(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 45962, 1474] NotebookOptionsPosition[ 41180, 1319] NotebookOutlinePosition[ 41950, 1344] CellTagsIndexPosition[ 41907, 1341] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Curvature and the Einstein Equation", "Subtitle"], Cell[TextData[{ "This is the ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook ", StyleBox["Curvature and the Einstein Equation ", FontSlant->"Italic"], StyleBox["available from the book website. F", FontVariations->{"CompatibilityType"->0}], "rom a given metric ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["g", FontSlant->"Italic"], "\[Alpha]\[Beta]"], TraditionalForm]]], " , it computes the components of the following: the inverse metric, ", Cell[BoxData[ FormBox[ SuperscriptBox["g", "\[Lambda]\[Sigma]"], TraditionalForm]]], ", the Christoffel symbols or affine connection," }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Lambda]"], "\[Mu]\[Nu]"], "=", RowBox[{ FractionBox["1", "2"], RowBox[{ SuperscriptBox["g", "\[Lambda]\[Sigma]"], "(", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Mu]"], SubscriptBox["g", "\[Sigma]\[Nu]"]}], "+", RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], SubscriptBox["g", "\[Sigma]\[Mu]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], SubscriptBox["g", "\[Mu]\[Nu]"]}]}], ")"}]}]}], TraditionalForm]]], "," }], "Text"], Cell[TextData[{ " ( ", Cell[BoxData[ FormBox[ SubscriptBox["\[PartialD]", RowBox[{"\[Alpha]", " "}]], TraditionalForm]]], "stands for the partial derivative ", Cell[BoxData[ FormBox[ RowBox[{"\[PartialD]", RowBox[{"/", RowBox[{"\[PartialD]", SuperscriptBox["x", "\[Alpha]"]}]}]}], TraditionalForm]]], "), the Riemann tensor," }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ SuperscriptBox["R", "\[Lambda]"], "\[Mu]\[Nu]\[Sigma]"], "=", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], " ", SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Lambda]"], "\[Mu]\[Sigma]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], " ", SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Lambda]"], "\[Mu]\[Nu]"]}], "+", RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Eta]"], "\[Mu]\[Sigma]"], SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Lambda]"], "\[Eta]\[Nu]"]}], "-", RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Eta]"], "\[Mu]\[Nu]"], SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Lambda]"], "\[Eta]\[Sigma]"]}]}]}], TraditionalForm]]], "," }], "Text"], Cell["the Ricci tensor", "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["R", "\[Mu]\[Nu]"], "=", SubscriptBox[ SuperscriptBox["R", "\[Lambda]"], "\[Mu]\[Lambda]\[Nu]"]}], TraditionalForm]]], "," }], "Text"], Cell["the scalar curvature,", "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"R", "=", RowBox[{ SuperscriptBox["g", "\[Mu]\[Nu]"], SubscriptBox["R", "\[Mu]\[Nu]"]}]}], TraditionalForm]]], "," }], "Text"], Cell["and the Einstein tensor,", "Text"], Cell[TextData[{ " ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["G", "\[Mu]\[Nu]"], "=", RowBox[{ SubscriptBox["R", "\[Mu]\[Nu]"], "-", RowBox[{ FractionBox["1", "2"], SubscriptBox["g", "\[Mu]\[Nu]"], "R"}]}]}], TraditionalForm]]], "." }], "Text"], Cell[TextData[{ "You must input the covariant components of the metric tensor ", Cell[BoxData[ FormBox[ SubscriptBox["g", "\[Mu]\[Nu]"], TraditionalForm]]], " by editing the relevant input line in this ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. 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", FontVariations->{"CompatibilityType"->0}], "For example, if ", StyleBox["m", FontWeight->"Bold"], " were used as a summation or table index below, then you would get the \ wrong answer for the present metric because the ", StyleBox["m", FontWeight->"Bold"], " in the metric would be treated as an index, rather than as the mass." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Calculating the inverse metric:", "Subsubsection"], Cell["The inverse metric is obtained through matrix inversion.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"inversemetric", "=", RowBox[{"Simplify", "[", RowBox[{"Inverse", "[", "metric", "]"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ FractionBox["r", RowBox[{ RowBox[{"2", " ", "G", " ", "M"}], "-", "r"}]], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"1", "-", FractionBox[ RowBox[{"2", " ", "G", " ", "M"}], "r"]}], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox["1", SuperscriptBox["r", "2"]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", FractionBox[ SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "2"], SuperscriptBox["r", "2"]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4929439919538*^9, 3.4975746502539997`*^9, 3.4975747600699997`*^9, 3.497574948059*^9, 3.497575058486*^9, 3.498820597259665*^9, 3.49882075278828*^9, 3.498943136079843*^9, 3.525831645862734*^9, 3.5258316813647647`*^9, 3.5258318314583497`*^9, 3.525847797936096*^9, 3.682171016264431*^9, 3.682219417919497*^9}] }, Open ]], Cell["This can also be displayed in matrix form:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"inversemetric", "//", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { FractionBox["r", RowBox[{ RowBox[{"2", " ", "G", " ", "M"}], "-", "r"}]], "0", "0", "0"}, {"0", RowBox[{"1", "-", FractionBox[ RowBox[{"2", " ", "G", " ", "M"}], "r"]}], "0", "0"}, {"0", "0", FractionBox["1", SuperscriptBox["r", "2"]], "0"}, {"0", "0", "0", FractionBox[ SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "2"], SuperscriptBox["r", "2"]]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.492943994311088*^9, 3.497574650269*^9, 3.49757476008*^9, 3.497574948075*^9, 3.497575058499*^9, 3.498820597310922*^9, 3.498820755062775*^9, 3.498943139306797*^9, 3.5258316479708548`*^9, 3.5258316825168304`*^9, 3.52583183251941*^9, 3.5258478001902246`*^9, 3.68217101848934*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Calculating the Christoffel symbols:", "Subsubsection"], Cell[TextData[{ "The calculation of the components of the Christoffel symbols is done by \ transcribing the definition given earlier into the notation of ", StyleBox["Mathematica", FontSlant->"Italic"], " and using the ", StyleBox["Mathematica", FontSlant->"Italic"], " functions ", StyleBox["D", FontWeight->"Bold"], " for taking partial derivatives, ", StyleBox["Sum", FontWeight->"Bold"], " for summing over repeated indices, ", StyleBox["Table", FontWeight->"Bold"], " for forming a list of components, and ", StyleBox["Simplify", FontWeight->"Bold"], " for simplifying the result." }], "Text"], Cell[BoxData[ RowBox[{"affine", ":=", RowBox[{"affine", "=", RowBox[{"Simplify", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}], "*", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"inversemetric", "[", RowBox[{"[", RowBox[{"i", ",", "s"}], "]"}], "]"}], ")"}], "*", "\[IndentingNewLine]", RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"metric", "[", RowBox[{"[", RowBox[{"s", ",", "j"}], "]"}], "]"}], ",", RowBox[{"coord", "[", RowBox[{"[", "k", "]"}], "]"}]}], " ", "]"}], "+", "\[IndentingNewLine]", RowBox[{"D", "[", RowBox[{ RowBox[{"metric", "[", RowBox[{"[", RowBox[{"s", ",", "k"}], "]"}], "]"}], ",", RowBox[{"coord", "[", RowBox[{"[", "j", "]"}], "]"}]}], " ", "]"}], "-", RowBox[{"D", "[", RowBox[{ RowBox[{"metric", "[", RowBox[{"[", RowBox[{"j", ",", "k"}], "]"}], "]"}], ",", RowBox[{"coord", "[", RowBox[{"[", "s", "]"}], "]"}]}], " ", "]"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"s", ",", "1", ",", "n"}], "}"}]}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"i", ",", "1", ",", "n"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", "n"}], "}"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}], " ", "]"}]}]}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Displaying the Christoffel symbols:", "Subsubsection"], Cell[TextData[{ "The nonzero Christoffel symbols are displayed below. 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