Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Conformal Transformations with ", FontSize->36], ButtonBox["xTensor", BaseStyle->"Hyperlink", ButtonData->{ URL["http://xact.es/"], None}, ButtonNote->"http://xact.es/"], StyleBox[" ", FontSize->36] }], "Subtitle", CellDingbat->"\[FilledSquare]"], Cell[TextData[StyleBox["Cyril Pitrou (pitrou@iap.fr)\nMarch 2018", FontSize->18]], "Subsubtitle"], Cell[TextData[{ StyleBox["Summary: A self-contained notebook that provides general functions \ to perform Conformal transformations on xTensor expressions.", FontSize->18], "\n\nThis is adapted from the corresponding definitions found in ", ButtonBox["xPand", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www2.iap.fr/users/pitrou/xpand.htm"], None}, ButtonNote->"http://www2.iap.fr/users/pitrou/xpand.htm"], " package.\n\nLimitations : \nWorks only for Levi-Civita connections.\nAny \ generalization or cleaning of definitions and implementations is welcome\nAn \ application to Brans-Dicke theory would be nice. TODO" }], "Subsubtitle"], Cell[CellGroupData[{ Cell[" Loading xTensor", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"<<", "xAct`xTensor`"}]], "Input", CellFrame->{{0, 0}, {0, 0.5}}], Cell[CellGroupData[{ Cell[BoxData["\<\"------------------------------------------------------------\ \"\>"], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Package xAct`xPerm` version \"\>", "\[InvisibleSpace]", "\<\"1.2.3\"\>", "\[InvisibleSpace]", "\<\", \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"2015", ",", "8", ",", "23"}], "}"}]}], SequenceForm["Package xAct`xPerm` version ", "1.2.3", ", ", {2015, 8, 23}], Editable->False]], "Print"], Cell[BoxData["\<\"CopyRight (C) 2003-2018, Jose M. 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Furthermore for a given \ tensor it also depends on the position of indices \n2) Relate the covariant \ derivative (Levi-Civita) CD of the original metric ", Cell[BoxData[ FormBox[ SubscriptBox["g", "\[Mu]\[Nu]"], TraditionalForm]]], " and the covariant derivative (Levi-Civita) of the the conformally related \ metric ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["S", "2"], SubscriptBox["g", "\[Mu]\[Nu]"]}], TraditionalForm]]], ".\nFor this step central we use a function provided in xTensor : \ ChristoffelToGradConformal" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "ChristoffelToGradConformal"}]], "Input"], Cell[BoxData[ StyleBox["\<\"ChristoffelToGradConformal[expr, metric1, metric2] expands \ expr expressing all Christoffel[covd1, covd2], where covdi is the Levi-Civita \ connection of metrici, in terms of derivatives (using covd2) of the conformal \ factor relating those metrics. The covariant derivatives covdi can be used in \ input instead of the metrici.\"\>", "MSG"]], "Print", "PrintUsage", CellTags->"Info23731262770-6848657"] }, Open ]], Cell[TextData[{ "The algorithm works in several steps\n1) When applying a conformal \ transformation from g1 to g2 = ", Cell[BoxData[ FormBox[ SuperscriptBox["S", "2"], TraditionalForm]], FormatType->"TraditionalForm"], " g1 (g1 and g2 are metrics and S a Scalar tensor), it formally replaces all \ tensors\nT by ConfHead[g1,g2][T] .\n2) Then it replaces ConfHead[g1,g2][T] by \ the conformally transformed version, that is multiplying by some power of the \ conformal factor ( -> ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["S", "n"], " ", "T"}], TraditionalForm]], FormatType->"TraditionalForm"], ").\n Curvature tensors need to be treated in a special manner. 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