We find an exact static solution in four dimensions to the field equations of the gravity by using a cylindrically static spacetime with two different ansatz, and . This solution is derived without imposing any conditions on . The black hole solution involves four constants: , , , and . Among these, is linked to the cosmological constant, to the black hole's mass, while and are responsible for the deviation of the solution from the linear form of . We demonstrate how the analytical function relies on . When is zero, becomes a constant function, leading to the non-metricity case. We investigate the singularity of this solution and show that the Kretschmann invariant has a much milder singularity compared to the non-metricity case. We produce a black hole that rotates with non-vanishing values of and by using a coordinate transformation. Then, we analyze the laws of thermodynamics to determine the physical characteristics of this black hole solution and demonstrate that it is locally thermodynamically stable.

中文翻译：

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使用圆柱静态时空的函数的一般形式[公式省略]


通过使用具有两个不同 ansatz 和 的圆柱静态时空，我们找到了引力场方程在四个维度上的精确静态解。该解是在不施加任何条件的情况下导出的。黑洞解涉及四个常数： 、 、 和 。其中， 与宇宙常数、黑洞质量有关，而 和 则导致解偏离 的线性形式。我们演示了分析函数如何依赖于 。当 为零时，变为常数函数，导致非度量情况。我们研究了该解的奇异性，并表明与非度量情况相比，Kretschmann 不变量具有更温和的奇异性。我们通过使用坐标变换产生一个以 和 的非零值旋转的黑洞。然后，我们分析热力学定律，以确定该黑洞溶液的物理特性，并证明其局部热力学稳定。