It has been recently shown in Chatterjee and Ghosh (Phys Rev Lett 125:041302, 2020, https://doi.org/10.1103/PhysRevLett.125.041302) that microstate counting carried out for quantum states residing on the horizon of a black hole leads to a correction of the form \(\exp (-A/4l_p^2)\) in the Bekenstein-Hawking form of the black hole entropy. In this paper, we develop a novel approach to obtain the possible form of the spacetime geometry from the entropy of the black hole for a given horizon radius. The uniqueness of this solution for a given energy-momentum tensor has also been discussed. Remarkably, the black hole geometry reconstructed has striking similarities to that of noncommutative-inspired Schwarzschild black holes (Nicolini et al. in Phys Lett B 632:547, 2006). We also obtain the matter density functions using Einstein field equations for the geometries we reconstruct from the thermodynamics of black holes. These also have similarities to that of the matter density function of a noncommutative-inspired Schwarzschild black hole. The conformal structure of the metric is briefly discussed and the Penrose–Carter diagram is drawn. We then compute the Komar energy and the Smarr formula for the effective black hole geometry and compare it with that of the noncommutative-inspired Schwarzschild black hole. We also discuss some astrophysical implications of the solutions. Finally, we propose a set of quantum Einstein vacuum field equations, as a solution of which we obtain one of the spacetime solutions obtained in this work. We then show a direct connection between the quantum Einstein vacuum field equations and the first law of black hole thermodynamics.

Chatterjee 和 Ghosh 最近在 (Phys Rev Lett 125:041302, 2020, https://doi.org/10.1103/PhysRevLett.125.041302) 中表明，对黑洞视界上的量子态进行微态计数会导致黑洞熵的 Bekenstein-Hawking 形式的\(\exp (-A/4l_p^2)\)形式的修正。在本文中，我们开发了一种新颖的方法，可以根据给定视界半径的黑洞熵来获取时空几何的可能形式。还讨论了该解决方案对于给定能量动量张量的独特性。值得注意的是，重建的黑洞几何结构与非交换性史瓦西黑洞的几何结构有着惊人的相似之处（Nicolini 等人，in Phys Lett B 632:547, 2006）。我们还使用爱因斯坦场方程获得了根据黑洞热力学重建的几何形状的物质密度函数。这些也与非交换史瓦西黑洞的物质密度函数有相似之处。简要讨论了度量的共形结构并绘制了彭罗斯-卡特图。然后，我们计算有效黑洞几何形状的科马尔能量和斯马尔公式，并将其与非交换性史瓦西黑洞的能量和斯马尔公式进行比较。我们还讨论了解决方案的一些天体物理学含义。最后，我们提出了一组量子爱因斯坦真空场方程，作为其解，我们获得了本工作中获得的时空解之一。然后，我们展示了量子爱因斯坦真空场方程与黑洞热力学第一定律之间的直接联系。