Two incompatible types of invariants in the octonion spaces

TL;DR

This paper explores invariants and conservation laws in electromagnetic and gravitational fields using octonion transformations, revealing two incompatible types of invariants and their implications for physical laws.

Contribution

It introduces a novel analysis of invariants in octonion spaces, distinguishing two incompatible groups of invariants related to electromagnetic and gravitational fields.

Findings

Identifies invariants under octonion rotational transformations such as speed of light and norm of radius vector.

Shows mass and energy invariants can coexist, but charge invariants are incompatible with mass invariants.

Reveals two distinct groups of invariants corresponding to different octonion spaces.

Abstract

The paper aims to study some invariants and conservation laws relevant to electromagnetic and gravitational fields, by means of the rotational transformations of octonion coordinate systems. The scholars utilize the octonions to analyze the electromagnetic and gravitational fields simultaneously, including the octonion field potential, field strength, field source, linear momentum, angular momentum, torque and force. When the octonion coordinate system transforms rotationally, the vector part of one octonion may alter, while the scalar part of the octonion will remain unchanged. This property allows one to deduce a few invariants, such as the scalar part of octonion radius vector, speed of light, and norm of octonion radius vector. These invariants are the basic postulates for the Galilean transformation and Lorentz transformation. Further, from the rotation transform of octonion coordinate systems, it is capable of deducing several invariants, including the mass, energy and power relevant to gravitational fields, in one octonion space $\mathbb{O}$. And the term relevant to the electric charge will transform with the rotation of coordinate systems. In another octonion space $\mathbb{O}_u$, it is capable of inferring a few invariants, including the electric charge related to electromagnetic fields. And the terms relevant to the mass and energy will vary with the rotation of octonion coordinate systems. So the invariants are divided into two different groups. In particular, the mass conservation law and energy conservation law can be effective simultaneously. But the charge conservation law and mass conservation law are unable to be valid simultaneously, in the strict sense. It is beneficial to further understand the laws of conservation.