The correspondence between sonic points of ideal photon gas accretion and photon spheres

In an accretion of fluid, its velocity may transit from subsonic to supersonic. The point at which such transition occurs is called sonic point and often mathematically special. We consider a steady-state and spherically symmetric accretion problem of ideal photon gas in general static spherically symmetric spacetime neglecting back reaction. Our main result is that the EOS of ideal photon gas leads to correspondence between its sonic point and the photon sphere of the spacetime in general situations. Despite of the dependence of the EOS on the dimension of spacetime, this correspondence holds for spacetimes of arbitrary dimensions.