Elastic stiffness of three-dimensional black holes and wormholes from Liouville line defects

We study elastic deformations of thin-shell black holes and wormholes in AdS$_3$ gravity. These geometries are sourced by line defects in the dual conformal field theory, and their shape and mass distribution define elastic moduli of the gravitational saddle. We compute the quadratic response of the partition function to these deformations, defining stiffness kernels for both transverse shape fluctuations and inhomogeneous mass-density fluctuations. The computation of the stiffness kernels can be realized as a hyperbolic response to a conformal welding problem which reduces to the universal Schwarzian response in the heavy-shell limit. The stiffness kernels are two-point functions of defect-local operators in CFT: the displacement operator, which measures the response to shape deformations, and a mass-density operator, which measures the response to local changes in the shell density. We compute the spectrum of these operators in the semiclassical limit, in various black hole and wormhole backgrounds. The spectrum can be discrete or continuous depending on the existence of a non-compact direction transverse to the shell in the geometry. We also provide a Lorentzian interpretation for the stiffness kernels using linear response theory and compute the relaxation time scales towards the corresponding transient deformations in the dual holographic CFT. Lastly, we compute the effect of these elastic deformations on black hole microstate statistics and black hole entropy.