(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 51089, 1476]*) (*NotebookOutlinePosition[ 82456, 2534]*) (* CellTagsIndexPosition[ 82412, 2530]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Hello Lubos: This notebook sets out to detail the unified field proposal. In effect, it \ serves as a \"mathematical grammar checker.\"\ \>", "Text"], Cell[CellGroupData[{ Cell["The Lagrange density", "Section"], Cell["\<\ These are the units of the components of the Lagrange \ density.\ \>", "Text"], Cell[BoxData[ \(\(units = {\[Gamma] \[Rule] 1, J \[Rule] m\^\(1/2\)\/\(\(L\^\(3/2\)\) t\), V \[Rule] L\^3, q \[Rule] \(m\^\(1/2\)\ L\^\(3/2\)\)\/t, U \[Rule] L\/t, A\^\[Mu] \[Rule] m\^\(1/2\)\/L\^\(1/2\), A\^\[Mu]\[Nu] \[Rule] m\^\(1/2\)\/\(t\ L\^\(1/2\)\), \@G -> L\^\(3/2\)\/\(m\^\(1/2\)\ t\), G \[Rule] L\^3\/\(m\ t\^2\), c \[Rule] L\/t, h \[Rule] \(m\ L\^2\)\/t};\)\)], "Input", CellLabel->"In[94]:="], Cell[TextData[{ "Evaluate the units of the three parts of the Lagrange density: the kinetic \ energy, the mass and electric charge in motion, and the asymmetric field \ strength tensor. If you are unfamiliar with ", StyleBox["Mathematica", FontSlant->"Italic"], ", \"/.\" indicates rules for substitution, so /. units means the unit \ rules will be used to substitute into the preceding expression. It is a way \ to check that all the parts of the Lagrange density have units of ", Cell[BoxData[ \(m\/L\^3\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(J\ A\^\[Mu]\)\/c /. units\)], "Input", CellLabel->"In[95]:="], Cell[BoxData[ \(m\/L\^3\)], "Output", CellLabel->"Out[95]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((A\^\[Mu]\[Nu]\/c\^2 /. units)\) A\^\[Mu]\[Nu] /. units\)], "Input", CellLabel->"In[96]:="], Cell[BoxData[ \(m\/L\^3\)], "Output", CellLabel->"Out[96]="] }, Open ]], Cell["Define the potential of the Lagrange density.", "Text"], Cell[BoxData[ \(\(A = {\[Phi][t, x, y, z], Ax[t, x, y, z], Ay[t, x, y, z], Az[t, x, y, z]};\)\)], "Input", CellLabel->"In[17]:="], Cell["\<\ The following functions are required to check if the symmetric plus \ antisymmetric field strength tensor is identical with the asymmetric tensor. \ One function flips the signs for covariant vectors. Four functions are \ required to take the contravariant and covariant derivative of a 4-vector \ with indices in uv or vu order. A function is needed to contract two rank 2 \ tensors: \ \>", "Text"], Cell[BoxData[ \(covariant[ A_] := {A[\([1]\)], \(-A[\([2]\)]\), \(-A[\([3]\)]\), \ \(-A[\([4]\)]\)}\)], "Input", CellLabel->"In[1]:=", InitializationCell->True], Cell[BoxData[ RowBox[{\(contraD[A_]\), ":=", RowBox[{"(", GridBox[{ {\(D[A[\([1]\)], t]\), \(D[A[\([2]\)], t]\), \(D[A[\([3]\)], t]\), \(D[A[\([4]\)], t]\)}, {\(\(-c\)\ D[A[\([1]\)], x]\), \(\(-c\)\ D[A[\([2]\)], x]\), \(\(-c\)\ D[A[\([3]\)], x]\), \(\(-c\)\ D[A[\([4]\)], x]\)}, {\(\(-c\)\ D[A[\([1]\)], y]\), \(\(-c\)\ D[A[\([2]\)], y]\), \(\(-c\)\ D[A[\([3]\)], y]\), \(\(-c\)\ D[A[\([4]\)], y]\)}, {\(\(-c\)\ D[A[\([1]\)], z]\), \(\(-c\)\ D[A[\([2]\)], z]\), \(\(-c\)\ D[A[\([3]\)], z]\), \(\(-c\)\ D[A[\([4]\)], z]\)} }], ")"}]}]], "Input", CellLabel->"In[2]:=", InitializationCell->True], Cell[BoxData[ RowBox[{\(coD[A_]\), ":=", RowBox[{"(", GridBox[{ {\(D[A[\([1]\)], t]\), \(D[A[\([2]\)], t]\), \(D[A[\([3]\)], t]\), \(D[A[\([4]\)], t]\)}, {\(c\ D[A[\([1]\)], x]\), \(c\ D[A[\([2]\)], x]\), \(c\ D[ A[\([3]\)], x]\), \(c\ D[A[\([4]\)], x]\)}, {\(c\ D[A[\([1]\)], y]\), \(c\ D[A[\([2]\)], y]\), \(c\ D[ A[\([3]\)], y]\), \(c\ D[A[\([4]\)], y]\)}, {\(c\ D[A[\([1]\)], z]\), \(c\ D[A[\([2]\)], z]\), \(c\ D[ A[\([3]\)], z]\), \(c\ D[A[\([4]\)], z]\)} }], ")"}]}]], "Input", CellLabel->"In[3]:=", InitializationCell->True], Cell[BoxData[ RowBox[{\(contraDvu[A_]\), ":=", RowBox[{"(", GridBox[{ {\(D[A[\([1]\)], t]\), \(\(-c\)\ D[A[\([1]\)], x]\), \(\(-c\)\ D[A[\([1]\)], y]\), \(\(-c\)\ D[A[\([1]\)], z]\)}, {\(D[A[\([2]\)], t]\), \(\(-c\)\ D[A[\([2]\)], x]\), \(\(-c\)\ D[A[\([2]\)], y]\), \(\(-c\)\ D[A[\([2]\)], z]\)}, {\(D[A[\([3]\)], t]\), \(\(-c\)\ D[A[\([3]\)], x]\), \(\(-c\)\ D[A[\([3]\)], y]\), \(\(-c\)\ D[A[\([3]\)], z]\)}, {\(D[A[\([4]\)], t]\), \(\(-c\)\ D[A[\([4]\)], x]\), \(\(-c\)\ D[A[\([4]\)], y]\), \(\(-c\)\ D[A[\([4]\)], z]\)} }], ")"}]}]], "Input", CellLabel->"In[4]:=", InitializationCell->True], Cell[BoxData[ RowBox[{\(coDvu[A_]\), ":=", RowBox[{"(", GridBox[{ {\(D[A[\([1]\)], t]\), \(c\ D[A[\([1]\)], x]\), \(c\ D[A[\([1]\)], y]\), \(c\ D[A[\([1]\)], z]\)}, {\(D[A[\([2]\)], t]\), \(c\ D[A[\([2]\)], x]\), \(c\ D[A[\([2]\)], y]\), \(c\ D[A[\([2]\)], z]\)}, {\(D[A[\([3]\)], t]\), \(c\ D[A[\([3]\)], x]\), \(c\ D[A[\([3]\)], y]\), \(c\ D[A[\([3]\)], z]\)}, {\(D[A[\([4]\)], t]\), \(c\ D[A[\([4]\)], x]\), \(c\ D[A[\([4]\)], y]\), \(c\ D[A[\([4]\)], z]\)} }], ")"}]}]], "Input", CellLabel->"In[5]:=", InitializationCell->True], Cell[BoxData[ \(contractMM[A_, B_] := Sum[A[\([i, j]\)]\ B[\([i, j]\)], {i, 1, 4}, {j, 1, 4}]\)], "Input", CellLabel->"In[6]:=", InitializationCell->True], Cell["\<\ Test the assertion that the contraction of the symmetric and \ antisymmetric tensors is equal to the contraction of the asymmetric tensor \ (will result in 0 if true):\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ contractMM[contraD[A] + contraDvu[A], coD[covariant[A]] + \ coDvu[covariant[A]]]\/\(8\ c\^2\) + contractMM[contraD[A] - contraDvu[A], coD[covariant[A]] - \ coDvu[covariant[A]]]\/\(8\ c\^2\) - contractMM[contraD[A], coD[covariant[A]]]\/\(2\ c\^2\)]\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[24]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Field equations", "Section"], Cell["Define the electric and mass current densities:", "Text"], Cell[BoxData[ \(\(J\_q = {\[Rho]\_q, J\_qx, J\_qy, J\_qz};\)\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(\(J\_m = {\[Rho]\_m, J\_mx, J\_my, J\_mz};\)\)], "Input", CellLabel->"In[19]:="], Cell["\<\ This package makes partial derivatives look like partial \ derivatives (it is not part of the standard set of packages).\ \>", "Text"], Cell[BoxData[ \(<< FormatPD.m\)], "Input", CellLabel->"In[7]:=", InitializationCell->True], Cell["Define the GEM Lagrange density:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((LGEM = \(-covariant[J\_q - J\_m] . A\)/c - Expand[contractMM[contraD[A], coD[covariant[A]]]\/\(2\ c\^2\)])\ \) /. {\[Phi][t, x, y, z] \[Rule] \[Phi], Ax[t, x, y, z] \[Rule] Ax, Ay[t, x, y, z] \[Rule] Ay, Az[t, x, y, z] \[Rule] Az}]\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \(\(-\(\(Ax\ J\_mx\)\/c\)\) - \(Ay\ J\_my\)\/c - \(Az\ J\_mz\)\/c + \(Ax\ \ J\_qx\)\/c + \(Ay\ J\_qy\)\/c + \(Az\ J\_qz\)\/c + \(\[Phi]\ \[Rho]\_m\)\/c - \ \(\[Phi]\ \[Rho]\_q\)\/c - 1\/2\ \((\[PartialD]Ax\/\[PartialD]z)\)\^2 - 1\/2\ \((\[PartialD]Ay\/\[PartialD]z)\)\^2 - 1\/2\ \((\[PartialD]Az\/\[PartialD]z)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]z)\)\^2 - 1\/2\ \((\[PartialD]Ax\/\[PartialD]y)\)\^2 - 1\/2\ \((\[PartialD]Ay\/\[PartialD]y)\)\^2 - 1\/2\ \((\[PartialD]Az\/\[PartialD]y)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]y)\)\^2 - 1\/2\ \((\[PartialD]Ax\/\[PartialD]x)\)\^2 - 1\/2\ \((\[PartialD]Ay\/\[PartialD]x)\)\^2 - 1\/2\ \((\[PartialD]Az\/\[PartialD]x)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]x)\)\^2 + \((\[PartialD]Ax\/\ \[PartialD]t)\)\^2\/\(2\ c\^2\) + \((\[PartialD]Ay\/\[PartialD]t)\)\^2\/\(2\ \ c\^2\) + \((\[PartialD]Az\/\[PartialD]t)\)\^2\/\(2\ c\^2\) - \((\[PartialD]\ \[Phi]\/\[PartialD]t)\)\^2\/\(2\ c\^2\)\)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell["\<\ Define functions to apply the Euler-Lagrange equations to a \ Lagrange density. The function potentialD takes the derivative of a Lagrange \ density with respect to the potential. 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\[Rho]\_q, \(-J\_mx\) + J\_qx, \(-J\_my\) + J\_qy, \(-J\_mz\) + J\_qz} == {c\ \[PartialD]\^2 \[Phi]\/\[PartialD]z\^2 + c\ \[PartialD]\^2 \[Phi]\/\[PartialD]y\^2 + c\ \[PartialD]\^2 \[Phi]\/\[PartialD]x\^2 - \(\[PartialD]\^2 \[Phi]\ \/\[PartialD]t\^2\)\/c, \(-c\)\ \[PartialD]\^2 Ax\/\[PartialD]z\^2 - c\ \[PartialD]\^2 Ax\/\[PartialD]y\^2 - c\ \[PartialD]\^2 Ax\/\[PartialD]x\^2 + \(\[PartialD]\^2 Ax\/\ \[PartialD]t\^2\)\/c, \(-c\)\ \[PartialD]\^2 Ay\/\[PartialD]z\^2 - c\ \[PartialD]\^2 Ay\/\[PartialD]y\^2 - c\ \[PartialD]\^2 Ay\/\[PartialD]x\^2 + \(\[PartialD]\^2 Ay\/\ \[PartialD]t\^2\)\/c, \(-c\)\ \[PartialD]\^2 Az\/\[PartialD]z\^2 - c\ \[PartialD]\^2 Az\/\[PartialD]y\^2 - c\ \[PartialD]\^2 Az\/\[PartialD]x\^2 + \(\[PartialD]\^2 Az\/\ \[PartialD]t\^2\)\/c}\)], "Output", CellLabel->"Out[40]="] }, Open ]], Cell["Rewrite as a 4D wave equation.", "Text"], Cell[BoxData[ \(\(J\_q - J\_m == \(\[EmptySquare]\^2\) A\^\[Mu];\)\)], "Input", CellLabel->"In[22]:="], Cell["\<\ Isolate Newton's field equation for gravity for the first 4D wave \ equation for the situation where q approaches zero and the second time \ derivative of phi is zero.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\((c\ \(potentialD[LGEM]\)[\([1]\)] /. \[Rho]\_q \[Rule] 0)\), "\[Equal]", RowBox[{"(", RowBox[{\(Expand[c\ \(fieldD[LGEM]\)[\([1]\)]]\), "/.", " ", RowBox[{ RowBox[{ SuperscriptBox["\[Phi]", TagBox[\((2, 0, 0, 0)\), Derivative], MultilineFunction->None], "[", \(t, x, y, z\), "]"}], "\[Rule]", "0"}]}], ")"}]}]], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \(\[Rho]\_m == c\ \[PartialD]\^2 \[Phi]\/\[PartialD]z\^2 + c\ \[PartialD]\^2 \[Phi]\/\[PartialD]y\^2 + c\ \[PartialD]\^2 \[Phi]\/\[PartialD]x\^2\)], "Output", CellLabel->"Out[41]="] }, Open ]], Cell["\<\ Isolate Gauss' law for the first 4D wave equation where m \ approaches zero.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((c\ \(potentialD[LGEM]\)[\([1]\)] /. \[Rho]\_m \[Rule] 0)\) \[Equal] Expand[c\ \(fieldD[LGEM]\)[\([1]\)]]\)], "Input", CellLabel->"In[42]:="], Cell[BoxData[ \(\(-\[Rho]\_q\) == c\ \[PartialD]\^2 \[Phi]\/\[PartialD]z\^2 + c\ \[PartialD]\^2 \[Phi]\/\[PartialD]y\^2 + c\ \[PartialD]\^2 \[Phi]\/\[PartialD]x\^2 - \(\[PartialD]\^2 \[Phi]\/\ \[PartialD]t\^2\)\/c\)], "Output", CellLabel->"Out[42]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Classical fields", "Section"], Cell["\<\ The long name \"EField\" for E must be used since E means 2.718... \ to Mathematica. Define the five classical fields that constitute the \ asymmetric tensor A^u,v\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Efield = {\(-D[Ax[t, x, y, z], t]\) - D[\[Phi][t, x, y, z], x], \(-D[Ay[t, x, y, z], t]\) - D[\[Phi][t, x, y, z], y], \(-D[Az[t, x, y, z], t]\) - D[\[Phi][t, x, y, z], z]}\), "\[IndentingNewLine]", \(e = {D[Ax[t, x, y, z], t] - D[\[Phi][t, x, y, z], x], D[Ay[t, x, y, z], t] - D[\[Phi][t, x, y, z], y], D[Az[t, x, y, z], t] - D[\[Phi][t, x, y, z], z]}\), "\[IndentingNewLine]", \(B = Curl[{Ax[t, x, y, z], Ay[t, x, y, z], Az[t, x, y, z]}]\), "\[IndentingNewLine]", \(b = {\(-D[Ay[t, x, y, z], z]\) - D[Az[t, x, y, z], y], \(-D[Ax[t, x, y, z], z]\) - D[Az[t, x, y, z], x], \(-D[Ax[t, x, y, z], y]\) - D[Ay[t, x, y, z], x]}\), "\[IndentingNewLine]", \(g = {D[\[Phi][t, x, y, z], t], \(-D[Ax[t, x, y, z], x]\), \(-D[Ay[t, x, y, z], y]\), \(-D[ Az[t, x, y, z], z]\)}\)}], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \({\(-\(\[PartialD]\[Phi]\/\[PartialD]x\)\) - \ \[PartialD]Ax\/\[PartialD]t, \(-\(\[PartialD]\[Phi]\/\[PartialD]y\)\) - \ \[PartialD]Ay\/\[PartialD]t, \(-\(\[PartialD]\[Phi]\/\[PartialD]z\)\) - \ \[PartialD]Az\/\[PartialD]t}\)], "Output", CellLabel->"Out[26]="], Cell[BoxData[ \({\(-\(\[PartialD]\[Phi]\/\[PartialD]x\)\) + \ \[PartialD]Ax\/\[PartialD]t, \(-\(\[PartialD]\[Phi]\/\[PartialD]y\)\) + \ \[PartialD]Ay\/\[PartialD]t, \(-\(\[PartialD]\[Phi]\/\[PartialD]z\)\) + \ \[PartialD]Az\/\[PartialD]t}\)], "Output", CellLabel->"Out[27]="], Cell[BoxData[ \({\(-\(\[PartialD]Ay\/\[PartialD]z\)\) + \[PartialD]Az\/\[PartialD]y, \ \[PartialD]Ax\/\[PartialD]z - \[PartialD]Az\/\[PartialD]x, \(-\(\[PartialD]Ax\ \/\[PartialD]y\)\) + \[PartialD]Ay\/\[PartialD]x}\)], "Output", CellLabel->"Out[28]="], Cell[BoxData[ \({\(-\(\[PartialD]Ay\/\[PartialD]z\)\) - \[PartialD]Az\/\[PartialD]y, \ \(-\(\[PartialD]Ax\/\[PartialD]z\)\) - \[PartialD]Az\/\[PartialD]x, \(-\(\ \[PartialD]Ax\/\[PartialD]y\)\) - \[PartialD]Ay\/\[PartialD]x}\)], "Output", CellLabel->"Out[29]="], Cell[BoxData[ \({\[PartialD]\[Phi]\/\[PartialD]t, \(-\(\[PartialD]Ax\/\[PartialD]x\)\), \ \(-\(\[PartialD]Ay\/\[PartialD]y\)\), \ \(-\(\[PartialD]Az\/\[PartialD]z\)\)}\)], "Output", CellLabel->"Out[30]="] }, Open ]], Cell["\<\ These functions are needed to determine the components of the \ asymmetric and symmetric field strength tensors.\ \>", "Text"], Cell[BoxData[{ RowBox[{\(contraDvu[A_]\), ":=", RowBox[{"(", GridBox[{ {\(D[A[\([1]\)], t]\), \(\(-c\)\ D[A[\([1]\)], x]\), \(\(-c\)\ D[A[\([1]\)], y]\), \(\(-c\)\ D[A[\([1]\)], z]\)}, {\(D[A[\([2]\)], t]\), \(\(-c\)\ D[A[\([2]\)], x]\), \(\(-c\)\ D[A[\([2]\)], y]\), \(\(-c\)\ D[A[\([2]\)], z]\)}, {\(D[A[\([3]\)], t]\), \(\(-c\)\ D[A[\([3]\)], x]\), \(\(-c\)\ D[A[\([3]\)], y]\), \(\(-c\)\ D[A[\([3]\)], z]\)}, {\(D[A[\([4]\)], t]\), \(\(-c\)\ D[A[\([4]\)], x]\), \(\(-c\)\ D[A[\([4]\)], y]\), \(\(-c\)\ D[A[\([4]\)], z]\)} }], ")"}]}], "\[IndentingNewLine]", \(symmetricD[A_] := contraD[A] + contraDvu[A]\), "\[IndentingNewLine]", \(antisymmetricD[ A_] := contraD[A] - contraDvu[A]\)}], "Input", CellLabel->"In[10]:=", InitializationCell->True], Cell["\<\ Write out the antisymmetric (E + B), symmetric (e + b + g), and a \ symmetric tensors (all) in terms of the individual components.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[antisymmetricD[A]]\)], "Input", CellLabel->"In[42]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { "0", \(c\ \[PartialD]\[Phi]\/\[PartialD]x + \[PartialD]Ax\/\ \[PartialD]t\), \(c\ \[PartialD]\[Phi]\/\[PartialD]y + \[PartialD]Ay\/\ \[PartialD]t\), \(c\ \[PartialD]\[Phi]\/\[PartialD]z + \[PartialD]Az\/\ \[PartialD]t\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]x - \[PartialD]Ax\/\ \[PartialD]t\), "0", \(c\ \[PartialD]Ax\/\[PartialD]y - c\ \[PartialD]Ay\/\[PartialD]x\), \(c\ \[PartialD]Ax\/\ \[PartialD]z - c\ \[PartialD]Az\/\[PartialD]x\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]y - \[PartialD]Ay\/\ \[PartialD]t\), \(\(-c\)\ \[PartialD]Ax\/\[PartialD]y + c\ \[PartialD]Ay\/\[PartialD]x\), "0", \(c\ \[PartialD]Ay\/\[PartialD]z - c\ \[PartialD]Az\/\[PartialD]y\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]z - \[PartialD]Az\/\ \[PartialD]t\), \(\(-c\)\ \[PartialD]Ax\/\[PartialD]z + c\ \[PartialD]Az\/\[PartialD]x\), \(\(-c\)\ \[PartialD]Ay\/\ \[PartialD]z + c\ \[PartialD]Az\/\[PartialD]y\), "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[42]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[symmetricD[A]]\)], "Input", CellLabel->"In[43]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(2\ \[PartialD]\[Phi]\/\[PartialD]t\), \(\(-c\)\ \[PartialD]\ \[Phi]\/\[PartialD]x + \[PartialD]Ax\/\[PartialD]t\), \(\(-c\)\ \[PartialD]\ \[Phi]\/\[PartialD]y + \[PartialD]Ay\/\[PartialD]t\), \(\(-c\)\ \[PartialD]\ \[Phi]\/\[PartialD]z + \[PartialD]Az\/\[PartialD]t\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]x + \[PartialD]Ax\/\ \[PartialD]t\), \(\(-2\)\ c\ \[PartialD]Ax\/\[PartialD]x\), \(\(-c\)\ \ \[PartialD]Ax\/\[PartialD]y - c\ \[PartialD]Ay\/\[PartialD]x\), \(\(-c\)\ \[PartialD]Ax\/\ \[PartialD]z - c\ \[PartialD]Az\/\[PartialD]x\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]y + \[PartialD]Ay\/\ \[PartialD]t\), \(\(-c\)\ \[PartialD]Ax\/\[PartialD]y - c\ \[PartialD]Ay\/\[PartialD]x\), \(\(-2\)\ c\ \ \[PartialD]Ay\/\[PartialD]y\), \(\(-c\)\ \[PartialD]Ay\/\[PartialD]z - c\ \[PartialD]Az\/\[PartialD]y\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]z + \[PartialD]Az\/\ \[PartialD]t\), \(\(-c\)\ \[PartialD]Ax\/\[PartialD]z - c\ \[PartialD]Az\/\[PartialD]x\), \(\(-c\)\ \[PartialD]Ay\/\ \[PartialD]z - c\ \[PartialD]Az\/\[PartialD]y\), \(\(-2\)\ c\ \ \[PartialD]Az\/\[PartialD]z\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[43]//MatrixForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\n\)\(Simplify[ MatrixForm[\((symmetricD[A] + antisymmetricD[A])\)/2]]\)\)\)], "Input", CellLabel->"In[44]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\[PartialD]\[Phi]\/\[PartialD]t\), \ \(\[PartialD]Ax\/\[PartialD]t\), \(\[PartialD]Ay\/\[PartialD]t\), \(\ \[PartialD]Az\/\[PartialD]t\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]x\), \(\(-c\)\ \ \[PartialD]Ax\/\[PartialD]x\), \(\(-c\)\ \[PartialD]Ay\/\[PartialD]x\), \ \(\(-c\)\ \[PartialD]Az\/\[PartialD]x\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]y\), \(\(-c\)\ \ \[PartialD]Ax\/\[PartialD]y\), \(\(-c\)\ \[PartialD]Ay\/\[PartialD]y\), \ \(\(-c\)\ \[PartialD]Az\/\[PartialD]y\)}, {\(\(-c\)\ \[PartialD]\[Phi]\/\[PartialD]z\), \(\(-c\)\ \ \[PartialD]Ax\/\[PartialD]z\), \(\(-c\)\ \[PartialD]Ay\/\[PartialD]z\), \ \(\(-c\)\ \[PartialD]Az\/\[PartialD]z\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[44]//MatrixForm="] }, Open ]], Cell["\<\ To do div, grad, curl and all that requires the vector analysis \ package:\ \>", "Text"], Cell[BoxData[ \(<< Calculus`VectorAnalysis`\)], "Input", CellLabel->"In[13]:=", InitializationCell->True], Cell[BoxData[ \(\(SetCoordinates[Cartesian[x, y, z]];\)\)], "Input", CellLabel->"In[14]:=", InitializationCell->True], Cell["GEM version of a unified Gauss' law:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(c\/2\) \((Div[Efield] + Div[e])\) + D[g[\([1]\)], t]\/c] \[Equal] \((J\_q - J\_m)\)[\([1]\)]\)], "Input",\ CellLabel->"In[48]:="], Cell[BoxData[ \(\(-c\)\ \[PartialD]\^2 \[Phi]\/\[PartialD]z\^2 - c\ \[PartialD]\^2 \[Phi]\/\[PartialD]y\^2 - c\ \[PartialD]\^2 \[Phi]\/\[PartialD]x\^2 + \(\[PartialD]\^2 \[Phi]\/\ \[PartialD]t\^2\)\/c == \(-\[Rho]\_m\) + \[Rho]\_q\)], "Output", CellLabel->"Out[48]="] }, Open ]], Cell["\<\ These sets of substitution rules are required to set DivE = 0 and \ Div e = 0:\ \>", "Text"], Cell[BoxData[ \(\(noEfield = {D[Ax[t, x, y, z], t] -> D[\[Phi][t, x, y, z], x], D[Ay[t, x, y, z], t] -> D[\[Phi][t, x, y, z], y], D[Az[t, x, y, z], t] -> D[\[Phi][t, x, y, z], z]};\)\)], "Input", CellLabel->"In[32]:="], Cell[BoxData[ \(\(noe = {D[Ax[t, x, y, z], t] \[Rule] \(-D[\[Phi][t, x, y, z], x]\), D[Ay[t, x, y, z], t] \[Rule] \(-D[\[Phi][t, x, y, z], y]\), D[Az[t, x, y, z], t] \[Rule] \(-D[\[Phi][t, x, y, z], z]\)};\)\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \(\(nogt = {D[\[Phi][t, x, y, z], t] \[Rule] 0};\)\)], "Input", CellLabel->"In[34]:="], Cell["\<\ If there is no divergence of the E field, no dynamic g, and no \ electric charge density, Newton's field equations for gravity results.\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(c\/2\) \((Div[Efield /. noEfield] + Div[e /. noEfield])\) + D[g[\([1]\)] /. nogt]\/c] \[Equal] \((J\_q - J\_m)\)[\([1]\)] /. \[Rho]\_q \[Rule] 0\)], "Input", CellLabel->"In[47]:="], Cell[BoxData[ \(\(-c\)\ \((\[PartialD]\^2 \[Phi]\/\[PartialD]z\^2 + \[PartialD]\^2 \ \[Phi]\/\[PartialD]y\^2 + \[PartialD]\^2 \[Phi]\/\[PartialD]x\^2)\) == \(-\ \[Rho]\_m\)\)], "Output", CellLabel->"Out[47]="] }, Open ]], Cell["\<\ If there is no divergence of the symmetric e filed and m is zero, \ Gauss' law results.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(c\/2\) \((Div[Efield /. noe] + Div[e /. noe])\) + D[g[\([1]\)], t]\/c] \[Equal] \((J\_q - J\_m)\)[\([1]\)] /. \[Rho]\_m \[Rule] 0\)], "Input", CellLabel->"In[46]:="], Cell[BoxData[ \(\(-c\)\ \[PartialD]\^2 \[Phi]\/\[PartialD]z\^2 - c\ \[PartialD]\^2 \[Phi]\/\[PartialD]y\^2 - c\ \[PartialD]\^2 \[Phi]\/\[PartialD]x\^2 + \(\[PartialD]\^2 \[Phi]\/\ \[PartialD]t\^2\)\/c == \[Rho]\_q\)], "Output", CellLabel->"Out[46]="] }, Open ]], Cell["The homogeneous Maxwell equations", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Div[B]\)], "Input", CellLabel->"In[53]:="], Cell[BoxData[ \(0\)], "Output", CellLabel->"Out[53]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Curl[Efield] + D[B, t]\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \({0, 0, 0}\)], "Output", CellLabel->"Out[54]="] }, Open ]], Cell[CellGroupData[{ Cell["A physically relevant 4D solution", "Subsection"], Cell["\<\ A function to test for solutions to the vacuum field \ equations:\ \>", "Text"], Cell[BoxData[ \(test[potential_] := Simplify[{D[potential[\([1]\)], {t, 2}] - D[potential[\([1]\)], {x, 2}] - D[potential[\([1]\)], {y, 2}] - D[potential[\([1]\)], {z, 2}], D[potential[\([2]\)], {t, 2}] - D[potential[\([2]\)], {x, 2}] - D[potential[\([2]\)], {y, 2}] - D[potential[\([2]\)], {z, 2}], D[potential[\([3]\)], {t, 2}] - D[potential[\([3]\)], {x, 2}] - D[potential[\([3]\)], {y, 2}] - D[potential[\([3]\)], {z, 2}], D[potential[\([4]\)], {t, 2}] - D[potential[\([4]\)], {x, 2}] - D[potential[\([4]\)], {y, 2}] - D[potential[\([4]\)], {z, 2}]}]\)], "Input", CellLabel->"In[16]:=", InitializationCell->True], Cell[TextData[{ "Test the 1/R potential. The units of a potential are ", Cell[BoxData[ \(\@m\/\@L\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(test[{\@\(h\/c\)\/\@\(x\^2 + y\^2 + z\^2\), 0, 0, 0}]\)], "Input", CellLabel->"In[56]:="], Cell[BoxData[ \({0, 0, 0, 0}\)], "Output", CellLabel->"Out[56]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(1\/L\) \@\(h\/c\) /. units\)], "Input", CellLabel->"In[57]:="], Cell[BoxData[ \(\@\(L\ m\)\/L\)], "Output", CellLabel->"Out[57]="] }, Open ]], Cell["Test a potential that is an inverse distance squared.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(test[{\(\(\@G\) h/c\^2\)\/\(x\^2 + y\^2 + z\^2 - t\^2\), 0, 0, 0}]\)], "Input", CellLabel->"In[58]:="], Cell[BoxData[ \({0, 0, 0, 0}\)], "Output", CellLabel->"Out[58]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\@G\) h\)\/\(\(c\^2\) L\^2\) /. units\)], "Input", CellLabel->"In[59]:="], Cell[BoxData[ \(\@m\/\@L\)], "Output", CellLabel->"Out[59]="] }, Open ]], Cell["\<\ The units imply this potential would involve relativistic quantum \ gravity.\ \>", "Text"], Cell["Test a potential that is a normalized linear perturbation.", "Text"], Cell[BoxData[ \(\(a = {\(c\/\@G\)\/\(\((1 + \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \ \((\(k\ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 \ - \((1 + \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\), 0, 0, 0};\)\)], "Input", CellLabel->"In[60]:="], Cell[CellGroupData[{ Cell[BoxData[ \(test[a]\)], "Input", CellLabel->"In[61]:="], Cell[BoxData[ \({0, 0, 0, 0}\)], "Output", CellLabel->"Out[61]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c\/\@G /. units\)], "Input", CellLabel->"In[62]:="], Cell[BoxData[ \(L\/\(\@\(L\^3\/\(m\ t\^2\)\)\ t\)\)], "Output", CellLabel->"Out[62]="] }, Open ]], Cell["\<\ These units imply relativistic gravity, but not quantum mechanics. \ Variations on this potential are the focus of the following work.\ \>", "Text"], Cell["Look at the derivatives to first order in k.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[D[a, t]] /. k\^2 \[Rule] 0\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[ \({\(2\ c\ k\)\/\(\@G\ \((\(-\((1 + \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \((1 \ + \(k\ x\)\/\[Sigma]\^2)\)\^2 + \((1 + \(k\ y\)\/\[Sigma]\^2)\)\^2 + \((1 + \ \(k\ z\)\/\[Sigma]\^2)\)\^2)\)\^2\ \[Sigma]\^2\), 0, 0, 0}\)], "Output", CellLabel->"Out[63]="] }, Open ]], Cell[TextData[{ "For a small oscillation, the denominator will be approximately ", Cell[BoxData[ \(2 \[Sigma]\^2\)]], ". This substitution list will make it so (and for other cases to come)." }], "Text"], Cell[BoxData[ \(\(sublist = {k\^2 \[Rule] 0, \(-\((1 + \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \((1 + \(k\ x\)\/\ \[Sigma]\^2)\)\^2 + \((1 + \(k\ y\)\/\[Sigma]\^2)\)\^2 + \((1 + \(k\ z\)\/\ \[Sigma]\^2)\)\^2 \[Rule] 2, \(-\((1 + \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \((1 - \(k\ x\)\/\ \[Sigma]\^2)\)\^2 + \((1 - \(k\ y\)\/\[Sigma]\^2)\)\^2 + \((1 - \(k\ z\)\/\ \[Sigma]\^2)\)\^2 \[Rule] 2, \(-\((1 - \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \((1 + \(k\ x\)\/\ \[Sigma]\^2)\)\^2 + \((1 - \(k\ y\)\/\[Sigma]\^2)\)\^2 + \((1 - \(k\ z\)\/\ \[Sigma]\^2)\)\^2 \[Rule] 2, \(-\((1 - \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \((1 - \(k\ x\)\/\ \[Sigma]\^2)\)\^2 + \((1 + \(k\ y\)\/\[Sigma]\^2)\)\^2 + \((1 - \(k\ z\)\/\ \[Sigma]\^2)\)\^2 \[Rule] 2, \[IndentingNewLine]\(-\((1 - \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \ \((1 - \(k\ x\)\/\[Sigma]\^2)\)\^2 + \((1 - \(k\ y\)\/\[Sigma]\^2)\)\^2 + \ \((1 + \(k\ z\)\/\[Sigma]\^2)\)\^2 \[Rule] 2, \[IndentingNewLine]\(-\((1 - \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \ \((1 + \(k\ x\)\/\[Sigma]\^2)\)\^2 + \((1 + \(k\ y\)\/\[Sigma]\^2)\)\^2 + \ \((1 + \(k\ z\)\/\[Sigma]\^2)\)\^2 \[Rule] 2, \[IndentingNewLine]\(-\((1 - \(k\ t\)\/\[Sigma]\^2)\)\^2\) + \ \((1 - \(k\ x\)\/\[Sigma]\^2)\)\^2 + \((1 - \(k\ y\)\/\[Sigma]\^2)\)\^2 + \ \((1 - \(k\ z\)\/\[Sigma]\^2)\)\^2 \[Rule] 2};\)\)], "Input", CellLabel->"In[63]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[D[a, t]] /. sublist\)], "Input", CellLabel->"In[65]:="], Cell[BoxData[ \({\(c\ k\)\/\(2\ \@G\ \[Sigma]\^2\), 0, 0, 0}\)], "Output", CellLabel->"Out[65]="] }, Open ]], Cell["\<\ Write a potential that solves the field equations, but only has \ derivatives along the diagonal.\ \>", "Text"], Cell[BoxData[ \(\(diagonalSHO = {1\/\(\((1 + \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \ \((\(k\ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 \ - \((1 + \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\) + 1\/\(\((1 - \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 - \((\(k\ \ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 - 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0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica"}, Hyphenation->True, ParagraphSpacing->{0, 8}, CounterIncrements->"Text"], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MathCaption"], CellFrame->{{4, 0}, {0, 0}}, CellMargins->{{47, 62}, {0, 14}}, CellFrameMargins->5, CellFrameColor->RGBColor[0, 0.2, 1], Hyphenation->True, LineSpacing->{1, 1}, ParagraphSpacing->{0, 8}, FontColor->RGBColor[0, 0, 0.6]], Cell[StyleData["MathCaption", "Printout"], CellMargins->{{34, 62}, {0, 14}}, CellFrameColor->GrayLevel[0.700008], FontSize->10, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Input/Output", "Subsection", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names. \ \>", "Text", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{56, 4}, {3, 9}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->StandardForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], Cell[StyleData["Input", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {3, 8}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{57, 4}, {5, 2}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->StandardForm, CounterIncrements->"Output", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], Cell[StyleData["Output", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {4, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{56, 4}, {3, 8}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0.2, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], Cell[StyleData["Message", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {4, 2}}, FontSize->10, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{56, 4}, {3, 8}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{21, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], Cell[StyleData["Print", "Printout"], ShowCellBracket->False, CellMargins->{{42, 4}, {4, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{56, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{40, 4}, {4, 2}}, ImageSize->{250, 250}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0.2, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], Cell[StyleData["CellLabel", "Printout"], FontSize->7, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Unique Styles", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["TextTop"], CellFrame->{{0, 0}, {0, 2}}, CellMargins->{{10, 4}, {2, 80}}, CellHorizontalScrolling->True, CellFrameMargins->4, ShowSpecialCharacters->Automatic, ParagraphSpacing->{0, 8}, CounterIncrements->"Text"], Cell[StyleData["TextTop", "Printout"], CellMargins->{{2, 4}, {2, 80}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ItemizedText"], CellMargins->{{20, 4}, {0, 8}}, ShowSpecialCharacters->Automatic, Hyphenation->True, ParagraphSpacing->{0, 8}, ParagraphIndent->-15, CounterIncrements->"Text"], Cell[StyleData["ItemizedText", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ItemizedTextNote"], CellMargins->{{35, 4}, {0, 4}}, ShowSpecialCharacters->Automatic, Hyphenation->True, ParagraphSpacing->{0, 4}, CounterIncrements->"Text"], Cell[StyleData["ItemizedTextNote", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["IndentedText"], CellMargins->{{20, 4}, {0, 6}}, ShowSpecialCharacters->Automatic, Hyphenation->True, ParagraphSpacing->{0, 8}, CounterIncrements->"Text"], Cell[StyleData["IndentedText", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Note"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ShowSpecialCharacters->Automatic, ParagraphSpacing->{0, 8}, CounterIncrements->"Text", FontFamily->"Helvetica", FontSize->10], Cell[StyleData["Note", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["1ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnSpacings->1}], Cell[StyleData["1ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["2ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->{0.39, 0.59}}], Cell[StyleData["2ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["2ColumnSmallBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnSpacings->1.5, ColumnWidths->0.35, ColumnAlignments->{Right, Left}}], Cell[StyleData["2ColumnSmallBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["3ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->0.325}], Cell[StyleData["3ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["3ColumnSmallBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnSpacings->1.5, ColumnWidths->0.23, ColumnAlignments->{Right, Center, Left}}], Cell[StyleData["3ColumnSmallBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["4ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->{0.145, 0.345, 0.145, 0.345}}], Cell[StyleData["4ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["5ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->0.195}], Cell[StyleData["5ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["6ColumnBox"], CellFrame->True, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, LineIndent->0, Background->GrayLevel[0.8], FrameBoxOptions->{BoxMargins->{{1, 1}, {1.5, 1.5}}}, GridBoxOptions->{ColumnWidths->{0.13, 0.23, 0.13, 0.13, 0.23, 0.13}}], Cell[StyleData["6ColumnBox", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Picture"], CellMargins->{{10, Inherited}, {0, 8}}, CellHorizontalScrolling->True], Cell[StyleData["Picture", "Printout"], CellMargins->{{2, Inherited}, {0, 8}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Caption"], CellMargins->{{10, 50}, {0, 3}}, PageBreakAbove->False, Hyphenation->True, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["Caption", "Printout"], CellMargins->{{2, 50}, {2, 4}}, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Tables", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["2ColumnTable"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, GridBoxOptions->{ColumnWidths->{0.39, 0.59}, ColumnAlignments->{Left}}], Cell[StyleData["2ColumnTable", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["3ColumnTable"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, StyleMenuListing->None, GridBoxOptions->{ColumnWidths->0.325, ColumnAlignments->{Left}}], Cell[StyleData["3ColumnTable", "Printout"], CellMargins->{{2, 4}, {0, 8}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["ChemicalFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Printout"], CellMargins->{{34, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, SpanMaxSize->Infinity, UnderoverscriptBoxOptions->{LimitsPositioning->True}, GridBoxOptions->{ColumnWidths->Automatic}], Cell[StyleData["DisplayFormula", "Printout"], CellMargins->{{34, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Printout"], CellMargins->{{2, Inherited}, {Inherited, Inherited}}, FontSize->9.5] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Subsection"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Subsection"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1717, 49, 160, 4, 48, "Text"], Cell[CellGroupData[{ Cell[1902, 57, 39, 0, 85, "Section"], Cell[1944, 59, 88, 3, 24, "Text"], Cell[2035, 64, 479, 8, 85, "Input"], Cell[2517, 74, 548, 12, 60, "Text"], Cell[CellGroupData[{ Cell[3090, 90, 85, 2, 45, "Input"], Cell[3178, 94, 66, 2, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3281, 101, 115, 2, 45, "Input"], Cell[3399, 105, 66, 2, 36, "Output"] }, Open ]], Cell[3480, 110, 61, 0, 24, "Text"], Cell[3544, 112, 146, 3, 27, "Input"], Cell[3693, 117, 412, 7, 56, "Text"], Cell[4108, 126, 176, 5, 27, "Input", InitializationCell->True], Cell[4287, 133, 799, 16, 81, "Input", InitializationCell->True], Cell[5089, 151, 678, 13, 81, "Input", InitializationCell->True], Cell[5770, 166, 814, 17, 81, "Input", InitializationCell->True], Cell[6587, 185, 750, 17, 81, "Input", InitializationCell->True], Cell[7340, 204, 169, 4, 27, "Input", InitializationCell->True], Cell[7512, 210, 191, 4, 24, "Text"], Cell[CellGroupData[{ Cell[7728, 218, 339, 7, 118, "Input"], Cell[8070, 227, 60, 2, 22, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[8179, 235, 34, 0, 85, "Section"], Cell[8216, 237, 63, 0, 24, "Text"], Cell[8282, 239, 102, 2, 28, "Input"], Cell[8387, 243, 102, 2, 28, "Input"], Cell[8492, 247, 144, 3, 24, "Text"], Cell[8639, 252, 98, 3, 27, "Input", InitializationCell->True], Cell[8740, 257, 48, 0, 24, "Text"], Cell[CellGroupData[{ Cell[8813, 261, 325, 5, 70, "Input"], Cell[9141, 268, 1084, 18, 127, "Output"] }, Open ]], Cell[10240, 289, 302, 5, 40, "Text"], Cell[10545, 296, 232, 5, 27, "Input", InitializationCell->True], Cell[10780, 303, 7384, 171, 135, "Input", InitializationCell->True], Cell[18167, 476, 50, 0, 24, "Text"], Cell[CellGroupData[{ 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