The renormalization group flows from high energy to low energy. In flat spacetime, this flow is irreversible — the number of effective degrees of freedom decreases monotonically as you coarse-grain from the ultraviolet to the infrared. This is Zamolodchikov's c-theorem in two dimensions, its higher-dimensional generalizations, and the entropic proofs based on strong subadditivity of entanglement entropy. The arrow is well-established.

Does the arrow survive in curved spacetime?

Anti-de Sitter space has negative curvature and a timelike boundary at spatial infinity. Both features fundamentally alter infrared dynamics compared to flat space. The boundary provides a natural IR regulator — fields in AdS don't spread to infinity in the same way. The curvature modifies the density of states and the structure of entanglement. It was not obvious that the flat-space irreversibility proofs would transfer.

They do. The key ingredients are the same: strong subadditivity of entanglement entropy and the symmetries of the background spacetime. The authors define RG charges — differences in entanglement entropy between a theory at a given scale and its ultraviolet fixed point — and prove these charges decrease monotonically under RG flow in 2, 3, and 4 spacetime dimensions. The proof is geometric: it uses the isometry group of AdS rather than the Poincaré group, but the logical structure is unchanged.

The lattice calculations in AdS bring new technical challenges — AdS lattices don't have the translation invariance that simplifies flat-space computations — but the continuum limit recovers the expected AdS isometries and the monotonicity holds.

The irreversibility of the RG is not a feature of flat spacetime that might be modified by curvature. It is a consequence of entanglement structure that holds wherever the entanglement entropy obeys strong subadditivity. The arrow is geometric, not environmental.