A smooth BTZ black bounce with an extremal null throat

Abstract

We study a static, circularly symmetric deformation of the non-rotating BTZ black hole obtained by inserting a smooth transition function into the \emph{inverse} radial metric component, $g^{rr}=S_\delta(r)F(r)$ with $S_\delta=\tanh[(r-r_h)/\delta]$, leaving $g_{tt}=-F$ untouched. This was motivated by the proposal that such a construction realizes a Lorentzian-to-Riemannian signature change at the horizon; we show that it does not. In coordinates $r-r_h=q^2$ with an advanced time, the metric extends real-analytically across $r=r_h$, and the extension is Lorentzian: $q=0$ is a regular null hypersurface, a degenerate Killing horizon with vanishing surface gravity, beyond which lies a second, isometric copy of the exterior. The areal radius has a minimum there, so the geometry is a black bounce; the would-be Riemannian branch is a separate geometry the Lorentzian sector never reaches. We give the effective source in closed form, an invariant account of the energy conditions, and identify the near-throat geometry as AdS$_2\times S^1$. The scalar effective potential is proven strictly positive for every mode, and the throat circle is a minimal surface whose length gives an entropy $\pi r_h/2G$, reproduced independently by the Wald--Noether charge and by a Cardy estimate from the computed Brown--York mass -- concordant results for which no first law is available since $\kappa=0$. The throat carries an Aretakis-type instability, with a conserved leading transverse derivative and a linearly growing subleading one. We also record a negative result: smoothing $g_{tt}$ instead, as in the Lorentzian-Euclidean Schwarzschild proposal, is singular at the horizon for any finite smoothing width. We state explicitly what the construction does not establish.