(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 184848, 5308] NotebookOptionsPosition[ 173918, 5138] NotebookOutlinePosition[ 174412, 5157] CellTagsIndexPosition[ 174369, 5154] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Cartesian Kerr-Schild Cartesian Tetrad", "Section", CellChangeTimes->{{3.842964040830364*^9, 3.8429640478229427`*^9}, { 3.843640590004999*^9, 3.843640597899662*^9}, {3.860845382616249*^9, 3.860845384875581*^9}},ExpressionUUID->"a4c93185-a179-494a-a3c2-\ 359519993679"], Cell[CellGroupData[{ Cell["CKS Metric", "Subsection", CellChangeTimes->{{3.843642137780231*^9, 3.843642175541808*^9}},ExpressionUUID->"d788c344-5890-447c-8611-\ bae561abb97c"], Cell[BoxData[{ RowBox[{"ClearAll", "[", "\"\\"", "]"}], "\n", RowBox[{ RowBox[{"$Assumptions", " ", "=", " ", RowBox[{"_", " ", "\[Element]", " ", "Reals"}]}], ";"}]}], "Code", CellChangeTimes->{{3.842529254666725*^9, 3.8425292695235643`*^9}, 3.8425436046609497`*^9, {3.842627421429356*^9, 3.842627422084736*^9}, { 3.8426283441556177`*^9, 3.8426283466838503`*^9}, {3.842632589192568*^9, 3.842632589856708*^9}}, CellLabel-> "In[300]:=",ExpressionUUID->"b410c9ac-994c-4670-b6ad-614b14de306a"], Cell["\<\ Mathematica does not begin indexing at 0; I find it most straightforward to \ produce fake indices for looping over metrics, four vectors, spatial \ metrics, and three vectors that are consistent with the usual notation (e.g., \ the time component of a four vector being the 0 index). In the following let \ f0-f3 be the loop limits for four row and/or column data structures and t1-t3 \ be the loop limits for three row and/or column data structures. \ \>", "Text", CellChangeTimes->{{3.843640646092126*^9, 3.8436407177084627`*^9}, { 3.843640751476488*^9, 3.843640828052985*^9}, {3.843640889709468*^9, 3.8436409229974194`*^9}, {3.8436537607121572`*^9, 3.8436537914252777`*^9}, 3.843657538528639*^9, {3.843657584026084*^9, 3.8436575854168987`*^9}},ExpressionUUID->"3842ca6d-05e9-41bf-bdc0-\ 2deb0c088d67"], Cell[BoxData[{ RowBox[{ RowBox[{"f0", "=", "1"}], ";", " ", RowBox[{"f1", "=", "2"}], ";", " ", RowBox[{"f2", "=", "3"}], ";", " ", RowBox[{"f3", "=", "4"}], ";"}], "\n", RowBox[{ RowBox[{"t1", "=", "1"}], ";", " ", RowBox[{"t2", "=", "2"}], ";", " ", RowBox[{"t3", "=", "3"}], ";"}]}], "Code", CellChangeTimes->{{3.842537698596817*^9, 3.842537833217692*^9}, { 3.84254119958043*^9, 3.8425412040747747`*^9}, {3.8425423837358418`*^9, 3.842542424240418*^9}, {3.84254322733712*^9, 3.8425432695495768`*^9}, { 3.842543428414287*^9, 3.842543428774683*^9}, {3.8425434606310167`*^9, 3.842543469990225*^9}, {3.8425449373381557`*^9, 3.842544937579175*^9}, 3.84364074383735*^9}, CellLabel-> "In[302]:=",ExpressionUUID->"ec504711-674c-4043-8995-9e05c651976e"], Cell[TextData[{ "We begin the with covariant form of the Cartesian Kerr-Schild metric \ (Visser 2008), \n\n", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["g", "\[Mu]\[Nu]"], "=", RowBox[{ SubscriptBox["\[Eta]", "\[Mu]\[Nu]"], "+", RowBox[{"f", " ", SubscriptBox["l", "\[Mu]"], SubscriptBox["l", "\[Nu]"]}]}]}], TraditionalForm]],ExpressionUUID-> "f690820a-0a20-48b3-a392-42c8876ae450"], "\n\nwhere ", Cell[BoxData[ FormBox[ SubscriptBox["\[Eta]", "\[Mu]\[Nu]"], TraditionalForm]],ExpressionUUID-> "29279aa4-49f4-4423-b506-57bb1f6394da"], "is the Minkowski metric, and\n\n", Cell[BoxData[ FormBox[ RowBox[{"f", "=", FractionBox[ RowBox[{"2", " ", "m", " ", SuperscriptBox["r", "3"]}], RowBox[{ SuperscriptBox["r", "4"], "+", RowBox[{ SuperscriptBox["a", "2"], SuperscriptBox["z", "2"]}]}]]}], TraditionalForm]],ExpressionUUID-> "0953a93f-98ee-424d-8927-e4f59421fe34"], ",\n\nand ", Cell[BoxData[ FormBox[ SubscriptBox["l", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "5aebec06-a8a2-474a-a01b-e5b459b7b2a8"], " is a null vector with respect to ", Cell[BoxData[ FormBox[ SubscriptBox["g", "\[Mu]\[Nu]"], TraditionalForm]],ExpressionUUID-> "7dc613dd-4724-4b66-8f31-574cc83e1ab6"], "and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Eta]", "\[Mu]\[Nu]"], TraditionalForm]],ExpressionUUID-> "a942bcf9-b25b-46f6-92fb-675c117a02d4"], "\n\n", Cell[BoxData[ FormBox[ SubscriptBox["l", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "3d48ea18-01ee-4bc0-b15e-b9723ededdca"], "= (1, ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"r", " ", "x"}], " ", "+", " ", "ay"}], RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["a", "2"]}]], ",", " ", FractionBox[ RowBox[{ RowBox[{"r", " ", "y"}], "-", RowBox[{"a", " ", "x"}]}], RowBox[{ SuperscriptBox["r", "2"], "+", " ", SuperscriptBox["a", "2"]}]], ",", FractionBox["z", "r"]}], TraditionalForm]],ExpressionUUID-> "de838e60-4bd8-48ca-b7ca-ceb13cd908d4"], ")\n\nand\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"r", "(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], "=", SqrtBox[ FractionBox[ RowBox[{ SuperscriptBox["R", "2"], "-", SuperscriptBox["a", "2"], "+", SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["R", "2"], "-", SuperscriptBox["a", "2"]}], ")"}], "2"], "+", RowBox[{"4", SuperscriptBox["a", "2"], SuperscriptBox["z", "2"]}]}]]}], "2"]]}], TraditionalForm]], ExpressionUUID->"99f7a1b4-afce-444b-8edd-37d2b661618c"], ",\n\nwhere ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["R", "2"], "=", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}]}], TraditionalForm]],ExpressionUUID-> "32f983eb-2313-4613-a006-4cd81f3c6e5f"], " and ", Cell[BoxData[ FormBox["m", TraditionalForm]],ExpressionUUID-> "785f4c87-af43-44cf-8c1b-42a6e6251fa5"], " and ", Cell[BoxData[ FormBox["a", TraditionalForm]],ExpressionUUID-> "9774d715-0aba-4052-8d2c-0c5df7ec4aa2"], " are the mass and spin of the black hole respectively. We also have\n\n", Cell[BoxData[ FormBox[ SuperscriptBox["l", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "eabea67b-a4dc-44af-a3a1-6d2465436bbe"], "= (-1, ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"r", " ", "x"}], " ", "+", " ", "ay"}], RowBox[{ SuperscriptBox["r", "2"], "+", SuperscriptBox["a", "2"]}]], ",", " ", FractionBox[ RowBox[{ RowBox[{"r", " ", "y"}], "-", RowBox[{"a", " ", "x"}]}], RowBox[{ SuperscriptBox["r", "2"], "+", " ", SuperscriptBox["a", "2"]}]], ",", FractionBox["z", "r"]}], TraditionalForm]],ExpressionUUID-> "b8bd89b6-df68-46b0-9b8a-bd11c1fc976d"], ").\n\nBy definition, \n\n ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox[ SubscriptBox["l", "1"], "2"], "+", SuperscriptBox[ SubscriptBox["l", "2"], "2"], "+", SuperscriptBox[ SubscriptBox["l", "3"], "2"]}], "=", "0"}], TraditionalForm]], ExpressionUUID->"298ea757-b15f-4b1d-b62d-44c55d52b628"], "." }], "Text", CellChangeTimes->{{3.843640845100853*^9, 3.843640853884677*^9}, { 3.843640978950141*^9, 3.843641397912587*^9}, {3.8436414564725122`*^9, 3.8436416038052998`*^9}, {3.843643909742943*^9, 3.843643981106103*^9}},ExpressionUUID->"0fe4122f-d02b-40ef-be3a-\ 4094623244ac"], Cell[BoxData[{ RowBox[{ RowBox[{"ll", "=", RowBox[{"{", RowBox[{"ll0", ",", "l1", ",", "l2", ",", "l3"}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"lu", "=", RowBox[{"{", RowBox[{"lu0", ",", "l1", ",", "l2", ",", "l3"}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"\[Eta]", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "1", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "1"}], "}"}]}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"gcov", "=", RowBox[{ RowBox[{ RowBox[{"FullSimplify", "@", RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"f", "*", RowBox[{"ll", "[", RowBox[{"[", "\[Mu]", "]"}], "]"}], "*", RowBox[{"ll", "[", RowBox[{"[", "\[Nu]", "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Mu]", ",", "f0", ",", "f3"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Nu]", ",", "f0", ",", "f3"}], "}"}]}], "]"}]}], "+", "\[Eta]"}], " ", "/.", RowBox[{"{", RowBox[{"ll0", "->", " ", "1"}], "}"}]}]}], ";"}], "\n", RowBox[{ RowBox[{"gcon", "=", RowBox[{ RowBox[{ RowBox[{"FullSimplify", "@", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"-", "f"}], "*", RowBox[{"lu", "[", RowBox[{"[", "\[Mu]", "]"}], "]"}], "*", RowBox[{"lu", "[", RowBox[{"[", "\[Nu]", "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Mu]", ",", "f0", ",", "f3"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Nu]", ",", "f0", ",", "f3"}], "}"}]}], "]"}]}], "+", "\[Eta]"}], " ", "/.", RowBox[{"{", RowBox[{"lu0", "->", RowBox[{"-", "1"}]}], "}"}]}]}], ";"}]}], "Code", CellChangeTimes->{{3.8426263947611313`*^9, 3.842626403097658*^9}, { 3.842626512973634*^9, 3.842626533701805*^9}, {3.8426266096084337`*^9, 3.842626633993639*^9}, {3.842626801198398*^9, 3.84262680176581*^9}, { 3.8426270064090643`*^9, 3.842627010544999*^9}, {3.842630193961521*^9, 3.842630197021923*^9}, {3.842630559938671*^9, 3.842630561810848*^9}, { 3.842630707394599*^9, 3.842630712667458*^9}, {3.843641614767501*^9, 3.843641617911425*^9}, {3.8436439941829967`*^9, 3.843644015727935*^9}, { 3.86325404340514*^9, 3.863254045076572*^9}, {3.863254077159136*^9, 3.8632540793678303`*^9}, {3.863254165247023*^9, 3.863254194452548*^9}, { 3.863254230048059*^9, 3.8632542524399652`*^9}}, CellLabel-> "In[304]:=",ExpressionUUID->"b9fb364a-58d1-4caf-9cfe-33470d3c82e2"], Cell[TextData[{ "Let\[CloseCurlyQuote]s confirm a few things about the metric. First, let\ \[CloseCurlyQuote]s check that the inverse of ", Cell[BoxData[ FormBox[ SubscriptBox["g", "\[Mu]\[Nu]"], TraditionalForm]],ExpressionUUID-> "6a0859fc-2220-4d14-a3a4-c4ebe08b5b9c"], "gives ", Cell[BoxData[ FormBox[ SuperscriptBox["g", "\[Mu]\[Nu]"], TraditionalForm]],ExpressionUUID-> "4595f824-0eef-4af7-98a8-dc1280df239f"] }], "Text", CellChangeTimes->{{3.843641643361092*^9, 3.843641725222241*^9}, 3.843641947093843*^9, {3.8436439866877832`*^9, 3.8436439871349373`*^9}, { 3.843653921901956*^9, 3.843653927812873*^9}},ExpressionUUID->"8b145923-8e6b-478a-a086-\ 0d0a8a3c3732"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{ RowBox[{"Inverse", "[", "gcov", "]"}], "==", "gcon"}], "//.", RowBox[{"{", RowBox[{ SuperscriptBox["l1", "2"], "->", RowBox[{"1", "-", SuperscriptBox["l2", "2"], "-", SuperscriptBox["l3", "2"]}]}], "}"}]}], "]"}]], "Code", CellChangeTimes->{{3.843641726304533*^9, 3.843641911559039*^9}}, CellLabel-> "In[309]:=",ExpressionUUID->"aff6b2a5-cbd7-4183-b6e6-f5964791e119"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.843641729923274*^9, 3.84364177894167*^9}, 3.843641818387711*^9, {3.84364185307088*^9, 3.843641911943725*^9}, 3.843644024370935*^9, 3.859449883734166*^9, 3.859455070494961*^9, 3.863254268451927*^9, 3.878462004307782*^9}, CellLabel-> "Out[309]=",ExpressionUUID->"f3c6e976-4442-4b4a-b621-7105073d10b7"] }, Open ]], Cell[TextData[{ "Second, let\[CloseCurlyQuote]s check that the ", Cell[BoxData[ FormBox[ RowBox[{ SqrtBox[ RowBox[{"-", "g"}]], "=", "1"}], TraditionalForm]],ExpressionUUID-> "1c4bf57d-68b7-4c2f-adf0-34f22af4af8c"] }], "Text", CellChangeTimes->{{3.8436419191178083`*^9, 3.843641943854261*^9}},ExpressionUUID->"cd17adaa-45a6-40b3-8986-\ 4362b28a1020"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{"-", RowBox[{"Det", "[", "gcov", "]"}]}], "]"}], "//.", RowBox[{"{", RowBox[{ SuperscriptBox["l1", "2"], "->", RowBox[{"1", "-", SuperscriptBox["l2", "2"], "-", SuperscriptBox["l3", "2"]}]}], "}"}]}], "]"}]], "Code", CellChangeTimes->{{3.8436419834376907`*^9, 3.84364201048777*^9}}, CellLabel-> "In[310]:=",ExpressionUUID->"e938eed3-980a-4411-9228-860c1ee61ca5"], Cell[BoxData["1"], "Output", CellChangeTimes->{{3.843641991988236*^9, 3.843642010824299*^9}, 3.8436440257244797`*^9, 3.85944988435321*^9, 3.8594550713051233`*^9, 3.8632542703383512`*^9, 3.878462005224976*^9}, CellLabel-> "Out[310]=",ExpressionUUID->"06e98e75-7aa5-472c-b130-935e3a3babd9"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Equations Solved", "Subsection", CellChangeTimes->{{3.843642137780231*^9, 3.843642188109624*^9}},ExpressionUUID->"6ea94518-8140-4102-bcfb-\ e88f869300ff"], Cell[TextData[{ "Next, I\[CloseCurlyQuote]ll recap @cwhite\[CloseCurlyQuote]s notes on the \ equations solved in Athena++. For now, we will restrict ourselves to vacuum \ gray transfer. The equations solved (in discretized form) are\n\n", Cell[BoxData[ FormBox[ SubsuperscriptBox[ RowBox[{ RowBox[{ FractionBox["1", "\[CapitalDelta]t"], " ", "[", SubscriptBox[ RowBox[{"\[LeftAngleBracket]", RowBox[{ SuperscriptBox["n", "0"], SubscriptBox["n", "0"], " ", "I"}], " ", "\[RightAngleBracket]"}], RowBox[{"V", " ", "\[CapitalOmega]"}]], "]"}], " "}], SubsuperscriptBox["x", "-", "0"], SubsuperscriptBox["x", "+", "0"]], TraditionalForm]],ExpressionUUID-> "99f2283a-0d9c-4ba3-aaeb-9a11c4c17e03"], " + ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["\[Sum]", RowBox[{" ", RowBox[{"i", "\[Element]", RowBox[{"{", RowBox[{"1", ",", "2", ",", "3"}], "}"}]}]}]], " ", SubsuperscriptBox[ RowBox[{ FractionBox["1", "\[CapitalDelta]V"], "[", RowBox[{"\[CapitalDelta]", " ", SubscriptBox["A", "i"], SubscriptBox[ RowBox[{"\[LeftAngleBracket]", RowBox[{ SuperscriptBox["n", "i"], SubscriptBox["n", "0"], " ", "I"}], " ", "\[RightAngleBracket]"}], RowBox[{"t", " ", SubscriptBox["A", "i"], " ", "\[CapitalOmega]"}]]}], "]"}], SubsuperscriptBox["x", "-", "i"], SubsuperscriptBox["x", "+", "i"]]}], TraditionalForm]],ExpressionUUID-> "71a00a7a-2ba7-4192-b6dc-b42cb36ec104"], " + ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["\[Sum]", RowBox[{" ", RowBox[{"a", "\[Element]", RowBox[{"{", RowBox[{"\[Zeta]", ",", "\[Psi]"}], "}"}]}]}]], " ", SubsuperscriptBox[ RowBox[{ FractionBox["1", "\[CapitalDelta]\[CapitalOmega]"], "[", RowBox[{"\[CapitalDelta]", " ", SubscriptBox["\[Lambda]", "a"], SubscriptBox[ RowBox[{"\[LeftAngleBracket]", RowBox[{ SuperscriptBox["n", "a"], SubscriptBox["n", "0"], " ", "I"}], " ", "\[RightAngleBracket]"}], RowBox[{"t", " ", "V", " ", SubscriptBox["\[Lambda]", "a"]}]]}], "]"}], SubscriptBox["a", "-"], SubscriptBox["a", "+"]]}], TraditionalForm]],ExpressionUUID-> "85bac88c-8318-4d7d-852f-0b36408eab75"], " = 0\n\nwhere \[CapitalDelta]t is the numerical timestep, I is the \ frequency integrated intensity, ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "a701a9f2-faa4-41a9-b608-a306fe39f0cf"], " and ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "a"], TraditionalForm]],ExpressionUUID-> "68ee24e3-d584-443f-8908-c41c1ab6aca6"], " are normal vector coordinate coordinate components for the spatial and \ angular meshes, the second term corresponds to spatial fluxes on the \ Cartesian mesh, and the final term corresponds to angular fluxes on the \ angular mesh defined for each numerical cell. Spatial fluxes are dependent \ on areas at cell faces ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalDelta]A", "i"], TraditionalForm]],ExpressionUUID-> "6d9608d1-37aa-479b-aa48-1a3651b3d674"], " and cell volumes \[CapitalDelta]V. Angular fluxes are dependent on arc \ lengths ", Cell[BoxData[ FormBox[ SubscriptBox["\[CapitalDelta]\[Lambda]", "a"], TraditionalForm]], ExpressionUUID->"1356002b-9eec-40e4-9a36-7c51c77f23b9"], " and solid angles \[CapitalDelta]\[CapitalOmega]. \n\nThe infrastructure \ to compute arc lengths, solid angles, face areas, and cell volumes are \ already implemented in @cwhite\[CloseCurlyQuote]s version of GR radiation in \ Athena++. However, we must make some extensions to compute ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "e753e05f-c99e-4cf7-bfe4-d8fa22b66ec6"], " and ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "a"], TraditionalForm]],ExpressionUUID-> "ac9419e8-dcdd-46a7-bb62-ef6c84201b71"], " when working with CKS. \n\nTo move forward, we describe an implementation \ of an orthonormal tetrad for CKS." }], "Text", CellChangeTimes->{{3.8436420400164967`*^9, 3.843642049030167*^9}, { 3.843642190710965*^9, 3.843642214861657*^9}, {3.8436423351496677`*^9, 3.843642362523979*^9}, {3.8436424233071213`*^9, 3.843642450869409*^9}, { 3.8436425342450333`*^9, 3.843642777871765*^9}, {3.8436428333822117`*^9, 3.843643184249979*^9}, {3.843643247003167*^9, 3.843643345373102*^9}, { 3.843643375437419*^9, 3.843643566923902*^9}, {3.843643632501686*^9, 3.8436437301273003`*^9}, 3.8436444961157207`*^9, {3.843653966934011*^9, 3.843653970581749*^9}, {3.843654033863277*^9, 3.843654036040642*^9}},ExpressionUUID->"ba090234-58ed-4f37-ad3c-\ 6ad6b5419cd6"] }, Open ]], Cell[CellGroupData[{ Cell["Tetrad", "Subsection", CellChangeTimes->{{3.843642137780231*^9, 3.843642188109624*^9}, { 3.8436437467924967`*^9, 3.843643750120428*^9}},ExpressionUUID->"d4453618-2014-4c7b-9890-\ 1c19046341b0"], Cell[TextData[{ Cell[BoxData[ RowBox[{ FormBox[ SuperscriptBox["n", "\[Mu]"], TraditionalForm], " "}]],ExpressionUUID-> "1dbc4fdc-a333-4a12-b3ad-bd12762d6de7"], " and ", Cell[BoxData[ FormBox[ SubscriptBox["n", "0"], TraditionalForm]],ExpressionUUID-> "d476aa8a-be0d-4ef4-99c4-c77c5a7c4da1"], " are computed following\n\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["n", "\[Mu]"], " ", "=", " ", RowBox[{ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Mu]"], " ", SuperscriptBox["n", OverscriptBox["\[Alpha]", "^"]], " "}]}], TraditionalForm]], ExpressionUUID->"2be15a7f-8684-41d1-8a9f-94dfefac0cce"], "\n", Cell[BoxData[ FormBox[ SubscriptBox["n", "0"], TraditionalForm]],ExpressionUUID-> "09829261-b187-4380-ac06-fb5353c01df6"], " = ", Cell[BoxData[ FormBox[ SubscriptBox["g", RowBox[{"0", "\[Mu]", " "}]], TraditionalForm]],ExpressionUUID-> "886512ec-2bb7-454c-b480-8b8d9ddf309e"], Cell[BoxData[ FormBox[ SuperscriptBox["n", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "42a44201-1425-478a-adfd-56b0d6927aab"], "\n\nwhere ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Mu]"], TraditionalForm]], ExpressionUUID->"96eeb511-4699-4094-a370-c27b56f26c5a"], " are orthonormal tetrad basis components and ", Cell[BoxData[ FormBox[ SuperscriptBox["n", OverscriptBox["\[Alpha]", "^"]], TraditionalForm]],ExpressionUUID-> "1388f4aa-530d-4c57-854d-664e5bb6dc18"], " is defined in Davis & Gammie (2019) as\n\n", Cell[BoxData[ FormBox[ SuperscriptBox["n", OverscriptBox["\[Alpha]", "^"]], TraditionalForm]],ExpressionUUID-> "797fc7cb-92e8-4ff0-8f4d-967f6154d29c"], " = ", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Delta]", "0", "\[Alpha]"], TraditionalForm]], ExpressionUUID->"1d181554-6693-47db-ac13-260c0b3d9071"], " + sin\[Zeta] cos \[Psi]", Cell[BoxData[ FormBox[ RowBox[{" ", SubsuperscriptBox["\[Delta]", "1", "\[Alpha]"]}], TraditionalForm]], ExpressionUUID->"b453c9da-999b-429a-a789-1c5b6730d00c"], " + sin\[Zeta] sin \[Psi]", Cell[BoxData[ FormBox[ RowBox[{" ", SubsuperscriptBox["\[Delta]", "2", "\[Alpha]"]}], TraditionalForm]], ExpressionUUID->"6e88d58b-8435-455d-8ab8-ffd6a21a37ef"], " + cos\[Zeta] ", Cell[BoxData[ FormBox[ RowBox[{" ", SubsuperscriptBox["\[Delta]", "3", "\[Alpha]"]}], TraditionalForm]], ExpressionUUID->"4d641ce3-e30f-4239-8f3a-b0457ad67f07"], ".\n\n@cwhite\[CloseCurlyQuote]s implementation computes ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "a"], TraditionalForm]],ExpressionUUID-> "697ec0cf-ef7b-4630-a6d3-0881c6019392"], " as\n\n", Cell[BoxData[ FormBox[ SuperscriptBox["n", "\[Zeta]"], TraditionalForm]],ExpressionUUID-> "9fd58a1c-0b49-4b31-a00e-72d77e73786d"], "= ", Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{"sin", " ", "\[Zeta]"}]], TraditionalForm]],ExpressionUUID-> "fb6ec3b4-b9d6-4eed-91ba-3c438eb8a43b"], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["n", OverscriptBox["\[Alpha]", "^"]]}], TraditionalForm]],ExpressionUUID-> "ab02d800-8784-4e49-b885-768aa3e9ab9b"], Cell[BoxData[ FormBox[ SuperscriptBox["n", OverscriptBox["\[Beta]", "^"]], TraditionalForm]],ExpressionUUID-> "94de9e0c-9fed-49e4-9849-49ff2afb5561"], " ", Cell[BoxData[ FormBox[ RowBox[{"(", SuperscriptBox["n", OverscriptBox["0", "^"]]}], TraditionalForm]],ExpressionUUID-> "e56d02f9-05dd-44f2-94a8-26dba05906e3"], " ", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Omega]", RowBox[{ OverscriptBox["\[Alpha]", "^"], " ", OverscriptBox["\[Beta]", "^"]}], OverscriptBox[ RowBox[{" ", "3"}], "^"]], TraditionalForm]],ExpressionUUID-> "79716d50-64dd-4408-8b5b-0a9df373279b"], " -", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["n", OverscriptBox["3", "^"]]}], TraditionalForm]],ExpressionUUID-> "f6e9b467-1ef4-496d-9f40-787758007c98"], " ", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Omega]", RowBox[{ OverscriptBox["\[Alpha]", "^"], " ", OverscriptBox["\[Beta]", "^"]}], OverscriptBox[ RowBox[{" ", "0"}], "^"]], TraditionalForm]],ExpressionUUID-> "372cd123-1bea-4e48-9ab2-85702e790698"], "). (**Note:** there appears to be some discrepancy between this and \ Equation 8 of DG2019, particularly by an additional entry of ", Cell[BoxData[ FormBox[ SuperscriptBox["n", OverscriptBox["0", "^"]], TraditionalForm]],ExpressionUUID-> "58462a5b-6c48-4424-acbb-d507d4f5d256"], "). \n\n", Cell[BoxData[ FormBox[ SuperscriptBox["n", "\[Zeta]"], TraditionalForm]],ExpressionUUID-> "8efae983-4dbd-4289-bc80-db8a100a100c"], "=", Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{ SuperscriptBox[ RowBox[{"sin", " "}], "2"], "\[Zeta]"}]], TraditionalForm]], ExpressionUUID->"ffc8e93d-3583-4b95-88ec-9e34a1dee529"], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["n", OverscriptBox["\[Alpha]", "^"]]}], TraditionalForm]],ExpressionUUID-> "a40dc86b-8c41-491b-b91b-c577c0264eff"], Cell[BoxData[ FormBox[ SuperscriptBox["n", OverscriptBox["\[Beta]", "^"]], TraditionalForm]],ExpressionUUID-> "808d9d06-dd36-4078-88a8-aa078f699fa7"], " ", Cell[BoxData[ FormBox[ RowBox[{"(", SuperscriptBox["n", OverscriptBox["2", "^"]]}], TraditionalForm]],ExpressionUUID-> "87e9a5e1-848a-4117-abaf-1743785cc810"], " ", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Omega]", RowBox[{ OverscriptBox["\[Alpha]", "^"], " ", OverscriptBox["\[Beta]", "^"]}], OverscriptBox[ RowBox[{" ", "1"}], "^"]], TraditionalForm]],ExpressionUUID-> "2e7b1f97-caa7-4557-8ec3-2a725920f45f"], " -", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["n", OverscriptBox["1", "^"]]}], TraditionalForm]],ExpressionUUID-> "f85be2f8-6151-4f26-b555-877c091c5e53"], " ", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Omega]", RowBox[{ OverscriptBox["\[Alpha]", "^"], " ", OverscriptBox["\[Beta]", "^"]}], OverscriptBox[ RowBox[{" ", "2"}], "^"]], TraditionalForm]],ExpressionUUID-> "bf518d77-7d5a-4aac-b900-6b53d7ee86fe"], "). \n\nwhere \n\n", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Omega]", RowBox[{ OverscriptBox["\[Alpha]", "^"], " ", OverscriptBox["\[Beta]", "^"]}], OverscriptBox[ RowBox[{" ", "\[Gamma]"}], "^"]], TraditionalForm]],ExpressionUUID-> "deac6913-d4ec-4452-8a0a-ec49ae4b67b6"], " = ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", "\[Rho]", RowBox[{"(", OverscriptBox["\[Gamma]", "^"], ")"}]], TraditionalForm]],ExpressionUUID-> "d6976573-e3cf-4bd5-ac92-3d6c0b06d802"], " ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Beta]", "^"], ")"}], "\[Sigma]"], TraditionalForm]], ExpressionUUID->"fdfe9fdd-1a5a-4a21-8561-4afa1c3079f3"], " ", Cell[BoxData[ FormBox[ RowBox[{"(", SubscriptBox["\[PartialD]", RowBox[{"\[Sigma]", " "}]]}], TraditionalForm]],ExpressionUUID-> "99836127-e24f-46a8-a5a4-e016864d1a63"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Rho]"], " ", "+", " ", SubsuperscriptBox["\[CapitalGamma]", "\[Sigma]\[Mu]", RowBox[{" ", "\[Rho]"}]]}], TraditionalForm]],ExpressionUUID-> "a84a29c1-d84f-4689-bfd0-4631722e379d"], " ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Mu]"], TraditionalForm]], ExpressionUUID->"8aac3faa-46f6-4390-8207-1378b8315b04"], "),\n\n", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", "\[Rho]", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}]], TraditionalForm]],ExpressionUUID-> "3de92571-ab56-4383-9e7f-e7c49fcdf83e"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Eta]", "\[Alpha]\[Beta]"], " ", SubscriptBox["g", RowBox[{"\[Rho]\[Sigma]", " "}]]}], TraditionalForm]],ExpressionUUID-> "7c7905de-caad-434f-bf0d-d8ed1e174aae"], Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Beta]", "^"], ")"}], "\[Sigma]"], TraditionalForm]], ExpressionUUID->"a3fda21e-5943-494a-89f7-07ac63151a51"], " ,\n\n", Cell[BoxData[ FormBox[ SubsuperscriptBox["\[CapitalGamma]", "\[Rho]\[Sigma]", RowBox[{" ", "\[Mu]"}]], TraditionalForm]],ExpressionUUID-> "1c3bed79-61f8-4955-af03-5fd6da590ac8"], " = ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["1", "2"], RowBox[{ SuperscriptBox["g", "\[Mu]\[Lambda]"], "(", SubscriptBox["\[PartialD]", "\[Rho]"]}]}], TraditionalForm]], ExpressionUUID->"b1177701-7c20-4a8c-bded-5b1c564119da"], " ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["g", "\[Lambda]\[Sigma]"], " "}], TraditionalForm]], ExpressionUUID->"762fa843-8def-4c3a-9460-a86a66b2bbfe"], "+ ", Cell[BoxData[ FormBox[ SubscriptBox["\[PartialD]", "\[Sigma]"], TraditionalForm]],ExpressionUUID-> "c5565b3f-feed-40c8-b8b5-99a497787c50"], " ", Cell[BoxData[ FormBox[ SubscriptBox["g", "\[Lambda]\[Rho]"], TraditionalForm]],ExpressionUUID-> "d73b5a81-2a52-4cd7-bf3c-29405b8a9183"], " - ", Cell[BoxData[ FormBox[ SubscriptBox["\[PartialD]", "\[Lambda]"], TraditionalForm]],ExpressionUUID-> "cfc46019-be9f-4e62-b502-e6c3a4b3b673"], " ", Cell[BoxData[ FormBox[ SubscriptBox["g", "\[Rho]\[Sigma]"], TraditionalForm]],ExpressionUUID-> "207b6270-42b4-4829-911e-43eebc8b9f7d"], ")\n\nand \[PartialD] g and \[PartialD] e correspond to derivatives of the \ metric and tetrad components, respectively. Given an angular patch \ specified by \[Zeta] and \[Psi], we can compute ", Cell[BoxData[ FormBox[ SuperscriptBox["n", OverscriptBox["\[Alpha]", "^"]], TraditionalForm]],ExpressionUUID-> "2fce6781-96a1-4154-b8b7-6a33c3a02da4"], ", and therefore , after specifying ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Mu]"], TraditionalForm]], ExpressionUUID->"e08f77fd-44ae-4962-90b8-668bb7032bfe"], " we can readily compute ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "7ab544bd-dbfc-4758-9b2f-e730797c18f1"], " and ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "a"], TraditionalForm]],ExpressionUUID-> "d5efc072-c12f-48d2-9373-48d8fb02d34e"], ". How should we assign ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Mu]"], TraditionalForm]], ExpressionUUID->"6e8a83f0-9c02-4762-97fa-84fbe77cd830"], "?\n\nAgain, following @cwhite\[CloseCurlyQuote]s notes, we seek orthonormal \ tetrad basis components ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Mu]"], TraditionalForm]], ExpressionUUID->"d4eccef2-ad6f-4f1f-80e6-c8507058fb4e"], " satisfying\n\n", Cell[BoxData[ FormBox[ SubscriptBox["g", RowBox[{"\[Mu]\[Nu]", " "}]], TraditionalForm]],ExpressionUUID-> "0ac948fd-54dd-40df-92a0-38e132c8d167"], Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "\[Mu]"], TraditionalForm]], ExpressionUUID->"60bebb67-ea76-45cd-884e-68e0eb1b80c6"], " ", Cell[BoxData[ FormBox[ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["\[Beta]", "^"], ")"}], "\[Nu]"], TraditionalForm]], ExpressionUUID->"3352c4cd-8f53-494f-8cb3-35ef766527c4"], "=", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["\[Eta]", "\[Alpha]\[Beta]"]}], TraditionalForm]], ExpressionUUID->"e6fccc78-8213-445f-b68f-b1aa9a514f9a"], " \n\nWe require that the timelike direction in the tetrad should be \ orthogonal to surfaces of constant time coordinate ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "0"], TraditionalForm]],ExpressionUUID-> "1e6b71bc-d619-4246-9b74-040b13d069c9"], " and this direction should be future pointing:\n\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["e", RowBox[{"(", OverscriptBox["0", "^"], ")"}], "\[Mu]"], "=", RowBox[{ RowBox[{"-", "\[Alpha]"}], " ", SuperscriptBox["g", RowBox[{"0", "\[Mu]"}]]}]}], TraditionalForm]],ExpressionUUID-> "b31e33aa-ad61-4a1d-aa42-35806d103453"], "\n\nwhere \[Alpha] = ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", SuperscriptBox["g", "00"]}], ")"}], RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], TraditionalForm]],ExpressionUUID-> "74c41781-3cad-4676-bb18-f91a37285bcb"], ". The remaining components are computed via the Gram-Schmidt procedure. \ We will consider a \[OpenCurlyDoubleQuote]Cartesian\[CloseCurlyDoubleQuote] \ tetrad obtained via Gram-Schmidt on vectors orthogonal to surfaces of \ constant (t,z,x,y), in that order. The Gram-Schmidt procedure follows,\n\n\ Let the spatial 3-metric be defined as \n\n", Cell[BoxData[ FormBox[ SubscriptBox["\[Gamma]", "ij"], TraditionalForm]],ExpressionUUID-> "3b1bc4e5-1d69-40d4-82df-a774ae8e9328"], " = ", Cell[BoxData[ FormBox[ SubscriptBox["g", "ij"], TraditionalForm]],ExpressionUUID-> "e97990e8-1d68-436f-9303-a0cfab77133d"], "\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Gamma]", "ij"], "=", RowBox[{ SuperscriptBox["\[Alpha]", "2"], "(", RowBox[{ RowBox[{ SuperscriptBox["g", RowBox[{"0", "i"}]], SuperscriptBox["g", RowBox[{"0", "j"}]]}], "-", RowBox[{ SuperscriptBox["g", "00"], " ", SuperscriptBox["g", "ij"]}]}], ")"}]}], TraditionalForm]], ExpressionUUID->"c623078c-e3a1-4206-a2e1-7f198a33ee62"], ". \n\nSelect coordinates ", Cell[BoxData[ FormBox[ SuperscriptBox["q", "a"], TraditionalForm]],ExpressionUUID-> "488564bb-7c75-4d57-8218-07e32ea334f0"], " that are a function of the spatial coordinates ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "305d4aaa-f1a6-4bd8-ae2c-a618ac242f20"], ". For a \[OpenCurlyDoubleQuote]Cartesian\[CloseCurlyDoubleQuote] tetrad, \ select ", Cell[BoxData[ FormBox[ SuperscriptBox["q", "a"], TraditionalForm]],ExpressionUUID-> "9a876e27-e259-4779-9e95-9d6f64b8c16e"], "=(x,y,z) and ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "\[Mu]"], TraditionalForm]],ExpressionUUID-> "6435645a-e182-4471-85ec-160be8a06ee0"], "=(t,x,y,z). Define\n\n", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["A", RowBox[{ RowBox[{"(", OverscriptBox["\[Alpha]", "^"], ")"}], "i"}]], "=", " ", FractionBox[ RowBox[{"\[PartialD]", " ", SuperscriptBox["q", "a"]}], RowBox[{"d", " ", SuperscriptBox["x", "i"]}]]}], TraditionalForm]],ExpressionUUID-> "f9c450a1-e3fe-426e-9e17-189e80db8638"], "\n\nCalculate \n\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "i"], "=", RowBox[{ SuperscriptBox["\[Gamma]", "ij"], SubscriptBox["A", RowBox[{ RowBox[{"(", OverscriptBox["3", "^"], ")"}], "j"}]]}]}], TraditionalForm]], ExpressionUUID->"5045d374-f9db-41c1-ad96-ba8629da445f"], "\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "i"], "=", SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "i"]}], TraditionalForm]], ExpressionUUID->"82443183-3418-4b5b-9da6-5878355e24fd"], "\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["C", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "i"], "=", " ", RowBox[{"(", SubscriptBox["\[Gamma]", "jk"], " "}]}], TraditionalForm]], ExpressionUUID->"13aac39a-2602-4b78-be43-f3b1ed779ca8"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "j"], TraditionalForm]],ExpressionUUID-> "cb4b10d2-4c8b-46ff-872d-c55d1287fecd"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "k"], TraditionalForm]],ExpressionUUID-> "d030be1e-9d68-468c-8884-bf138eeafac7"], Cell[BoxData[ FormBox[ SuperscriptBox[")", RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], TraditionalForm]],ExpressionUUID-> "4c7fb240-6214-4aa6-a514-8b993c36c806"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "i"], TraditionalForm]],ExpressionUUID-> "ae9dab6e-4a3b-4a38-b2d3-2320bc7bb479"], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "i"], "=", RowBox[{ SuperscriptBox["\[Gamma]", "ij"], SubscriptBox["A", RowBox[{ RowBox[{"(", OverscriptBox["1", "^"], ")"}], "j"}]]}]}], TraditionalForm]], ExpressionUUID->"9795f699-8515-44e6-92f8-5356a1293218"], "\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "i"], "=", SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "i"]}], TraditionalForm]], ExpressionUUID->"f67f6311-52b8-4b82-a435-be73784e766c"], "- ", Cell[BoxData[ FormBox[ SubscriptBox["\[Gamma]", "jk"], TraditionalForm]],ExpressionUUID-> "1eef761c-5deb-455b-8eb8-4800b273fba3"], Cell[BoxData[ FormBox[ SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "j"], TraditionalForm]],ExpressionUUID-> "dcc50083-5a14-441c-a10f-86a205331bb5"], Cell[BoxData[ FormBox[ SubsuperscriptBox["C", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "k"], TraditionalForm]],ExpressionUUID-> "e883729b-9617-4e10-8fd1-afd6ac56d8cd"], Cell[BoxData[ FormBox[ SubsuperscriptBox["C", RowBox[{"(", OverscriptBox["3", "^"], ")"}], "i"], TraditionalForm]],ExpressionUUID-> "f61570cc-fd12-4c89-a18e-4e4549d7223e"], "\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["C", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "i"], "=", " ", RowBox[{"(", SubscriptBox["\[Gamma]", "jk"], " "}]}], TraditionalForm]], ExpressionUUID->"94618373-e38b-4498-9655-b2bed677c017"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "j"], TraditionalForm]],ExpressionUUID-> "2f90c680-674d-461d-93e6-e5a95ffaa16c"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "k"], TraditionalForm]],ExpressionUUID-> "bd76f015-879a-492f-8ca5-368d67ae2f91"], Cell[BoxData[ FormBox[ SuperscriptBox[")", RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], TraditionalForm]],ExpressionUUID-> "326414df-447f-4fe1-b6f2-f2b4f201e1a2"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "i"], TraditionalForm]],ExpressionUUID-> "f450b710-2c6c-4035-a5e8-a379ae85c841"], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["2", "^"], ")"}], "i"], "=", RowBox[{ SuperscriptBox["\[Gamma]", "ij"], SubscriptBox["A", RowBox[{ RowBox[{"(", OverscriptBox["2", "^"], ")"}], "j"}]]}]}], TraditionalForm]], ExpressionUUID->"7bc799a7-534c-4aa6-8f4b-8a014acdb6ff"], "\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["2", "^"], ")"}], "i"], "=", SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["2", "^"], ")"}], "i"]}], TraditionalForm]], ExpressionUUID->"2dbd4aea-6786-45a9-987e-19fceabf397f"], "- ", Cell[BoxData[ FormBox[ SubscriptBox["\[Gamma]", "jk"], TraditionalForm]],ExpressionUUID-> "b8754609-a546-4fa7-8f3e-abb86b09eab6"], Cell[BoxData[ FormBox[ SubsuperscriptBox["A", RowBox[{"(", OverscriptBox["2", "^"], ")"}], "j"], TraditionalForm]],ExpressionUUID-> "276d2f2e-a6ee-4a25-9f5f-9a8b8fed6f5a"], Cell[BoxData[ 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Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["C", RowBox[{"(", OverscriptBox["1", "^"], ")"}], "i"], "=", " ", RowBox[{"(", SubscriptBox["\[Gamma]", "jk"], " "}]}], TraditionalForm]], ExpressionUUID->"643ea263-d1d6-4a7c-9f2c-3d6cdb3bd12d"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["2", "^"], ")"}], "j"], TraditionalForm]],ExpressionUUID-> "9eb8d1f0-b756-48f2-9ea0-9a3f1dc44818"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["2", "^"], ")"}], "k"], TraditionalForm]],ExpressionUUID-> "79a43723-ad06-438d-8aa9-25271f795d46"], Cell[BoxData[ FormBox[ SuperscriptBox[")", RowBox[{ RowBox[{"-", "1"}], "/", "2"}]], TraditionalForm]],ExpressionUUID-> "6ca27c00-ee0d-435f-bb30-c306b499fcb3"], Cell[BoxData[ FormBox[ SubsuperscriptBox["B", RowBox[{"(", OverscriptBox["2", "^"], ")"}], "i"], TraditionalForm]],ExpressionUUID-> "cc2b1d7b-a9b0-4477-8915-d2656071626c"], "\n\nThen set the orthonormal tetrad components following...\n\n", 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