Spin-0 fields and the NP-constants close to spatial infinity in Minkowski spacetime

Abstract

Newman–Penrose (NP) constants of massless spin-0 fields propagating in Minkowski spacetime are computed close to spatial and null infinity by means of Friedrich’s i0-cylinder. Assuming a certain regularity condition on the initial data ensuring that the field extends analytically to critical sets, it is shown that the NP constants at future I+ and past null infinity I− are independent of each other. In other words, the classical NP constants at I± stem from different parts of the initial data given on a Cauchy hypersurface. In contrast, it is shown that, using a slight generalization of the classical NP constants, the associated quantities (i0-cylinder NP constants) do not require the regularity condition being satisfied and give rise to conserved quantities at I± that are determined by the same piece of initial data, which, in turn, correspond to the terms controlling the regularity of the field. Additionally, it is shown how the conservation laws associated with the NP constants can be exploited to construct, in flat space, heuristic asymptotic-system expansions, which are sensitive to the logarithmic terms at the critical sets.

Polyhomogeneous spin-0 fields in Minkowski space–time

“…This review article gives an abridged discussion of the behaviour close to spatial and null infinity of a system of equations called the GBU model in Minkowski space-time based on [35,36]. The GBU system is a simple model that mimics some of the nonlinearities appearing in a recent reformulation of the Einstein field equations in generalized harmonic gauge well adapted to the hyperboloidal initial value problem using the dual-foliation formalism for numerical relativity (HypDF)-see [27,29,30].…”

“…Hence, the asymptotic expansions as presented in [33] are insensitive to the appearance of such logarithmic terms. Nonetheless, more recently, a possible avenue to obtain a heuristic expansion that does not make use of conformal methods but still captures the missing i 0 -cylinder logarithmic terms (in flat space-time) was found in [36]. The alternative expansion method of Gasperín & Pinto [36] departs from that of Duarte et al [33] as it exploits the existence of conservation laws related to the Newman-Penrose constants-see [36,52].…”

“…For further discussion on the i 0 -cylinder see [11,35,42,43] and references therein, and for the relation between the frames see [36,44,45].…”

“…Therefore, the natural question is if some asymptotic expansion capturing the i 0 -cylinder logs can be derived from a suitably modified asymptotic system notion. Such an alternative notion of asymptotic system was put forward in [36] and it is presented here to give a round discussion on how to recover the missed logarithmic terms in theorem 1 of Duarte et al [33]. To do this, recall that the (first order) asymptotic system is obtained by disregarding the terms that (…”

“…Nonetheless, to alleviate this shortcoming, a method to retrieve the leading order logarithmic term within the asymptotic system heuristics was laid out in [35]. Later, a method to recover all the missed higher-order logarithmic terms (in flat space-time) was obtained as a subproduct of the analysis of Gasperín & Pinto [36] for the calculation of the Newman-Penrose constants of spin-0 fields. In this review article, the discussion of the GBU-system close to spatial and null infinity of Duarte et al [35] and the higher order asymptotic system approach trough conservation laws in flat space-time discussed in [36] are revisited and presented in a coherent and contiguous way.…”