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

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.

References in corpus (1)

• Dark Matter in Universe as the Geometry of Empty Space