  We find general deformations of BTZ spacetime and identify the corresponding
thermofield initial states of the dual CFT. We deform the geometry by
introducing bulk fields dual to primary operators and find the back-reacted
gravity solutions to the quadratic order of the deformation parameter. The dual
thermofield initial states can be deformed by inserting arbitrary linear
combination of operators at the mid-point of the Euclidean time evolution that
appears in the construction of the thermofield initial states. The deformed
geometries are dual to thermofield states without deforming the boundary
Hamiltonians in the CFT side. We explicitly demonstrate that the AdS/CFT
correspondence is not a linear correspondence in the sense that the linear
structure of Hilbert space of the underlying CFT is realized nonlinearly in the
gravity side. We also find that their Penrose diagrams are no longer a square
but elongated horizontally due to deformation. These geometries describe a
relaxation of generic initial perturbation of thermal system while fixing the
total energy of the system. The coarse-grained entropy grows and the relaxation
time scale is of order $\beta/2\pi$. We clarify that the gravity description
involves coarse-graining inevitably missing some information of nonperurbative
degrees.
