Bulk Motion in Global AdS$_3$ from the Boundary Energy-Density Perspective

We study how bulk propagation in global AdS$_3$ is encoded in boundary energy densities of the dual CFT$_2$ on the cylinder. A key feature of the global geometry, which is absent in the Poincar\'e-patch description, is that a null excitation emitted from the boundary reaches the antipodal boundary point after $\Delta\tau=\pi$ and returns to the original boundary point after $\Delta\tau=2\pi$. We show that this periodic boundary-to-boundary propagation is reflected in the CFT energy density as chiral peaks that meet at the antipodal point and reappear after one global period. For a wave-packet state, the leading energy density consists of two chiral peaks moving along the boundary light-cone directions; their relative weights encode the impact parameter of the corresponding bulk null ray. For a state constructed from an operator of large conformal dimension with a Euclidean time regulator (LCD state), the exact cylinder stress tensor gives two periodic chiral pulses moving along the boundary light-cone directions. These pulses reach the antipodal point and return to the original point at the same global times as the radial null geodesic in the bulk. Thus the boundary energy density captures the boundary-to-boundary propagation and periodicity of localized bulk excitations in global AdS$_3$.