  It is not currently known how to put the Kerr spacetime metric into the
so-called Gordon form, although the closely related Kerr-Schild form of the
Kerr metric is well known. A Gordon form for the Kerr geometry, if it could be
found, would be particularly useful in developing analogue models for the Kerr
spacetime, since the Gordon form is explicitly given in terms of the 4-velocity
and "refractive index" of an effective medium. In the current article we report
progress toward this goal. First we present the Gordon form for an
approximation to Kerr spacetime in the slow-rotation limit, obtained by
suitably modifying the well-known Lense-Thirring form of the slow-rotation
metric. Second we present the Gordon form for the Kerr spacetime in the
near-null limit, (the 4-velocity of the medium being close to null). That these
two perturbative approximations to the Kerr spacetime in Gordon form exist
gives us some confidence that ultimately one might be able to write the exact
Kerr spacetime in this form.
