Photon ring structure of rotating regular black holes and no-horizon spacetimes
Rahul Kumar, Sushant G Ghosh
Abstract
Abstract The Kerr black holes possess a photon region with prograde and retrograde orbits radii, respectively, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>M</mml:mi> <mml:mo>⩽</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>⩽</mml:mo> <mml:mn>3</mml:mn> <mml:mi>M</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>3</mml:mn> <mml:mi>M</mml:mi> <mml:mo>⩽</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>⩽</mml:mo> <mml:mn>4</mml:mn> <mml:mi>M</mml:mi> </mml:math> , and thereby always cast a closed photon ring or a shadow silhouette for a ⩽ M . For a > M , it is a no-horizon spacetime (naked singularity) wherein prograde orbits spiral into the central singularity, and retrograde orbits produce an arc-like shadow with a dark spot at the center. We compare Kerr black holes’ photon ring structure with those produced by three rotating regular spacetimes, viz Bardeen, Hayward, and nonsingular. These are non-Kerr black hole metrics with an additional deviation parameter of g related to the nonlinear electrodynamics charge. It turns out that for a given a , there exists a critical value of g , g E such that Δ = 0 has no zeros for g > g E , one double zero at r = r E for g = g E , respectively, corresponding to a no-horizon regular spacetime and extremal black hole with degenerate horizon. We demonstrate that, unlike the Kerr naked singularity, no-horizon regular spacetimes can possess closed photon ring when g E < g ⩽ g c , e.g. for a = 0.10 M , Bardeen ( g E = 0.763 332 M < g ⩽ g c = 0.816 792 M ), Hayward ( g E = 1.052 97 M < g ⩽ g c = 1.164 846 M ) and nonsingular ( g E = 1.2020 M < g ⩽ g c = 1.222 461 M ) no-horizon spacetimes have closed photon ring. These results confirm that the mere existence of a closed photon ring does not prove that the compact object is necessarily a black hole. The ring circularity deviation observable Δ C for the three no-horizon rotating spacetimes satisfy Δ C ⩽ 0.10 as per the M87 * black hole shadow observations. We have also appended the case of Kerr–Newman no-horizon spacetimes (naked singularities) with similar features.