In a very interesting paper , Cunha , Berti , and Herdeiro have recently claimed that ultra-compact objects , self-gravitating horizonless solutions of the Einstein field equations which have a light ring , must possess at least two ( and , in general , an even number of ) light rings , of which the inner one is stable .
In the present compact paper we explicitly prove that , while this intriguing theorem is generally true , there is an important exception in the presence of degenerate light rings which , in the spherically symmetric static case , are characterized by the simple dimensionless relation 8 \pi r ^ { 2 } _ { \gamma } ( \rho + p _ { \text { T } } ) = 1 [ here r _ { \gamma } is the radius of the light ring and \ { \rho,p _ { \text { T } } \ } are respectively the energy density and tangential pressure of the matter fields ] .
Ultra-compact objects which belong to this unique family can have an odd number of light rings .
As a concrete example , we show that spherically symmetric constant density stars with dimensionless compactness M / R = 1 / 3 possess only one light ring which , interestingly , is shown to be unstable .