Properties of general stationary axisymmetric spacetimes: circularity and beyond

We analyse properties of general stationary and axisymmetric spacetimes, with a particular focus on circularity -- an accidental symmetry enjoyed by the Kerr metric, and therefore widely assumed when searching for rotating black hole solutions in alternative theories of gravity as well as when constructing models of Kerr mimickers. Within a gauge specified by seven (or six) free functions, the local existence of which we prove, we solve the differential circularity conditions and translate them into algebraic relations among the metric components. This result opens the way to investigating the consequences of circularity breaking in a controlled manner. In particular, we construct two simple analytical examples of non-circular deformations of the Kerr spacetime. The first one is "minimal", since the horizon and the ergosphere are identical to their Kerr counterparts, except for the fact that the horizon is not Killing and its surface gravity is therefore not constant. The second is "not so minimal", as the horizon's profile can be chosen arbitrarily and the difference between the horizon and the so-called rotosurface can be appreciated. Our findings thus pave the way for further research into the phenomenology of non-circular stationary and axisymmetric spacetimes.