The On-shell Gravity Action and Linear Dilaton Holography

Computing the Euclidean spacetime action on-shell provides a useful way of both testing holographic proposals and determining the string theory sphere partition function. We consider families of three-dimensional linear dilaton spacetimes for which there are holographic proposals that share features of a $T\bar{T}$-deformed CFT. We extend the holographic renormalization program beyond AdS to this class of geometries by identifying the boundary terms needed for a well-defined variational principle and a finite on-shell action. We show that the spacetime energy or mass determined from the on-shell action matches the $T\bar{T}$-deformed two-dimensional CFT energy. This provides more evidence for the role of the $T\bar{T}$ deformation in this holographic correspondence.