Notebook[{ Cell[CellGroupData[{ Cell["The Raychaudhuri equation", "Section"], Cell["\<\ Author: Leo Stein Date: Sept. 2011 This notebook follows section 9.2 of Wald.\ \>", "Text"], Cell[CellGroupData[{ Cell["0. Package, options, and overloading IndexSolve", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Needs", "[", "\"\\"", "]"}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData["\<\"------------------------------------------------------------\ \"\>"], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Package xAct`xPerm` version \"\>", "\[InvisibleSpace]", "\<\"1.1.2\"\>", "\[InvisibleSpace]", "\<\", \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"2011", ",", "7", ",", "15"}], "}"}]}], SequenceForm["Package xAct`xPerm` version ", "1.1.2", ", ", {2011, 7, 15}], Editable->False]], "Print"], Cell[BoxData["\<\"CopyRight (C) 2003-2011, Jose M. Martin-Garcia, under the \ General Public License.\"\>"], "Print"], Cell[BoxData["\<\"Connecting to external mac executable...\"\>"], "Print"], Cell[BoxData["\<\"Connection established.\"\>"], "Print"], Cell[BoxData["\<\"------------------------------------------------------------\ \"\>"], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Package xAct`xTensor` version \"\>", "\[InvisibleSpace]", "\<\"1.0.2\"\>", "\[InvisibleSpace]", "\<\", \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"2011", ",", "7", ",", "15"}], "}"}]}], SequenceForm[ "Package xAct`xTensor` version ", "1.0.2", ", ", {2011, 7, 15}], Editable->False]], "Print"], Cell[BoxData["\<\"CopyRight (C) 2002-2011, Jose M. Martin-Garcia, under the \ General Public License.\"\>"], "Print"], Cell[BoxData["\<\"------------------------------------------------------------\ \"\>"], "Print"], Cell[BoxData["\<\"These packages come with ABSOLUTELY NO WARRANTY; for \ details type Disclaimer[]. This is free software, and you are welcome to \ redistribute it under certain conditions. See the General Public License for \ details.\"\>"], "Print"], Cell[BoxData["\<\"------------------------------------------------------------\ \"\>"], "Print"] }, Open ]] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"$PrePrint", "=", "ScreenDollarIndices"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"SetOptions", "[", RowBox[{"ContractMetric", ",", RowBox[{"AllowUpperDerivatives", "\[Rule]", "True"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"SetOptions", "[", RowBox[{"MakeRule", ",", RowBox[{"MetricOn", "\[Rule]", "All"}], ",", RowBox[{"ContractMetrics", "\[Rule]", "True"}]}], "]"}], ";"}]], "Input"], Cell["\<\ This is a modified version of IndexSolve to allow for contracted indices; it \ does not work in general, but it works below.\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"Unprotect", "[", "IndexSolve", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"IndexSolve", "[", RowBox[{ RowBox[{"lhs_", "\[Equal]", "rhs_"}], ",", "object_", ",", "options___"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"seq", "=", RowBox[{"Simplification", "[", RowBox[{"Expand", "[", RowBox[{"lhs", "-", "rhs"}], "]"}], "]"}]}], ",", "frees", ",", "inds", ",", "dummies", ",", "shadow", ",", "shadrule", ",", RowBox[{"(*", RowBox[{"syms", ","}], "*)"}], "scalar", ",", "righths", ",", RowBox[{"verb", "=", RowBox[{ RowBox[{"Verbose", "/.", RowBox[{"CheckOptions", "[", "options", "]"}]}], "/.", RowBox[{"Options", "[", "MakeRule", "]"}]}]}]}], "}"}], ",", RowBox[{ RowBox[{"frees", "=", RowBox[{"FindFreeIndices", "[", RowBox[{"Evaluate", "[", "seq", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"inds", "=", RowBox[{"FindFreeIndices", "[", "object", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"dummies", "=", RowBox[{"FindDummyIndices", "[", "object", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{"verb", ",", RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", "frees", ",", "\"\<, inds: \>\"", ",", "inds", ",", "\"\<, dummies: \>\"", ",", "dummies"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"Sort", "@", "frees"}], "=!=", RowBox[{"Sort", "@", "inds"}]}], ",", RowBox[{"Throw", "[", RowBox[{"Message", "[", RowBox[{ RowBox[{"IndexSolve", "::", "free"}], ",", "inds"}], "]"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ "Need", " ", "to", " ", "do", " ", "something", " ", "with", " ", "symmetries", " ", RowBox[{"here", "!"}]}], "*)"}], RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"shadow", "@@", "frees"}], ",", RowBox[{"ManifoldsOf", "[", "object", "]"}]}], RowBox[{"(*", RowBox[{",", "syms"}], "*)"}], "]"}], ";", RowBox[{"(*", RowBox[{ "I", " ", "am", " ", "only", " ", "dealing", " ", "with", " ", "Tensors", " ", RowBox[{"here", ".", " ", "This"}], " ", "is", " ", RowBox[{"wrong", "."}]}], "*)"}], RowBox[{"shadrule", "=", RowBox[{"MakeRule", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Evaluate", "@", "object"}], ",", RowBox[{"Evaluate", "[", RowBox[{"shadow", "@@", "frees"}], "]"}]}], "}"}], ",", RowBox[{"MetricOn", "\[Rule]", "All"}], ",", RowBox[{"PatternIndices", "\[Rule]", "All"}]}], "]"}]}], ";", RowBox[{"(*", RowBox[{ "this", " ", "should", " ", "only", " ", "match", " ", "those", " ", "cases", " ", "with", " ", "the", " ", "same", " ", "free", " ", "indices", " ", "as", " ", "specified", " ", "in", " ", "object"}], "*)"}], RowBox[{"If", "[", RowBox[{"verb", ",", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"shadow", "@@", "frees"}], ",", "\"\<, shadrule: \>\"", ",", "shadrule"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"scalar", "=", RowBox[{"ToCanonical", "@", RowBox[{"Scalar", "@", RowBox[{"Coefficient", "[", RowBox[{ RowBox[{"Simplification", "[", RowBox[{"seq", "/.", "shadrule"}], "]"}], ",", RowBox[{"shadow", "@@", "frees"}]}], "]"}]}]}]}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{"verb", ",", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "scalar"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{ RowBox[{"!", RowBox[{"ScalarQ", "[", "scalar", "]"}]}], ",", RowBox[{"Throw", "[", RowBox[{"Message", "[", RowBox[{ RowBox[{"IndexSolve", "::", "error"}], ",", "\"\<\>\""}], "]"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"righths", "=", RowBox[{"Simplification", "[", RowBox[{"NoScalar", "[", RowBox[{"seq", "-", RowBox[{"scalar", " ", "object"}]}], "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{"verb", ",", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "righths"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Undef", "[", "shadow", "]"}], ";", RowBox[{"(*", RowBox[{ RowBox[{ RowBox[{ "would", " ", "prefer", " ", "if", " ", "DefTensor", " ", "and", " ", "Undef", " ", "were", " ", "quiet"}], "..."}], " ", RowBox[{"don", "'"}], "t", " ", "know", " ", "how", " ", "to", " ", "do", " ", "that"}], "*)"}], RowBox[{ RowBox[{"MakeRule", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Evaluate", "@", "object"}], ",", RowBox[{"Evaluate", "[", RowBox[{ RowBox[{"-", "righths"}], "/", "scalar"}], "]"}]}], "}"}], ",", RowBox[{"PatternIndices", "\[Rule]", "All"}], ",", "options"}], "]"}], "//", "Flatten"}]}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"Protect", "[", "IndexSolve", "]"}], ";"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["1. Manifold and metric", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefConstantSymbol", "[", "dim", "]"}], ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefConstantSymbol", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"constant symbol \"\>", "\[InvisibleSpace]", "dim", "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefConstantSymbol, ": Defining ", "constant symbol ", $CellContext`dim, ". ", ""], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefManifold", "[", RowBox[{"M", ",", "dim", ",", RowBox[{"IndexRange", "[", RowBox[{"a", ",", "q"}], "]"}]}], "]"}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefManifold", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"manifold \"\>", "\[InvisibleSpace]", "M", "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefManifold, ": Defining ", "manifold ", $CellContext`M, ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefVBundle", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"vbundle \"\>", "\[InvisibleSpace]", "TangentM", "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefVBundle, ": Defining ", "vbundle ", $CellContext`TangentM, ". ", ""], Editable->False]], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefMetric", "[", RowBox[{ RowBox[{"-", "1"}], ",", RowBox[{"met", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "CD", ",", RowBox[{"PrintAs", "\[Rule]", "\"\\""}], ",", RowBox[{"WeightedWithBasis", "\[Rule]", "AIndex"}]}], "]"}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric metric tensor \"\>", "\[InvisibleSpace]", RowBox[{"met", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric metric tensor ", $CellContext`met[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"antisymmetric tensor \"\>", "\[InvisibleSpace]", RowBox[{"epsilonmet", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "antisymmetric tensor ", $CellContext`epsilonmet[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefCovD", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"covariant derivative \"\>", "\[InvisibleSpace]", RowBox[{"CD", "[", RowBox[{"-", "a"}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefCovD, ": Defining ", "covariant derivative ", $CellContext`CD[-$CellContext`a], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"vanishing torsion tensor \"\>", "\[InvisibleSpace]", RowBox[{"TorsionCD", "[", RowBox[{"a", ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "vanishing torsion tensor ", $CellContext`TorsionCD[$CellContext`a, -$CellContext`b, -$CellContext`c], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric Christoffel tensor \"\>", "\[InvisibleSpace]", RowBox[{"ChristoffelCD", "[", RowBox[{"a", ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric Christoffel tensor ", $CellContext`ChristoffelCD[$CellContext`a, -$CellContext`b, \ -$CellContext`c], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Riemann tensor \"\>", "\[InvisibleSpace]", RowBox[{"RiemannCD", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}], ",", RowBox[{"-", "d"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "Riemann tensor ", $CellContext`RiemannCD[-$CellContext`a, -$CellContext`b, -$CellContext`c, \ -$CellContext`d], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric Ricci tensor \"\>", "\[InvisibleSpace]", RowBox[{"RicciCD", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric Ricci tensor ", $CellContext`RicciCD[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData["\<\"** DefCovD: Contractions of Riemann automatically replaced \ by Ricci.\"\>"], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Ricci scalar \"\>", "\[InvisibleSpace]", RowBox[{"RicciScalarCD", "[", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "Ricci scalar ", $CellContext`RicciScalarCD[], ". ", ""], Editable->False]], "Print"], Cell[BoxData["\<\"** DefCovD: Contractions of Ricci automatically replaced \ by RicciScalar.\"\>"], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric Einstein tensor \"\>", "\[InvisibleSpace]", RowBox[{"EinsteinCD", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric Einstein tensor ", $CellContext`EinsteinCD[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Weyl tensor \"\>", "\[InvisibleSpace]", RowBox[{"WeylCD", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}], ",", RowBox[{"-", "d"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "Weyl tensor ", $CellContext`WeylCD[-$CellContext`a, -$CellContext`b, -$CellContext`c, \ -$CellContext`d], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric TFRicci tensor \"\>", "\[InvisibleSpace]", RowBox[{"TFRicciCD", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric TFRicci tensor ", $CellContext`TFRicciCD[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Kretschmann scalar \"\>", "\[InvisibleSpace]", RowBox[{"KretschmannCD", "[", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "Kretschmann scalar ", $CellContext`KretschmannCD[], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** DefCovD: Computing RiemannToWeylRules for dim \"\>", "\[InvisibleSpace]", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}], SequenceForm[ "** DefCovD: Computing RiemannToWeylRules for dim ", $CellContext`dim], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** DefCovD: Computing RicciToTFRicci for dim \"\>", "\[InvisibleSpace]", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}], SequenceForm[ "** DefCovD: Computing RicciToTFRicci for dim ", $CellContext`dim], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** DefCovD: Computing RicciToEinsteinRules for dim \"\>", "\[InvisibleSpace]", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}], SequenceForm[ "** DefCovD: Computing RicciToEinsteinRules for dim ", $CellContext`dim], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"weight +2 density \"\>", "\[InvisibleSpace]", RowBox[{"Detmet", "[", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"Determinant.\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "weight +2 density ", $CellContext`Detmet[], ". ", "Determinant."], Editable->False]], "Print"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["2. The unit timelike geodesic congruence", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"\[Xi]", "[", "a", "]"}], ",", "M"}], "]"}], ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"tensor \"\>", "\[InvisibleSpace]", RowBox[{"\[Xi]", "[", "a", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "tensor ", $CellContext`\[Xi][$CellContext`a], ". ", ""], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"AutomaticRules", "[", RowBox[{"\[Xi]", ",", RowBox[{"MakeRule", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"\[Xi]", "[", "a", "]"}], RowBox[{"\[Xi]", "[", RowBox[{"-", "a"}], "]"}]}], ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"MetricOn", "\[Rule]", "All"}]}], "]"}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"AutomaticRules", "[", RowBox[{"\[Xi]", ",", RowBox[{"MakeRule", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"\[Xi]", "[", "a", "]"}], RowBox[{"\[Xi]", "[", "b", "]"}], RowBox[{"met", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}], ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"MetricOn", "\[Rule]", "All"}]}], "]"}]}], "]"}], ";"}]}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" Rules \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], "\[InvisibleSpace]", "\<\" have been declared as UpValues for \"\>", "\[InvisibleSpace]", "\[Xi]", "\[InvisibleSpace]", "\<\".\"\>"}], SequenceForm[" Rules ", Shallow[{1, 2}], " have been declared as UpValues for ", $CellContext`\[Xi], "."], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" Rules \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ "1", ",", "2", ",", "3", ",", "4", ",", "5", ",", "6", ",", "7", ",", "8"}], "}"}], "\[InvisibleSpace]", "\<\" have been declared as UpValues for \"\>", "\[InvisibleSpace]", "\[Xi]", "\[InvisibleSpace]", "\<\".\"\>"}], SequenceForm[" Rules ", Shallow[{1, 2, 3, 4, 5, 6, 7, 8}], " have been declared as UpValues for ", $CellContext`\[Xi], "."], Editable->False]], "Print"] }, Open ]] }, Open ]], Cell["\<\ \[Xi] is not just any timelike unit normal -- it is parallel transported \ along itself.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Xi]GeodesicRule", "=", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"\[Xi]", "[", "c", "]"}], RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "c"}], "]"}], "@", RowBox[{"\[Xi]", "[", "a", "]"}]}]}], ",", "0"}], "}"}], "]"}]}], ";"}]], "Input"], Cell["\<\ The above property can be used to convert a double derivative of \[Xi] into \ the product of two first derivatives of \[Xi]. The following is an identity:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DD\[Xi]Eq", "=", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "b"}], "]"}], "[", RowBox[{ RowBox[{"\[Xi]", "[", "c", "]"}], RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "c"}], "]"}], "@", RowBox[{"\[Xi]", "[", RowBox[{"-", "a"}], "]"}]}]}], "]"}], "\[Equal]", RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "b"}], "]"}], "[", RowBox[{ RowBox[{ RowBox[{"\[Xi]", "[", "c", "]"}], RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "c"}], "]"}], "@", RowBox[{"\[Xi]", "[", RowBox[{"-", "a"}], "]"}]}]}], "/.", "\[Xi]GeodesicRule"}], "]"}]}], ")"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "b"], SubscriptBox["\[EmptyDownTriangle]", "c"], GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {" "}, {"a"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`b][ $CellContext`CD[-$CellContext`c][ $CellContext`\[Xi][-$CellContext`a]]], Editable->False], ")"}]}], "+", RowBox[{ RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "b"], GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`b][ $CellContext`\[Xi][$CellContext`c]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "c"], GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {" "}, {"a"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`c][ $CellContext`\[Xi][-$CellContext`a]], Editable->False], ")"}]}]}], "\[Equal]", "0"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DD\[Xi]Eq2", "=", RowBox[{ RowBox[{"DD\[Xi]Eq", "//", "SortCovDs"}], "//", "Expand"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", InterpretationBox[ StyleBox[GridBox[{ { RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], StyleBox[GridBox[{ {" ", " ", " ", "d"}, {"c", "b", "a", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`RiemannCD[-$CellContext`c, -$CellContext`b, \ -$CellContext`a, $CellContext`d], Editable->False]}], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {" "}, {"d"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][-$CellContext`d], Editable->False]}], "+", RowBox[{ RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "b"], GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`b][ $CellContext`\[Xi][$CellContext`c]], Editable->False], ")"}], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "c"], GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {" "}, {"a"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`c][ $CellContext`\[Xi][-$CellContext`a]], Editable->False], ")"}]}], "+", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "c"], SubscriptBox["\[EmptyDownTriangle]", "b"], GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {" "}, {"a"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`c][ $CellContext`CD[-$CellContext`b][ $CellContext`\[Xi][-$CellContext`a]]], Editable->False], ")"}]}]}], "\[Equal]", "0"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"DD\[Xi]Rule1", "=", RowBox[{"IndexSolve", "[", RowBox[{"DD\[Xi]Eq", ",", RowBox[{ RowBox[{"\[Xi]", "[", "c", "]"}], RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "b"}], "]"}], "@", RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "c"}], "]"}], "@", RowBox[{"\[Xi]", "[", RowBox[{"-", "a"}], "]"}]}]}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DD\[Xi]Rule2", "=", RowBox[{"IndexSolve", "[", RowBox[{"DD\[Xi]Eq2", ",", RowBox[{ RowBox[{"\[Xi]", "[", "c", "]"}], RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "c"}], "]"}], "@", RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "b"}], "]"}], "@", RowBox[{"\[Xi]", "[", RowBox[{"-", "a"}], "]"}]}]}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DD\[Xi]Rule", "=", RowBox[{"Join", "[", RowBox[{"DD\[Xi]Rule1", ",", "DD\[Xi]Rule2"}], "]"}]}], ";"}]}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"tensor \"\>", "\[InvisibleSpace]", RowBox[{"shadow$1855", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "tensor ", $CellContext`shadow$1855[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "UndefTensor", "\[InvisibleSpace]", "\<\": Undefined \"\>", "\[InvisibleSpace]", "\<\"tensor\"\>", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "shadow$1855"}], SequenceForm[ "** ", xAct`xTensor`UndefTensor, ": Undefined ", "tensor", " ", $CellContext`shadow$1855], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"tensor \"\>", "\[InvisibleSpace]", RowBox[{"shadow$1907", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "tensor ", $CellContext`shadow$1907[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "UndefTensor", "\[InvisibleSpace]", "\<\": Undefined \"\>", "\[InvisibleSpace]", "\<\"tensor\"\>", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", "shadow$1907"}], SequenceForm[ "** ", xAct`xTensor`UndefTensor, ": Undefined ", "tensor", " ", $CellContext`shadow$1907], Editable->False]], "Print"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["3. Hypersurface decomposition", "Subsection"], Cell[BoxData[{ RowBox[{ RowBox[{"$ExtrinsicKSign", "=", RowBox[{"-", "1"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"$AccelerationSign", "=", RowBox[{"-", "1"}]}], ";"}]}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefMetric", "[", RowBox[{"1", ",", RowBox[{"meth", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "cd", ",", RowBox[{"SymbolOfCovD", "->", RowBox[{"{", RowBox[{"\"\<|\>\"", ",", "\"\\""}], "}"}]}], ",", RowBox[{"InducedFrom", "\[Rule]", RowBox[{"{", RowBox[{"met", ",", "\[Xi]"}], "}"}]}], ",", RowBox[{"PrintAs", "->", "\"\\""}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"DefMetric", "::", "\<\"old\"\>"}], ":", " ", "\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"There are already metrics \ \\\\\\\"\\\", \\\"MT\\\"]\\)\[NoBreak]\\!\\(\\*StyleBox[\\!\\({met}\\), \ \\\"MT\\\"]\\)\[NoBreak]\\!\\(\\*StyleBox[\\\"\\\\\\\" in vbundle \ \\\\\\\"\\\", \\\"MT\\\"]\\)\[NoBreak]\\!\\(\\*StyleBox[\\!\\(TangentM\\), \\\ \"MT\\\"]\\)\[NoBreak]\\!\\(\\*StyleBox[\\\"\\\\\\\".\\\\\\\"\\\", \ \\\"MT\\\"]\\)\"\>"}]], "Message", "MSG"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric metric tensor \"\>", "\[InvisibleSpace]", RowBox[{"meth", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric metric tensor ", $CellContext`meth[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"antisymmetric tensor \"\>", "\[InvisibleSpace]", RowBox[{"epsilonmeth", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "antisymmetric tensor ", $CellContext`epsilonmeth[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefCovD", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"covariant derivative \"\>", "\[InvisibleSpace]", RowBox[{"cd", "[", RowBox[{"-", "a"}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefCovD, ": Defining ", "covariant derivative ", $CellContext`cd[-$CellContext`a], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"vanishing torsion tensor \"\>", "\[InvisibleSpace]", RowBox[{"Torsioncd", "[", RowBox[{"a", ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "vanishing torsion tensor ", $CellContext`Torsioncd[$CellContext`a, -$CellContext`b, -$CellContext`c], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric Christoffel tensor \"\>", "\[InvisibleSpace]", RowBox[{"Christoffelcd", "[", RowBox[{"a", ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric Christoffel tensor ", $CellContext`Christoffelcd[$CellContext`a, -$CellContext`b, \ -$CellContext`c], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Riemann tensor \"\>", "\[InvisibleSpace]", RowBox[{"Riemanncd", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}], ",", RowBox[{"-", "d"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "Riemann tensor ", $CellContext`Riemanncd[-$CellContext`a, -$CellContext`b, -$CellContext`c, \ -$CellContext`d], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric Ricci tensor \"\>", "\[InvisibleSpace]", RowBox[{"Riccicd", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric Ricci tensor ", $CellContext`Riccicd[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData["\<\"** DefCovD: Contractions of Riemann automatically replaced \ by Ricci.\"\>"], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Ricci scalar \"\>", "\[InvisibleSpace]", RowBox[{"RicciScalarcd", "[", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "Ricci scalar ", $CellContext`RicciScalarcd[], ". ", ""], Editable->False]], "Print"], Cell[BoxData["\<\"** DefCovD: Contractions of Ricci automatically replaced \ by RicciScalar.\"\>"], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric Einstein tensor \"\>", "\[InvisibleSpace]", RowBox[{"Einsteincd", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric Einstein tensor ", $CellContext`Einsteincd[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Weyl tensor \"\>", "\[InvisibleSpace]", RowBox[{"Weylcd", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}], ",", RowBox[{"-", "d"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "Weyl tensor ", $CellContext`Weylcd[-$CellContext`a, -$CellContext`b, -$CellContext`c, \ -$CellContext`d], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"symmetric TFRicci tensor \"\>", "\[InvisibleSpace]", RowBox[{"TFRiccicd", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "symmetric TFRicci tensor ", $CellContext`TFRiccicd[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"Kretschmann scalar \"\>", "\[InvisibleSpace]", RowBox[{"Kretschmanncd", "[", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "Kretschmann scalar ", $CellContext`Kretschmanncd[], ". ", ""], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** DefCovD: Computing RiemannToWeylRules for dim \"\>", "\[InvisibleSpace]", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}], SequenceForm[ "** DefCovD: Computing RiemannToWeylRules for dim ", $CellContext`dim], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** DefCovD: Computing RicciToTFRicci for dim \"\>", "\[InvisibleSpace]", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}], SequenceForm[ "** DefCovD: Computing RicciToTFRicci for dim ", $CellContext`dim], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** DefCovD: Computing RicciToEinsteinRules for dim \"\>", "\[InvisibleSpace]", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}], SequenceForm[ "** DefCovD: Computing RicciToEinsteinRules for dim ", $CellContext`dim], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"weight +2 density \"\>", "\[InvisibleSpace]", RowBox[{"Detmeth", "[", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"Determinant.\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "weight +2 density ", $CellContext`Detmeth[], ". ", "Determinant."], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"extrinsic curvature tensor \"\>", "\[InvisibleSpace]", RowBox[{"ExtrinsicKmeth", "[", RowBox[{"a", ",", "b"}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"Associated to vector \[Xi]\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "extrinsic curvature tensor ", $CellContext`ExtrinsicKmeth[$CellContext`a, $CellContext`b], ". ", "Associated to vector \[Xi]"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"acceleration vector \"\>", "\[InvisibleSpace]", RowBox[{"Acceleration\[Xi]", "[", "a", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"Associated to vector \[Xi]\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefTensor, ": Defining ", "acceleration vector ", $CellContext`Acceleration\[Xi][$CellContext`a], ". ", "Associated to vector \[Xi]"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefInertHead", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"projector inert-head \"\>", "\[InvisibleSpace]", "Projectormeth", "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm[ "** ", xAct`xTensor`DefInertHead, ": Defining ", "projector inert-head ", $CellContext`Projectormeth, ". ", ""], Editable->False]], "Print"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["4. Quantities in the Raychaudhuri equation", "Subsection"], Cell["This follows the conventions of Wald, sec. 9.2.", "Text"], Cell["\<\ This is the gradient of \[Xi]. Orthogonal to \[Xi] itself (Eq. 9.2.2)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "M", ",", RowBox[{"OrthogonalTo", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"\[Xi]", "[", "a", "]"}], ",", RowBox[{"\[Xi]", "[", "b", "]"}]}], "}"}]}], ",", RowBox[{"ProjectedWith", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"meth", "[", RowBox[{"a", ",", RowBox[{"-", "c"}]}], "]"}], ",", RowBox[{"meth", "[", RowBox[{"b", ",", RowBox[{"-", "c"}]}], "]"}]}], "}"}]}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"tensor \"\>", "\[InvisibleSpace]", RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "tensor ", $CellContext`B[-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"] }, Open ]], Cell["\<\ How to go between B and the explicit gradient of \[Xi] (Eq. 9.2.1)\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"BToGrad\[Xi]Rule", "=", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "b"}], "]"}], "@", RowBox[{"\[Xi]", "[", RowBox[{"-", "a"}], "]"}]}]}], "}"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Grad\[Xi]ToBRule", "=", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "b"}], "]"}], "@", RowBox[{"\[Xi]", "[", RowBox[{"-", "a"}], "]"}]}], ",", RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}], "}"}], "]"}]}], ";"}]}], "Input"], Cell["This is the \"expansion\" scalar", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"\[Theta]", "[", "]"}], ",", "M"}], "]"}], ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"tensor \"\>", "\[InvisibleSpace]", RowBox[{"\[Theta]", "[", "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "tensor ", $CellContext`\[Theta][], ". ", ""], Editable->False]], "Print"] }, Open ]], Cell["How to write it in terms of B (Eq. 9.2.6)", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Expand\[Theta]Rule", "=", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{"\[Theta]", "[", "]"}], ",", RowBox[{ RowBox[{"B", "[", RowBox[{"a", ",", "b"}], "]"}], RowBox[{"meth", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}]}], "}"}], "]"}]}], ";"}]], "Input"], Cell["This is the symmetric shear tensor", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"\[Sigma]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "M", ",", RowBox[{"Symmetric", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "}"}], "]"}], ",", RowBox[{"OrthogonalTo", "\[Rule]", RowBox[{"{", RowBox[{"\[Xi]", "[", "a", "]"}], "}"}]}], ",", RowBox[{"ProjectedWith", "\[Rule]", RowBox[{"{", RowBox[{"meth", "[", RowBox[{"a", ",", RowBox[{"-", "c"}]}], "]"}], "}"}]}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"tensor \"\>", "\[InvisibleSpace]", RowBox[{"\[Sigma]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "tensor ", $CellContext`\[Sigma][-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"] }, Open ]], Cell["\[Sigma] is tracefree under contraction with h.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"AutomaticRules", "[", RowBox[{"\[Sigma]", ",", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{"\[Sigma]", "[", RowBox[{"a", ",", RowBox[{"-", "a"}]}], "]"}], ",", "0"}], "}"}], "]"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" Rules \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], "\[InvisibleSpace]", "\<\" have been declared as DownValues for \"\>", "\[InvisibleSpace]", "\[Sigma]", "\[InvisibleSpace]", "\<\".\"\>"}], SequenceForm[" Rules ", Shallow[{1, 2}], " have been declared as DownValues for ", $CellContext`\[Sigma], "."], Editable->False]], "Print"] }, Open ]], Cell["This is how to write it in terms of B (Eq. 9.2.7)", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Expand\[Sigma]Rule", "=", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{"\[Sigma]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Symmetrize", "[", RowBox[{ RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "}"}]}], "]"}], "-", RowBox[{ FractionBox["1", RowBox[{"dim", "-", "1"}]], RowBox[{"\[Theta]", "[", "]"}], " ", RowBox[{"meth", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}]}], "//", "Evaluate"}]}], "}"}], "]"}]}], ";"}]], "Input"], Cell["This is the antisymmetric twist tensor.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DefTensor", "[", RowBox[{ RowBox[{"\[Omega]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "M", ",", RowBox[{"Antisymmetric", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "}"}], "]"}], ",", RowBox[{"OrthogonalTo", "\[Rule]", RowBox[{"{", RowBox[{"\[Xi]", "[", "a", "]"}], "}"}]}], ",", RowBox[{"ProjectedWith", "\[Rule]", RowBox[{"{", RowBox[{"meth", "[", RowBox[{"a", ",", RowBox[{"-", "c"}]}], "]"}], "}"}]}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefTensor", "\[InvisibleSpace]", "\<\": Defining \"\>", "\[InvisibleSpace]", "\<\"tensor \"\>", "\[InvisibleSpace]", RowBox[{"\[Omega]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "\[InvisibleSpace]", "\<\". \"\>", "\[InvisibleSpace]", "\<\"\"\>"}], SequenceForm["** ", xAct`xTensor`DefTensor, ": Defining ", "tensor ", $CellContext`\[Omega][-$CellContext`a, -$CellContext`b], ". ", ""], Editable->False]], "Print"] }, Open ]], Cell["How to write it in terms of B (Eq. 9.2.8)", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Expand\[Omega]Rule", "=", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{"\[Omega]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"Antisymmetrize", "[", RowBox[{ RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "}"}]}], "]"}], "//", "Evaluate"}]}], "}"}], "]"}]}], ";"}]], "Input"], Cell["The three preceding rules in one", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"ExpandOpticalTensorsRule", "=", RowBox[{"FoldedRule", "[", RowBox[{ "Expand\[Omega]Rule", ",", "Expand\[Sigma]Rule", ",", "Expand\[Theta]Rule"}], "]"}]}], ";"}]], "Input"], Cell["\<\ This is the opposite of the previous rule -- how to decompose B into the \ trace, symmetric tracefree, and antisymmetric parts.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"BDecompositionRule", "=", RowBox[{"MakeRule", "[", RowBox[{"{", RowBox[{ RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ FractionBox["1", RowBox[{"dim", "-", "1"}]], RowBox[{"\[Theta]", "[", "]"}], RowBox[{"meth", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}], "+", RowBox[{"\[Sigma]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "+", RowBox[{"\[Omega]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}]}], "}"}], "]"}]}], ";"}]], "Input"], Cell["They are really inverses of each other:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], "/.", "BDecompositionRule"}], "/.", "ExpandOpticalTensorsRule"}]], "Input"], Cell[BoxData[ InterpretationBox[ StyleBox[GridBox[{ {"B", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`B[-$CellContext`a, -$CellContext`b], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"\[Theta]", "[", "]"}], ",", RowBox[{"\[Sigma]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}], ",", RowBox[{"\[Omega]", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}], "}"}], "/.", "ExpandOpticalTensorsRule"}], "/.", "BDecompositionRule"}], "//", "ToCanonical"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ InterpretationBox[ StyleBox["\[Theta]", ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Theta][], Editable->False], ",", InterpretationBox[ StyleBox[GridBox[{ {"\[Sigma]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Sigma][-$CellContext`a, -$CellContext`b], Editable->False], ",", InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][-$CellContext`a, -$CellContext`b], Editable->False]}], "}"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["5. The Raychaudhuri equation", "Subsection"], Cell["This is Wald's Eq. 9.2.10, the left hand side", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eq10LHS", "=", RowBox[{ RowBox[{"\[Xi]", "[", "c", "]"}], RowBox[{ RowBox[{"CD", "[", RowBox[{"-", "c"}], "]"}], "@", RowBox[{"B", "[", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "]"}]}]}]}]], "Input"], Cell[BoxData[ RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "c"], GridBox[{ {"B", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`c][ $CellContext`B[-$CellContext`a, -$CellContext`b]], Editable->False], ")"}]}]], "Output"] }, Open ]], Cell["\<\ The right hand side comes from writing B explicitly as the gradient of \[Xi], \ commuting derivatives, then using the geodesic property of \[Xi] (see sec. 2 \ above) to rewrite the double derivative of \[Xi] as the product of two first \ derivatives, and finally rewriting those as B.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eq10RHS", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"CommuteCovDs", "[", RowBox[{ RowBox[{"Eq10LHS", "/.", "BToGrad\[Xi]Rule"}], ",", "CD", ",", RowBox[{"{", RowBox[{ RowBox[{"-", "b"}], ",", RowBox[{"-", "c"}]}], "}"}]}], "]"}], "//", "Expand"}], ")"}], "/.", "DD\[Xi]Rule"}], "/.", "Grad\[Xi]ToBRule"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", InterpretationBox[ StyleBox[GridBox[{ {"B", StyleBox[GridBox[{ {" ", "c"}, {"a", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`B[-$CellContext`a, $CellContext`c], Editable->False]}], " ", InterpretationBox[ StyleBox[GridBox[{ {"B", StyleBox[GridBox[{ {" ", " "}, {"c", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`B[-$CellContext`c, -$CellContext`b], Editable->False]}], "-", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ { RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], StyleBox[GridBox[{ {" ", " ", " ", "d"}, {"b", "c", "a", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`RiemannCD[-$CellContext`b, -$CellContext`c, -$CellContext`a, \ $CellContext`d], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {" "}, {"d"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][-$CellContext`d], Editable->False]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eq10", "=", RowBox[{"(", RowBox[{"Eq10LHS", "\[Equal]", "Eq10RHS"}], ")"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "c"], GridBox[{ {"B", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`c][ $CellContext`B[-$CellContext`a, -$CellContext`b]], Editable->False], ")"}]}], "\[Equal]", RowBox[{ RowBox[{ RowBox[{"-", InterpretationBox[ StyleBox[GridBox[{ {"B", StyleBox[GridBox[{ {" ", "c"}, {"a", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`B[-$CellContext`a, $CellContext`c], Editable->False]}], " ", InterpretationBox[ StyleBox[GridBox[{ {"B", StyleBox[GridBox[{ {" ", " "}, {"c", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`B[-$CellContext`c, -$CellContext`b], Editable->False]}], "-", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ { RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], StyleBox[GridBox[{ {" ", " ", " ", "d"}, {"b", "c", "a", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`RiemannCD[-$CellContext`b, -$CellContext`c, \ -$CellContext`a, $CellContext`d], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {" "}, {"d"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][-$CellContext`d], Editable->False]}]}]}]], "Output"] }, Open ]], Cell["\<\ To go to Eq. 9.2.11, the trace of Eq. 9.2.10 is taken, first on the LHS and \ then on the RHS.\ \>", "Text"], Cell["Cast B in terms of its trace, STF, and antisymmetric parts.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RaychaudhuriEq", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"#", " ", RowBox[{"met", "[", RowBox[{"a", ",", "b"}], "]"}]}], "//", "ContractMetric"}], ")"}], "/.", "BDecompositionRule"}], "//", "ToCanonical"}], "//", "ReplaceDummies"}], ")"}], "&"}], "/@", "Eq10"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"a"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`a], Editable->False], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "a"], "\[Theta]"}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`a][ $CellContext`\[Theta][]], Editable->False], ")"}]}], "\[Equal]", RowBox[{ RowBox[{"-", FractionBox[ SuperscriptBox[ InterpretationBox[ StyleBox["\[Theta]", ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Theta][], Editable->False], "2"], RowBox[{ RowBox[{"-", "1"}], "+", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}]]}], "-", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ { RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`RicciCD[-$CellContext`a, -$CellContext`b], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"a"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`a], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"b"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`b], Editable->False]}], "-", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Sigma]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Sigma][-$CellContext`a, -$CellContext`b], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Sigma]", StyleBox[GridBox[{ {"a", "b"}, {" ", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Sigma][$CellContext`a, $CellContext`b], Editable->False]}], "+", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][-$CellContext`a, -$CellContext`b], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {"a", "b"}, {" ", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][$CellContext`a, $CellContext`b], Editable->False]}]}]}]], "Output"] }, Open ]], Cell["This is \"the\" Raychaudhuri equation (in 3+1 dimensions):", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RaychaudhuriEq4", "=", RowBox[{"RaychaudhuriEq", "/.", RowBox[{"dim", "\[Rule]", "4"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"a"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`a], Editable->False], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "a"], "\[Theta]"}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`a][ $CellContext`\[Theta][]], Editable->False], ")"}]}], "\[Equal]", RowBox[{ RowBox[{"-", FractionBox[ SuperscriptBox[ InterpretationBox[ StyleBox["\[Theta]", ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Theta][], Editable->False], "2"], "3"]}], "-", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ { RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`RicciCD[-$CellContext`a, -$CellContext`b], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"a"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`a], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"b"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`b], Editable->False]}], "-", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Sigma]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Sigma][-$CellContext`a, -$CellContext`b], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Sigma]", StyleBox[GridBox[{ {"a", "b"}, {" ", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Sigma][$CellContext`a, $CellContext`b], Editable->False]}], "+", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][-$CellContext`a, -$CellContext`b], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {"a", "b"}, {" ", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][$CellContext`a, $CellContext`b], Editable->False]}]}]}]], "Output"] }, Open ]], Cell["\<\ The antisymmetric equation is simple to construct. This is Eq. 9.2.14\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eq14", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Antisymmetrize", "[", RowBox[{"#", ",", RowBox[{"{", RowBox[{ RowBox[{"-", "a"}], ",", RowBox[{"-", "b"}]}], "}"}]}], "]"}], "/.", "BDecompositionRule"}], "//", "ToCanonical"}], ")"}], "&"}], "/@", "Eq10"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Xi]", StyleBox[GridBox[{ {"c"}, {" "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Xi][$CellContext`c], Editable->False], " ", RowBox[{"(", InterpretationBox[ StyleBox[ RowBox[{ SubscriptBox["\[EmptyDownTriangle]", "c"], GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`CD[-$CellContext`c][ $CellContext`\[Omega][-$CellContext`a, -$CellContext`b]], Editable->False], ")"}]}], "\[Equal]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"2", " ", InterpretationBox[ StyleBox["\[Theta]", ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Theta][], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {" ", " "}, {"a", "b"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][-$CellContext`a, -$CellContext`b], Editable->False]}], RowBox[{ RowBox[{"-", "1"}], "+", InterpretationBox[ StyleBox["dim", ShowAutoStyles->False, AutoSpacing->False], $CellContext`dim, Editable->False]}]]}], "-", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Sigma]", StyleBox[GridBox[{ {" ", "c"}, {"b", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Sigma][-$CellContext`b, $CellContext`c], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {" ", " "}, {"a", "c"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][-$CellContext`a, -$CellContext`c], Editable->False]}], "+", RowBox[{ InterpretationBox[ StyleBox[GridBox[{ {"\[Sigma]", StyleBox[GridBox[{ {" ", "c"}, {"a", " "} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Sigma][-$CellContext`a, $CellContext`c], Editable->False], " ", InterpretationBox[ StyleBox[GridBox[{ {"\[Omega]", StyleBox[GridBox[{ {" ", " "}, {"b", "c"} }, GridBoxSpacings->{"Columns" -> { Offset[0.], { Offset[0.034999999999999996`]}, Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ Offset[0.]}}, "RowsIndexed" -> {}}], FontSize->9]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.034999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], ShowAutoStyles->False, AutoSpacing->False], $CellContext`\[Omega][-$CellContext`b, -$CellContext`c], Editable->False]}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "Equation 9.2.12 can also be derived with some more effort, using STFPart, \ RiemannToWeyl, and by defining an auxilliary field ", Cell[BoxData[ FormBox[ OverscriptBox["R", "~"], TraditionalForm]]], "." }], "Text"] }, Open ]] }, Open ]] }, WindowSize->{1385, 651}, WindowMargins->{{10, Automatic}, {Automatic, 12}}, PrivateNotebookOptions->{"FileOutlineCache"->False}, ShowSelection->True, TrackCellChangeTimes->False, FrontEndVersion->"10.4 for Mac OS X x86 (32-bit, 64-bit Kernel) (April 11, \ 2016)", StyleDefinitions->"Default.nb" ]