Stationary Strings in the Spacetime of Rotating Black Holes in Five-Dimensional Minimal Gauged Supergravity

TL;DR

This paper investigates the separability and integrability of stationary string equations near rotating charged black holes in five-dimensional supergravity, revealing special cases where full separability occurs and identifying conformal symmetries in the general case.

Contribution

It demonstrates the conditions under which the Hamilton-Jacobi equation for stationary strings is separable in this complex spacetime, providing explicit Killing tensors for special cases and a conformal Killing tensor in the general case.

Findings

Complete separability occurs only for zero charge or equal angular momenta cases.

Explicit Killing tensors are derived for these special cases.

A conformal Killing tensor exists in the general black hole spacetime.

Abstract

We examine the separability properties of the equation of motion for a stationary string near a rotating charged black hole with two independent angular momenta in five-dimensional minimal gauged supergravity. It is known that the separability problem for the stationary string in a general stationary spacetime is reduced to that for the usual Hamilton-Jacobi equation for geodesics of its quotient space with one dimension fewer. Using this fact, we show that the "effective metric" of the quotient space does not allow the complete separability for the Hamilton-Jacobi equation, albeit such a separability occurs in the original spacetime of the black hole. We also show that only for two special cases of interest the Hamilton-Jacobi equation admits the complete separation of variables and therefore the integrability for the stationary string motion in the original background: namely, when the black hole has zero electric charge or it has an arbitrary electric charge, but two equal angular momenta. We give the explicit expressions for the Killing tensors corresponding to these cases. However, for the general black hole spacetime the effective metric of the quotient space admits a conformal Killing tensor. We construct the explicit expression for this tensor.