$T\bar T$ deformations and the pp wave correspondence

In this paper we consider $T\bar T$ deformations in the context of pp waves obtained from gravity duals. We propose a deformation of $AdS_5\times S^5$ similar to the deformation of the single trace $T\bar T$ deformation of $AdS_3\times S^3\times T^4$ with NS-NS flux, and study it through the Penrose limit, concluding that it must correspond to some dipole theory, probably noncommutative. We $T\bar T$ deform the worldsheet string for the $AdS_5\times S^5$ pp wave, and find a corresponding spin chain Hamiltonian. Finally, directly $T\bar T$ deforming the spin chain Hamiltonian obtained from the pp wave, we find that it corresponds to an equivalent BMN sector of the ${\cal N}=4$ SYM.