Null infinity as an inverted extremal horizon: Matching an infinite set of conserved quantities for gravitational perturbations

Abstract

Every spacetime that is asymptotically flat near null infinity can be conformally mapped via a spatial inversion onto the geometry around an extremal, non-rotating and non-expanding horizon. We set up a dictionary for this geometric duality, connecting the geometry and physics near null infinity to those near the dual horizon. We then study its physical implications for conserved quantities for extremal black holes, extending previously known results to the case of gravitational perturbations. In particular, we derive a tower of near-horizon gravitational charges that are exactly conserved and show their one-to-one matching with Newman-Penrose conserved quantities associated with gravitational perturbations of the extremal Reissner-Nordström black hole geometry. We furthermore demonstrate the physical relevance of spatial inversions for extremal Kerr-Newman black holes, even if the latter are notoriously not conformally isometric under such inversions.

Figures (2)

• Figure 1: Penrose diagram representation of the conformal isomorphism between an asymptotically flat spacetime and the geometry near a dual extremal horizon. The spatial inversion in Eq. \ref{['eq:GenCT']} (black arrows) conformally maps the geometry near $\mathscr{I}^{+}$ of an asymptotically flat spacetime (red partial Penrose diagram) to that near a future horizon $\mathscr{H}^{+}$ that is extremal, non-expanding and non-rotating (blue partial Penrose diagram), and vice versa. An exactly analogous spatial inversion maps the geometry near $\mathscr{I}^{-}$ to that near a past horizon $\mathscr{H}^{-}$ with the same properties.
• Figure 2: Part of the Penrose diagram of an extremal Reissner-Nordström black hole that describes the causally connected patch in the exterior geometry. As opposed to what happens with a generic pair of conformally related asymptotically flat spacetime and a dual geometry near an extremal horizon (see Fig. \ref{['fig:PenroseDiagramInversion']}), the existence of the Couch-Torrence inversion (black arrows) for this geometry can be understood as the manifestation that the two null surfaces reside in the same spacetime.