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Obtaining solutions of the Einstein field equations describing spinningcompact bodies is typically challenging. The Newman-Janis algorithm provides aprocedure to obtain rotating spacetimes from a static, spherically symmetric,seed metric. It is not guaranteed, however, that the resulting rotatingspacetime solves the same field equations as the seed. Moreover, the former maynot be circular, and thus expressible in Boyer-Lindquist-like coordinates.Amongst the variations of the original procedure, a modified Newman-Janisalgorithm (MNJA) has been proposed that, by construction, originates acircular, spinning spacetime, expressible in Boyer-Lindquist-like coordinates.As a down side, the procedure introduces an ambiguity, that requires extraassumptions on the matter content of the model. In this paper we observe thatthe rotating spacetimes obtained through the MNJA always admit separability ofthe Hamilton-Jacobi equation for the case of null geodesics, in which case,moreover, the aforementioned ambiguity has no impact, since it amounts to anoverall metric conformal factor. We also show that the Hamilton-Jacobi equationfor light rays propagating in a plasma admits separability if the plasmafrequency obeys a certain constraint. As an illustration, we compute the shadowand lensing of some spinning black holes obtained by the MNJA.

Spinning black holes with a separable Hamilton-Jacobi equation from a modified Newman-Janis algorithm