Figure 21

Figure 21: The region $𝒦$ , defined by Equation (128 ), in the Kerr spacetime. The picture is purely spatial and shows a meridional section $φ = constant$ , with the axis of symmetry at the left-hand boundary. Through each point of $𝒦$ there is a spherical geodesic. Along each of these spherical geodesics, the coordinate $𝜗$ oscillates between extremal values, corresponding to boundary points of $𝒦$ . The region $𝒦$ meets the axis at radius $rc$ , given by $r3c − 3mr2c + a2rc + ma2 = 0$ . Its boundary intersects the equatorial plane in circles of radius $rph +$ (corotating circular light ray) and $ph r−$ (counter-rotating circular light ray). $ph r±$ are determined by $ph ph 2 2 r± (r± − 3m ) = 4ma$ and $r+ < rp+h< 3m < rp−h< 4m$ . In the Schwarzschild limit $a → 0$ the region $𝒦$ shrinks to the light sphere $r = 3m$ . In the extreme Kerr limit $a → m$ the region $𝒦$ extends to the horizon because in this limit both $rph→ m +$ and $r → m +$ ; for a caveat, as to geometric misinterpretations of this limit (see Figure 3 in [16]).