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      "title": "Maxwell Stress Tensor [*].",
      "body": "That the current laws of electromagnetism can be encapsulated within a syntactically dense equation is delightful, and also that this tensor fits within the paradigm of special relativity. I have commented that a larger tensor should be constructed with which to account for ∃magnetic monopole, especially within Black Hole. Today I will focus mainly on the rank-2 version that is more commonly understood. \nThis can be thought of as a matrix, however it is with the additional structure such that both indices transform under the rotation of a Lorentz. One must transform both ranks to perform a change into a different observer frame. In this sense, although seperately, the electric field **E** and magnetic field **B** are subjective to the observer, when combined as to **E**⊗**B**, this combined object is then one field which is preserved via flat changes in velocity and position (or a pure change of observer). \nBy this I mean that an observer is changed sufficiently slowly such that the acceleration be negligible upon this curvature such that we might remain within the special relativity paradigm (It has been effectively argued that acceleration can be accounted for within S.R., however this is not central to my discussion at this moment). \nMathematically, one can perform a pure change of observer from one frame to another without disrupting the physical system. However, such a physical change of observation frame would likely disrupt the curvature, thus causing a ripple in the purely flat approximation and falling under general relativity and more (a new theory or everything containing the observer effect). The Maxwell equations (here the permittivity and permeability are set to unity):\n\n<a href=\"https://www.codecogs.com/eqnedit.php?latex=\\large&space;\\left\\{&space;\\nabla&space;\\cdot&space;\\textbf&space;E=0=\\nabla&space;\\cdot&space;\\textbf&space;B;\\\\&space;\\nabla&space;\\times&space;\\textbf&space;E=-\\partial&space;\\textbf&space;B/\\partial&space;t;&space;\\nabla&space;\\textbf&space;E=\\partial&space;\\times&space;\\textbf&space;E/\\partial&space;t\\right\\}\" target=\"_blank\"><img src=\"https://latex.codecogs.com/gif.latex?\\large&space;\\left\\{&space;\\nabla&space;\\cdot&space;\\textbf&space;E=0=\\nabla&space;\\cdot&space;\\textbf&space;B;\\\\&space;\\nabla&space;\\times&space;\\textbf&space;E=-\\partial&space;\\textbf&space;B/\\partial&space;t;&space;\\nabla&space;\\textbf&space;E=\\partial&space;\\times&space;\\textbf&space;E/\\partial&space;t\\right\\}\" title=\"\\large \\left\\{ \\nabla \\cdot \\textbf E=0=\\nabla \\cdot \\textbf B;\\\\ \\nabla \\times \\textbf E=-\\partial \\textbf B/\\partial t; \\nabla \\textbf E=\\partial \\times \\textbf B/\\partial t\\right\\}\" /></a>\n\nOne sees that one could construct a certain something that would make ∇.(**E**⊗**B**)=0 and ∇x(**E**⊗**B**)=±∂(**B**⊗**E**)/∂t . This certain something is represented as the vector potential:\n<a href=\"https://www.codecogs.com/eqnedit.php?latex=\\large&space;A^\\mu&space;\\equiv&space;(\\textbf&space;E\\otimes&space;\\textbf&space;B)\" target=\"_blank\"><img src=\"https://latex.codecogs.com/gif.latex?\\large&space;A^\\mu&space;\\equiv&space;(\\textbf&space;E\\otimes&space;\\textbf&space;B)\" title=\"\\large A^\\mu \\equiv (\\textbf E\\otimes \\textbf B)\" /></a>\n\nSuch that under the Einstein summation convention:\n<a href=\"https://www.codecogs.com/eqnedit.php?latex=\\large&space;\\sum&space;_\\mu&space;\\limits&space;\\left(&space;\\frac{\\partial&space;A^\\mu&space;}{\\partial&space;x^\\mu&space;}\\equiv&space;\\partial&space;_\\mu&space;A^\\mu&space;\\right&space;)\\\\&space;\\equiv&space;\\left[\\partial&space;_\\mu&space;A^\\mu\\right](*)\" target=\"_blank\"><img src=\"https://latex.codecogs.com/gif.latex?\\large&space;\\sum&space;_\\mu&space;\\limits&space;\\left(&space;\\frac{\\partial&space;A^\\mu&space;}{\\partial&space;x^\\mu&space;}\\equiv&space;\\partial&space;_\\mu&space;A^\\mu&space;\\right&space;)\\\\&space;\\equiv&space;\\left[\\partial&space;_\\mu&space;A^\\mu\\right](*)\" title=\"\\large \\sum _\\mu \\limits \\left( \\frac{\\partial A^\\mu }{\\partial x^\\mu }\\equiv \\partial _\\mu A^\\mu \\right )\\\\ \\equiv \\left[\\partial _\\mu A^\\mu\\right](*)\" /></a>\n\nTherefore the flat electrodynamics reduces to [*], a single symbol for an entire theory.",
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      "body": "This is a lecture note. The flipping of electric dipoles can communicate informations between two points in space-time (Q--->P). Be aware that a sec0nd-order differential equation requires two initial conditions (boundary?). The membrane paradigm is equivalent by definition or otherwise (≡) to a BlackHole EventHorizon being like a conducting membrane such that Black Holes are really perfectly spherically symmetric (? ¿)/\n\nTherefore, the entire physics of Black Hole (B.H.) might be described by (M,L,Q), where Q is known as the electric charge. See also ``Black Hole Zoo''. The questions stilll remain; for example; what happens to  the collapsing entropy? Do Black Holes have/are magnetic monopoles? We have the following equations (which are not important to the thesis):\n![CodeCogsEqn.gif](https://steemitimages.com/DQmRhanp1sLxpVn96REKbqMTeXJUD3gcgAQuujVFct3xh2g/CodeCogsEqn.gif)\n\nThis is because the Action on Q.P=2 Pi n for some integer n.\n\nDoes there exist a curl (∇ x *) version of the DiracDelta: δ(P-Q).\n\nThe laws of Electromagnetism(E.M.) must therefore be rederived. At the location of the monopole, no vector potential exists!\n\nDoes stellar collapse not form a parallel universe? Try throwing a goat into the B.H. from a spacecraft to see what happens. Each point on the Penrose Diagram is surrounded by a perfect sphere, but at its center is a singularity. At the moment, naked singularities are not believed to be physical, here we have a \"vacuum breakdown\" in the equations.\n\n ![IMG_20180516_151326068.jpg](https://steemitimages.com/DQmeiPVEaASmENnQK93My2vc9zC5NLE696KZ5Ty3bX3u3Hr/IMG_20180516_151326068.jpg)\n\nRecall that within General Relativity all forms of energy gravitate. See also the \"Cauchy Horizon.\"\n\n\"The nabla is a triangular symbol like an inverted Greek delta: or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp, and was suggested by the encyclopedist William Robertson Smith to Peter Guthrie Tait in correspondence.\nNabla symbol - Wikipedia\"\nhttps://en.wikipedia.org/wiki/Nabla_symbol\n\n![download.jpg](https://steemitimages.com/DQmQJCzQuKrAGbb3AtjGQciSYfkH8gSGSGesKgtdQuBQWN6/download.jpg)",
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      "title": "Why does a magnetic circuit conduct magnetic field lines.",
      "body": "I was pondering this question as to why magnetic circuits conduct magnetic field lines in Fall 2017. \nThis is a progress report (rough idea), I have not peer reviewed these ideas nor completed the calculations.\nI plan to invest more work (and program some real graphs) if the question keeps coming up in my researches,\nor if this post generates interest from the community. If the dear reader have any feedback, suggestions, or questions, feel free to comment.\n\n<!-- Prevalence of atomic orbitals, or intrinsic spin angular momentums? -->\n![dipole_field1_resized.jpg](https://steemitimages.com/DQmZRat7PuusnSBY4QfWd8fuwEPgAMP4aGXqVX5nH2VhTft/dipole_field1_resized.jpg)\n![dipole_field2_resized.jpg](https://steemitimages.com/DQmQ4TKZZzmopZRzsy8oPbKr3B1FE3B73y2qZwc1rmv3Mg6/dipole_field2_resized.jpg)\n\n\nThe claim is simply that the south of the magnetic dipoles in the circuit have a statistical tendency to point towards the north of nearby dipoles. \nA magnetic field line passing through the circuit will tend to (probabilistically) polarize the dipoles. We have the Bohr megneton:\n\n\n<img src=\"https://www.codecogs.com/gif.latex?\n\\mu _B=\\frac{e\\hbar }{2 m}\n\" />\n\n\nWe also have the energies of non-interacting spins of electrons (spin up or spin down) (Where B denotes the magnetic field strength):\n\n<img src=\"https://www.codecogs.com/gif.latex?\n\\varepsilon _\\pm =\\mp \\mu _B B\n\" />\n\nNext, we have that the applied magnetic field B has Boltzmann statistics (and tau=k_B T is the temperature in Boltzmann units) on alligned spins: \n\n\n<img src=\"https://www.codecogs.com/gif.latex?\\mathbb P(\\pm )=\\frac{e^{\\varepsilon _\\mp /\\tau }}{e^{\\varepsilon _+/\\tau }+e^{\\varepsilon _-/\\tau }}\"\n/>\n\n\nThis indicates that the probability of the spin aligning is more than the probability of anti-aligning.\n Each dipole moment then goes on to generate a magnetic field of its own, which dominates nearby dipoles:\n\n\n<img src=\"https://www.codecogs.com/gif.latex?\n\\mathbf{B}_{\\mu_B}=(3 (\\mathbf{\\mu }_B\\cdot \\hat r) \\hat r-\\mathbf{\\mu }_B)/r^3\n\" />\n<!-- Provide reference -->\n\n\nThis is how magnetic circuits can round-corners:\n![dipole_field3_resized.jpg](https://steemitimages.com/DQmXn2yADTgURG3AaLtMUJzrCkSNW5mWxw3My9SyhKLpVD2/dipole_field3_resized.jpg)\n\n\n<img src=\"https://www.codecogs.com/gif.latex?\n|\\psi >=\\sqrt{\\mathbb P_+}\\ |\\uparrow >-\\sqrt{\\mathbb P_-}\\ |\\downarrow >\n\" />\n \nAnother question that has been occupying my thoughts at night is: does B_applied constitute an observer?\n\nThe magnetization density will depend on the metal. See also: the Pauli exclusion principle. Also, in paramagnetic materials orbitals occupied with odd numbers of electrons can easily change spin. In diamagnetic materials, filled orbitals resist magnetic flux.",
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