Null states and time evolution in a toy model of black hole dynamics

A bstract Spacetime wormholes can provide non-perturbative contributions to the gravitational path integral that make the actual number of states e S in a gravitational system much smaller than the number of states $$ {e}^{S_{\textrm{p}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:msup> </mml:math> predicted by perturbative semiclassical effective field theory. The effects on the physics of the system are naturally profound in contexts in which the perturbative description actively involves N = O ( e S ) of the possible $$ {e}^{S_{\textrm{p}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:msup> </mml:math> perturbative states; e.g., in late stages of black hole evaporation. Such contexts are typically associated with the existence of non-trivial quantum extremal surfaces. However, by forcing a simple topological gravity model to evolve in time, we find that such effects can also have large impact for N ≪ e S (in which case no quantum extremal surfaces can arise). In particular, even for small N , the insertion of generic operators into the path integral can cause the non-perturbative time evolution to differ dramatically from perturbative expectations. On the other hand, this discrepancy is small for the special case where the inserted operators are non-trivial only in a subspace of dimension D ≪ e S . We thus study this latter case in detail. We also discuss potential implications for more realistic gravitational systems.