Killing tensors and photon surfaces in foliated spacetimes

TL;DR

This paper introduces a geometric method to construct nontrivial rank two Killing tensors in foliated spacetimes, linking them to photon surfaces, and demonstrates the technique on various complex spacetime metrics.

Contribution

It provides a novel algebraic approach to generate Killing tensors from spacetime foliations, avoiding differential equations, and reveals their connection to photon surfaces.

Findings

Constructed Killing tensors in Kerr and related metrics.

Linked Killing tensors to photon surfaces in spacetimes with two Killing vectors.

Demonstrated the method's applicability to complex supergravity solutions.

Abstract

We present a purely geometric method for constructing a rank two Killing tensor in a spacetime with a codimension one foliation that lifts the trivial Killing tensors from slices to the entire manifold. The resulting Killing tensor can be nontrivial. A deep connection is found between the existence of such a Killing tensor and the presence of generalized photon surfaces in spacetime with two Killing vector fields. This technique generates Killing tensors in a purely algebraic way, without solving differential equations. The use of our method is demonstrated for Kerr, Kerr-Newman-NUT-AdS metrics and Kerr-NUT-AdS multicharge gauged supergravity solution.