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He \ had to rely on his old math teacher Minkowski to learn the theory of \ transformations (I do not know the details of Einstein's education, but it \ could make an interesting discussion :-) Eventually, Einstein understood \ general transformations, embodied in the work of Riemann, well enough to \ formulate general relativity.\ \>", "Text"], Cell["\<\ A. W. Conway and L. Silberstein proposed a different mathematical \ structure behind special relativity in 1911 and 1912 respectively (a copy of \ Silberstein's work is on the web. Henry Baker has made it available at \ ftp://ftp.netcom.com/pub/hb/hbaker/quaternions/). Cayley had observed back \ in 1854 that rotations in 3D could be achieved using a pair of quaternions \ with a norm of one:\ \>", "Text"], Cell[BoxData[{ \(q'\ = \ a\ q\ b\), \(where\ \ \(a\^*\)\ a\ = \ \(\(b\^*\) b\ = \ 1\)\)}], "Input", FontSize->16], Cell["\<\ If this works in 3D space, why not do the 4D transformations of \ special relativity? It turns out that a and b must be complex-valued \ quaternions, or biquaternions. Is this so bad? Let me quote P.A.M. Dirac \ (Proc. Royal Irish Academy A, 1945, 50, p. 261):\ \>", "Text"], Cell["\<\ \"Quaternions themselves occupy a unique place in mathematics in \ that they are the most general quantities that satisfy the division axiom--that the product of two factors cannot vanish without either factor \ vanishing. Biquaternions do not satisfy this axiom, and do not have any \ fundamental property which distinguishes them from other hyper-complex \ numbers. Also, they have eight components, which is rather too many for a \ simple scheme for describing quantities in space-time.\"\ \>", "Text"], Cell["\<\ Just for the record: plenty of fine work has been done with \ biquaternions, and I do not deny the validity of any of it. Much effort has \ been directed toward \"other hyper-complex numbers\", such as Clifford \ algebras. For the record, I am making a choice to focus on quaternions for \ reasons outlined by Dirac.\ \>", "Text"], Cell["\<\ Dirac took a Mobius transformation from complex analysis and tried \ to develop a quaternion analog. The approach is too general, and must be \ restricted to graft the results to the Lorentz group. I personally have \ found this approach hard to follow, and have yet to build a working model of \ it in Mathematica. I needed something simpler :-)\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Rotation + Dilation", "Subsection"], Cell["\<\ Multiplication of complex numbers can be thought of as a rotation \ and a dilation. Conway and Silberstein's proposals only have the rotation \ component. An additional dilation term might allow quaternions to do the \ necessary work.\ \>", "Text"], Cell[TextData[ "C. M\[ODoubleDot]ller wrote a general form for a Lorentz transformation \ using vectors (\"The Theory of Relativity\", QC6 F521, 1952, eq. 25). For \ fixed collinear coordinate systems:"], "Text"], Cell[BoxData[{ RowBox[{\(\(X\& \[RightVector] \)'\), " ", "=", " ", RowBox[{\(X\& \[RightVector] \), " ", "+", " ", RowBox[{ \((\[Gamma]\ - \ 1)\), \((\(V\& \[RightVector] \).\(X\& \[RightVector] \))\), FractionBox[\(V\& \[RightVector] \), RowBox[{"|", \(V\& \[RightVector] \), SuperscriptBox[ StyleBox["|", FontSize->16], "2"]}]]}], " ", "-", " ", \(\[Gamma]\ t\ \(V\& \[RightVector] \)\)}]}], \(t'\ = \ \[Gamma]\ t\ - \ \[Gamma] \((\(V\& \[RightVector] \).\(X\& \[RightVector] \))\)\), \(where\ c = 1, \ \[Gamma]\ = \ 1\/\@\(1\ - \ \((v/c)\)\^2\)\)}], "Input",\ FontSize->16], Cell["If V is only in the i direction, then", "Text"], Cell[BoxData[{ \(\(X\& \[RightVector] \)'\ = \ \((\[Gamma] \( X\& \[RightVector] \)\ - \ \[Gamma]\ t\ \(V\& \[RightVector] \)\ )\) \(i\&^\) + \ y\ \(j\&^\)\ + \ z\ \(k\&^\)\), \(t'\ = \ \[Gamma]\ t\ - \ \[Gamma] \((\(V\& \[RightVector] \).\(X\& \[RightVector] \))\)\)}], "Input", FontSize->16], Cell["\<\ The additional complication to the X' equation handles velocities \ in different directions than i.\ \>", "Text"], Cell["\<\ This has a vector equation and a scalar equation. A quaternion \ equation that would generate these terms must be devoid of any terms \ involving cross products. The symmetric product (anti-commutator) lacks the \ cross product;\ \>", "Text"], Cell[BoxData[ \({q, \ q'}\ = \ \(\(q\ q'\ + \ q'\ q\)\/2\ = \ \((t\ t'\ - \ \(X\& \[RightVector] \).\(X\& \[RightVector] \)', \ t\ \(X\& \[RightVector] \)\ + \ \(X\& \[RightVector] \)\ t') \)\)\)], "Input", FontSize->16], Cell[TextData[ "M\[ODoubleDot]ller's equation looks like it should involve two terms, one of \ the form AqA (a rotation), the other Bq (a dilation)."], "Text"], Cell[BoxData[ RowBox[{\(q'\), " ", "=", " ", RowBox[{ RowBox[{"q", "+", " ", RowBox[{\((\[Gamma]\ - \ 1)\), FractionBox[ \({{\(\(V\& \[RightVector] \)\^*\), q}, \(V\& \[RightVector] \)} \ \), RowBox[{"|", \(V\& \[RightVector] \), SuperscriptBox[ StyleBox["|", FontSize->16], "2"]}]]}], " ", "+", " ", \(\[Gamma] {\(\(V\& \[RightVector] \)\^*\), \(q\^*\)}\)}], "\n", "=", " ", RowBox[{ RowBox[{"q", "+", " ", RowBox[{\((\[Gamma]\ - \ 1)\), FractionBox[ \({\((\(V\& \[RightVector] \).\(X\& \[RightVector] \), \(-t\)\ V)\), \((0, \(V\& \[RightVector] \))\)}\ \), RowBox[{"|", \(V\& \[RightVector] \), SuperscriptBox[ StyleBox["|", FontSize->16], "2"]}]]}], " ", "+", " ", \(\[Gamma] {\((0, \(-\(V\& \[RightVector] \)\))\), \((t, \(-\(X\& \[RightVector] \)\))\)}\)}], "\n", "=", " ", RowBox[{\((t, \(X\& \[RightVector] \))\), " ", "+", " ", RowBox[{\((\[Gamma]\ - \ 1)\), RowBox[{"(", RowBox[{"t", ",", RowBox[{ \((\(V\& \[RightVector] \).\(X\& \[RightVector] \))\), FractionBox[\(V\& \[RightVector] \), RowBox[{"|", \(V\& \[RightVector] \), SuperscriptBox[ StyleBox["|", FontSize->16], "2"]}]]}]}], ")"}]}], " ", "-", " ", \(\[Gamma] \((\((\(V\& \[RightVector] \).\(X\& \[RightVector] \))\), \ t\ \(V\& \[RightVector] \))\)\)}]}]}]}]], "Input", FontSize->16], Cell[TextData[ "This is the general form of the Lorentz transformation presented by M\ \[ODoubleDot]ller. Real quaternions are used in a rotation and a dialation \ to perform the work of the Lorentz group."], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Implications", "Subsection"], Cell["\<\ Is this result at all interesting? A straight rewrite of Moller's \ equation would have been dull. What is interesting is the equation which \ generates the Lorentz transformation. Notice how the Lorentz transformation \ depends linearly on q, but the generator depends on q and q*. That may have \ interesting interpretations. The generator involves only symmetric products. \ There has been some question in the literature about whether special \ relativity handles rotations correctly. This is probably one of the more \ confusing topics in physics, so I will just let the observation stand by \ itself.\ \>", "Text"], Cell["\<\ Two ways exist to use quaternions to do Lorentz transformations (to \ be discussed in the next web page). The other technique relies on the \ property of a division algebra. There exists a quaternion L such that:\ \>", "Text"], Cell[BoxData[{ \(q'\ = \ L\ q\), \(such\ that\ \ \ scalar \((q', \ q')\)\ = \ \(scalar \((q, \ q)\)\ = \ t\^2 - \ \(X\& \[RightVector] \).\(X\& \[RightVector] \)\)\)}], "Input", FontSize->16], Cell["For a boost along the i direction,", "Text"], Cell[BoxData[{ \(L\ = \(\(\ q'\)\/q\ = \ \(\(\((\[Gamma]\ t\ - \ \[Gamma]\ v\ x, \ \(-\[Gamma]\)\ v\ t\ + \ \[Gamma]\ x, \ y, \ z)\) \((t, \(-x\), \(-y\), \(-z\))\)\)\/\(( t\^2 + \ x\^2 + \ y\^2 + \ z\^2)\)\n = \((\[Gamma]\ t\^2 - \ 2 \[Gamma]\ t\ v\ x + \ \[Gamma]\ x\^2\ + y\^2 + z\^2, \ \[Gamma]\ v \((\(-t\^2\) + \ x\^2)\), \n \t\t\t\t\t\t t\ y\ - \ x\ z\ - \ \[Gamma]\ t \((y\ + \ v\ z)\)\ + \ \[Gamma]\ x \((v\ y\ + \ z)\), \n\t\t\t\t\t\t t\ z\ + \ xy\ + \ \[Gamma]\ t \((v\ y\ - \ z)\)\ + \ \[Gamma]\ x \((\(-y\)\ + \ v\ z)\))\)/ \((t\^2 + \ x\^2 + \ y\^2 + \ z\^2)\)\)\)\n\t\t\t\t\t\), \(if\ x\ = \ \(y\ = \ \(z\ = \ 0\)\), \ then\ \ L\ = \ \((\[Gamma], \ \(-\[Gamma]\)\ v, 0, 0)\)\), \(if\ t\ = \ \(y\ = \ \(z\ = \ 0\)\), \ then\ \ L\ = \((\[Gamma], \ \[Gamma]\ v, 0, 0)\)\)}], "Input", FontSize->16], Cell["\<\ The quaternion L depends on the velocity and can depend on location \ in spacetime (85% of the type of problems assigned undergraduates in special \ relativity use an L that does not depend on location in spacetime). Some \ people view that as a bug, but I see it as a modern feature found in the \ standard model and general relativity as the demand that all symmetry is \ local. The existence of two approaches may be of interest in itself.\ \>", "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 800}, {0, 566}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{680, 357}, WindowMargins->{{16, Automatic}, {33, Automatic}}, 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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