General form of the function [formula omitted] using cylindrically static spacetime

Abstract

We find an exact static solution in four dimensions to the field equations of the f(Q) gravity by using a cylindrically static spacetime with two different ansatz, ν(r) and μ(r). This solution is derived without imposing any conditions on f(Q). The black hole solution involves four constants: c1, c2, c3, and c4. Among these, c1 is linked to the cosmological constant, c2 to the black hole's mass, while c3 and c4 are responsible for the deviation of the solution from the linear form of f(Q). We demonstrate how the analytical function f(Q) relies on c3. When c3 is zero, f(Q) 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 Q and f(Q) 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.