Cosmological singularities, entanglement and quantum extremal surfaces

We study aspects of entanglement and extremal surfaces in various families of spacetimes exhibiting cosmological, Big-Crunch, singularities, in particular isotropic $AdS$ Kasner. The classical extremal surface dips into the bulk radial and time directions. Explicitly analysing the extremization equations in the semiclassical region far from the singularity, we find the surface bends in the direction away from the singularity. In the 2-dim cosmologies obtained by dimensional reduction of these and other singularities, we have studied quantum extremal surfaces by extremizing the generalized entropy. The resulting extremization shows the quantum extremal surfaces to always be driven to the semiclassical region far from the singularity. We give some comments and speculations on our analysis.