On the relation between metric and spin--2 form. of lin. Ein. theory

TL;DR

This paper explores the relationship between metric and spin-2 forms in linearized Einstein theory, proposing a 20-charge solution space, analyzing global conditions for non-singular metrics, and introducing asymptotic conformal Yano--Killing tensors for inertial frame definition.

Contribution

It introduces a 20-dimensional charged solution space for spin-2 equations and connects it with extended Poincaré symmetry, providing new insights into global metric existence and asymptotic structures.

Findings

Non-singular metrics exist if and only if 10 charges vanish.

The solution space is 20-dimensional, analogous to electromagnetic charges.

Asymptotic conformal Yano--Killing tensors help define inertial frames in GR.

Abstract

A twenty--dimensional space of charged solutions of spin--2 equations is proposed. The relation with extended (via dilatation) Poincar\'e group is analyzed. Locally, each solution of the theory may be described in terms of a potential, which can be interpreted as a metric tensor satisfying linearized Einstein equations. Globally, the non--singular metric tensor exists if and only if 10 among the above 20 charges do vanish. The situation is analogous to that in classical electrodynamics, where vanishing of magnetic monopole implies the global existence of the electro--magnetic potentials. The notion of {\em asymptotic conformal Yano--Killing tensor} is defined and used as a basic concept to introduce an inertial frame in General Relativity via asymptotic conditions at spatial infinity. The introduced class of asymptotically flat solutions is free of supertranslation ambiguities.