Newman-Penrose-like exact and approximate conservation laws: a covariant and conformal formulation

TL;DR

This paper develops a covariant and conformal framework to derive and generalize Newman-Penrose constants and Aretakis charges, applicable to various massless fields and space-times, including asymptotically flat and extremal horizons.

Contribution

It introduces a conformal GHP formalism to obtain covariant expressions for NP constants and Aretakis charges, extending their applicability to diverse fields and space-times.

Findings

Derived covariant expressions for NP constants in Minkowski space.

Generalized NP constants to massless fields of arbitrary spin.

Proved existence of Aretakis charges on extremal horizons.

Abstract

Using a conformal extension of the Geroch-Held-Penrose (GHP) formalism I derive a manifestly covariant and conformal expression of Newman-Penrose (NP) constants, which are a set of conserved quantities associated to solutions to the wave equation on light cones in Minkowski space. The resulting expression generalizes to massless fields of arbitrary spin -- including the electromagnetic field, Weyl fermions, and the linearized Weyl tensor -- on conformally flat space-times. In some non-conformally flat space-times there may exist conserved charges on very special null hypersurfaces. Using the conformal GHP formalism I prove the existence of conserved Aretakis charges on extremal Killing horizons. In the absence of exact conservation laws it is still useful to extend the definition for NP constants to some asymptotically flat curved space-times, where the conservation laws become approximate conservation laws which rapidly approach exact conservation laws at null infinity. I derive explicit expressions for spherically symmetric space-times.