AdS/CFT correspondence in a Friedmann-Lemaitre-Robertson-Walker brane

According to the AdS/CFT correspondence conjecture, the Randall-Sundrum infinite braneworld is equivalent to four dimensional Einstein gravity with $\mathcal{N}=4$ super Yang-Mills fields at low energies. Here we derive a four dimensional effective equation of motion for tensor-type perturbations in two different pictures, and demonstrate their equivalence.