(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 238523, 7428] NotebookOptionsPosition[ 228675, 7179] NotebookOutlinePosition[ 229616, 7213] CellTagsIndexPosition[ 229449, 7205] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Modified Gravity", "Title", CellChangeTimes->{{3.547011934480917*^9, 3.547011946807622*^9}}], Cell["A list of modified gravity for reference use", "Subtitle", CellChangeTimes->{{3.5470119493467674`*^9, 3.547011977812395*^9}}], Cell[CellGroupData[{ Cell["MA Lei", "Author", CellChangeTimes->{{3.547012031694477*^9, 3.5470120602691116`*^9}}], Cell["\<\ Well, since I moved all my documents and softwares to a organization called \ CosmologyTaskForce on github, I should use CosmologyTaskForce as the author. \ However, there is no members but myself untill now. That\[CloseCurlyQuote]s \ why I just wrote my name here.\ \>", "Text", CellChangeTimes->{{3.5506986897873755`*^9, 3.5506987010480194`*^9}, { 3.550698731379754*^9, 3.550698824949106*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Abstract", "AbstractSection", CellChangeTimes->{{3.5470120653894043`*^9, 3.5470120669424934`*^9}, { 3.547012209886669*^9, 3.547012293102429*^9}, {3.5470123692837863`*^9, 3.547012369611805*^9}, {3.5470124039067664`*^9, 3.5470124257240143`*^9}}], Cell["\<\ This is an article for myself. I listed some basic knowledge about modified \ gravity.\ \>", "Abstract", CellChangeTimes->{{3.5470124295782347`*^9, 3.5470125262147617`*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["f(R) Gravity", "Section", CellChangeTimes->{{3.547012530377*^9, 3.5470125487520514`*^9}}, FontColor->RGBColor[1, 0.5, 0]], Cell[TextData[{ StyleBox["Keypoint", FontWeight->"Bold"], ": Theories that changes R to f(R) in Hillbert-Einstein action.\n", Cell[BoxData[ FormBox[ RowBox[{"\[ScriptCapitalL]", "=", RowBox[{ SqrtBox[ RowBox[{"-", "g"}]], "R"}]}], TraditionalForm]]], " -> ", Cell[BoxData[ FormBox[ RowBox[{"\[ScriptCapitalL]", "=", RowBox[{ SqrtBox[ RowBox[{"-", "g"}]], RowBox[{"f", "(", "R", ")"}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.547012617804001*^9, 3.5470126501058483`*^9}}], Cell["A standard action of f(R) gravity is", "Text", CellChangeTimes->{{3.5470126537940593`*^9, 3.5470126547111115`*^9}, { 3.547012687008959*^9, 3.5470127091022224`*^9}}], Cell[BoxData[ FormBox[ RowBox[{"S", "=", RowBox[{ RowBox[{ FractionBox["1", RowBox[{"2", "\[Kappa]"}]], RowBox[{"\[Integral]", RowBox[{ SuperscriptBox["d", "4"], "x", " ", SqrtBox[ RowBox[{"-", "g"}]], RowBox[{"f", "(", "R", ")"}]}]}]}], " ", "+", " ", RowBox[{"\[Integral]", RowBox[{ SuperscriptBox["d", "4"], "x", " ", RowBox[{ SubscriptBox["\[ScriptCapitalL]", "M"], "(", RowBox[{ SubscriptBox["g", "\[Mu]\[Nu]"], ",", SubscriptBox["\[CapitalPsi]", "M"]}], ")"}]}]}]}]}], TraditionalForm]], "EquationNumbered", CellChangeTimes->{{3.54701280871392*^9, 3.5470128094019594`*^9}, { 3.54703169771923*^9, 3.547031697730202*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell["\<\ Thus the field equation can be calculated using the variation principle.\ \>", "Text", CellChangeTimes->{{3.547031459106856*^9, 3.547031462757839*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Field Equation", FontWeight->"Bold"]], "Subsubsection", CellChangeTimes->{{3.54703166089777*^9, 3.547031672203014*^9}, { 3.547032302259251*^9, 3.547032302272423*^9}, 3.5471963779363403`*^9}], Cell["Variation of action", "Text", CellChangeTimes->{{3.547032304717725*^9, 3.54703231160878*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"\[Delta]S", "\[AlignmentMarker]", "\[Congruent]", FractionBox[ RowBox[{"d", " ", RowBox[{"S", "[", "\[Lambda]", "]"}]}], RowBox[{"d", " ", "\[Lambda]"}]]}], SubscriptBox["|", RowBox[{"\[Lambda]", "=", "0"}]]}], "\n", "\[AlignmentMarker]", "=", RowBox[{"\[Integral]", RowBox[{ SuperscriptBox["d", "4"], "x", " ", RowBox[{"(", RowBox[{ SubscriptBox["\[Delta]\[ScriptCapitalL]", "G"], "+", SubscriptBox["\[Delta]\[ScriptCapitalL]", "m"]}], ")"}]}]}]}], TraditionalForm]], "EquationNumbered", CellChangeTimes->{{3.547031677643144*^9, 3.547031677731772*^9}, { 3.547031717097259*^9, 3.547031739764551*^9}, {3.547031829140755*^9, 3.547031863505499*^9}, {3.54703208900036*^9, 3.547032113752619*^9}, 3.5471180743187866`*^9, {3.547193990470785*^9, 3.547193995236058*^9}}, TextAlignment->AlignmentMarker, Background->RGBColor[0.88, 1, 0.88]], Cell[TextData[{ "Calculate the terms seperately. 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