What is the significance of the photon sphere radius in the Schwarzschild metric?

• A. It is the surface from which light cannot escape.
• B. It marks the radius where photons can orbit the black hole; inside it, photons must move inward.
• C. It determines the maximum possible orbital speed of matter.
• D. It has no relevance to the shadow.

Answer: (B)

In the Schwarzschild spacetime, there is a special radius where light can in principle travel in a circle around the black hole—the photon sphere. At r = 3GM/c^2 (which is 1.5 times the Schwarzschild radius), null geodesics can form a circular orbit. This orbit is unstable: a small nudge outward or inward causes the photon to leave the circular path—outward it can escape, inward it inevitably falls toward the black hole. Inside this radius, circular photon orbits don’t exist and photons are drawn inward. Outside, light paths can bend dramatically but don’t stay in a stable circle. This radius is closely tied to the black hole’s shadow, since light near this sphere largely determines the boundary of the dark region seen by distant observers. So the statement that it’s the radius where photons can orbit the black hole, with inner photons forced to move inward, captures the essential physics.