We study closed photon orbits in spherically symmetric static solutions of supergravity theories, a Horndeski theory, and a theory of quintessence. These orbits lie in what we shall call a photon sphere (antiphoton sphere) if the orbit is unstable (stable). We show that in all the asymptotically flat solutions we examine that admit a regular event horizon, and whose energy-momentum tensor satisfies the strong energy condition, there is one and only one photon sphere outside the event horizon. We give an example of a Horndeski theory black hole (whose energy-momentum tensor violates the strong energy condition) whose metric admits both a photon sphere and an antiphoton sphere. The uniqueness and nonexistence also holds for asymptotically anti–de Sitter solutions in gauged supergravity. The latter also exhibits the projective symmetry that was first discovered for the Schwarzschild–de Sitter metrics: the unparametrized null geodesics are the same as when the cosmological or gauge coupling constant vanishes. We also study the closely related problem of accretion flows by perfect fluids in these metrics. For a radiation fluid, Bondi’s sonic horizon coincides with the photon sphere. For a general polytropic equation of state this is not the case. Finally we exhibit counterexamples to a conjecture of Hod’s.

• Received 29 September 2016

DOI:https://doi.org/10.1103/PhysRevD.94.106005