├── README.md ├── Lagrangian-variation-xPert-VarD.nb └── Warped-product.nb /README.md: -------------------------------------------------------------------------------- 1 | Example xAct notebooks 2 | ======== 3 | 4 | This is a collection of example xAct notebooks that contain common xAct usage patterns. Right click on a file name below and hit "Save link as..." to download a notebook. 5 | 6 | [`ActionVariation_Metric_Fields.nb`](../../raw/master/ActionVariation_Metric_Fields.nb): 7 | General function to vary a Lagrangian 8 | 9 | 10 | [`Angular-integration.nb`](../../raw/master/Angular-integration.nb): 11 | Example of angular integration on multiple direction vectors (adapted from Post-Newtonian computations) 12 | 13 | 14 | [`Bianchi_I_xCoba.nb`](../../raw/master/Bianchi_I_xCoba.nb): 15 | Bianchi I Einstein equation using the CTensor framework of xCoba 16 | 17 | [`CCE-system-public.nb`](../../raw/master/CCE-system-public.nb): 18 | Equations for Cauchy-characteristic extraction, in 1+1+2 form. The 19 | angular sector is treated with spin-weighted scalars and 20 | raising/lowering operators eth (or edth). A large number of features 21 | are demonstrated, including product manifolds and metrics, and multiple 22 | metrics on the same manifold. Needs more documentation. 23 | 24 | [`Clifford_Algebra_Traces.nb`](../../raw/master/Clifford_Algebra_Traces.nb): 25 | Clifford algebra construction in all integer dimensions and fast evaluations of traces of products of Dirac matrices. 26 | The tracing function is applied to weak-interaction reaction rates. 27 | 28 | [`ConformalEinsteinEqs.nb`](../../raw/master/ConformalEinsteinEqs.nb): 29 | Implementation of the Friedrich's conformal field equations in dimension 4 30 | (metric equations and extended equations). 31 | 32 | 33 | [`Conformal_Transformation.nb`](../../raw/master/Conformal_Transformation.nb): 34 | Tools to perform conformal transformations and examples. Adapted from a subpart of xPand's code. 35 | 36 | [`CTensorChangeCoords.nb`](../../raw/master/CTensorChangeCoords.nb): 37 | Simple example of coordinate change using the CTensor framework. 38 | 39 | [`EB-decomp-and-3+1-split.nb`](../../raw/master/EB-decomp-and-3+1-split.nb): 40 | Examples of 3+1 decomposition applied to the Weyl tensor 41 | 42 | 43 | [`EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb`](../../raw/master/EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb): 44 | Getting equations of motion and simplifying "C-tensors" for Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons theories 45 | 46 | 47 | [`FriedmannLemaitreMetric_CoordinatesApproach_xCoba.nb`](../../raw/master/FriedmannLemaitreMetric_CoordinatesApproach_xCoba.nb): 48 | It shows how to define charts and specify the components of the metric so as to obtain the components of the related curvature tensors. Shows also how to change between different charts. Applied to homogeneous cosmology. 49 | 50 | 51 | [`Gamma matrices (Dirac algebra).nb`](../../raw/master/Gamma%20matrices%20(Dirac%20algebra).nb): 52 | How to treat the algebra of gamma matrices with DefProduct, and canonicalize them to derive many standard identities. 53 | 54 | [`KerrMetric.nb`](../../raw/master/KerrMetric.nb): 55 | Some computations for the Kerr metric in Boyer Lindqvist coordinates and the null tetrad. Computation of the surface gravity. 56 | 57 | [`Lagrangian-variation-xPert-VarD.nb`](../../raw/master/Lagrangian-variation-xPert-VarD.nb): 58 | A common usage pattern for varying a Lagrangian with respect to the metric to get Einstein equations 59 | 60 | 61 | [`Raychaudhuri.nb`](../../raw/master/Raychaudhuri.nb): 62 | Follows section 9.2 of Wald's book (General Relativity) on the Raychaudhuri equation for a congruence of timelike geodesics 63 | 64 | 65 | [`SymManipulatorLovelockExample.nb`](../../raw/master/SymManipulatorLovelockExample.nb): 66 | Application of the Spinors and SymManipulator package to show the Lovelock tensor is non-dynamical. 67 | 68 | 69 | [`Schwarzschild_Metric_Birkhoff_Theorem.nb`](../../raw/master/Schwarzschild_Metric_Birkhoff_Theorem.nb): 70 | Construction of the Schwarzschild metric. 71 | Eddington Finkelstein coordinates and surface gravity as simple applications of the CTensor tools of xCoba. 72 | 73 | [`Spherical-vs-Cartesian-Coordinates`](../../raw/master/Spherical-vs-Cartesian-Coordinates.nb): 74 | Change of coordinates between Cartesian and spherical coordinates in flat 3D. 75 | Can be used as a template for setting up change of coordinates. 76 | 77 | 78 | [`TimelikeCongruence.nb`](../../raw/master/TimelikeCongruence.nb): 79 | 1+3 decomposition of the Ricci and Bianchi identities in dimension 4 with respect to an 80 | arbitrary unit timelike vector field. The kinematical quantities and the spatial parts of the Riemann 81 | tensor are used to obtain the decompositions. 82 | 83 | 84 | [`Warped-product.nb`](../../raw/master/Warped-product.nb): 85 | Minimal example 1) creating a warped product geometry and 2) breaking down the Einstein-Hilbert Lagrangian into the effective lower-dimensional Lagrangian. 86 | -------------------------------------------------------------------------------- /Lagrangian-variation-xPert-VarD.nb: -------------------------------------------------------------------------------- 1 | Notebook[{ 2 | 3 | Cell[CellGroupData[{ 4 | Cell["Variation of Lagrangian with xPert and VarD", "Subtitle"], 5 | 6 | Cell["\<\ 7 | Leo C. Stein (leo.stein@gmail.com) 8 | August 2013\ 9 | \>", "Subsubtitle"], 10 | 11 | Cell["\<\ 12 | Summary: A common usage pattern for varying a Lagrangian. 13 | VarD doesn\[CloseCurlyQuote]t know more complicated math like the variation \ 14 | of a curvature tensor with respect to the metric. However, xPert does, so \ 15 | combining them makes it straightforward to vary Lagrangian.\ 16 | \>", "Subsubtitle"], 17 | 18 | Cell[CellGroupData[{ 19 | 20 | Cell["0. Setup", "Subsubsection"], 21 | 22 | Cell[CellGroupData[{ 23 | 24 | Cell[BoxData[ 25 | RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input"], 26 | 27 | Cell[CellGroupData[{ 28 | 29 | Cell[BoxData["\<\"------------------------------------------------------------\ 30 | \"\>"], "Print"], 31 | 32 | Cell[BoxData[ 33 | InterpretationBox[ 34 | RowBox[{"\<\"Package xAct`xPerm` version \"\>", 35 | "\[InvisibleSpace]", "\<\"1.2.0\"\>", "\[InvisibleSpace]", "\<\", \"\>", 36 | "\[InvisibleSpace]", 37 | RowBox[{"{", 38 | RowBox[{"2013", ",", "1", ",", "27"}], "}"}]}], 39 | SequenceForm["Package xAct`xPerm` version ", "1.2.0", ", ", {2013, 1, 27}], 40 | 41 | Editable->False]], "Print"], 42 | 43 | Cell[BoxData["\<\"CopyRight (C) 2003-2013, Jose M. Martin-Garcia, under the \ 44 | General Public License.\"\>"], "Print"], 45 | 46 | Cell[BoxData["\<\"Connecting to external mac executable...\"\>"], "Print"], 47 | 48 | Cell[BoxData["\<\"Connection established.\"\>"], "Print"], 49 | 50 | Cell[BoxData["\<\"------------------------------------------------------------\ 51 | \"\>"], "Print"], 52 | 53 | Cell[BoxData[ 54 | InterpretationBox[ 55 | RowBox[{"\<\"Package xAct`xTensor` version \"\>", 56 | "\[InvisibleSpace]", "\<\"1.0.5\"\>", "\[InvisibleSpace]", "\<\", \"\>", 57 | "\[InvisibleSpace]", 58 | RowBox[{"{", 59 | RowBox[{"2013", ",", "1", ",", "27"}], "}"}]}], 60 | SequenceForm[ 61 | "Package xAct`xTensor` version ", "1.0.5", ", ", {2013, 1, 27}], 62 | Editable->False]], "Print"], 63 | 64 | Cell[BoxData["\<\"CopyRight (C) 2002-2013, Jose M. 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This is free software, and you are welcome to \ 90 | redistribute it under certain conditions. See the General Public License for \ 91 | details.\"\>"], "Print"], 92 | 93 | Cell[BoxData["\<\"------------------------------------------------------------\ 94 | \"\>"], "Print"], 95 | 96 | Cell[BoxData[ 97 | InterpretationBox[ 98 | RowBox[{"\<\"** Variable \"\>", "\[InvisibleSpace]", "$PrePrint", 99 | "\[InvisibleSpace]", "\<\" assigned value \"\>", "\[InvisibleSpace]", 100 | "ScreenDollarIndices"}], 101 | SequenceForm[ 102 | "** Variable ", $PrePrint, " assigned value ", 103 | xAct`xTensor`ScreenDollarIndices], 104 | Editable->False]], "Print"], 105 | 106 | Cell[BoxData[ 107 | InterpretationBox[ 108 | RowBox[{"\<\"** Variable \"\>", "\[InvisibleSpace]", "$CovDFormat", 109 | "\[InvisibleSpace]", "\<\" changed from \"\>", 110 | "\[InvisibleSpace]", "\<\"Prefix\"\>", "\[InvisibleSpace]", "\<\" to \"\>", 111 | "\[InvisibleSpace]", "\<\"Postfix\"\>"}], 112 | SequenceForm[ 113 | "** Variable ", xAct`xTensor`$CovDFormat, " changed from ", "Prefix", 114 | " to ", "Postfix"], 115 | Editable->False]], "Print"], 116 | 117 | Cell[BoxData[ 118 | InterpretationBox[ 119 | RowBox[{"\<\"** Option \"\>", "\[InvisibleSpace]", "AllowUpperDerivatives", 120 | "\[InvisibleSpace]", "\<\" of \"\>", "\[InvisibleSpace]", "ContractMetric", 121 | "\[InvisibleSpace]", "\<\" changed from \"\>", "\[InvisibleSpace]", 122 | "False", "\[InvisibleSpace]", "\<\" to \"\>", "\[InvisibleSpace]", "True"}], 123 | SequenceForm[ 124 | "** Option ", xAct`xTensor`AllowUpperDerivatives, " of ", 125 | xAct`xTensor`ContractMetric, " changed from ", False, " to ", True], 126 | Editable->False]], "Print"], 127 | 128 | Cell[BoxData[ 129 | InterpretationBox[ 130 | RowBox[{"\<\"** Option \"\>", "\[InvisibleSpace]", "MetricOn", 131 | "\[InvisibleSpace]", "\<\" of \"\>", "\[InvisibleSpace]", "MakeRule", 132 | "\[InvisibleSpace]", "\<\" changed from \"\>", "\[InvisibleSpace]", "None", 133 | "\[InvisibleSpace]", "\<\" to \"\>", "\[InvisibleSpace]", "All"}], 134 | SequenceForm[ 135 | "** Option ", xAct`xTensor`MetricOn, " of ", xAct`xTensor`MakeRule, 136 | " changed from ", None, " to ", All], 137 | Editable->False]], "Print"], 138 | 139 | Cell[BoxData[ 140 | InterpretationBox[ 141 | RowBox[{"\<\"** Option \"\>", "\[InvisibleSpace]", "ContractMetrics", 142 | "\[InvisibleSpace]", "\<\" of \"\>", "\[InvisibleSpace]", "MakeRule", 143 | "\[InvisibleSpace]", "\<\" changed from \"\>", "\[InvisibleSpace]", 144 | "False", "\[InvisibleSpace]", "\<\" to \"\>", "\[InvisibleSpace]", "True"}], 145 | SequenceForm[ 146 | "** Option ", xAct`xTensor`ContractMetrics, " of ", xAct`xTensor`MakeRule, 147 | " changed from ", False, " to ", True], 148 | Editable->False]], "Print"], 149 | 150 | Cell[BoxData[ 151 | InterpretationBox[ 152 | RowBox[{"\<\"** \"\>", "\[InvisibleSpace]", "DefInertHead", 153 | "\[InvisibleSpace]", "\<\": Defining \"\>", 154 | "\[InvisibleSpace]", "\<\"inert head \"\>", "\[InvisibleSpace]", 155 | "Perturbation", "\[InvisibleSpace]", "\<\". \"\>", 156 | "\[InvisibleSpace]", "\<\"\"\>"}], 157 | SequenceForm[ 158 | "** ", xAct`xTensor`DefInertHead, ": Defining ", "inert head ", 159 | xAct`xPert`Perturbation, ". ", ""], 160 | Editable->False]], "Print"] 161 | }, Open ]] 162 | }, Open ]], 163 | 164 | Cell["Turn off extra info", "Text"], 165 | 166 | Cell[BoxData[ 167 | RowBox[{ 168 | RowBox[{"$DefInfoQ", "=", "False"}], ";"}]], "Input"] 169 | }, Open ]], 170 | 171 | Cell[CellGroupData[{ 172 | 173 | Cell["1. Example: GR in 4 dimensions", "Subsection"], 174 | 175 | Cell["Define a 4-manifold with Lorentzian metric.", "Text"], 176 | 177 | Cell[BoxData[{ 178 | RowBox[{"DefManifold", "[", 179 | RowBox[{"M4", ",", "4", ",", 180 | RowBox[{"IndexRange", "[", 181 | RowBox[{"a", ",", "f"}], "]"}]}], "]"}], "\[IndentingNewLine]", 182 | RowBox[{"DefMetric", "[", 183 | RowBox[{ 184 | RowBox[{"-", "1"}], ",", 185 | RowBox[{"g", "[", 186 | RowBox[{ 187 | RowBox[{"-", "a"}], ",", 188 | RowBox[{"-", "b"}]}], "]"}], ",", "CD"}], "]"}]}], "Input"], 189 | 190 | Cell["Define the metric perturbation", "Text"], 191 | 192 | Cell[BoxData[ 193 | RowBox[{"DefMetricPerturbation", "[", 194 | RowBox[{"g", ",", "h", ",", "\[Epsilon]"}], "]"}]], "Input"], 195 | 196 | Cell["\<\ 197 | Here\[CloseCurlyQuote]s the Lagrangian density for GR:\ 198 | \>", "Text"], 199 | 200 | Cell[CellGroupData[{ 201 | 202 | Cell[BoxData[ 203 | RowBox[{"LGR", "=", 204 | RowBox[{ 205 | RowBox[{"Sqrt", "[", 206 | RowBox[{"-", 207 | RowBox[{"Detg", "[", "]"}]}], "]"}], 208 | RowBox[{"RicciScalarCD", "[", "]"}]}]}]], "Input"], 209 | 210 | Cell[BoxData[ 211 | RowBox[{ 212 | SqrtBox[ 213 | RowBox[{"-", 214 | InterpretationBox[ 215 | StyleBox[ 216 | OverscriptBox["g", 217 | OverscriptBox[ 218 | StyleBox["~", 219 | FontColor->RGBColor[0, 0, 0]], 220 | StyleBox["~", 221 | FontColor->RGBColor[0, 0, 0]]]], 222 | ShowAutoStyles->False, 223 | AutoSpacing->False], 224 | $CellContext`Detg[], 225 | Editable->False]}]], " ", 226 | InterpretationBox[ 227 | StyleBox[ 228 | RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], 229 | ShowAutoStyles->False, 230 | AutoSpacing->False], 231 | $CellContext`RicciScalarCD[], 232 | Editable->False]}]], "Output"] 233 | }, Open ]], 234 | 235 | Cell["\<\ 236 | First find the linear perturbation, expanding it, then \ 237 | contract/canonicalize/simplify.\ 238 | \>", "Text"], 239 | 240 | Cell[CellGroupData[{ 241 | 242 | Cell[BoxData[ 243 | RowBox[{"LGRpert", "=", 244 | RowBox[{ 245 | RowBox[{ 246 | 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-$CellContext`a]]], 438 | Editable->False]}]}], ")"}]}]], "Output"] 439 | }, Open ]], 440 | 441 | Cell["\<\ 442 | Now perform (minus) the variation with respect to h[LI[1],a,b]. This will \ 443 | introduce delta[-LI[1],LI[1]] which is sometimes useful but here we just set \ 444 | it to 1.\ 445 | \>", "Text"], 446 | 447 | Cell[CellGroupData[{ 448 | 449 | Cell[BoxData[ 450 | RowBox[{"EOMGR0", "=", 451 | RowBox[{ 452 | RowBox[{ 453 | RowBox[{"0", "==", 454 | RowBox[{ 455 | FractionBox[ 456 | RowBox[{"-", "1"}], 457 | RowBox[{"Sqrt", "[", 458 | RowBox[{"-", 459 | RowBox[{"Detg", "[", "]"}]}], "]"}]], 460 | RowBox[{ 461 | RowBox[{"VarD", "[", 462 | RowBox[{ 463 | RowBox[{"h", "[", 464 | RowBox[{ 465 | RowBox[{"LI", "[", "1", "]"}], ",", "a", ",", "b"}], "]"}], ",", 466 | "CD"}], "]"}], "@", "LGRpert"}]}]}], "/.", 467 | RowBox[{ 468 | RowBox[{"delta", "[", 469 | RowBox[{ 470 | RowBox[{"-", 471 | RowBox[{"LI", "[", "1", "]"}]}], ",", 472 | RowBox[{"LI", "[", "1", "]"}]}], "]"}], "\[Rule]", "1"}]}], "//", 473 | "Expand"}]}]], "Input"], 474 | 475 | Cell[BoxData[ 476 | RowBox[{"0", "\[Equal]", 477 | RowBox[{ 478 | InterpretationBox[ 479 | StyleBox[GridBox[{ 480 | { 481 | RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], 482 | StyleBox[GridBox[{ 483 | {" ", " "}, 484 | {"a", "b"} 485 | }, 486 | GridBoxSpacings->{"Columns" -> { 487 | Offset[0.], { 488 | Offset[0.034999999999999996`]}, 489 | Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ 490 | Offset[0.]}}, "RowsIndexed" -> {}}], 491 | FontSize->Rational[39, 4]]} 492 | }, 493 | GridBoxAlignment->{ 494 | "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, 495 | "RowsIndexed" -> {}}, 496 | GridBoxSpacings->{"Columns" -> { 497 | Offset[0.27999999999999997`], { 498 | Offset[0.034999999999999996`]}, 499 | Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { 500 | Offset[0.2], { 501 | Offset[0.4]}, 502 | Offset[0.2]}, "RowsIndexed" -> {}}], 503 | ShowAutoStyles->False, 504 | AutoSpacing->False], 505 | $CellContext`RicciCD[-$CellContext`a, -$CellContext`b], 506 | Editable->False], "-", 507 | RowBox[{ 508 | FractionBox["1", "2"], " ", 509 | InterpretationBox[ 510 | StyleBox[GridBox[{ 511 | {"g", 512 | StyleBox[GridBox[{ 513 | {" ", " "}, 514 | {"a", "b"} 515 | }, 516 | GridBoxSpacings->{"Columns" -> { 517 | Offset[0.], { 518 | Offset[0.034999999999999996`]}, 519 | Offset[0.]}, "ColumnsIndexed" -> {}, "Rows" -> {{ 520 | Offset[0.]}}, "RowsIndexed" -> {}}], 521 | FontSize->Rational[39, 4]]} 522 | }, 523 | GridBoxAlignment->{ 524 | "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, 525 | "RowsIndexed" -> {}}, 526 | GridBoxSpacings->{"Columns" -> { 527 | Offset[0.27999999999999997`], { 528 | Offset[0.034999999999999996`]}, 529 | Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { 530 | Offset[0.2], { 531 | Offset[0.4]}, 532 | Offset[0.2]}, "RowsIndexed" -> {}}], 533 | ShowAutoStyles->False, 534 | AutoSpacing->False], 535 | $CellContext`g[-$CellContext`a, -$CellContext`b], 536 | Editable->False], " ", 537 | InterpretationBox[ 538 | StyleBox[ 539 | RowBox[{"R", "[", "\[EmptyDownTriangle]", "]"}], 540 | ShowAutoStyles->False, 541 | AutoSpacing->False], 542 | $CellContext`RicciScalarCD[], 543 | Editable->False]}]}]}]], "Output"] 544 | }, Open ]] 545 | }, Open ]], 546 | 547 | Cell[CellGroupData[{ 548 | 549 | Cell["2. Example: Scalar-Tensor theory", "Subsection"], 550 | 551 | Cell["\<\ 552 | Let\[CloseCurlyQuote]s add a scalar field \[Phi] with potential U(\[Phi]) and \ 553 | coupling function A(\[Phi]) in the Jordan frame. Define these objects:\ 554 | \>", "Text"], 555 | 556 | Cell[BoxData[{ 557 | RowBox[{"DefTensor", "[", 558 | RowBox[{ 559 | RowBox[{"\[Phi]", "[", "]"}], ",", "M4"}], "]"}], "\[IndentingNewLine]", 560 | RowBox[{ 561 | RowBox[{"DefScalarFunction", "/@", 562 | RowBox[{"{", 563 | RowBox[{"U", ",", "A"}], "}"}]}], ";"}]}], "Input"], 564 | 565 | Cell["\<\ 566 | We also need a tensor perturbation \ 567 | \[Delta]\[Phi][LI[\[OpenCurlyDoubleQuote]order\[CloseCurlyDoubleQuote]]]\ 568 | \>", "Text"], 569 | 570 | Cell[BoxData[ 571 | RowBox[{"DefTensorPerturbation", "[", 572 | RowBox[{ 573 | RowBox[{"\[Delta]\[Phi]", "[", 574 | RowBox[{"LI", "[", "\"\\"", "]"}], "]"}], ",", 575 | RowBox[{"\[Phi]", "[", "]"}], ",", 576 | RowBox[{"{", "M4", "}"}]}], "]"}]], "Input"], 577 | 578 | Cell["Here is the Lagrangian for this theory", "Text"], 579 | 580 | Cell[CellGroupData[{ 581 | 582 | Cell[BoxData[ 583 | RowBox[{"LST", "=", 584 | RowBox[{ 585 | RowBox[{"Sqrt", "[", 586 | RowBox[{"-", 587 | RowBox[{"Detg", 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"Section"], 3 | 4 | Cell[CellGroupData[{ 5 | 6 | Cell[TextData[{ 7 | "Author: Leo C. Stein (", 8 | Cell[BoxData[ 9 | FormBox[ 10 | ButtonBox[ 11 | RowBox[{"leo", ".", 12 | RowBox[{"stein", "@", "gmail"}], ".", "com"}], 13 | BaseStyle->"Hyperlink", 14 | ButtonData->{ 15 | URL["mailto:leo.stein@gmail.com"], None}, 16 | ButtonNote->"mailto:leo.stein@gmail.com"], TraditionalForm]]], 17 | ")\nDate: May 2014\nThis notebook gives a minimal example of creating a \ 18 | warped product geometry and breaks down the Einstein-Hilbert Lagrangian into \ 19 | the effective lower-dimensional Lagrangian, a la ", 20 | Cell[BoxData[ 21 | FormBox[ 22 | ButtonBox[ 23 | RowBox[{"Randall", "-", 24 | RowBox[{"Sundrum", " ", "[", 25 | RowBox[{ 26 | RowBox[{"http", ":"}], "//", 27 | RowBox[{ 28 | RowBox[{ 29 | RowBox[{ 30 | RowBox[{"arxiv", ".", "org"}], "/", "abs"}], "/", "hep"}], "-", 31 | RowBox[{"ph", "/", "9905221"}]}]}], "]"}]}], 32 | BaseStyle->"Hyperlink", 33 | ButtonData->{ 34 | URL["http://arxiv.org/abs/hep-ph/9905221"], None}, 35 | ButtonNote->"http://arxiv.org/abs/hep-ph/9905221"], TraditionalForm]]], 36 | "." 37 | }], "Subsubtitle"], 38 | 39 | Cell[CellGroupData[{ 40 | 41 | Cell["0. 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Defining manifolds, metrics, and fields\ 114 | \>", "Subsection"], 115 | 116 | Cell["Make two manifolds and combine them into a product manifols", "Text"], 117 | 118 | Cell[BoxData[ 119 | RowBox[{"DefManifold", "[", 120 | RowBox[{"M4", ",", " ", "4", ",", 121 | RowBox[{"IndexRange", "[", 122 | RowBox[{"a", ",", "q"}], "]"}]}], "]"}]], "Input"], 123 | 124 | Cell[CellGroupData[{ 125 | 126 | Cell[BoxData[ 127 | RowBox[{"DefManifold", "[", 128 | RowBox[{"M1", ",", "1", ",", 129 | RowBox[{"IndexRange", "[", 130 | RowBox[{"A", ",", "D"}], "]"}]}], "]"}]], "Input"], 131 | 132 | Cell[BoxData[ 133 | RowBox[{ 134 | StyleBox[ 135 | RowBox[{"ValidateSymbol", "::", "capital"}], "MessageName"], ":", 136 | " ", "\<\"System name \[NoBreak]\\!\\(\\\"C\\\"\\)\[NoBreak] is overloaded \ 137 | as an abstract index.\"\>"}]], "Message", "MSG"], 138 | 139 | Cell[BoxData[ 140 | RowBox[{ 141 | StyleBox[ 142 | RowBox[{"ValidateSymbol", "::", "capital"}], "MessageName"], ":", 143 | " ", "\<\"System name \[NoBreak]\\!\\(\\\"D\\\"\\)\[NoBreak] is overloaded \ 144 | as an abstract index.\"\>"}]], "Message", "MSG"] 145 | }, Open ]], 146 | 147 | Cell[BoxData[ 148 | RowBox[{"DefManifold", "[", 149 | RowBox[{"M5", ",", 150 | RowBox[{"{", 151 | RowBox[{"M4", ",", "M1"}], "}"}], ",", 152 | RowBox[{"{", 153 | RowBox[{ 154 | "\[Mu]", ",", " ", "\[Nu]", ",", " ", "\[Lambda]", ",", " ", "\[Sigma]", 155 | ",", " ", "\[Eta]", ",", " ", "\[Rho]"}], "}"}]}], "]"}]], "Input"], 156 | 157 | Cell["Two lower-dimensional metrics", "Text"], 158 | 159 | Cell[BoxData[ 160 | RowBox[{"DefMetric", "[", 161 | RowBox[{ 162 | RowBox[{"-", "1"}], ",", 163 | RowBox[{"g4", "[", 164 | RowBox[{ 165 | RowBox[{"-", "a"}], ",", 166 | RowBox[{"-", "b"}]}], "]"}], ",", "CD", ",", 167 | RowBox[{ 168 | "PrintAs", "\[Rule]", "\"\<\!\(\*SuperscriptBox[\(g\), \((4)\)]\)\>\""}]}], 169 | "]"}]], "Input"], 170 | 171 | Cell[BoxData[ 172 | RowBox[{"DefMetric", "[", 173 | RowBox[{"1", ",", 174 | RowBox[{"g1", "[", 175 | RowBox[{ 176 | RowBox[{"-", "A"}], ",", 177 | RowBox[{"-", "B"}]}], "]"}], ",", "CD1", ",", 178 | RowBox[{"{", 179 | RowBox[{"\"\<:\>\"", ",", "\"\\""}], "}"}], ",", 180 | RowBox[{ 181 | "PrintAs", "\[Rule]", "\"\<\!\(\*SuperscriptBox[\(g\), \((1)\)]\)\>\""}]}], 182 | "]"}]], "Input"], 183 | 184 | Cell[CellGroupData[{ 185 | 186 | Cell["Warp factors", "Subsubsection"], 187 | 188 | Cell["\<\ 189 | We have a coordinate function (y) on the 1-dimensional manifold, some \ 190 | constant (K), and a warp function (T) which acts like a dilaton\ 191 | \>", "Text"], 192 | 193 | Cell[BoxData[ 194 | RowBox[{"DefTensor", "[", 195 | RowBox[{ 196 | RowBox[{"y", "[", "]"}], ",", 197 | RowBox[{"{", "M1", "}"}]}], "]"}]], "Input"], 198 | 199 | Cell[BoxData[ 200 | RowBox[{"DefConstantSymbol", "[", "K", "]"}]], "Input"], 201 | 202 | Cell[BoxData[ 203 | RowBox[{"DefTensor", "[", 204 | RowBox[{ 205 | RowBox[{"T", "[", "]"}], ",", 206 | RowBox[{"{", "M4", "}"}]}], "]"}]], "Input"] 207 | }, Open ]], 208 | 209 | Cell[CellGroupData[{ 210 | 211 | Cell["Product metric", "Subsubsection"], 212 | 213 | Cell["\<\ 214 | Finally, combine all of these things to make a product metric. 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