Mariem Magdy Ali Mohamed

The purpose of this article is to compute the asymptotic charges of a vacuum solution to the Einstein field equations describing a naked singularity with a non-vanishing quadrupole moment, known in the literature as the Zipoy-Voorhees spacetime (q-metric). In addition to the well-known asymptotic quantities such as the Bondi-Sachs energy-momentum, the BMS charges and NP constants of this spacetime are computed. Explicit calculations of the latter are relatively scarce in the literature. Moreover, it has been proven that the NP constants of asymptotically flat, stationary, vacuum, and algebraically special spacetimes vanish (for instance, those of the Kerr spacetime). A by-product of the present analysis is to show that the algebraically special condition in the aforementioned result appears to be crucial, since the q-metric provides a counterexample to the conjecture that all asymptotically flat, stationary, vacuum, and asymptotically algebraically special spacetimes (a weaker version of the algebraically special condition) have vanishing NP constants.

We show how the space spinor formalism for 2-component spinors can be used to construct estimates for spinor fields satisfying first order equations. We discuss the connection of the approach presented in this article with other strategies for the construction of estimates. In addition, we recast several concepts related to the notion of hyperbolicity in the context of spinor equations. The approach described in this article can be regarded as an adaptation to first order equations of the method of positive commutators for second order hyperbolic equations.

For asymptotically flat spacetimes, a conjecture by Strominger states that asymptotic BMS-supertranslations and their associated charges at past null infinity I− can be related to those at future null infinity I+ via an antipodal map at spatial infinity i0. We analyze the validity of this conjecture using Friedrich’s formulation of spatial infinity, which gives rise to a regular initial value problem for the conformal field equations at spatial infinity. A central structure in this analysis is the cylinder at spatial infinity I representing a blow-up of the standard spatial infinity point i0 to a 2-sphere. The cylinder I touches past and future null infinities I± at the critical sets I±. We show that for a generic class of asymptotically Euclidean and regular initial data, BMS-supertranslation charges are not well-defined at I± unless the initial data satisfies an extra regularity condition. We also show that given initial data that satisfy the regularity condition, BMS-supertranslation charges at I± are fully determined by the initial data and that the relation between the charges at I− and those at I+ directly follows from our regularity condition.

The asymptotic structure of null and spatial infinities of asymptotically flat spacetimes plays an essential role in discussing gravitational radiation, gravitational memory effect, and conserved quantities in General Relativity (GR). Bondi, Metzner and Sachs (BMS) established that the asymptotic symmetry group for asymptotically simple spacetimes is the infinite-dimensional BMS group. Given that null infinity is divided into two sets: past null infinity I − and future null infinity I + , one can identify two independent symmetry groups: BMS − at I − and BMS + at I + . Associated with these symmetries are the so-called BMS charges. A recent conjecture by Strominger suggests that the generators of BMS − and BMS + and their associated charges are related via an antipodal reflection map near spatial infinity. To verify this matching, an analysis of the gravitational field near spatial infinity is required. This task is complicated due to the singular nature of spatial infinity for spacetimes with non-vanishing ADM mass. Different frameworks have been introduced in the literature to address this singularity, e.g. Friedrich’s cylinder, Ashtekar-Hansen’s hyperboloid and Ashtekar-Romano’s asymptote at spatial infinity. This paper reviews the role of Friedrich’s formulation of spatial infinity in the investigation of the matching of the spin-2 charges on Minkowski spacetime and in the full GR setting. This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial infinity. We evaluate the charges at the critical sets where spatial infinity meets null infinity with the aim of finding the relation between the charges at future and past null infinity. To this end, we make use of Friedrich's framework of the cylinder at spatial infinity to obtain asymptotic expansions of the Maxwell and spin-2 fields near spatial infinity, which are fully determined in terms of initial data on a Cauchy hypersurface. Expanding the initial data in terms of spin-weighted spherical harmonics, it is shown that only a subset of the initial data, that satisfies certain regularity conditions, gives rise to well-defined charges at the point where future (past) infinity meets spatial infinity. Given such initial data, the charges are shown to be fully expressed in terms of the freely specifiable part of the data. Moreover, it is shown that there exists a natural correspondence between the charges defined at future and past null infinity.

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial infinity. We evaluate the charges at the critical sets where spatial infinity meets null infinity with the aim of finding the relation between the charges at future and past null infinities. To this end, we make use of Friedrich’s framework of the cylinder at spatial infinity to obtain asymptotic expansions of the Maxwell and spin-2 fields near spatial infinity, which are fully determined in terms of initial data on a Cauchy hypersurface. With expanding the initial data in terms of spin-weighted spherical harmonics, it is shown that only a subset of the initial data, which satisfy certain regularity conditions, gives rise to well-defined charges at the point where future (past) infinity meets spatial infinity. Given such initial data, the charges are shown to be fully expressed in terms of the freely specifiable part of the data. Moreover, it is shown that there exists a natural correspondence between the charges defined at future and past null infinities.