Multipole-based framework for Kerr black hole mimickers. From general construction to a specific case of study

We present a novel systematic framework for engineering energy-momentum tensors in linearized gravity that generate gravitational fields with a prescribed multipolar structure, aimed at modeling black hole mimickers. As a concrete example, we analyze an anisotropic rotating fluid with a Gaussian-like energy-density profile which, at linearized order, satisfies all energy and causality conditions while generating a gravitational field that exactly reproduces the multipolar structure of a Kerr black hole. We then investigate the phenomenology of both the source and the induced metric, comparing our findings with the corresponding black hole limit.