AI 中文总结
本文构造带有极端零喉的光滑BTZ黑洞反弹，研究其几何性质、熵、不稳定性，否定了洛伦兹-黎曼号差改变及$g_{tt}$光滑处理的可行性。
AI 中文摘要
我们研究非旋转BTZ黑洞的静态、圆对称形变，该形变通过在逆径向度规分量中插入一个光滑过渡函数得到，即$g^{rr}=S_\delta(r)F(r)$，其中$S_\delta=\tanh[(r-r_h)/\delta]$，同时保持$g_{tt}=-F$不变。该工作的动机是这类构造可实现视界处的洛伦兹-黎曼号差改变，但我们证明其无法实现。在超前时间的坐标$r-r_h=q^2$中，度规在$r=r_h$处可实解析延拓，且延拓后为洛伦兹型：$q=0$是正则零超曲面，是表面引力为零的退化 Killing 视界，其外存在一个等距复制的外部区域。这里的面积半径取最小值，因此该几何是黑洞反弹；原本的黎曼分支是洛伦兹区永远无法到达的独立几何。我们给出闭合形式的有效源、能量条件的不变描述，并确定近喉几何为AdS$_2\times S^1$。已证明标量有效势对所有模严格为正，喉圆是极小曲面，其长度给出熵为$\pi r_h/2G$，该熵通过Wald-Noether荷和由Brown-York质量计算的Cardy估计独立重现，由于$\kappa=0$，不存在热力学第一定律。喉带有Aretakis型不稳定性，存在守恒的主导横向导数和线性增长的次主导导数。我们还记录了一个否定结果：若像洛伦兹-欧氏Schwarzschild方案那样对$g_{tt}$进行光滑处理，在任意有限光滑宽度下视界处均为奇异。我们明确说明该构造未确立的内容。
英文摘要
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, $g^{rr}=S_δ(r)F(r)$ with $S_δ=\tanh[(r-r_h)/δ]$, 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 $πr_h/2G$, 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 $κ=0$. 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.
Comments26 pages, 6 Figures, 5 appendices, and 33 references
