(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 40769, 950]*) (*NotebookOutlinePosition[ 70201, 1987]*) (* CellTagsIndexPosition[ 70157, 1983]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The Maxwell equations for gravity", "Subtitle"], Cell[TextData[StyleBox["doug "]], "Subsubtitle"], Cell[TextData[{ StyleBox[ "Introduction\nConstraints on a gravity theory\nThe Lorentz gravitational \ force and the Maxwell equations of gravity\nLinks to Newton gravity and \ general relativity\nQuantum gravity\n"], StyleBox["Implications"] }], "Text"], Cell["Based on a post to sci.physics.research", "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction"]], "Subsection"], Cell["\<\ In this post, I will propose a new theory of gravity based on quaternion \ operators and potentials. Because the constraints on any theory of gravity \ are so tight,I feel I have no choices to make, just tight spots to slip \ through. Some of the consequences will be examined that make this proposal \ different from general relativity even in the classical region (good-bye dark \ matter??).\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Constraints on a gravity theory"]], "Subsection"], Cell["\<\ A proposal for gravity should satisfy the following three constraints: \ \>", "Text"], Cell[TextData[{ "1. For a classical point mass, reduces to Newton's law of gravity, F = - G \ ", Cell[BoxData[ \(TraditionalForm\`\(m\ m'\)\/r\^2\)]], ". \n2. The form should be manifestly covariant under a Lorentz \ transformation. \n3. The law should work with quantum mechanics. " }], "Text", CellMargins->{{35, 14}, {Inherited, Inherited}}], Cell["\<\ Electromagnetism can pass through classical, relativistic, and quantum \ constraints. Therefore, to achieve similar goals, a theory of gravity could \ clone the structure of electromagnetic theory to the letter. The laws \ governing electromagnetism--the Lorentz force and the Maxwell equations--can \ be written as operators acting on the same 4-potential.\ \>", "Text"], Cell[BoxData[{ \(The\ Lorentz\ force\), \(\(\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\)\+\(-- \(--\(--\(---\)\)\)\)\) \((\[Phi], \(-A\&\[RightVector]\))\) \(\(\((\[Gamma], \(-\[Gamma]\[Beta]\))\) - \ \((\[Gamma], \(-\[Gamma]\[Beta]\))\) \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\)\)\+\(-- \(--\(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(---- \)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\) \((\[Phi], A\&\[RightVector])\)\n = \(\[Gamma] \((\(-\[Beta]\)\[CenterDot]\(( \[EmptyDownTriangle]\&\[RightVector]\[InvisibleComma] \[Phi] + \[PartialD]A\&\[RightVector]\/\[PartialD]t)\), \(-\((\[EmptyDownTriangle]\&\[RightVector]\[InvisibleComma] \[Phi] + \[PartialD]A\&\[RightVector]\/\[PartialD]t) \)\) + \ \[Beta]\ X \((\(\[EmptyDownTriangle]\&\[RightVector]\) X A\&\[RightVector])\)\[InvisibleComma] )\)\n = \ \[Gamma]\ e \((\ \[Beta].E, E\ + \[Beta]\ \[Times]\ B)\)\)\)}], "Input", CellMargins->{{1, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{\(The\ Maxwell\ equations\), " ", "\n", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ UnderscriptBox[ RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \)}]}], \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)], ",", \((\[Phi], A\&\[RightVector])\)}]}]}], StyleBox["]", FontWeight->"Bold"]}], StyleBox["}", FontWeight->"Bold"]}], " ", "\n", "+", RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ UnderscriptBox[ RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"vector", RowBox[{ StyleBox["{", FontWeight->"Bold"], \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \)}]}]}], \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(--\(-- \(--\(--\(--- \)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)], ",", \((\[Phi], \(-A\&\[RightVector]\))\)}]}]}], StyleBox["}", FontWeight->"Bold"]}], StyleBox["]", FontWeight->"Bold"]}], "\n", "=", RowBox[{ RowBox[{"(", RowBox[{ \(\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]\), " ", ",", RowBox[{ \(-\[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", \(\[EmptyDownTriangle]\&\[RightVector]\), " ", SuperscriptBox["\[Phi]", TagBox["", Derivative], MultilineFunction->None]}]}], ")"}], "=", RowBox[{ \((\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]B \&\[RightVector], \[EmptyDownTriangle]\&\[RightVector]\ X\ E\&\[RightVector] + \[PartialD]B\&\[RightVector]\/\[PartialD]t)\), "=", RowBox[{\((0, \ 0\&\[RightVector])\), "\n", "\n", "\t\t", StyleBox["[", FontWeight->"Bold"], RowBox[{ UnderscriptBox[ RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\)}]}], \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(-\(---\)\)\)\)\)\)\)\)\)\)\)\)], ",", \((\[Phi], A\&\[RightVector])\)}], StyleBox["]", FontWeight->"Bold"]}]}]}]}], StyleBox["]", FontWeight->"Bold"]}], "\n", "\t", "-", " ", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ UnderscriptBox[ RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"vector", RowBox[{ StyleBox["{", FontWeight->"Bold"], \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\)}]}]}], \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(-- \(--\(- \(--\(---- \)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)], ",", \((\[Phi], \(-A\&\[RightVector]\))\)}], StyleBox["}", FontWeight->"Bold"]}]}], StyleBox["}", FontWeight->"Bold"]}], "\n", "=", \(\((\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ \[Phi]\ - \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\[PartialD]A \&\[RightVector]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]\ X\ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]\ + \[PartialD]\^2 A\&\[RightVector]\/\[PartialD]t\^2 + \(\[PartialD]\[EmptyDownTriangle]\&\[RightVector]\[InvisibleComma]\ \[Phi]\)\/\[PartialD]t)\) = 4\ \[Pi]\ \((\[Rho], J\&\[RightVector])\)\)}]], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ If the underlined operators act on a different potential, then equations with \ the same properties would be created that apply to that different potential. \ A potential for gravity should satisfy the following three constraints.\ \>", "Text"], Cell["\<\ 1. Apply to all particles, including light, reducing to mass in the classical \ limit. 2. Be unidirectional. 3. Involve curvature in some way. \ \>", "Text", CellMargins->{{36, Inherited}, {Inherited, Inherited}}], Cell["\<\ Every particle has a 4-momentum, composed of the scalar energy E and the \ 3-momentum P. The mass of a particle in flat spacetime is the scalar part of \ the 4-momentum squared. \ \>", "Text"], Cell[BoxData[ \(\((E, \ \(P\& \[RightVector] \))\)\^2\ = \ \((E\^2\ - \ \(P\& \[RightVector] \)\^2, \ 2\ E\ \ \(P\& \[RightVector] \))\)\)], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ Define flat spacetime operationally: when the scalar of the square of a \ quaternion--scalar(q^2)--is the scalar squared minus the 3-vector \ squared--scalar(q)^2 - vector(q)^2--then spacetime is flat. In flat \ spacetime, the mass of two particles is the sum. \ \>", "Text"], Cell[BoxData[ \(\((E, P\&\[RightVector])\)\^2 + \((E\^\[Prime], P\&\[RightVector]\^\[Prime])\)\^2 = \(\((E\^2 - P\&\[RightVector]\^2, 2\ E\ P\&\[RightVector])\) + \((\(E\^\[Prime]\)\^2 - \(P\&\[RightVector]\^\[Prime]\)\^2, 2\ E\^\[Prime]\ P\&\[RightVector]\^\[Prime])\)\ \n = \ \ \((m\^2\ + \ \(m\^\[Prime]\)\^2, \ 2\ \((E\ P\&\[RightVector]\ + \ E\^\[Prime]\ P\&\[RightVector]\^\[Prime])\))\)\)\)], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ If we want to be consistent in calculating the mass, add the two 4- momenta \ together first and then square.\ \>", "Text"], Cell[BoxData[ \(\(\((E + E\^\[Prime], \ P\&\[RightVector] + P\&\[RightVector]\^\[Prime]) \)\^2\ = \ \n \((E\^2 - P\&\[RightVector]\^2 + \(E\^\[Prime]\)\^2 - \(P\&\[RightVector]\^\[Prime]\)\^2 + \(2 \((E\ E\^\[Prime] - P\&\[RightVector]\ P\&\[RightVector]\^\[Prime])\)\)\+\(-- \(--\(--\(--\(--\(--\(---\)\)\)\)\)\)\), 2 \((E\ P\&\[RightVector] + E\^\[Prime]\ P\&\[RightVector]\^\[Prime] + \(E\ P\&\[RightVector]\^\[Prime] + \(E\^\[Prime]\) P\&\[RightVector]\)\+\(-- \(--\(--\(--\(---\)\)\)\)\))\))\)\ \)\)], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The underlined scalar term indicates that spacetime containing two particles \ in it is no longer flat. The scalar term is unidirectional. For real \ particles, \ \>", "Text"], Cell[BoxData[ \(\(E\ > \(P\& \[RightVector] \)\ \ \ \ and\ \ \ \ E\^\[Prime]\ > \ \(P\& \[RightVector] \)\^\[Prime], \ so\ \ \ \ E\ E\^\[Prime] - \(P\& \[RightVector] \)\ \(P\& \[RightVector] \)\^\[Prime]\ > \ 0\ \)\)], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["If E >> P and E' >> P', then", "Text"], Cell[BoxData[ \(E\ E\^\[Prime] - \(P\& \[RightVector] \)\ \(P\& \[RightVector] \)\^\[Prime]\ \[TildeFullEqual] \ m\ m\^\[Prime]\)], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The Lorentz gravitational force and the Maxwell equations of gravity"]], "Subsection"], Cell["\<\ The quaternion field 2 (E E' - P P', E P' + P E' + PxP') has the three \ properties required of a gravitational field (reduces to m m', is \ unidirectional, and involves curvature). Generate the quaternion laws of \ gravity by having the operators act on the gravitational field.\ \>", "Text"], Cell[BoxData[{\(The\ Lorentz\ gravitational\ force\), RowBox[{ RowBox[{ RowBox[{ RowBox[{ \((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\), RowBox[{"(", RowBox[{ \(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\), ",", RowBox[{ RowBox[{\(-E\), " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "-", \(\(E\^\[Prime]\) P\&\[RightVector]\)}]}], ")"}], \((\[Gamma], \(-\[Gamma]\[Beta]\))\)}], "\n", "\t\t\t", "-", " ", RowBox[{ \((\[Gamma], \(-\[Gamma]\[Beta]\))\), \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), RowBox[{"(", RowBox[{ \(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\), ",", RowBox[{ RowBox[{"E", " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}]}], ")"}]}]}], "\n", "=", RowBox[{"\[Gamma]", RowBox[{"(", RowBox[{ \(\(-\[Beta]\)\[CenterDot]\[EmptyDownTriangle]\&\[RightVector]\), "\[InvisibleComma]", RowBox[{ \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime])\), "-", RowBox[{"\[Beta]", "\[CenterDot]", FractionBox[ RowBox[{ RowBox[{\(\[PartialD]E\), " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}], \(\[PartialD]t\)]}]}]}], ")"}]}]}], ",", "\n", "\t\t\t", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ \(\[EmptyDownTriangle]\&\[RightVector]\), "\[InvisibleComma]", RowBox[{ \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime])\), "+", FractionBox[ RowBox[{ RowBox[{\(\[PartialD]E\), " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}], \(\[PartialD]t\)]}]}], ")"}]}], "+", " ", RowBox[{"\[Beta]", " ", "X", RowBox[{"(", RowBox[{ RowBox[{\(\[EmptyDownTriangle]\&\[RightVector]\), "X", RowBox[{"(", RowBox[{ RowBox[{"E", " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}], ")"}]}], "\[InvisibleComma]"}], ")"}]}]}], "\n", "=", " ", \(\[Gamma]\ \((\ \[Beta].G\&\[RightVector], G\&\[RightVector]\ + \[Beta]\ \[Times]\ F\&\[RightVector]) \)\)}]}]}], "Input", CellMargins->{{1, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ RowBox[{\(The\ Maxwell\ equations\ of\ gravitational\), " "}], RowBox[{ RowBox[{ RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"(", RowBox[{ \(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\), ",", RowBox[{ RowBox[{"E", " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}]}], ")"}]}], StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["}", FontWeight->"Bold"]}], " ", "\n", "+", RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{"vector", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"(", RowBox[{ \(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\), ",", RowBox[{ RowBox[{\(-E\), " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "-", \(\(E\^\[Prime]\) P\&\[RightVector]\)}]}], ")"}]}], StyleBox["}", FontWeight->"Bold"]}]}]}], StyleBox["]", FontWeight->"Bold"]}]}], "\n", "=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ \(\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", RowBox[{"(", RowBox[{ RowBox[{"E", " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}], ")"}]}], " ", ",", RowBox[{ \(-\[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", \(\[EmptyDownTriangle]\&\[RightVector]\), " ", RowBox[{"(", RowBox[{\(E\ E\^\[Prime]\), " ", "-", RowBox[{\(P\&\[RightVector]\), ".", SuperscriptBox[\(P\&\[RightVector]\^\[Prime]\), TagBox["", Derivative], MultilineFunction->None]}]}], ")"}]}]}], ")"}], "\n", "=", RowBox[{ \((\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]F \&\[RightVector], \[EmptyDownTriangle]\&\[RightVector]\ X\ G\&\[RightVector] + \[PartialD]F\&\[RightVector]\/\[PartialD]t)\), "=", RowBox[{ RowBox[{ RowBox[{\((0, \ 0\&\[RightVector])\), "\n", "\n", "\t\t", StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"(", RowBox[{ \(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\), ",", RowBox[{ RowBox[{"E", " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}]}], ")"}]}], StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["]", FontWeight->"Bold"]}], "\n", "\t", "-", " ", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"vector", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{"(", RowBox[{ \(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\), ",", RowBox[{ RowBox[{\(-E\), " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "-", \(\(E\^\[Prime]\) P\&\[RightVector]\)}]}], ")"}]}], StyleBox["}", FontWeight->"Bold"]}]}]}], StyleBox["}", FontWeight->"Bold"]}]}], "\n", "=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ \(\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ E\ E\^\[Prime]\), " ", "-", \(P\&\[RightVector].P\&\[RightVector]\^\[Prime]\), "-", RowBox[{ \(\[EmptyDownTriangle]\&\[RightVector]\), "\[CenterDot]", FractionBox[ RowBox[{ RowBox[{\(\[PartialD]E\), " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}], \(\[PartialD]t\)]}]}], ",", "\n", "\t\t\t\t\t", RowBox[{ RowBox[{ \(\[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", \(\[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", RowBox[{"(", RowBox[{ RowBox[{"E", " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{\(\[PartialD]\^2 E\), " ", SuperscriptBox[\(P\&\[RightVector]\), "\[Prime]", MultilineFunction->None]}], "+", \(\(E\^\[Prime]\) P\&\[RightVector]\)}], \(\[PartialD]t\^2\)], "+", \(\(\[PartialD]\[EmptyDownTriangle]\&\[RightVector]\ \[InvisibleComma] \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]) \)\)\/\[PartialD]t\)}]}], ")"}], "\n", "=", \(\((\ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]G \&\[RightVector]\ \ , \ \[EmptyDownTriangle]\&\[RightVector]\ X\ F\&\[RightVector]\ - \[PartialD]G\&\[RightVector]\/\[PartialD]t)\) = 8\ \[Pi]\ \((mass\ density, \(mass\ flow\)\&\[RightVector]) \)\)}]}]}]}]}]}], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ Looks complicated, but it was only an exercise in cutting and pasting new \ fields into the Lorentz force and the Maxwell equations.\ \>", "Text"], Cell["The key question is how well this proposal fits the data.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Links to Newton gravity and general relativity"]], "Subsection"], Cell["\<\ Newton's law of gravity works well, but where exactly is it hiding in that \ collection of symbols? Let electromagnetic theory be the guide. Coulomb's law \ is found in the covariant Lorentz force law if beta = A = 0 and the scalar \ potential for a point charge is q/r. For the Lorentz gravitational force, \ beta = P = P' = 0, so E E' ~= m m', gamma ~1, and the scalar potential for a \ point mass is m m'/r. \ \>", "Text"], Cell[BoxData[ \(\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\)) \) \((\(m\ m\^\[Prime]\)\/r, 0\&\[RightVector])\) \((1, 0\&\[RightVector])\) - \ \((1, 0\&\[RightVector])\) \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \) \((\(m\ m\^\[Prime]\)\/r, 0\&\[RightVector])\)\n\t = \(\((0, \(-\[EmptyDownTriangle]\&\[RightVector]\) \(m\ m\^\[Prime]\)\/r) \) = \((0, \(m\ m\^\[Prime]\)\/r\^2)\)\)\)], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["Multiply by -G, and that is Newton's law.", "Text"], Cell["\<\ Newton's law describes a 3-vector. How does one \"repair\" the law so that it \ gets along with the 4-force formulations of special relativity? This \ seemingly small job actually involves general relativity (see Misner, Thorne \ and Wheeler, chapter 7). No repairs are required for the Lorentz \ gravitational force law, since its form is every bit as covariant as the \ Lorentz force of electromagnetism. For a point source, the law becomes: \ \>", "Text"], Cell[BoxData[ \(\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\)) \) \((\(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\)\/r, 0\&\[RightVector])\) \((\[Gamma], \(-\[Gamma]\[Beta]\))\)\n\t\t\t - \ \((\[Gamma], \(-\[Gamma]\[Beta]\))\) \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \) \((\(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\)\/r, 0\&\[RightVector])\)\n = \(\[Gamma] \((\(-\[Beta]\)\[CenterDot]\[EmptyDownTriangle]\&\[RightVector] \((\(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\)\/r)\), \(-\[EmptyDownTriangle]\&\[RightVector]\)\[InvisibleComma] \((\(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\)\/r) \)\[InvisibleComma] )\)\n = \[Gamma] \((\[Beta]\[CenterDot]\(( \(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\)\/r\^2) \), \((\(E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\)\/r\^2) \)\[InvisibleComma] )\)\)\)], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ I do not know yet whether this form of the Lorentz gravitational force is \ consistent with the experimental tests designed for the Schwarzschild metric \ of general relativity, but I do have high hopes :-)\ \>", "Text"], Cell["\<\ Here's a subtle but fundamental distinction between electromagnetism and \ gravity. For the Lorentz force of electromagnetism, the test charge q is \ independent of the 4-potential. Varying q does nothing to the 4- potential. \ For gravity, there is no way to separate the two momentum 4-vectors that \ together compose the gravitational source. Varying a test mass varies the \ 4-potential. This may play a similar function to the non-linearity seen in \ general relativity.\ \>", "Text"], Cell["\<\ Now to the big difference: what is gravitational analogue of magnetism? \ Magnetism involves moving electric charges. The gravitational analogue must \ involve moving pairs of masses. Consider a spiral galaxy spinning around an \ axis. These equations predict a force generated by moving pairs of masses \ that cannot be accounted for by a scalar potential. The velocity profile of \ spiral galaxies is flat (V(r) = k) as seen using neutral atomic hydrogen \ spectral lines. That is the data.\ \>", "Text"], Cell["\<\ Newton's law predicts a Keplerian decline, V(r) = (k M(r)/r)^.5. Since this \ equation fails, people have proposed the dark matter hypothesis, where most \ of the matter in the universe is composed of \"no-see-'ems\", a type of bug \ that cannot be seen, but leaves a bite from its visit.\ \>", "Text"], Cell["\<\ The B field is generated by DelxA. The gravitational analogue, let's call it \ the ME field since that sounds like B and implies the non-linearity, is \ Delx(E P' + P E' + PxP'). Do I know how to use the ME field to predict the \ velocity profile of a spiral galaxy? No. I will start playing with \ electromagnetic analogies, specifically electric charges on disks. [note \ added later: I have heard that this sort of calculation has been made for \ general relativity, and the effect is not of the scale necessary to account \ for the effect seen.]\ \>", "Text"], Cell["\<\ There is a small amount of data about gravity waves. Some of the data is \ based on the rate of decay of signals from binary pulsars due to energy \ emitted as gravity waves. The Maxwell gravitational wave equation also \ predicts such waves. I have yet to start doing calculations with this set of \ equations. One point I can make is that there are no ME field monopoles, just \ as there are no B field monopoles. However, the ME field is really more \ analogous to a dipole since it necessarily involves two 4-momenta. Therefore \ there are no ME field dipoles. Gravity waves may require a quadrupole source \ for this reason. General relativity makes a similar statement.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Quantum gravity"]], "Subsection"], Cell["\<\ I have not broached the topic of quantum mechanics, yet the topic is critical \ for a modern gravitational theory. Here again, I must be brief since I am \ still formulating how to think about quantum mechanics with quaternions. My \ hope is to clone quantum electrodynamics, creating quantum gravitational \ dynamics, since the equations have identical forms, something like so:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime], E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime])\)\), StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["}", FontWeight->"Bold"]}], " ", "\n", "+", RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\), \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime], \(-E\)\ E\^\[Prime]\ + P\&\[RightVector].P\&\[RightVector]\^\[Prime])\)\), StyleBox["}", FontWeight->"Bold"]}]}], StyleBox["]", FontWeight->"Bold"]}]}], "\n", "=", RowBox[{ RowBox[{"(", RowBox[{ \(\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ X \((\ E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime])\)\), " ", ",", RowBox[{ \(-\[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", \(\[EmptyDownTriangle]\&\[RightVector]\), " ", SuperscriptBox[ \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime])\), TagBox["", Derivative], MultilineFunction->None]}]}], ")"}], "=", RowBox[{ RowBox[{ RowBox[{\((0, \ 0\&\[RightVector])\), "\n", "\t", "\n", StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime], E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime])\)\), StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["]", FontWeight->"Bold"]}], "\n", "\t", "+", " ", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), ",", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\), \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime], \(-E\)\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime])\)\), StyleBox["}", FontWeight->"Bold"]}]}], StyleBox["}", FontWeight->"Bold"]}]}], "\n", "=", \(\((\ \(\[PartialD]\^2 E\ E\^\[Prime]\ - P\&\[RightVector].P \&\[RightVector]\^\[Prime]\)\/\[PartialD]t\^2 + \[EmptyDownTriangle]\&\[RightVector]\^2\ \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\ )\), \n \t\t\t\t\(- \(\(\[PartialD]\^2 E\ E\^\[Prime]\ - P\&\[RightVector].P \&\[RightVector]\^\[Prime]\)\/\[PartialD]t\^2 \)\) - \ \[EmptyDownTriangle]\&\[RightVector]\^2\ \((E\ E\^\[Prime]\ - P\&\[RightVector].P\&\[RightVector]\^\[Prime]\ )\))\)\n \t\t = 8 \[Pi]\ \((mass, \(mass\ flow\)\&\[RightVector])\)\)}]}]}]], "Input", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ That sounds like a reasonable, though unsupported, line of logic.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Implications"]], "Subsection"], Cell["\<\ What has been accomplished? I have generated two equations for gravity--the \ Lorentz gravitational force and the Maxwell gravitational wave equation--with \ as many implications as their electromagnetic counterparts. The only solid \ connection to data has involved Newton's law of gravity. A link to the \ experimental tests of general relativity is merely a hope. A connection to \ quantum mechanics is more tenuous. The proof for this proposal, if it ever \ comes, will involve the ability to explain the velocity profiles of systems \ with masses in motion.\ \>", "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 451}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{532, 198}, WindowMargins->{{35, Automatic}, {Automatic, 9}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PrintingOptions->{"PrintingMargins"->{{72, 57.5625}, {57.5625, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False, "FirstPageHeader"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Style Definitions"]], "Subtitle"], Cell[TextData[StyleBox[ "Modify the definitions below to change the default appearance of all cells \ in a given style. Make modifications to any definition using commands in the \ Format menu."]], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Style Environment Names"]], "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, ScriptMinSize->12, FontSize->16], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellBracketOptions->{"Margins"->{1, 1}, "Widths"->{0, 5}}, ScriptMinSize->8, FontSize->11], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ScriptMinSize->5, FontSize->10, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Notebook Options"]], "Section", FontFamily->"New York"], Cell[TextData[StyleBox[ "The options defined for the style below will be used at the Notebook \ level."]], "Text", FontFamily->"New York"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None, FontFamily->"New York"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Headings"]], "Section", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{7, Inherited}, {8, 40}}, Evaluatable->False, CellGroupingRules->{"TitleGrouping", 0}, CellHorizontalScrolling->False, PageBreakBelow->False, TextAlignment->Center, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->24, FontWeight->"Bold"], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontFamily->"Times New Roman", FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {12, 30}}, FontFamily->"Times New Roman", FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{7, Inherited}, {6, 15}}, Evaluatable->False, CellGroupingRules->{"TitleGrouping", 10}, CellHorizontalScrolling->False, PageBreakBelow->False, TextAlignment->Center, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->18], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->36], Cell[StyleData["Subtitle", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontFamily->"Times New Roman", FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{7, Inherited}, {6, 15}}, Evaluatable->False, CellGroupingRules->{"TitleGrouping", 20}, CellHorizontalScrolling->False, PageBreakBelow->False, TextAlignment->Center, CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->14, FontSlant->"Italic"], Cell[StyleData["Subsubtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 8}}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontFamily->"Times New Roman", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[GraySquare]", CellMargins->{{22, Inherited}, {8, 20}}, Evaluatable->False, CellGroupingRules->{"SectionGrouping", 30}, CellHorizontalScrolling->False, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->18, FontWeight->"Bold", FontVariations->{"Underline"->True}], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontFamily->"Times New Roman", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{19, Inherited}, {8, 15}}, Evaluatable->False, CellGroupingRules->{"SectionGrouping", 40}, CellHorizontalScrolling->False, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontFamily->"Times New Roman", FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[EmptySquare]", CellMargins->{{18, Inherited}, {8, 12}}, Evaluatable->False, CellGroupingRules->{"SectionGrouping", 50}, CellHorizontalScrolling->False, PageBreakBelow->False, CounterIncrements->"Subsubsection", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontFamily->"Times New Roman", FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontFamily->"Times New Roman", FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Body Text"]], "Section", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{7, 10}, {7, 7}}, Evaluatable->False, CellHorizontalScrolling->False, PageBreakWithin->Automatic, LineSpacing->{1, 3}, CounterIncrements->"Text", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontFamily->"Times New Roman"], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontFamily->"Times New Roman"], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}, FontFamily->"Times New Roman"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{7, 10}, {6, 6}}, Evaluatable->False, CellHorizontalScrolling->False, PageBreakWithin->Automatic, LineSpacing->{1, 3}, CounterIncrements->"SmallText", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->10], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontFamily->"Times New Roman", FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, FontFamily->"Times New Roman", FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Input/Output"]], "Section", FontSize->14, FontWeight->"Plain"], Cell[TextData[StyleBox[ "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", FontSize->14], Cell[CellGroupData[{ Cell[StyleData["Input"], PageWidth->Infinity, CellMargins->{{42, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", AspectRatioFixed->True, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontSize->14, FontWeight->"Plain"], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}, FontSize->14, FontWeight->"Plain"], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, FontSize->14, FontWeight->"Plain"] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", StyleMenuListing->None, FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[StyleData["Output"], PageWidth->Infinity, CellMargins->{{42, 10}, {7, 5}}, CellEditDuplicate->True, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output", AspectRatioFixed->True, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}, FontSize->14], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}, FontSize->14], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], PageWidth->Infinity, CellMargins->{{42, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->14], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->14], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->14, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], PageWidth->Infinity, CellMargins->{{42, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->14], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->14], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], PageWidth->Infinity, CellMargins->{{7, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", AspectRatioFixed->True, ImageSize->{387, 393}, ImageMargins->{{34, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}, FontSize->14], Cell[StyleData["Graphics", "Condensed"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}, FontSize->14], Cell[StyleData["Graphics", "Printout"], ImageSize->{250, 250}, ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Condensed"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->14, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Formulas and Programming"]], "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Headers and Footers"]], "Section"], Cell[StyleData["Header"], CellMargins->{{7, 0}, {4, 1}}, Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{7, 0}, {0, 4}}, Evaluatable->False, PageBreakWithin->Automatic, TextAlignment->Center, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Palette Styles"]], "Section"], Cell[TextData[StyleBox[ "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]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Hyperlink Styles"]], "Section"], Cell[TextData[StyleBox[ "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", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[TextData[StyleBox[ "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", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], 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", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], 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[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], 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", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], 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", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Placeholder Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles useful for making placeholder objects in \ palette templates."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->Primary, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["FormatType Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles that are mixed in with the styles of most \ cells. If a cell's FormatType matches the name of one of the styles defined \ below, then that style is applied between the cell's style and its own \ options."]], "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell[TextData[StyleBox[ "The style defined below is mixed in to any cell that is in an inline cell \ within another."]], "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. 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[CellGroupData[{ Cell[1731, 51, 53, 0, 43, "Subtitle"], Cell[1787, 53, 69, 0, 39, "Subsubtitle"], Cell[1859, 55, 259, 6, 128, "Text"], Cell[2121, 63, 55, 0, 33, "Text"], Cell[CellGroupData[{ Cell[2201, 67, 54, 0, 42, "Subsection"], Cell[2258, 69, 417, 7, 90, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[2712, 81, 73, 0, 42, "Subsection"], Cell[2788, 83, 95, 2, 33, "Text"], Cell[2886, 87, 359, 8, 74, "Text"], Cell[3248, 97, 381, 6, 90, "Text"], Cell[3632, 105, 1539, 31, 152, "Input"], Cell[5174, 138, 6842, 146, 347, "Input"], Cell[12019, 286, 254, 4, 71, "Text"], Cell[12276, 292, 226, 6, 71, "Text"], Cell[12505, 300, 202, 4, 52, "Text"], Cell[12710, 306, 244, 5, 37, "Input"], Cell[12957, 313, 286, 5, 71, "Text"], Cell[13246, 320, 564, 10, 60, "Input"], Cell[13813, 332, 133, 3, 52, "Text"], Cell[13949, 337, 740, 14, 71, "Input"], Cell[14692, 353, 185, 4, 52, "Text"], Cell[14880, 359, 334, 7, 32, "Input"], Cell[15217, 368, 44, 0, 33, "Text"], Cell[15264, 370, 234, 5, 32, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[15535, 380, 114, 2, 42, "Subsection"], Cell[15652, 384, 303, 5, 71, "Text"], Cell[15958, 391, 3580, 80, 228, "Input"], Cell[19541, 473, 8845, 194, 422, "Input"], Cell[28389, 669, 156, 3, 52, "Text"], Cell[28548, 674, 73, 0, 33, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[28658, 679, 89, 1, 42, "Subsection"], Cell[28750, 682, 435, 7, 90, "Text"], Cell[29188, 691, 596, 10, 85, "Input"], Cell[29787, 703, 57, 0, 33, "Text"], Cell[29847, 705, 468, 7, 109, "Text"], Cell[30318, 714, 1418, 26, 197, "Input"], Cell[31739, 742, 231, 4, 71, "Text"], Cell[31973, 748, 500, 8, 109, "Text"], Cell[32476, 758, 517, 8, 109, "Text"], Cell[32996, 768, 312, 5, 71, "Text"], Cell[33311, 775, 574, 9, 128, "Text"], Cell[33888, 786, 699, 10, 147, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[34624, 801, 57, 0, 42, "Subsection"], Cell[34684, 803, 403, 6, 90, "Text"], Cell[35090, 811, 4879, 115, 335, "Input"], Cell[39972, 928, 89, 2, 33, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[40098, 935, 54, 0, 42, "Subsection"], Cell[40155, 937, 586, 9, 128, "Text"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)