Holographic algebras at null infinity

Abstract

We construct an algebra of observables associated to a cut of null infinity in asymptotically flat spacetimes. The Bondi mass associated with a sharp cut of future null infinity is not a well-defined quantum operator: its fluctuations diverge even after smearing in retarded time. We instead introduce a finite-radius, time-smeared version of the Bondi mass and adjoin this operator to the matter and graviton observables in an arbitrarily small asymptotic neighborhood of the cut. We separately analyze spacetimes with and without black holes and find, in both cases, Type III$_1$ von Neumann algebras that satisfy non-trivial nesting relations. In Minkowski spacetime, the resulting algebra reconstructs the spacelike wedge associated with the cut. In a stationary black hole spacetime, it reconstructs the region bounded by the cut and the black hole bifurcation surface, providing an asymptotically flat analogue of an entanglement wedge. In the limit as the cut is moved to past infinity, we recover a Type I$_{\infty}$ algebra for spacetimes without black holes and a Type II$_{\infty}$ algebra for black hole spacetimes. For algebras at finite cuts of retarded time, we construct a Type II$_\infty$ regularization of the black hole algebra whose renormalized von Neumann entropy agrees with the generalized entropy. In appendices, we prove two technical results about quantum fields, in a Schwarzschild spacetime, that asymptote to the Minkowski vacuum at infinity: a split property for unbounded, spacelike separated regions and the construction of a faithful, normal, and semifinite Hartle-Hawking weight whose modular flow is Schwarzschild time evolution.