The effect of gravitational spin-orbit coupling on the circular photon orbit in the Schwarzschild geometry

The (1, 0)+(0, 1) representation of the group SL(2, C) provides a six-component spinor equivalent to the electromagnetic field tensor. By means of the (1, 0)+(0, 1) description, one can treat the photon field in curved spacetime via spin connection and the tetrad formalism, which is of great advantage to study the gravitational spin-orbit coupling of photons. Once the gravitational spin-orbit coupling is taken into account, the traditional radius of the circular photon orbit in the Schwarzschild geometry should be replaced with two different radiuses corresponding to the photons with the helicities of +1 and -1, respectively. Owing to the splitting of energy levels induced by the spin-orbit coupling, photons (from Hawking radiations, say) escaping from a Schwarzschild black hole are partially polarized, provided that their initial velocities possess nonzero tangential components.