(*********************************************************************** 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[ 59927, 1697]*) (*NotebookOutlinePosition[ 91294, 2755]*) (* CellTagsIndexPosition[ 91250, 2751]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Hello: This notebook sets out to be a companion for five SPR posts investigating a \ particular Lagrange density. In effect, it serves as a \"mathematical \ grammar checker.\"\ \>", "Text"], Cell[CellGroupData[{ Cell["Post I: 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 substition, so /. unts 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 idential 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 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["Post II: 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] . 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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["Post III: 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["Post IV: Quantization", "Subsection"], Cell["The classical EM Lagrange density:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((LEM = \(-covariant[J\_q] . A\)/c - Expand[\(1\/\(4\ c\^2\)\) \((contractMM[ contraD[A] - contraDvu[A], coD[covariant[A]] - coDvu[covariant[A]]])\)])\) /. {\[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[137]:="], Cell[BoxData[ \(\(Ax\ J\_qx\)\/c + \(Ay\ J\_qy\)\/c + \(Az\ J\_qz\)\/c - \(\[Phi]\ \ \[Rho]\_q\)\/c - 1\/2\ \((\[PartialD]Ax\/\[PartialD]z)\)\^2 - 1\/2\ \((\[PartialD]Ay\/\[PartialD]z)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]z)\)\^2 - 1\/2\ \((\[PartialD]Ax\/\[PartialD]y)\)\^2 + \ \[PartialD]Ay\/\[PartialD]z\ \[PartialD]Az\/\[PartialD]y - 1\/2\ \((\[PartialD]Az\/\[PartialD]y)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]y)\)\^2 + \[PartialD]Ax\/\ \[PartialD]y\ \[PartialD]Ay\/\[PartialD]x - 1\/2\ \((\[PartialD]Ay\/\[PartialD]x)\)\^2 + \ \[PartialD]Ax\/\[PartialD]z\ \[PartialD]Az\/\[PartialD]x - 1\/2\ \((\[PartialD]Az\/\[PartialD]x)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]x)\)\^2 + \(\[PartialD]\[Phi]\/\ \[PartialD]x\ \[PartialD]Ax\/\[PartialD]t\)\/c + \ \((\[PartialD]Ax\/\[PartialD]t)\)\^2\/\(2\ c\^2\) + \(\[PartialD]\[Phi]\/\ \[PartialD]y\ \[PartialD]Ay\/\[PartialD]t\)\/c + \ \((\[PartialD]Ay\/\[PartialD]t)\)\^2\/\(2\ c\^2\) + \(\[PartialD]\[Phi]\/\ \[PartialD]z\ \[PartialD]Az\/\[PartialD]t\)\/c + \ \((\[PartialD]Az\/\[PartialD]t)\)\^2\/\(2\ c\^2\)\)], "Output", CellLabel->"Out[137]="] }, Open ]], Cell[TextData[{ "A function to calculate the generalized momentum: The only tricky part \ here is that ", StyleBox["Mathematica", FontSlant->"Italic"], " does not want to treat a partial derivative as a dervitive variable, so \ substitutions need to be made, then undone." }], "Text"], Cell[BoxData[ \(momentum[L_] := Module[{noPd, toPd, mo}, noPd = {\[PartialD]\_t \[Phi][t, x, y, z] \[Rule] dphidt, \[PartialD]\_x \[Phi][t, x, y, z] \[Rule] dphidx, \[PartialD]\_y \[Phi][t, x, y, z] \[Rule] dphidy, \[PartialD]\_z \[Phi][t, x, y, z] \[Rule] dphidz, \[PartialD]\_t Ax[t, x, y, z] \[Rule] dAxdt, \[PartialD]\_x Ax[t, x, y, z] \[Rule] dAxdx, \[PartialD]\_y Ax[t, x, y, z] \[Rule] dAxdy, \[PartialD]\_z Ax[t, x, y, z] \[Rule] dAxdz, \[PartialD]\_t Ay[t, x, y, z] \[Rule] dAydt, \[PartialD]\_x Ay[t, x, y, z] \[Rule] dAydx, \[PartialD]\_y Ay[t, x, y, z] \[Rule] dAydy, \[PartialD]\_z Ay[t, x, y, z] \[Rule] dAydz, \[PartialD]\_t Az[t, x, y, z] \[Rule] dAzdt, \[PartialD]\_x Az[t, x, y, z] \[Rule] dAzdx, \[PartialD]\_y Az[t, x, y, z] \[Rule] dAzdy, \[PartialD]\_z Az[t, x, y, z] \[Rule] dAzdz}; \[IndentingNewLine]toPd = {dphidt \[Rule] \ \[PartialD]\_t \[Phi][t, x, y, z], dphidx \[Rule] \[PartialD]\_x \[Phi][t, x, y, z], dphidy \[Rule] \[PartialD]\_y \[Phi][t, x, y, z], dphidz \[Rule] \[PartialD]\_z \[Phi][t, x, y, z], dAxdt \[Rule] \[PartialD]\_t Ax[t, x, y, z], dAxdx \[Rule] \[PartialD]\_x Ax[t, x, y, z], dAxdy \[Rule] \[PartialD]\_y Ax[t, x, y, z], dAxdz \[Rule] \[PartialD]\_z Ax[t, x, y, z], dAydt \[Rule] \[PartialD]\_t Ay[t, x, y, z], dAydx \[Rule] \[PartialD]\_x Ay[t, x, y, z], dAydy \[Rule] \[PartialD]\_y Ay[t, x, y, z], dAydz \[Rule] \[PartialD]\_z Ay[t, x, y, z], dAzdt \[Rule] \[PartialD]\_t Az[t, x, y, z], dAzdx \[Rule] \[PartialD]\_x Az[t, x, y, z], dAzdy \[Rule] \[PartialD]\_y Az[t, x, y, z], dAzdz \[Rule] \[PartialD]\_z Az[t, x, y, z]}; \[IndentingNewLine]mo = {D[L /. noPd, dphidt], D[L /. noPd, dAxdt], D[L /. noPd, dAydt], D[L /. noPd, dAzdt]}; \[IndentingNewLine]mo /. toPd\[IndentingNewLine]]\)], "Input", CellLabel->"In[138]:=", InitializationCell->True], Cell["\<\ Calculate the generalized 4-momentum of the classical EM Lagrange \ density.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(momentum[LEM]\)], "Input", CellLabel->"In[139]:="], Cell[BoxData[ \({0, \(\[PartialD]\[Phi]\/\[PartialD]x\)\/c + \(\[PartialD]Ax\/\ \[PartialD]t\)\/c\^2, \(\[PartialD]\[Phi]\/\[PartialD]y\)\/c + \ \(\[PartialD]Ay\/\[PartialD]t\)\/c\^2, \(\[PartialD]\[Phi]\/\[PartialD]z\)\/c \ + \(\[PartialD]Az\/\[PartialD]t\)\/c\^2}\)], "Output", CellLabel->"Out[139]="] }, Open ]], Cell["\<\ Calculate the generalized 4-momentum of the GEM Lagrange density.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(momentum[LGEM]\)], "Input", CellLabel->"In[140]:="], Cell[BoxData[ \({\(-\(\(\[PartialD]\[Phi]\/\[PartialD]t\)\/c\^2\)\), \(\[PartialD]Ax\/\ \[PartialD]t\)\/c\^2, \(\[PartialD]Ay\/\[PartialD]t\)\/c\^2, \ \(\[PartialD]Az\/\[PartialD]t\)\/c\^2}\)], "Output", CellLabel->"Out[140]="] }, Open ]], Cell["Define the Gupta/Bleuler Lagrange density.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((LGB = \(-covariant[J\_q] . A\)/ c - \(1\/\(2 c\^2\)\) \((D[A[\([1]\)], t] + c\ D[A[\([2]\)], x] + c\ \ D[A[\([3]\)], y] + c\ D[A[\([4]\)], z])\)\^2 - Expand[\(1\/\(4\ c\^2\)\) \((contractMM[ contraD[A] - contraDvu[A], coD[covariant[A]] - coDvu[covariant[A]]])\)])\) /. {\[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[141]:="], Cell[BoxData[ \(\(Ax\ J\_qx\)\/c + \(Ay\ J\_qy\)\/c + \(Az\ J\_qz\)\/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 - \ \[PartialD]Az\/\[PartialD]z\ \[PartialD]Ay\/\[PartialD]y - 1\/2\ \((\[PartialD]Ay\/\[PartialD]y)\)\^2 + \ \[PartialD]Ay\/\[PartialD]z\ \[PartialD]Az\/\[PartialD]y - 1\/2\ \((\[PartialD]Az\/\[PartialD]y)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]y)\)\^2 - \[PartialD]Az\/\ \[PartialD]z\ \[PartialD]Ax\/\[PartialD]x - \[PartialD]Ay\/\[PartialD]y\ \ \[PartialD]Ax\/\[PartialD]x - 1\/2\ \((\[PartialD]Ax\/\[PartialD]x)\)\^2 + \ \[PartialD]Ax\/\[PartialD]y\ \[PartialD]Ay\/\[PartialD]x - 1\/2\ \((\[PartialD]Ay\/\[PartialD]x)\)\^2 + \ \[PartialD]Ax\/\[PartialD]z\ \[PartialD]Az\/\[PartialD]x - 1\/2\ \((\[PartialD]Az\/\[PartialD]x)\)\^2 + 1\/2\ \((\[PartialD]\[Phi]\/\[PartialD]x)\)\^2 + \(\[PartialD]\[Phi]\/\ \[PartialD]x\ \[PartialD]Ax\/\[PartialD]t\)\/c + \ \((\[PartialD]Ax\/\[PartialD]t)\)\^2\/\(2\ c\^2\) + \(\[PartialD]\[Phi]\/\ \[PartialD]y\ \[PartialD]Ay\/\[PartialD]t\)\/c + \ \((\[PartialD]Ay\/\[PartialD]t)\)\^2\/\(2\ c\^2\) + \(\[PartialD]\[Phi]\/\ \[PartialD]z\ \[PartialD]Az\/\[PartialD]t\)\/c + \ \((\[PartialD]Az\/\[PartialD]t)\)\^2\/\(2\ c\^2\) - \(\[PartialD]Az\/\ \[PartialD]z\ \[PartialD]\[Phi]\/\[PartialD]t\)\/c - \(\[PartialD]Ay\/\ \[PartialD]y\ \[PartialD]\[Phi]\/\[PartialD]t\)\/c - \(\[PartialD]Ax\/\ \[PartialD]x\ \[PartialD]\[Phi]\/\[PartialD]t\)\/c - \((\[PartialD]\[Phi]\/\ \[PartialD]t)\)\^2\/\(2\ c\^2\)\)], "Output", CellLabel->"Out[141]="] }, Open ]], Cell["Calculate its generalized 4-momentum.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(momentum[LGB]\)], "Input", CellLabel->"In[142]:="], Cell[BoxData[ \({\(-\(\(c\ \[PartialD]Az\/\[PartialD]z + c\ \[PartialD]Ay\/\[PartialD]y + c\ \[PartialD]Ax\/\[PartialD]x + \ \[PartialD]\[Phi]\/\[PartialD]t\)\/c\^2\)\), \ \(\[PartialD]\[Phi]\/\[PartialD]x\)\/c + \ \(\[PartialD]Ax\/\[PartialD]t\)\/c\^2, \(\[PartialD]\[Phi]\/\[PartialD]y\)\/c \ + \(\[PartialD]Ay\/\[PartialD]t\)\/c\^2, \ \(\[PartialD]\[Phi]\/\[PartialD]z\)\/c + \ \(\[PartialD]Az\/\[PartialD]t\)\/c\^2}\)], "Output", CellLabel->"Out[142]="] }, Open ]], Cell["\<\ Calculate the field equations for the Gupta/Bleuler Lagrange \ density using the Euler-Lagrange equation.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Expand[c\ potentialD[LGB]]] == Expand[c\ fieldD[LGB]]\)], "Input", CellLabel->"In[143]:="], Cell[BoxData[ \({\(-\[Rho]\_q\), J\_qx, J\_qy, 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[143]="] }, Open ]], Cell["This can be rewritten.", "Text"], Cell[BoxData[ \(\(J\_q\^\[Mu] == \(\[EmptySquare]\^2\) A\^\[Mu];\)\)], "Input", CellLabel->"In[22]:="] }, Open ]], Cell[CellGroupData[{ Cell["Post V: Why hasn't this been done before?", "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[15]:=", 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};\)\)], "Input", CellLabel->"In[64]:="], 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 - \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 - \((1 + \ \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\), 1\/\(\((1 + \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ \ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 - \((1 + \ \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\) + 1\/\(\((1 + \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 - \((\(k\ \ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 - \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 - \((1 - \ \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\), 1\/\(\((1 + \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ \ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 - \((1 + \ \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\) + 1\/\(\((1 - \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ \ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 - \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 - \((1 - \ \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\), 1\/\(\((1 + \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ \ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 - \((1 + \ \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\) + 1\/\(\((1 - \((\(k\ x\)\/\[Sigma]\^2)\))\)\^2 + \((1 - \((\(k\ \ y\)\/\[Sigma]\^2)\))\)\^2 + \((1 + \((\(k\ z\)\/\[Sigma]\^2)\))\)\^2 - \((1 - \ \((\(k\ t\)\/\[Sigma]\^2)\))\)\^2\)};\)\)], "Input", CellLabel->"In[66]:="], Cell[CellGroupData[{ Cell[BoxData[ \(test[diagonalSHO]\)], "Input", CellLabel->"In[67]:="], Cell[BoxData[ \({0, 0, 0, 0}\)], "Output", CellLabel->"Out[67]="] }, Open ]], Cell["Take the derivative of the potential.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[Expand[contraD[diagonalSHO]] /. sublist]\)], "Input", CellLabel->"In[68]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(k\/\[Sigma]\^2\), "0", "0", "0"}, {"0", \(\(c\ k\)\/\[Sigma]\^2\), "0", "0"}, {"0", "0", \(\(c\ k\)\/\[Sigma]\^2\), "0"}, {"0", "0", "0", \(\(c\ k\)\/\[Sigma]\^2\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", CellLabel->"Out[68]//MatrixForm="] }, Open ]], Cell["\<\ Use this derivative of a potential in a Lorentz-like force \ law\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(contractVM[V_, M_] := {Sum[V[\([i]\)]\ M[\([1, i]\)], {i, 1, 4}], Sum[V[\([i]\)]\ M[\([2, i]\)], {i, 1, 4}], Sum[V[\([i]\)]\ M[\([3, i]\)], {i, 1, 4}], Sum[V[\([i]\)]\ M[\([4, i]\)], {i, 1, 4}]}\)], "Input", CellLabel->"In[16]:=", InitializationCell->True], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(contractVM\)\" is \ similar to existing symbol \"\!\(contractMM\)\"."\)], "Message", CellLabel->"From In[16]:="] }, Open ]], Cell[BoxData[ \(\(\(v = {U\_0[\[Tau]], U\_1[\[Tau]], U\_2[\[Tau]], U\_3[\[Tau]]};\)\(\ \)\)\)], "Input", CellLabel->"In[70]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(-m\)\ c\ contractVM[covariant[v], Expand[contraD[diagonalSHO]] /. sublist]]\)], "Input", CellLabel->"In[71]:="], Cell[BoxData[ \({\(-\(\(c\ k\ m\ U\_0[\[Tau]]\)\/\[Sigma]\^2\)\), \(c\^2\ k\ m\ U\_1[\ \[Tau]]\)\/\[Sigma]\^2, \(c\^2\ k\ m\ U\_2[\[Tau]]\)\/\[Sigma]\^2, \(c\^2\ k\ \ m\ U\_3[\[Tau]]\)\/\[Sigma]\^2}\)], "Output", CellLabel->"Out[71]="] }, Open ]], Cell[TextData[{ "Replace ", Cell[BoxData[ \(\[Sigma]\^2\)]], "with ", Cell[BoxData[ \(\(-\[Tau]\^2\)\)]], ". 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CellElementSpacings->{"ClosedGroupTopMargin"->18}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 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"}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->18, FontWeight->"Bold"], Cell[StyleData["Section", "Printout"], CellMargins->{{2, 4}, {2, 80}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{24, 4}, {2, 18}}, CellElementSpacings->{"ClosedGroupTopMargin"->12}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameLabelMargins->6, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 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"}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->15, FontWeight->"Bold"], Cell[StyleData["Subsection", "Printout"], CellMargins->{{2, 4}, {2, 18}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellMargins->{{10, 4}, {2, 18}}, CellElementSpacings->{"ClosedGroupTopMargin"->12}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 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"}, CounterIncrements->"Subsubsection", FontFamily->"Helvetica", FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{2, 4}, {2, 18}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Text", "Subsection"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{10, 4}, {0, 8}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 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. 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