The Photon Sphere and the AdS/CFT Correspondence

The AdS/CFT correspondence connects bulk fields $\phi$ to boundary operators $\mathcal{O}$ characterized by source frequency $\omega$ and angular momentum $l$. Here we explore their connection to massless particles with an impact parameter $b=\omega/l$. In the AdS Schwarzschild spacetime, these particles follow unstable orbits around the photon sphere -- with Lyapunov exponent $\lambda$ -- when $b$ is near a critical value. The behavior of the bulk field is obtained numerically and then studied using an analytic approach, which leads to a precise approximate formula for the amplitude of the bulk field $\phi$. This gives the correct qualitative behavior for the system, with the amplitude of the field taking the shape of an arrowhead in tortoise coordinates. The field behaves analogously to the massless particles, and the amplitude of $\phi$ diverges at the critical impact parameter when the source frequency takes the value $\omega \approx \lambda l$, where $\lambda$ is the Lyapunov exponent of the null geodesics. We find this transition occurs when $b = \lambda$. We show this is precisely when the first QNM becomes available, and obtain an approximate formula for the first few overtones.