Aug 03 2023

## Ricci tensor field S

When a free electron accelerates, it may be able to increase in mass, charge, and diameter for the increase in work it must do. Again, we think of an emitting antenna.

We know that “s” can stand for spin, and that electrons have spin. The electrons in the antenna may impart spin into a “tensor space T_{s}^{r} “ ([1], pg. 209], where “r” is the vector away from the antenna, and “s” is the spin. What we can liken this to is a corkscrew in a gravitational field. Each corkscrew “s is a direct sum of simple ideals: s_{1} + … +s_{k}” ([2], Appendix 5, pg. 279)

To send these corkscrews out in all directions from an antenna is a phenomenal amount of work. It is not absolutely necessary that accelerated free electrons expand for this to occur, though they would at least absorb gravitons at a greater rate than a free electron at rest or traveling at constant velocity in a straight line.

It is not known what percentage of these corkscrews would be left-handed. When two electrons are near each other, “(β^{i}) is invariant by the left translation” ([2], pg. 207), and they repel each other.

Furthermore to the Ricci tensor field containing spin, there are the following two corollaries:

“Corollary 5.5 *If M is a compact Riemannian manifold with vanishing Ricci tensor field, then every infinitesimal isometry of M is a parallel vector field*.” ([2], pg. 251)

“Corollary 5.6 *If a connected compact homogeneous Riemannian manifold M has zero Ricci tensor, then M is a Euclidean torus*.” ([2], pg. 251)

As the output from an emitting antenna turns into tori, there is a “concatenation of paths” ([3], pg. 229].

Of course, as these tori wear out, they disintegrate, because of “a theorem of Weyl that any representation of a semisimple Lie algebra is completely reducible” ([2], Appendix 5, pg. 279]. We now know that these Lie algebras are reducible to gravitons.

[1] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society