Local Invariants Vanishing on Stationary Horizons

For a general black hole spacetime, the location of its surface (the event horizon, which is the boundary of the region from which causal curves can go to asymptotic future null infinity) depends on the future evolution of the spacetime and is not determined locally. However, for black holes that settle down to become stationary, one might ask whether one can find a local invariant that is generically nonzero off the horizon but vanishes on the horizon. In particular, one might look for a scalar polynomial curvature invariant [1–6], which is a scalar obtained by complete contraction of all the indices of a polynomial in the Riemann curvature tensor and its covariant derivatives. For example, Anders Karlhede, Ulf Lindstrom and Jan Aman [7] showed that RRαβγδ;ǫ crosses zero and switches sign as one crosses the horizon of the Schwarzschild metric, and one can easily show that this is also true for any smooth static spherically symmetric horizon. An invariant that vanishes on more general stationary horizons could be useful in numerical relativity for finding the approximate location of the horizon once the spacetime has settled down to become approximately stationary. Majd Abdelqader and Kayll Lake [8] have recently found a local scalar polynomial curvature invariant that vanishes at the horizon of the Kerr black hole. After casting this invariant into a simpler form that is proportional to the squared norm of the wedge product of two curvature-invariant gradients, we realized that the procedure generalizes to give a way to locate any nonsingular stationary horizon in terms of a local curvature invariant. Essentially, if one constructs as many gradients of independent curvature invariants as the cohomogeneity of a stationary spacetime, at a generic point in the spacetime these gradients will be linearly independent and spacelike, but at an horizon, a linear combination will become null. This implies that the squared norm of the wedge product of the gradients will vanish at a stationary horizon. Abdelqader and Lake [8] give the following six curvature invariants for the Kerr metric, which we here copy

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