The relation between metric and spin-2 formulations of linearized Einstein theory

Abstract

A twenty-dimensional space of charged solutions of spin-2 equations is proposed. The relation with extended (via dilatation) Poincaré 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 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.

CYK tensors, Maxwell field and conserved quantities for the spin-2 field

“…They lead to the notion of the Conformal Yano-Killing tensor (see [17] and [18]). A Conformal Yano-Killing (CYK) equation (3.3) possesses twenty-dimensional space of solutions for flat Minkowski metric in four-dimensional spacetime.…”

Mixed Ge-Si Dimer Formation in Ge/Si(100) and Si/Ge(100) Growth

“…One can also show that if W is a spin-2 field, then * W (and consequently W * ) is a spin-2 field as well. If we assume that W fulfils equations (25), then the following equivalences are satisfied…”

“…When W is the Weyl tensor of the linearized Einstein theory, p µ turns out to be its momentum, b µ is its dual momentum, and j µν is its angular momentum tensor 6 (see [25]). In general, there might also be non-vanishing charges w µν .…”

“…The first kind is linear and the second quadratic with respect to the Weyl tensor, but a basis for both of them is the Maxwell field. Conserved quantities which are linear with respect to the CYK tensor were investigated many times (cf [1,21,22,25,26,28,35,36]). On the other hand, quadratic charges are less known and have usually been examined in terms of the Bel-Robinson tensor (see e.g.…”

Conformal Yano–Killing tensors for the Taub–NUT metric

“…Conserved quantities which are linear with respect to CYK tensor were investigated many times (cf. [6], [7], [9], [10], [12], [15], [16]). On the other hand, quadratic charges are less known and have usually been examined in terms of the Bel-Robinson tensor (see e.g.…”