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Register a new userOptical caustics of Kerr spacetime: the full structure
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V. Bozza
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We present an exhaustive numerical investigation of the optical caustics in gravitational lensing by a spinning black hole for an observer at infinity. Besides the primary caustic, we examine higher order caustics, formed by photons performing one or several loops around the black hole. Our investigation covers the whole parameter space, including the black hole spin, its inclination with respect to the line of sight, the source distance, and the caustic order. By comparing our results with the available analytical approximations, we find perfect agreement in their respective domains of validity. We then prove that all caustics maintain their shape (a tube with astroidal cross-section) in the entire parameter space without suffering any transitions to different caustic shapes. For nearly extremal spin, however, higher order caustics grow so large that their cross-sections at fixed radii wind several times around the black hole. As a consequence, for each caustic order, the number of images ranges from 2 to 2(n+1), where n is the number of loops spanned by the caustic. As for the critical curves, we note that for high values of the spin they develop a small dip on the side corresponding to prograde orbits.
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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.
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