{"id":921,"date":"2022-11-17T23:04:24","date_gmt":"2022-11-18T04:04:24","guid":{"rendered":"http:\/\/www.fruechtetheory.com\/blog\/?p=921"},"modified":"2023-12-20T14:46:34","modified_gmt":"2023-12-20T19:46:34","slug":"the-coulomb-gauge","status":"publish","type":"post","link":"https:\/\/www.fruechtetheory.com\/blog\/2022\/11\/17\/the-coulomb-gauge\/","title":{"rendered":"The Coulomb Gauge\u00a0\u00a0\u00a0\u00a0"},"content":{"rendered":"\n

There is another name for a free graviton, – it is \u201cthe identity isomorphism idEx<\/sub>, here denoted 1x<\/sub>, and the elements 1x<\/sub>, x \u03f5 M, act as unities for any multiplication in which they can take part\u201d ([1], pg. 4). We see that unlike \u03c0, idEx<\/sub> has some degree of circular polarization and\/or skewed sine waves. In some writing instances \u03c0 is the same as idEx<\/sub> and I am not trying to dictate how they should be used.<\/p>\n\n\n\n

In the \u201cCoulomb<\/em>, radiation<\/em>, or transverse gauge<\/em>. This is the gauge in which \u2207 \u00b7 A<\/strong> = 0\u201d ([2], pg. 241), we have a classical description. In the tensor sense, we have the forms \u03a7ij<\/sub>. The direction we choose for \u03a7 is always transverse to the radial electric field at a chosen point, and the coordinate frame Ui<\/sub> is picked centered on the same point, creating a k-plane. We have that \u201cThe forms \u03a7ij<\/sub> are the transition forms<\/em> for the Lie algebroid atlas {Ui<\/sub>, \u03c8i<\/sub>, \u0398i<\/sup>}\u201d ([1], pg. 206), and \u0398i<\/sup> varies with the density of the gamma ray field:<\/p>\n\n\n\n

https:\/\/www.fruechtetheory.com\/blog\/2022\/10\/05\/the-vector-potential\/<\/a><\/p>\n\n\n\n

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 \u0398i<\/sup> decreases for a given charge and distance from the charge.<\/p>\n\n\n\n

In Jackson\u2019s problem 6.19 (b), \u201cthe original and space-inverted vector potential differ by a gauge transformation\u201d ([2], pg. 291). Though the earth catches some of the sun\u2019s gravitons all the time, the sun\u2019s gravitons during the day are greater at the face of the earth than at night, and inverted, changing the Coulomb gauge.<\/p>\n\n\n\n

With the \u201cLorenz condition<\/em> (1867), \u2207 \u00b7 A<\/strong> + (1\/c2<\/sup>) \u1e9f\u03c6\/dt = 0\u201d ([2], pg. 240), it is mathematically shown that the system {Ui<\/sub>, \u03c8i<\/sub>, \u0398i<\/sup>} acts fast compared to the gradient of A<\/strong>, and
           \u03b9X<\/sub> (\u03c6 \u02c4 \u03c8) = \u03b9X<\/sub>(\u03c6) \u02c4 \u03c8 + (-1)i<\/sup> \u03c6 \u02c4 \u03b9X<\/sub>(\u03c8)              ([1], pg. 306)
Also, as small as gravitons are, we may as well call the k-planes \u201cflat connections \u0398i<\/sup>\u201c ([1], pg. 206).<\/p>\n\n\n\n

Since we have \u201ct the fixed point set of \u03b8<\/em>\u201d ([3], pg. 401), t is on the center line of a gamma ray, and \u201cg0<\/sub> = t0<\/sub> + p0<\/sub> is a Cartan decomposition of<\/em> g0<\/sub>\u201c ([3], pg. 184). In certain situations the center can shift as well, in which case \u201cc0<\/sub> is the center of t0<\/sub>\u201d ([3], pg. 452) as t0<\/sub> moves back and forth.<\/p>\n\n\n\n

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 \u201cf: M<\/em> \u2192 H be a smooth map<\/em>\u201d ([1], pg. 183), and \u201cLet h be a proper subalgebra of g of maximum dimension\u201d ([3], pg. 160).<\/p>\n\n\n\n

Incidentally, the identity isomorphism reminds us of quantum 1:<\/p>\n\n\n\n

https:\/\/www.fruechtetheory.com\/blog\/2009\/09\/16\/the-fundamental-quantum-unit\/<\/a><\/p>\n\n\n\n

[1] Mackenzie, Kirill C. H., \u201cGeneral Theory of Lie Groupoids and Lie Algebroids\u201d, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society
[2] Jackson, J. D., \u201cClassical Electrodynamics, Third Edition\u201d, c. 1999 John David Jackson, John Wiley & Sons, Inc
[3] Helgason, Sigurdur, \u201cDifferential Geometry, Lie Groups, and Symmetric Spaces\u201d, American Mathematical Society, 2012<\/p>\n","protected":false},"excerpt":{"rendered":"

There is another name for a free graviton, – it is \u201cthe identity isomorphism idEx, here denoted 1x, and the elements 1x, x \u03f5 M, act as unities for any multiplication in which they can take part\u201d ([1], pg. 4). We see that unlike \u03c0, idEx has some degree of circular polarization and\/or skewed sine […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,18,16,5],"tags":[],"class_list":["post-921","post","type-post","status-publish","format-standard","hentry","category-classical-electrodynamics","category-mathematics","category-quantum-field-theory","category-quantum-mechanics"],"_links":{"self":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/921","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/comments?post=921"}],"version-history":[{"count":7,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/921\/revisions"}],"predecessor-version":[{"id":1046,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/921\/revisions\/1046"}],"wp:attachment":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/media?parent=921"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/categories?post=921"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/tags?post=921"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}