(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 28769, 742] NotebookOptionsPosition[ 25006, 683] NotebookOutlinePosition[ 25457, 700] CellTagsIndexPosition[ 25414, 697] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Conformal transformations", "Title", CellChangeTimes->{{3.955028494621468*^9, 3.955028504399789*^9}},ExpressionUUID->"a7cc0017-0e14-4698-97f7-\ 1ca4568c8542"], Cell["Brief usage example of the xAct and xTensor packages", "Subtitle", CellChangeTimes->{{3.96166048167594*^9, 3.961660526139915*^9}},ExpressionUUID->"60487e40-818c-474b-942e-\ b71cefe99067"], Cell["\<\ In this brief tutorial we derive expressions for the covariant derivative and \ Riemann curvature tensor associated with a metric resulting from a conformal \ transformation as shown in the Appendix D from the \ \[OpenCurlyDoubleQuote]General Relativity\[CloseCurlyDoubleQuote] book by \ Wald. \ \>", "Text", CellChangeTimes->{{3.9616605330197153`*^9, 3.961660649443626*^9}},ExpressionUUID->"1bf35833-fa02-4d70-a9e6-\ 590e9434fe1e"], Cell[CellGroupData[{ Cell["Load xAct", "Section", CellChangeTimes->{{3.955028511524665*^9, 3.9550285272672443`*^9}},ExpressionUUID->"0a49b301-96d5-47b7-b6f5-\ 7216576f4a24"], Cell["\<\ Once xAct has been successfully installed, the first step is including the \ xTensor package as follows\ \>", "Text", CellChangeTimes->{{3.961660667555468*^9, 3.961660731186514*^9}},ExpressionUUID->"49f5a5c6-6bf3-41ba-afab-\ 8e67d41a403e"], Cell[BoxData[{ RowBox[{ RowBox[{"<<", "xAct`xTensor`"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"<<", "xAct`TexAct`"}], ";"}]}], "Input", CellLabel-> "In[791]:=",ExpressionUUID->"2fd2f53a-4183-492a-a952-2189e90d6249"] }, Open ]], Cell[CellGroupData[{ Cell["Define manifold", "Section", CellChangeTimes->{{3.955028533311227*^9, 3.955028548332649*^9}, { 3.961653763818295*^9, 3.96165376647447*^9}},ExpressionUUID->"5f800243-43d9-465d-ab67-\ 0cecfec03dd2"], Cell[TextData[{ "After loading xTensor, we declare the manifold we\[CloseCurlyQuote]ll use. ", Cell[BoxData[ FormBox["DefManifold", TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "75155726-360e-4d25-8bc7-cdd881a3022e"], " expects a name for the manifold, its dimension and a list of the symbols \ we\[CloseCurlyQuote]ll used as dummy indices for tensors components" }], "Text", CellChangeTimes->{{3.961487272654566*^9, 3.961487298509144*^9}, { 3.9614877568120127`*^9, 3.961487812795907*^9}, {3.96166075537022*^9, 3.961660906730241*^9}},ExpressionUUID->"e4e69470-896c-4bf6-9c8b-\ 46362639e4e0"], Cell[BoxData[ RowBox[{"DefManifold", "[", RowBox[{"M4", ",", " ", "4", ",", " ", RowBox[{"{", RowBox[{ "a", ",", " ", "b", ",", " ", "c", ",", " ", "d", ",", " ", "e", ",", " ", "f"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.955023337613821*^9, 3.95502338066257*^9}, { 3.9616552660551167`*^9, 3.961655267710513*^9}}, CellLabel-> "In[793]:=",ExpressionUUID->"f2b73011-41da-4216-9ba8-67ef69975141"], Cell["\<\ Before going into the details of the metric and it\[CloseCurlyQuote]s \ conformal transformation, we introduce a vector, co-vector and scalar field \ for testing our definitions\ \>", "Text", CellChangeTimes->{{3.9616535906185923`*^9, 3.9616537085466223`*^9}, { 3.9616609275224543`*^9, 3.9616609315866623`*^9}},ExpressionUUID->"a0b2fea6-e354-4534-969f-\ a056e798ad2d"], Cell[BoxData[ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"\[Phi]", "[", "]"}], ",", "M4"}], "]"}]], "Input", CellChangeTimes->{{3.955028444706019*^9, 3.9550284556973457`*^9}}, CellLabel-> "In[794]:=",ExpressionUUID->"4037f682-2146-4cb7-8df8-cceb85259964"], Cell[BoxData[ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"\[Omega]", "[", RowBox[{"-", "a"}], "]"}], ",", "M4"}], "]"}]], "Input", CellChangeTimes->{{3.9550284575700207`*^9, 3.9550284705702467`*^9}, { 3.9550305728694563`*^9, 3.9550305963551073`*^9}}, CellLabel-> "In[795]:=",ExpressionUUID->"3ee17cea-4207-4e66-b654-65af81f6a349"], Cell[BoxData[ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"v", "[", "a", "]"}], ",", " ", "M4"}], "]"}]], "Input", CellChangeTimes->{{3.9550284728474894`*^9, 3.955028480628694*^9}}, CellLabel-> "In[796]:=",ExpressionUUID->"ba3c7918-f22f-4869-9596-ce88531e733e"] }, Open ]], Cell[CellGroupData[{ Cell["Metric", "Section", CellChangeTimes->{{3.955028618429727*^9, 3.9550286201239653`*^9}, { 3.961653779882298*^9, 3.961653783034212*^9}},ExpressionUUID->"50dae142-254a-4d0d-9b5b-\ 9baf99c1b6d0"], Cell[TextData[{ "With our manifold and objects defined, we continue by declaring a metric ", Cell[BoxData[ FormBox["g", TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "999bff23-53a7-47b6-b8ce-84f5876dc255"], " with negative determinant and associated covariant derivative operator ", Cell[BoxData[ FormBox["CD", TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "23d52a0d-63f2-420c-8f50-206b05ebeef0"], ". The ", Cell[BoxData[ FormBox["-", TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "21b55233-bf61-4997-8f55-9391753f3495"], " sign in front of the dummy indices represents the covariant nature of the \ component" }], "Text", CellChangeTimes->{{3.961487941874721*^9, 3.96148797762912*^9}, { 3.961653016089932*^9, 3.961653064478984*^9}, {3.961653219684231*^9, 3.961653223996208*^9}, {3.961661038057949*^9, 3.9616611218260117`*^9}},ExpressionUUID->"a0e45ebd-1aac-443d-bdf2-\ 1f41e0361f6b"], Cell[BoxData[ RowBox[{"DefMetric", "[", RowBox[{ RowBox[{"-", "1"}], ",", " ", RowBox[{"g", "[", RowBox[{ RowBox[{"-", "a"}], ",", " ", RowBox[{"-", "b"}]}], "]"}], ",", " ", "CD"}], "]"}]], "Input", CellChangeTimes->{3.961653211484542*^9}, CellLabel-> "In[797]:=",ExpressionUUID->"625b4323-6ac1-4727-8ae1-327a72d9a9f5"] }, Open ]], Cell[CellGroupData[{ Cell["Conformal transformation", "Section", CellChangeTimes->{{3.9550285859441147`*^9, 3.95502859199504*^9}, { 3.961653816569553*^9, 3.9616538204101954`*^9}},ExpressionUUID->"10b20b73-f6db-45f1-a2bd-\ 80d807952d8f"], Cell[TextData[{ "We are now ready to introduce a new metric resulting from the conformal \ transformation of our metric ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ StyleBox["g", "TI"], TraditionalForm], "errors" -> {}, "input" -> "g", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "057b9558-96c5-49fc-8065-d2ad383204df"], ". Here we introduce ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ RowBox[{ SubscriptBox[ StyleBox["h", "TI"], RowBox[{ StyleBox["a", "TI"], StyleBox["b", "TI"]}]], "\[LongEqual]", SuperscriptBox["\[CapitalOmega]", "2"], SubscriptBox[ StyleBox["g", "TI"], RowBox[{ StyleBox["a", "TI"], StyleBox["b", "TI"]}]]}], TraditionalForm], "errors" -> {}, "input" -> "h_{ab} = \\Omega^2 g_{ab}", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "aed78e47-f4f5-438e-8b3b-91b4952efddb"], " . Thus, we need first declare a scalar field ", Cell[BoxData[ FormBox[ TemplateBox[Association[ "boxes" -> FormBox["\[CapitalOmega]", TraditionalForm], "errors" -> {}, "input" -> "\\Omega", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "eec46500-da9d-4c05-a666-7748868acc1a"], " and a tensor ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ StyleBox["h", "TI"], TraditionalForm], "errors" -> {}, "input" -> "h", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "369261e5-2a79-4556-aca5-867bcca11699"], " for the transformed metric." }], "Text", CellChangeTimes->{{3.9616532502282257`*^9, 3.9616533321508017`*^9}, { 3.961653389204213*^9, 3.9616534547794333`*^9}, {3.961653828722475*^9, 3.961653829649892*^9}, {3.961653863946307*^9, 3.961653865218367*^9}, { 3.961661172065489*^9, 3.961661258921567*^9}},ExpressionUUID->"63372f94-6fc7-44ea-a50c-\ 3f6ef1684579"], Cell[BoxData[ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"\[CapitalOmega]", "[", "]"}], ",", " ", "M4"}], "]"}]], "Input", CellLabel-> "In[798]:=",ExpressionUUID->"ac2d360b-1a90-44ed-b927-1b5473cc21e9"], Cell[BoxData[ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"h", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", " ", "M4", ",", " ", RowBox[{"Symmetric", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "}"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.955023646962303*^9, 3.955023705041223*^9}}, CellLabel-> "In[799]:=",ExpressionUUID->"16648c0c-1ed0-4b16-b9c1-f3b7ad43a69e"], Cell[TextData[{ "Note we declared ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ SubscriptBox[ StyleBox["h", "TI"], RowBox[{ StyleBox["a", "TI"], StyleBox["b", "TI"]}]], TraditionalForm], "errors" -> {}, "input" -> "h_{ab}", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "c80cd3b2-baed-410f-83b6-33f7fecb3428"], " to be a symmetric tensor as a metric should be. We introduce a new \ covariant derivative which we\[CloseCurlyQuote]ll associated with our new \ metric and represented it ", Cell[BoxData[ FormBox["D", TraditionalForm]],ExpressionUUID-> "86943cda-7b9a-4feb-9fa3-266216418f38"] }], "Text", CellChangeTimes->{{3.961653476007193*^9, 3.961653543594967*^9}, { 3.9616538862421217`*^9, 3.961653925250005*^9}, {3.9616551580380707`*^9, 3.961655171648201*^9}, {3.9616612897055483`*^9, 3.961661304905047*^9}},ExpressionUUID->"77718993-841d-4f7e-8e9f-\ 98098ed816df"], Cell[BoxData[ RowBox[{"DefCovD", "[", RowBox[{ RowBox[{"hCD", "[", RowBox[{"-", "a"}], "]"}], ",", " ", RowBox[{"SymbolOfCovD", "->", RowBox[{"{", RowBox[{"\"\<|\>\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.95502937410616*^9, 3.955029404880175*^9}, { 3.955029559247651*^9, 3.955029578165971*^9}, {3.955029719879252*^9, 3.9550297214275827`*^9}, {3.9550297524681997`*^9, 3.955029796729156*^9}, { 3.955029875338228*^9, 3.955029913344639*^9}, {3.955029954129013*^9, 3.955030072451722*^9}, {3.955030321977878*^9, 3.9550303621462097`*^9}, { 3.955030397424735*^9, 3.955030398894948*^9}}, CellLabel-> "In[800]:=",ExpressionUUID->"b30fd8c3-90e7-4d04-be8f-5c0b420cde01"], Cell[TextData[{ "We now need to define ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ StyleBox["h", "TI"], TraditionalForm], "errors" -> {}, "input" -> "h", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "c2688b23-ff96-45a9-b141-bd8aa6e81c02"], " in terms of the original metric ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ StyleBox["g", "TI"], TraditionalForm], "errors" -> {}, "input" -> "g", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "cabcd713-3625-4207-887e-d20a677473e3"], ". As far as I know, xTensor does not provide an easy way to work with \ multiple metric tensors. There exits the option of \ \[OpenCurlyDoubleQuote]frozen\[CloseCurlyDoubleQuote] metrics, but the author \ himself claims this code is untested. For more details see section 7.4 of the \ xTensor documentation. Henceforth, we continue by defining our metric \ \[OpenCurlyDoubleQuote]by hand\[CloseCurlyDoubleQuote]" }], "Text", CellChangeTimes->{{3.961653843033733*^9, 3.961653870049555*^9}, { 3.961653950797401*^9, 3.9616539635414*^9}, {3.961661362648322*^9, 3.9616613918171377`*^9}, {3.9616615193367443`*^9, 3.961661658536003*^9}, { 3.9616616969442797`*^9, 3.961661727160232*^9}},ExpressionUUID->"b786c884-f420-4f76-890f-\ dc9a8e26c132"], Cell[BoxData[ RowBox[{ RowBox[{"h", "[", RowBox[{ RowBox[{"a_", "?", "DownIndexQ"}], ",", RowBox[{"b_", "?", "DownIndexQ"}]}], "]"}], ":=", " ", RowBox[{ RowBox[{ RowBox[{"\[CapitalOmega]", "[", "]"}], "^", "2"}], " ", RowBox[{"(", RowBox[{"g", "[", RowBox[{"a", ",", "b"}], "]"}], ")"}]}]}]], "Input", CellChangeTimes->{{3.9616539794362383`*^9, 3.961653981002342*^9}}, CellLabel-> "In[801]:=",ExpressionUUID->"57b245d8-888f-4d56-815e-76e1ae765e47"], Cell[BoxData[ RowBox[{ RowBox[{"h", "[", RowBox[{ RowBox[{"a_", "?", "UpIndexQ"}], ",", " ", RowBox[{"b_", "?", "UpIndexQ"}]}], "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"\[CapitalOmega]", "[", "]"}], "^", RowBox[{"-", "2"}]}], RowBox[{"g", "[", RowBox[{"a", ",", "b"}], "]"}]}]}]], "Input", CellLabel-> "In[802]:=",ExpressionUUID->"a14b0804-4485-4c6f-bca8-80c77ddc42d3"], Cell[TextData[{ "xTensor provides the expression ", Cell[BoxData[ FormBox[ RowBox[{"Christoffel", "[", RowBox[{"hCD", ",", "CD"}], "]"}], TraditionalForm]],ExpressionUUID-> "35d72d19-401a-422c-8080-67b9b96574db"], " to represent the Christoffel symbols. However, we want to explicitly \ express everything in terms of the original metric ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ StyleBox["g", "TI"], TraditionalForm], "errors" -> {}, "input" -> "g", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "5a0f78d0-da61-4aee-b0da-71d2b0a60677"], ". We introduce a rule to replace instances of the Christoffel symbols by \ their explicit value in terms of ", Cell[BoxData[ FormBox[ TemplateBox[Association["boxes" -> FormBox[ StyleBox["g", "TI"], TraditionalForm], "errors" -> {}, "input" -> "g", "state" -> "Boxes"], "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "e5f98aa5-2c02-4230-8971-d31550fbfb66"] }], "Text", CellChangeTimes->{{3.9616540380115747`*^9, 3.961654139169346*^9}, { 3.96165421293713*^9, 3.961654214689187*^9}, {3.961654256161004*^9, 3.961654307168804*^9}, {3.961654412440547*^9, 3.9616544936563997`*^9}, { 3.96166177101614*^9, 3.9616617764641647`*^9}},ExpressionUUID->"55db4314-6e4e-4107-846c-\ 119780d92cc9"], Cell[BoxData[ RowBox[{ RowBox[{"ChristoffelRule", "=", "\[IndentingNewLine]", RowBox[{"MakeRule", "[", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{ RowBox[{"Christoffel", "[", RowBox[{"hCD", ",", " ", "CD"}], "]"}], "[", RowBox[{"c", ",", RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"h", "[", RowBox[{"c", ",", "d"}], "]"}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "a"}], "]"}], "[", RowBox[{"h", "[", RowBox[{ RowBox[{"-", "b"}], ",", RowBox[{"-", "d"}]}], "]"}], "]"}], "+", RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "b"}], "]"}], "[", RowBox[{"h", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "d"}]}], "]"}], "]"}], "-", RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "d"}], "]"}], "[", RowBox[{"h", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "]"}]}], ")"}], "/", "2"}]}]}], "}"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.961661995185031*^9, 3.961662004832115*^9}}, CellLabel-> "In[803]:=",ExpressionUUID->"5b15a151-dee3-4ef4-a05b-6a16c28ba495"], Cell[TextData[{ "As mentioned, this rules replaces Christoffel symbols by their definition. ", Cell[BoxData[ FormBox["MakeRule", TraditionalForm]],ExpressionUUID-> "03970e64-52fa-4d6a-9b33-82d262679652"], " takes care of possible issues from accidental shadowing of the dummy \ indices. Hence, not interfering with other possible indices with the same \ name.\n\nThe function ", Cell[BoxData[ FormBox["ChangeCovD", TraditionalForm]],ExpressionUUID-> "97b652a0-ef75-4464-a01b-e03646e5ddbb"], " replaces application of the ", Cell[BoxData[ FormBox["hCD", TraditionalForm]],ExpressionUUID-> "cec36a52-80cc-425f-a9ec-e9e61bc5f922"], " derivatives with their form using the original derivative ", Cell[BoxData[ FormBox["CD", TraditionalForm]],ExpressionUUID-> "c628c73c-54f2-46e3-88d5-8ce5c89830d6"], ", though the result is expressed using ", Cell[BoxData[ FormBox[ RowBox[{"Christoffel", "[", RowBox[{"hCD", ",", " ", "CD"}], "]"}], TraditionalForm]],ExpressionUUID-> "96a19d5e-b427-4cec-8aef-bc48fa04dd0a"] }], "Text", CellChangeTimes->{{3.961654548363965*^9, 3.961654650759923*^9}, { 3.961655208117923*^9, 3.961655208509633*^9}, {3.961655344797076*^9, 3.961655497608226*^9}, {3.961662189278303*^9, 3.96166220940705*^9}, { 3.9616623248060207`*^9, 3.961662348766543*^9}},ExpressionUUID->"5658ccd9-8703-4df1-b021-\ 1694740ec340"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"hCD", "[", RowBox[{"-", "a"}], "]"}], "[", RowBox[{"\[Omega]", "[", RowBox[{"-", "b"}], "]"}], "]"}], "==", RowBox[{"ChangeCovD", "[", RowBox[{ RowBox[{ RowBox[{"hCD", "[", RowBox[{"-", "a"}], "]"}], "[", RowBox[{"\[Omega]", "[", RowBox[{"-", "b"}], "]"}], "]"}], ",", " ", "hCD", ",", " ", "CD"}], "]"}]}], "//", "ScreenDollarIndices"}]], "Input", CellChangeTimes->{ 3.961655306713554*^9, {3.961730795604987*^9, 3.96173081051075*^9}}, CellLabel-> "In[804]:=",ExpressionUUID->"a7c9b8c1-5f57-456c-81a6-45113a02e085"], Cell["\<\ we can change this by applying the rule we just defined. 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