Detecting horizons of symmetric black holes using relative differential invariants

Abstract Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">k</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> be a nontrivial finite-dimensional Lie algebra of vector fields on a manifold M , and consider the family of Lorentzian metrics on M whose Killing algebra contains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">k</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> . We show that scalar relative differential invariants of such metrics, with respect to a Lie algebra of vector fields on M preserving <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">k</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> , can be used to detect the horizons of several well-known black holes. In particular, using the Lie algebra structure of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">k</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> , we construct a general relative differential invariant of order 0 that always vanishes on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">k</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> -invariant Killing horizons. While the current work is meant to demonstrate the relevance of jet bundles and relative differential invariants in physical applications, we also provide a computationally simple approach for finding a relative differential invariant that detects Killing horizons. The computation and use of this relative differential invariant is comparable in difficulty to other horizon detection methods when there is an obvious Killing vector field that generates the Killing horizon, and often simpler when the preferred Killing vector field is not obvious.