(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 957076, 23407] NotebookOptionsPosition[ 934848, 23052] NotebookOutlinePosition[ 935259, 23069] CellTagsIndexPosition[ 935216, 23066] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Equations of motion for charged scalar and vector boson stars\ \>", "Title", CellChangeTimes->{{3.829222464150318*^9, 3.8292224672870407`*^9}, { 3.829222637960559*^9, 3.8292226443068485`*^9}, {3.829228262193406*^9, 3.8292282780423837`*^9}},ExpressionUUID->"c4b119b3-e0a5-4f72-b568-\ 697476d4355b"], Cell["\<\ Here we describe the equations to obtain spherically symmetric solutions of \ the system composed by a complex charged scalar and vector. We shall consider \ all terms combined to obtain the differential equations, using a control \ parameter to turn off eventually. We also add the perturbations, to illustrate how the procedure works to find \ the quasinormal modes of such configurations.\ \>", "Text", CellChangeTimes->{{3.8292282838467293`*^9, 3.829228313935773*^9}, { 3.8292284120197964`*^9, 3.8292284415114183`*^9}, {3.829228484811627*^9, 3.8292285251418943`*^9}, {3.8292290786294584`*^9, 3.8292290835886374`*^9}},ExpressionUUID->"cfd66165-ee4e-4fd2-b845-\ ffb5e7bd9264"], Cell["Let us first clear all the possible variables", "Text", CellChangeTimes->{{3.8292285354706016`*^9, 3.829228556713709*^9}},ExpressionUUID->"a4d23a75-f183-441f-97cf-\ 173415b0a55f"], Cell[BoxData[ RowBox[{ RowBox[{"Quit", "[", "]"}], ";"}]], "Input", CellLabel->"In[52]:=",ExpressionUUID->"8d847f62-0e38-4c09-8d7d-cd253b8cc0b8"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"LaunchKernels", "[", "]"}]], "Input", CellChangeTimes->{{3.833285132321178*^9, 3.8332851365043488`*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"bb1a9ed3-573b-4c15-9658-e223c346ca29"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"1", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"2", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"3", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"4", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"5", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"6", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"7", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"8", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"9", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"10", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"11", ",", "\<\"local\"\>"}], "]"}], ",", RowBox[{"\<\"KernelObject\"\>", "[", RowBox[{"12", ",", "\<\"local\"\>"}], "]"}]}], "}"}]], "Output", CellChangeTimes->{3.8332851402079825`*^9}, CellLabel->"Out[1]=",ExpressionUUID->"fd825cbb-9bf8-4266-aca1-6cb12fbd29ff"] }, Open ]], Cell["\<\ and we define the coordinate system to be used (we shall use cos=\[Chi] to \ better perform the algebraic computations)\ \>", "Text", CellChangeTimes->{{3.829228568087365*^9, 3.829228609649785*^9}},ExpressionUUID->"df881938-b1d6-41d6-8599-\ c024b76825db"], Cell[BoxData[{ RowBox[{ RowBox[{"$Assumptions", "=", RowBox[{"{", RowBox[{ RowBox[{"r", "\[GreaterEqual]", "0"}], ",", RowBox[{"1", "\[GreaterEqual]", "\[Chi]", "\[GreaterEqual]", RowBox[{"-", "1"}]}], ",", RowBox[{"Element", "[", RowBox[{ RowBox[{"{", RowBox[{"\[ScriptM]", ",", "\[ScriptL]"}], "}"}], ",", "Integers"}], "]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"x", "=", RowBox[{"{", RowBox[{"t", ",", "r", ",", "\[Chi]", ",", "\[Phi]"}], "}"}]}], ";"}]}], "Input", CellChangeTimes->{{3.829228740778304*^9, 3.8292287761934147`*^9}}, CellLabel->"In[2]:=",ExpressionUUID->"0fef8591-f155-425b-8ff8-3036daedb4b4"], Cell[CellGroupData[{ Cell["Metric, scalar field and vector field", "Subsection", CellChangeTimes->{{3.82922878859496*^9, 3.829228816583603*^9}},ExpressionUUID->"8af72f2b-37af-494d-9e16-\ 7bfdee2b1bb0"], Cell[CellGroupData[{ Cell["\<\ Metric for spherically symmetric spacetime, plus the axial and polar \ perturbations\ \>", "Subsubsection", CellChangeTimes->{{3.829228849984803*^9, 3.82922886030338*^9}, { 3.8292289038945694`*^9, 3.829228907838542*^9}, {3.829228941292822*^9, 3.829228944249112*^9}},ExpressionUUID->"68d61911-bdac-4d2f-b43f-\ 777e544df766"], Cell["\<\ Let us define the metric to be used in the equations of motion. The metric \ will be represented by a matrix\ \>", "Text", CellChangeTimes->{{3.8292291420070066`*^9, 3.8292291524714084`*^9}, { 3.829229229353677*^9, 3.8292292359256973`*^9}},ExpressionUUID->"a39f9792-9e39-4e19-b441-\ 4fefa74dd4e5"], Cell[BoxData[ RowBox[{ RowBox[{"gdd", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"g00", ",", "g01", ",", "g02", ",", "g03"}], "}"}], ",", RowBox[{"{", RowBox[{"g01", ",", "g11", ",", "g12", ",", "g13"}], "}"}], ",", RowBox[{"{", RowBox[{"g02", ",", "g12", ",", "g22", ",", "g23"}], "}"}], ",", RowBox[{"{", RowBox[{"g03", ",", "g13", ",", "g23", ",", "g33"}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.518601553379979*^9, 3.51860169326866*^9}, { 3.518601903749179*^9, 3.518601909118168*^9}, {3.518602110086179*^9, 3.518602114814166*^9}, 3.5186050422961903`*^9, {3.5187032191509953`*^9, 3.518703267685907*^9}, {3.539598979900779*^9, 3.539599059497171*^9}, { 3.541778573580955*^9, 3.54177863791438*^9}, 3.541785210980154*^9, { 3.542088401815126*^9, 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Note that because the field is complex, we are \ separating the perturbations into two. 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