Nov 17 2022

## The Coulomb Gauge

There is another name for a free graviton, – it is “the identity isomorphism id_{Ex}, here denoted 1_{x}, and the elements 1_{x}, x ϵ M, act as unities for any multiplication in which they can take part” ([1], pg. 4). We see that unlike π, id_{Ex} has some degree of circular polarization and/or skewed sine waves. In some writing instances π is the same as id_{Ex} and I am not trying to dictate how they should be used.

In the “*Coulomb*, *radiation*, or *transverse gauge*. This is the gauge in which **∇ · A** = 0” ([2], pg. 241), we have a classical description. In the tensor sense, we have the forms Χ_{ij}. The direction we choose for Χ is always transverse to the radial electric field at a chosen point, and the coordinate frame U_{i} is picked centered on the same point, creating a k-plane. We have that “The forms Χ_{ij} are the *transition forms* for the Lie algebroid atlas {U_{i}, ψ_{i}, Θ^{i}}” ([1], pg. 206), and Θ^{i} varies with the density of the gamma ray field:

https://www.fruechtetheory.com/blog/2022/10/05/the-vector-potential/

Considering the transition form TP/G [1], we may here call G the density of the gravitational field. It is seen that as the density goes up the transition angle Θ^{i} decreases for a given charge and distance from the charge.

In Jackson’s problem 6.19 (b), “the original and space-inverted vector potential differ by a gauge transformation” ([2], pg. 291). Though the earth catches some of the sun’s gravitons all the time, the sun’s gravitons during the day are greater at the face of the earth than at night, and inverted, changing the Coulomb gauge.

With the “*Lorenz condition* (1867), **∇ · A** + (1/c^{2}) ẟφ/dt = 0” ([2], pg. 240), it is mathematically shown that the system {U_{i}, ψ_{i}, Θ^{i}} acts fast compared to the gradient of **A**, and

ι_{X} (φ ˄ ψ) = ι_{X}(φ) ˄ ψ + (-1)^{i} φ ˄ ι_{X}(ψ) ([1], pg. 306)

Also, as small as gravitons are, we may as well call the k-planes “flat connections Θ^{i}“ ([1], pg. 206).

Since we have “t the fixed point set of *θ*” ([3], pg. 401), t is on the center line of a gamma ray, and “g_{0} = t_{0} + p_{0} *is a Cartan decomposition of* g_{0}“ ([3], pg. 184). In certain situations the center can shift as well, in which case “c_{0} is the center of t_{0}” ([3], pg. 452) as t_{0} moves back and forth.

With the polarization factor, it is interesting to call h the vector summation of two gamma ray electric fields. When a gravitational field is yet more compact, h is the summation of more than 2 electric fields, so that “f: *M* → *H be a smooth map*” ([1], pg. 183), and “Let h be a proper subalgebra of g of maximum dimension” ([3], pg. 160).

Incidentally, the identity isomorphism reminds us of quantum 1:

https://www.fruechtetheory.com/blog/2009/09/16/the-fundamental-quantum-unit/

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

[2] Jackson, J. D., “Classical Electrodynamics, Third Edition”, c. 1999 John David Jackson, John Wiley & Sons, Inc

[3] Helgason, Sigurdur, “Differential Geometry, Lie Groups, and Symmetric Spaces”, American Mathematical Society, 2012