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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 10969, 268]*) (*NotebookOutlinePosition[ 40587, 1334]*) (* CellTagsIndexPosition[ 40543, 1330]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The stress-energy tensor of the electomagnetic field", "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell["doug ", "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Introduction Generating a symmetric 2-tensor using quaternions Implications\ \>", "Text"], Cell[CellGroupData[{ Cell[" Introduction", "Subsection"], Cell[TextData[ "I will outline a way to generate the terms of the symmetric 2-rank \ stress-energy-momentum tensor of an electromagnetic field using quaternions. \ This method may provide some insight into what information the \ stress-energy-momentum tensor contains."], "Text"], Cell[TextData[ "Any equation written with 4-vectors can be rewritten with quaternions. A \ straight translation of terms could probably be automated with a computer \ program. What is more interesting is when an equation is generated by the \ product of operators acting on quaternion fields. I have found that \ generator equations often yield useful insights."], "Text"], Cell["\<\ A tensor is a bookkeeping device designed to keep together elements \ that transform in a similar way. People can choose alternative bookkeeping \ systems, so long as the tensor behaves the same way under transformations. \ Using the terms as defined in \"The classical theory of fields\" by Landau \ and Lifshitz, the antisymmetric 2-rank field tensor F is used to generate the \ stress-energy- momentum tensor T\ \>", "Text"], Cell[BoxData[ \(T\^ik\ = \ \(\(1\ \)\/\(4\ \[Pi]\)\) \((\(-\ F\^iL\) \(F\^k\)\_L\ + \ \(1\/4\) \(\[Delta]\^ik\) \(F\_LM\) F\^LM)\)\)], "Input"], Cell[TextData[ "I have a practical sense of an E field (the stuff that makes my hair stand \ on end) and a B field (the invisible hand directing a compass), but have \ little sense of the field tensor F, a particular combination of the other \ two. Therefore, express the stress-energy-momentum tensor T in terms of the \ E and B fields only:"], "Text"], Cell[BoxData[{ RowBox[{\(T\^ik\), " ", "=", RowBox[{"(", GridBox[{ {"W", "Sx", "Sy", "Sz"}, {"Sx", "mxx", "mxy", "myz"}, {"Sy", "myx", "myy", "myz"}, {"Sz", "mzx", "mzy", "mzz"} }], ")"}]}], \(where\ with\ a\ and\ b\ running\ thru\ x, \ y\ and\ z\), \(W\ = \ \(1\/\(8 \[Pi]\)\) \((\(E\& \[RightVector] \)\^2 + \ \(B\& \[RightVector] \)\^2)\)\ \ The \ energy\ density\), \(Sa\ = \ \(1\/\(4 \[Pi]\)\) \((\(E\& \[RightVector] \)\ x \( B\& \[RightVector] \))\)\ \ \ The\ Poynting\ vector\), \(mab\ = \ \(1\/\(4 \[Pi]\)\) \((\(-Ea\)\ Eb\ - \ Ba\ Bb + \ 0.5\ \(\[Delta]\_ab\) \((\(E\& \[RightVector] \)\^2 + \ \(B\& \[RightVector] \)\^2)\)) \)\ \ \ The\ Maxwell\ stress\ tensor\)}], "Input"], Cell["\<\ Together, the energy density, Poynting vector and the Maxwell \ stress tensor are all the components of the stress-energy-momentum tensor of \ the electromagnetic field.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Generating a symmetric 2-tensor using quaternions", "Subsection"], Cell[TextData[ "How should one rationally go about to find a generator equation that creates \ these terms instead of using the month-long hunt-and-peck technique actually \ used? Everything is symmetric, so use the symmetric product:"], "Text"], Cell[BoxData[ \({q, \ q'}\ = \(\(\ q\ q'\ + \ q'\ q\ \)\/2 = \ \((t\ t'\ - \ \(X\& \[RightVector] \).\(X\& \[RightVector] \)', \ t\ \(X\& \[RightVector] \)\ + \ \(X\& \[RightVector] \)\ t') \)\)\)], "Input"], Cell[TextData[ "The fields E and B are kept separate except for the cross product in the \ Poynting vector. Individual directions of a field can be selected by using a \ unit vector Ua:"], "Text"], Cell[BoxData[ \(\({\(E\& \[RightVector] \), \ Ux}\ = \ \(\((\(-Ex\), 0)\)\ \ \ where\ Ux\ = \ \((0, 1, 0, 0)\)\)\ \ \ \)\)], "Input"], Cell[TextData[ "The following double sum generates all the terms of the \ stress-energy-momentum tensor:"], "Text"], Cell[BoxData[ \(e = q[0, Ex, \ Ey, Ez]; \nb = q[0, Bx, By, Bz]; \nux = q[0, 1, 0, 0]; \n uy = q[0, 0, 1, 0]; \nuz = q[0, 0, 0, 1]; \n Simplify[\(( \((\((qxeven[ux, ux] + \ qxeven[uy, uy] + qxeven[uz, uz])\)/3 + q[\(-1\), 0, 0, 0])\)\ .\((e.e\ + \ b.b)\)/2\ \n - \ qxeven[e, ux].qxeven[e, ux] - \ qxeven[e, uy].qxeven[e, ux]\ - qxeven[e, uz].qxeven[e, ux]\ \n - \ qxeven[e, ux].qxeven[e, uy]\ - \ qxeven[e, uy].qxeven[e, uy] - \ qxeven[e, uz].qxeven[e, uy]\ \n - qxeven[e, ux].qxeven[e, uz]\ - \ qxeven[e, uy].qxeven[e, uz] - \ qxeven[e, uz].qxeven[e, uz]\n\t\t - \ qxeven[b, ux].qxeven[b, ux] - \ qxeven[b, uy].qxeven[b, ux]\ - qxeven[b, uz].qxeven[b, ux]\ \n - \ qxeven[b, ux].qxeven[b, uy] - \ qxeven[b, uy].qxeven[b, uy]\ - \ qxeven[b, uz].qxeven[b, uy]\ \n - qxeven[b, ux].qxeven[b, uz]\ - \ qxeven[b, uy].qxeven[b, uz] - \ qxeven[b, uz].qxeven[b, uz])\)\ [[1, 1]]]\n - 2\ \((By\ Bz + Bx\ \((By + Bz)\) + Ex\ Ey + Ex\ Ez + Ey\ Ez)\)\)], "Input"], Cell[BoxData[ \(T\^ik\ = \(\[CapitalSigma]\+\(a = x\)\%\(y, z\)\ \ \[CapitalSigma]\+\(b = x\)\%\(y, z\)\ \(1\/\(4 \[Pi]\)\) \((\(({Ua, Ub}\/3 - 1)\) \((\((0, E)\)\^2 + \((0, B)\)\^2)\)\/2\n - \ {E, \ Ua} {E, \ Ub}\ - \ {B, \ Ua} {B, \ Ub}\n - \ {\([E, \ B]\), \ Ua}\ - \ {\([E, \ B]\), \ Ub})\)\n = \ \((\(-Ex\)\ Ey\ - \ Ex\ Ez\ - \ Ey\ Ez\ - \ Bx\ By\ - \ Bx\ Bz\ - \ By\ Bz\n\ \ \ \ \ \ + \ Ey\ Bz\ - \ Ez\ By\ + \ Ez\ Bx\ - \ Ex\ Bz\ + \ Ex\ By\ - \ Ey\ Bx, \ 0)\)/2\ \[Pi]\)\)], "Input"], Cell[TextData[ "The first line generates the energy density W, and part of the +0.5 delta(a, \ b)(E^2 + B^2) term of the Maxwell stress tensor. The rest of that tensor is \ generated by the second line. The third line creates the Poynting vector. \ Using quaternions, the net sum of these terms ends up in the scalar."], "Text"], Cell[TextData[ "Does the generator equation have the correct properties? Switching the \ order of Ua and Ub leaves T unchanged, so it is symmetric. Check the trace, \ when Ua = Ub"], "Text"], Cell[BoxData[{ \(Simplify[ \((\((\((qxeven[ux, ux] + \ qxeven[uy, uy] + qxeven[uz, uz])\)/3 + q[\(-1\), 0, 0, 0])\)\ .\((e.e\ + \ b.b)\)/2\ \n - \ qxeven[e, ux].qxeven[e, ux]\n - \ qxeven[e, uy].qxeven[e, uy]\n - \ qxeven[e, uz].qxeven[e, uz]\n - \ qxeven[b, ux].qxeven[b, ux]\n - \ qxeven[b, uy].qxeven[b, uy]\ \ \n - \ qxeven[b, uz].qxeven[b, uz])\)\ [[1, 1]]]\), \(0\)}], "Input"], Cell[BoxData[ \(trace \((T\^ik)\)\ = \(\[CapitalSigma]\+\(a = x\)\%\(y, z\)\ \ \(1\/\(4 \[Pi]\)\) \((\(({Ua, Ua}\/3 - 1)\) \((\((0, E)\)\^2 + \((0, B)\)\^2)\)\/2 - \ {E, \ Ua}\^2\ - \ {B, \ Ua}\^2)\)\n = \ 0\)\)], "Input"], Cell[TextData["The trace equals zero, as it should."], "Text"], Cell[TextData[ "The generator is composed of three parts that have different dependencies on \ the unit vectors: those terms that involve Ua and Ub, those that involve Ua \ or Ub, and those that involve neither. These are the Maxwell stress tensor, \ the Poynting vector and the energy density respectively. Changing the basis \ vectors Ua and Ub will effect these three components differently."], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Implications", "Subsection"], Cell["\<\ So what does the stress-energy-momentum tensor represent? It looks \ like every combination of the 3-vectors E and B that avoids quadratics (like \ Ex^2) and over-counting cross terms. I like what I will call the \"net\" \ stress-energy-momentum quaternion:\ \>", "Text"], Cell[BoxData[ \(net \((T\^ik)\) = \ \((\(-Ex\)\ Ey\ - \ Ex\ Ez\ - \ Ey\ Ez\ - \ Bx\ By\ - \ Bx\ Bz\ - \ By\ Bz\n\ \ \ \ \ \ + \ Ey\ Bz\ - \ Ez\ By\ + \ Ez\ Bx\ - \ Ex\ Bz\ + \ Ex\ By\ - \ Ey\ Bx, \ 0)\)/2\ \[Pi]\)], "Input"], Cell[TextData[ "This has the same properties as an stress-energy-momentum tensor. Since the \ vector is zero, it commutes with any other quaternion (this may be a reason \ it is so useful). Switching x terms for y terms would flip the signs of the \ terms produced by the Poynting vector as required, but not the others. There \ are no terms of the form Ex^2, which is equivalent to the statement that the \ trace of the tensor is zero."], "Text"], Cell[TextData[ "On a personal note, I never thought I would understand what a symmetric \ 2-rank tensor was, even though I listen in on a discussion of the topic. \ Yes, I could nod along with the algebra, but without any sense of F, it felt \ hollow. Now that I have a generator and a net quaternion expression, it \ looks quite elegant and straightforward to me."], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 800}, {0, 566}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{685, 330}, WindowMargins->{{19, Automatic}, {Automatic, 34}}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{False, Inherited}, PrintingOptions->{"PrintingMargins"->{{72, 57.5625}, {57.5625, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False, "FirstPageHeader"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->False, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. 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If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. 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