Holographic Dual of Synge's Identity in AdS/CFT

We identify CFT counterparts of Synge’s identity for finite AdS geodesics in arbitrary dimensions. Starting from exact finite bulk–boundary pairs, the Synge equation induces endpoint scale equations for a Weyl-frame scalar two-point function and for the finite entanglement entropy between disjoint balls. In the planar frame, the common-scale correlator equation is the dilatation Ward identity. In the entanglement sector, the corresponding relative-scale equation is governed by a transverse hyperbolic area and reduces in two dimensions to the previously identified entanglement RG equation.