The Riemann Geometry of Space and Gravitational Waves With The Spin $s=1$

TL;DR

This paper explores the Riemann tensor's role in describing gravitational waves, proposing that inhomogeneous media can generate s=1 gravitational waves, expanding the traditional understanding limited to s=2 waves.

Contribution

It introduces a novel perspective that the Weyl tensor describes gravitational waves with spin s=1, differing from the conventional s=2 waves, and links this to inhomogeneous matter media.

Findings

Weyl tensor can describe s=1 gravitational waves.

Inhomogeneous media can generate s=1 gravitational waves.

Introduces tensor K_{ik} as an antisymmetric analog to Ricci tensor.

Abstract

It is taken into account that not the Ricci tensor ${R_{il}}$ (Einstein equation), but the Riemann tensor ${R_{iklm}}$ provides the most general description of the space geometry. If ${{R_{il}}=0}$ (the space empty with matter, but it can be occupied by gravitational waves) then ${{R_{iklm}}={C_{iklm}}} $ . The tensor ${C_{iklm}}$ is the Weyl tensor, which disappears by conversion:${{R_{il}}={g^{km}}{R_{iklm}}}$ and we lose all information about the space structure, which is described by ${C_{iklm}}$ .The symmetry of ${R_{il}}$ provides the existents of gravitational waves with the spin s=2. We show that ${C_{iklm}}$ describes gravitational waves with s=1. Such gravitational waves can be created in inhomogeneous media, where the selected directions are determined by derivates of the energy-momentum tensor ${T^m}_{i,k}$ of matter. It is taken into account that gravitation is described not only by the metric tensor $g^{ik} = (1/2)(\gamma^i \gamma^k + \gamma^k \gamma^i)$, but also by the anti-symmetric tensor $\sigma^{ik} = (i/2)({\gamma^i}{\gamma^k} - {\gamma^k}{\gamma^i})$, where ${\gamma^k}(x)$ are Clifford matrices. We show that the tensor ${K_{ik}} = (1/4){\sigma^{lm}} {R_{iklm}}$, is the anty-simmetric analog to the Ricci tensor. The ${K_{lm}}$ describes various kinds of space metrics, not described by the Ricci tensor. It includes the Weyl tensor ${C_{iklm}}$, because $\sigma^{lm} {C_{lmik}}$ is not zero.