Gravitational dressing: from the crossed product to more general algebraic and mathematical structure

Abstract

The crossed product, and consequent transition from von Neumann algebras of type III to II, is recovered from a truncation of more general gravitational dressing constructions, about certain spacetimes. This is done by extending "standard dressing" constructions previously used to give a perturbative definition of "gravitational splittings," defining approximate localization of information. This result appears to illustrate that this algebraic transition is a small piece of a more general algebraic, or other mathematical, structure associated with quantum gravity. The leading-order structure involves noncommutativity from separated regions, and at the nonperturbative level connects with a possible explanation of holographic behavior for gravity.

Figures (3)

• Figure 1: Illustration of a standard dressing construction. The dressing $V_S$ external to the neighborhood $U$ only depends on the neighborhood, e.g. through choice of a fixed point $x_0$ of the neighborhood. The full dressing arises by including an additional piece associated with the dressing of a general point.
• Figure 2: The spacetime of an eternal black hole, represented via Penrose or Eddington-Finkelstein diagrams, together with different spatial slicings. The top row shows a general slicing of the interior and right exterior. The second row shows one member of a family of stationary slicesBHQIUENVU; other members are found through translation by the Killing vector $\xi_t$. The third row is a member of a family of extremal slices, also related by translation by the Killing vector.
• Figure 3: The spatial geometry of an extremal slice of Fig. \ref{['slices']}c.