第一定律熵与最小长度Simpson--Visser型规则黑洞中的简并极端残余：几何热力学、相结构及观测判别

中文摘要 AI 辅助

我们构造了一个几何最小长度形变的史瓦西时空的第一定律一致熵，该时空通过面积半径替换$R(r)=\sqrt{r^{2}+\ell^{2}}$在$r\in[0,+\infty)$上获得，其中$f(R)=1-2M/R$，其非零爱因斯坦张量源出一个没有经典物质对应的有效几何流体。对第一定律积分得到$S=\pi[r_{h}R_{h}+\ell^{2}\ln((r_{h}+R_{h})/\ell)]$。该式在函数形式上与Joshi和Joshi独立发现的半经典项一致，但我们基于独立的物理理由固定其边界条件$S(r_h=0)=0$，并将其作为模型的完整熵，而非采用Bekenstein--Hawking面积定律，在此基础上构建自由能、几何热力学和模式稳定性分析。由亥姆霍兹自由能$F(M)$控制的蒸发在$M_{\min}=\ell/2$处终止于一个先前未被认识的端点：一个简并极端规则黑洞，其中规则中心与二次阶简并Killing视界重合，$f\approx r^{2}/(2\ell^{2})$，在面积半径$\ell$处，有$T_{H}\to 0$，$S\to 0$，$C\to 0^{+}$，且曲率处处有限；我们首次展示了其Penrose--Carter结构。相同的熵（含$\ell$）定义了一个Legendre不变的几何热力学（GTD）描述的平衡态空间，其曲率标量在Davies型相变$M^{*}=\ell/\sqrt{2}$和$M_{\min}$处独立发散，这种发散在最小长度黑洞文献中没有对应物，特定于$S=0$处的真实视界。我们将此残余嵌入Tsukamoto定性指出的观测判别中：精确的阴影简并结合可测量的光子环通量增强$r_{n}=e^{-2\pi/a}>e^{-2\pi}$，可供下一代甚长基线干涉测量获取。

英文摘要

We construct the first-law-consistent entropy of a geometrically minimal-length-deformed Schwarzschild spacetime, obtained via the areal-radius substitution $R(r)=\sqrt{r^{2}+\ell^{2}}$ on $r\in[0,+\infty)$ with $f(R)=1-2M/R$, whose nonvanishing Einstein tensor sources an effective geometric fluid with no classical matter counterpart. Integrating the first law gives $S=π[r_{h}R_{h}+\ell^{2}\ln((r_{h}+R_{h})/\ell)]$. This coincides in functional form with the semiclassical term found independently by Joshi and Joshi, but we fix its boundary condition $S(r_h=0)=0$ on independent physical grounds and adopt it, rather than the Bekenstein--Hawking area law, as the complete entropy of the model, building the free energy, geometrothermodynamics, and mode-stability analysis on it. Evaporation, governed by the Helmholtz free energy $F(M)$, terminates at $M_{\min}=\ell/2$ in a previously unrecognised endpoint: a degenerate extremal regular black hole, where the regular centre coincides with a degenerate Killing horizon of quadratic order, $f\approx r^{2}/(2\ell^{2})$, at areal radius $\ell$, with $T_{H}\to 0$, $S\to 0$, $C\to 0^{+}$, and finite curvature everywhere; we display its Penrose--Carter structure for the first time. The same entropy, with $\ell$, defines the equilibrium state space of a Legendre-invariant geometrothermodynamic (GTD) description whose curvature scalar diverges independently at the Davies-type transition $M^{*}=\ell/\sqrt{2}$ and at $M_{\min}$, a divergence with no counterpart in the minimal-length black hole literature, specific to a genuine horizon at $S=0$. We embed this remnant within the observational discriminant noted qualitatively by Tsukamoto: exact shadow degeneracy combined with a measurable photon-ring flux enhancement $r_{n}=e^{-2π/a}>e^{-2π}$, accessible to next-generation very-long-baseline interferometry.

发表机构

• University Moulay Ismail（穆莱·伊斯梅尔大学）
• Cadi Ayyad University（卡迪·阿亚德大学）

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