing tensor. As a result, the Hamilton-Jacobi equations of motion allow a separation of variables. Using first integrals we present the equations of motion in the first-order form. We describe different types of motion of particles and light and study some interesting special cases. We proved that there are no stable circular orbits in equatorial planes in the background of this metric.

Paper Structure

This paper contains 10 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Myers-Perry metric and its properties
  3. Equations of motion for particles and light. First integrals.
  4. General types of motion and special cases
  5. Radial motion
  6. Motion in $\theta$-direction
  7. Circular orbits
  8. Principal null congruences
  9. Conclusions
  10. Optical scalars for null congruences in a higher dimensional space-time

Figures (1)

  • Figure 1: The effective potential $V_S$ for radial motion in the five-dimensional Schwarzschild space-time for several values of the angular momentum $L$. We set $m=r_0=1$ and four different curves correspond to $L^2=1, 10, 100 {\ \rm and \ } 1000$ from the bottom up respectively.