A simple diagnosis of non-smoothness of black hole horizon: curvature singularity at horizons in extremal Kaluza–Klein black holes

Abstract

We propose a simple method to prove non-smoothness of a black hole horizon. The existence of a C 1 extension across the horizon implies that there is no C N +2 extension across the horizon if some components of N -th covariant derivative of Riemann tensor diverge at the horizon in the coordinates of the C 1 extension. In particular, the divergence of a component of the Riemann tensor at the horizon directly indicates the presence of a curvature singularity. By using this method, we can confirm the existence of a curvature singularity for several cases where the scalar invariants constructed from the Riemann tensor, e.g., the Ricci scalar and the Kretschmann invariant, take finite values at the horizon. As a concrete example of the application, we show that the Kaluza-Klein black holes constructed by Myers have a curvature singularity at the horizon if the spacetime dimension is higher than five.Recently, the case of multi-center coplanar black hole and membrane horizons were studied in [44]. 3 In D ≥ 6, KK black holes with a S 1 compactified extra dimension do not have C 2 horizon [37].

A pair of extremal charged black holes on Kerr–Taub-bolt space

“…Each black hole has an analytic Killing horizon, then there is no curvature singularity on and outside the black hole horizons. We note that although multi-black hole solutions in higher dimensions tend to have non-smooth event horizons [30][31][32][33][34][35][36], our solutions have smooth event horizons like multi-black holes with non-trivial asymptotic structure [37].…”

“…From (29), (36), and (37), we see that the lens parameters of the northern and the southern black holes and spatial infinity, p ± and p, are coprime to each other, and satisfy…”

All Higher-Dimensional Majumdar–Papapetrou Black Holes

“…Curiously, in higher dimensions n > 4, the solutions with multiple horizon components (N > 1) do not have smooth horizons and analytic extensions do not exist in general [9]. In particular, if n = 5 the metric at the horizon is generically C 2 and the Maxwell field is C 0 , whereas if n > 5 the metric is generically C 1 at the horizon and the Maxwell field is C 0 [10][11][12][13]. Therefore, as we explain below, we will allow for this lower differentiability in our analysis.…”

“…Let (x i ) be cartesian coordinates on R n−1 and p ∈ R n−1 correspond to a horizon component. The coordinate change (x i ) → (ρ, y a ) maps the euclidean metric to the general form for the base metric near the horizon (13) if and only if…”

All higher-dimensional Majumdar-Papapetrou black holes

“…Curiously, in higher dimensions n > 4, the solutions with multiple horizon components (N > 1) do not have smooth horizons and analytic extensions do not exist in general [9]. In particular, if n = 5 the metric at the horizon is generically C 2 and the Maxwell field is C 0 , whereas if n > 5 the metric is generically C 1 at the horizon and the Maxwell field is C 0 [10][11][12][13]. Therefore, as we explain below, we will allow for this lower differentiability in our analysis.…”