The action of active diffeomorphisms (diffs) $r\to \rho (r)$ on the Schwarzschild metric leads to metrics which are also static spherically symmetric solutions of the Einstein vacuum field equations. It is shown how in a $limiting$ case it allows to introduce a deformation of the manifold such that $\rho (r=0)=0$, and $\rho (r=0^{+})=2GM$ corresponding, respectively, to the spacelike singularity and horizon of the Schwarzschild metric. In doing so, one ends up with a spherical $void$ surrounding the singularity at $r=0$. In order to explore the “interior” region of this void, we introduce complex radial coordinates whose imaginary components have a direct link to the inverse Hawking temperature, and which furnish a path that provides access to interior region. In addition, we show that the black hole entropy $\frac{A}{4}$ (in Planck units) is equal to the $area$ of a rectangular strip in the $complex$ radial-coordinate plane associated to this path. The gist of the physical interpretation behind this construction is that there is an emergence of thermal dimensions which unfolds as one plunges into the interior void region via the use of complex coordinates, and whose imaginary components capture the span of the thermal dimensions. Namely, the filling of the void leads to an $emergent$ internal/ thermal dimension via the imaginary part ${\beta }_r$ of the complex radial variable $r=r+i{\beta }_r$.

主动微分同胚 (diffs) 的作用$r\to \rho (r)$在 Schwarzschild 度量上得出的度量也是爱因斯坦真空场方程的静态球对称解。它显示了如何在$升\mathrm{一世}米\mathrm{一世}吨\mathrm{一世}nG$情况下它允许引入歧管的变形使得$\rho (r=0)=0$， 和$\rho (r=0^{+})=\mathrm{2个}G米$分别对应于 Schwarzschild 度量的类空奇点和视界。在这样做的时候，一个人最终得到一个球形$vo\mathrm{一世}d$奇点周围$r=0$. 为了探索这个空洞的“内部”区域，我们引入了复杂的径向坐标，其虚部与霍金反温度有直接联系，并提供了一条通往内部区域的路径。此外，我们证明了黑洞熵$\frac{\mathrm{一个}}{\mathrm{4个}}$（以普朗克单位）等于$\mathrm{一个}r\mathrm{电子}\mathrm{一个}$中的矩形条带$Co米p升\mathrm{电子}X$与此路径关联的径向坐标平面。这种结构背后的物理解释的要点是，当一个人通过使用复坐标进入内部空隙区域时，会出现热维度，其虚部捕捉热维度的跨度。即，空隙的填充导致$\mathrm{电子}米\mathrm{电子}rG\mathrm{电子}n吨$通过虚部的内部/热尺寸${\beta }_r$复杂的径向变量$r=r+\mathrm{一世}{\beta }_r$.