Moving holographic boundaries

In this paper, we show that for one sign of the deformation coupling single-trace $T\bar{T}$ deformation moves the holographic screen in Gödel universe radially inward. For the other sign of the coupling it moves the holographic screen radially outward. We (thus) argue, on general grounds, that in holography (single-trace) $T\bar{T}$ deformation can be generally thought of as either moving the holographic boundary into the bulk or washing it away to infinity. We explain in what sense. In Anti-de Sitter this breaks the spacetime conformal symmetry. We further note that moving timelike holographic boundary into bulk creates (at onset) a curvature singularity. In the boundary the singularity is understood by states with imaginary energies. To define or make the bulk theory sensible we introduce an ultraviolet cutoff and thereby move the boundary into the bulk. However, it is not clear whether suitable boundary conditions that lead to a consistent string theory exist. In this paper we first obtain the Penrose limit of the single-trace $T\bar{T}$ deformed string background and then perform T-duality along a space-like isometry to obtain a class of deformed Gödel universes. The string background we consider is $Ad{S}_3\times S^3\times M_4$. The single-trace $T\bar{T}$ deformation is a particular example of the more general $O(d,d)$ transformations.