(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 922620, 24611] NotebookOptionsPosition[ 895596, 24032] NotebookOutlinePosition[ 896005, 24049] CellTagsIndexPosition[ 895962, 24046] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{"TextCell", "[", "\[IndentingNewLine]", RowBox[{ "\"\\"", ",", " ", "\"\\""}], " ", "]"}]}]], "Input", CellChangeTimes->{{3.517207454078125*^9, 3.517207455203125*^9}}], Cell[BoxData[ InterpretationBox[Cell[ "Even Parity Perturbations of the Self-Similar LTB Spacetime", "Title"], TextCell[ "Even Parity Perturbations of the Self-Similar LTB Spacetime", "Title"]]], "Output", CellChangeTimes->{3.517207460875*^9}] }, {2}]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"TextCell", "[", RowBox[{ "\"\\"", ",", " ", "\"\\"", ",", " ", RowBox[{"CellFrame", "\[Rule]", "True"}], ",", " ", RowBox[{"Background", "\[Rule]", RowBox[{"GrayLevel", "[", "0.9", "]"}]}]}], "]"}]], "Input", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.517138808375*^9, 3.51713881471875*^9}, { 3.517138879859375*^9, 3.51713903328125*^9}, {3.51720672284375*^9, 3.517206927765625*^9}, {3.517207044640625*^9, 3.5172071283125*^9}, { 3.51720720196875*^9, 3.517207224859375*^9}, {3.517207451484375*^9, 3.51720745871875*^9}, {3.5212629556875*^9, 3.521263339828125*^9}, { 3.52126340246875*^9, 3.5212634354375*^9}, {3.5212634759375*^9, 3.521263526265625*^9}, {3.521263697125*^9, 3.52126372859375*^9}, { 3.521263915265625*^9, 3.521263915828125*^9}, {3.52126451653125*^9, 3.5212645186875*^9}, {3.52126985975*^9, 3.52126986290625*^9}, { 3.521437170296875*^9, 3.521437187625*^9}}], Cell[BoxData[ InterpretationBox[Cell["\<\ In this notebook we study linear even parity perturbations of the \ self-similar LTB spacetime. Section 5.2 of the thesis discusses the initial \ system of equations and the procedure we follow in reducing them. The details \ of this procedure are provided here and Section 5.2 of the thesis should be \ read in conjunction with this notebook. In this notebook, we first input the Gerlach-Sengupta perturbations and the \ linearised Einstein equations. We introduce a first order reduction and \ identify the initial set of perturbation equations. This set consists of \ seven equations for six variables. We identify the Einstein constraint, and \ the trivial constraint which arises from the first order reduction. We then \ use these two constraints to eliminate two variables, thus reducing the \ initial system to a four dimensional system (that is, four equations in four \ variables). We do this in two stages, first reducing to a five dimensional, \ and then to a four dimensional system. In so doing, we identify the five \ dimensional symmetric hyperbolic system which is used in Chapters 5 and 6 of \ the thesis. \[IndentingNewLine]The notebook is divided into sections as follows: \ \[IndentingNewLine]Section 1: The Background Spacetime. In this section, we \ input the background metric and calculate the connection components and the \ Ricci tensor.\[IndentingNewLine]Section 2: The Perturbation Variables and \ Equations of Motion. Here we input the Gerlach-Sengupta equations and the \ perturbation terms for the even parity perturbations. \ \[IndentingNewLine]Section 3: The Initial System. In this section we take the \ initial equations and rewrite them as a vector equation with matrix \ coefficients. We identify two constraints, namely the Einstein constraint and \ the trivial constraint. \[IndentingNewLine]Section 4: Constraint \ Propagation. In this section, we show explicitly that the Einstein constraint \ is propagated by the equations of motion. \[IndentingNewLine]Section 5: \ Dimensional Reduction. In this section, we reduce the system first to a six \ dimensional, and then to a five dimensional system. \ \[IndentingNewLine]Section 6: The Four Dimensional System. We eliminate the \ final constraint and reduce to the four dimensional system. We note that this \ system is not symmetric hyperbolic. \ \>", "Text", CellFrame->True, Background->GrayLevel[0.9]], TextCell[ "In this notebook we study linear even parity perturbations of the \ self-similar LTB spacetime. Section 5.2 of the thesis discusses the initial \ system of equations and the procedure we follow in reducing them. The details \ of this procedure are provided here and Section 5.2 of the thesis should be \ read in conjunction with this notebook. \n\nIn this notebook, we first input \ the Gerlach-Sengupta perturbations and the linearised Einstein equations. We \ introduce a first order reduction and identify the initial set of \ perturbation equations. This set consists of seven equations for six \ variables. We identify the Einstein constraint, and the trivial constraint \ which arises from the first order reduction. We then use these two \ constraints to eliminate two variables, thus reducing the initial system to a \ four dimensional system (that is, four equations in four variables). We do \ this in two stages, first reducing to a five dimensional, and then to a four \ dimensional system. In so doing, we identify the five dimensional symmetric \ hyperbolic system which is used in Chapters 5 and 6 of the thesis. \n\ \[IndentingNewLine]The notebook is divided into sections as follows: \ \[IndentingNewLine]Section 1: The Background Spacetime. In this section, we \ input the background metric and calculate the connection components and the \ Ricci tensor.\[IndentingNewLine]Section 2: The Perturbation Variables and \ Equations of Motion. Here we input the Gerlach-Sengupta equations and the \ perturbation terms for the even parity perturbations. \ \[IndentingNewLine]Section 3: The Initial System. In this section we take the \ initial equations and rewrite them as a vector equation with matrix \ coefficients. We identify two constraints, namely the Einstein constraint and \ the trivial constraint. \[IndentingNewLine]Section 4: Constraint \ Propagation. In this section, we show explicitly that the Einstein constraint \ is propagated by the equations of motion. \[IndentingNewLine]Section 5: \ Dimensional Reduction. In this section, we reduce the system first to a six \ dimensional, and then to a five dimensional system. \ \[IndentingNewLine]Section 6: The Four Dimensional System. We eliminate the \ final constraint and reduce to the four dimensional system. We note that this \ system is not symmetric hyperbolic. \n ", "Text", CellFrame -> True, Background -> GrayLevel[0.9]]]], "Output", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{ 3.51720743740625*^9, 3.517216290265625*^9, 3.517216392234375*^9, { 3.517216428171875*^9, 3.517216448203125*^9}, {3.5172164809375*^9, 3.51721649209375*^9}, 3.517217676625*^9, 3.521263601359375*^9, 3.521263954046875*^9, 3.521264536859375*^9, 3.521269864171875*^9, 3.521437216078125*^9}] }, {2}]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.52126365415625*^9}], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[Cell["Section 1: The Background Spacetime", "Section"], TextCell["Section 1: The Background Spacetime", "Section"]]], "Output", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{3.521263667875*^9, 3.521271406828125*^9}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{ 3.51721626746875*^9, 3.517216392234375*^9, {3.517216428171875*^9, 3.517216448203125*^9}, 3.517217676625*^9}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{3.51729656803125*^9, 3.517311590625*^9}], Cell[BoxData[ InterpretationBox[Cell["\<\ See Chapter 2 of the thesis for a full discussion of these coordinates. 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To do this, we take the following steps:\ \[IndentingNewLine]- We can show (see point (2) in Section 5.2 of Chapter 5) \ that it is possible to replace equation 3.27 with equation 5.3 provided that \ equations 3.28 and 3.31 hold. We therefore use the stress-energy conservation \ equation (equation 5.3) in favour of equation 3.27. \[IndentingNewLine]- We \ select the two components of equation 3.28, that is E1 and E2. \ \[IndentingNewLine]- We have used the first component of equation 3.26 (that \ is, E4) to solve for \[Delta]\[Rho], so we select the other two components of \ equation 3.26 (that is, E5 and E6). \[IndentingNewLine]- These four equations \ produce EQ1, shown below. \[IndentingNewLine]- We note that in this list of \ equations only the scalar k (here called kksc) appears at second order so we \ introduce a first order reduction by promoting the two first order partial \ derivatives of k with respect to t and p to new variables. \ \[IndentingNewLine]- This produces EQ2. \[IndentingNewLine]- We next add in \ the stress-energy conservation equation, to produce EQ4. \[IndentingNewLine]- \ Having introduced a first order reduction, we must add two trivial evolution \ equations arising from the reduction, which produces EQ5 and EQ6. \ \[IndentingNewLine]- Finally, we define our initial state vector, X.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.9]], TextCell[ "We now wish to construct a useful system of equations from the \ Gerlach-Sengupta equations. To do this, we take the following steps:\ \[IndentingNewLine]- We can show (see point (2) in Section 5.2 of Chapter 5) \ that it is possible to replace equation 3.27 with equation 5.3 provided that \ equations 3.28 and 3.31 hold. We therefore use the stress-energy conservation \ equation (equation 5.3) in favour of equation 3.27. \[IndentingNewLine]- We \ select the two components of equation 3.28, that is E1 and E2. \ \[IndentingNewLine]- We have used the first component of equation 3.26 (that \ is, E4) to solve for \[Delta]\[Rho], so we select the other two components of \ equation 3.26 (that is, E5 and E6). \[IndentingNewLine]- These four equations \ produce EQ1, shown below. \[IndentingNewLine]- We note that in this list of \ equations only the scalar k (here called kksc) appears at second order so we \ introduce a first order reduction by promoting the two first order partial \ derivatives of k with respect to t and p to new variables. \ \[IndentingNewLine]- This produces EQ2. \[IndentingNewLine]- We next add in \ the stress-energy conservation equation, to produce EQ4. \[IndentingNewLine]- \ Having introduced a first order reduction, we must add two trivial evolution \ equations arising from the 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order terms (X) and source terms, and in \ particular contains no terms of the form (\[PartialD]X/\[PartialD]z), that \ is, no ''time'' derivatives. We note that the above system has two \ constraints. Firstly, it has a nontrivial Einstein constraint which we \ identify below. Additionally, there is a trivial constraint arising from the \ first order reduction. This constraint is simply \[IndentingNewLine]v - \ \[PartialD]u / \[PartialD]p = 0. \[IndentingNewLine]\[IndentingNewLine]Since \ a constraint by definition contains no time derivatives, we construct the \ Einstein constraint by looking for a linear combination of the equations in \ EQ6 which does not contain any z-derivatives. To see if such a combination \ exists, we examine the determinant of submatrices of A. \[IndentingNewLine]\ \[IndentingNewLine]We construct a matrix (here called checkA2) consisting of \ the first six rows of A. It has a vanishing determinant, indicating that the \ constraint can be found by taking a linear combination of the first six \ equations of EQ6.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.9]], TextCell[ "Recall that a constraint is a relation between spatial derivatives (\ \[PartialD]X/\[PartialD]p), zero order terms (X) and source terms, and in \ particular contains no terms of the form (\[PartialD]X/\[PartialD]z), that \ is, no ''time'' derivatives. We note that the above system has two \ constraints. Firstly, it has a nontrivial Einstein constraint which we \ identify below. Additionally, there is a trivial constraint arising from the \ first order reduction. This constraint is simply \[IndentingNewLine]v - \ \[PartialD]u / \[PartialD]p = 0. \[IndentingNewLine]\[IndentingNewLine]Since \ a constraint by definition contains no time derivatives, we construct the \ Einstein constraint by looking for a linear combination of the equations in \ EQ6 which does not contain any z-derivatives. To see if such a combination \ exists, we examine the determinant of submatrices of A. \[IndentingNewLine]\ \[IndentingNewLine]We construct a matrix (here called checkA2) consisting of \ the first six rows of A. 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We first produce a linear combination of \[PartialD] C/\ \[PartialD] z and \[PartialD] C/\[PartialD] p which contains no second order \ derivatives.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.9]], TextCell[ "We aim to show that there exists some functions f(z) and g(z) such that \ \[PartialD] C/\[PartialD] z + f(z) \[PartialD] C/\[PartialD] p + g(z) C = 0 \ is satisfied. 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and S is the similarity \ matrix newPsim, then K will obey the equation \[PartialD] K/\[PartialD] z + H \[PartialD] K/\[PartialD] p + G K = ksource, \ where the matrix H will be newPjord. This corresponds to equation 5.12 in \ Chapter 5. We see below that the matrix H is not symmetric and thus the K \ system is not symmetric hyperbolic. As symmetric hyperbolicity is such a \ useful property, we choose instead to work with the five dimensional system, \ which was determined in Section 5 above. We will therefore go no further with \ the four dimensional system. \ \>", "Text", CellFrame->True, Background->GrayLevel[0.9]], TextCell[ "So if we let N = S K, where K is our new state vector and S is the \ similarity matrix newPsim, then K will obey the equation \n\[PartialD] K/\ \[PartialD] z + H \[PartialD] K/\[PartialD] p + G K = ksource, where the \ matrix H will be newPjord. This corresponds to equation 5.12 in Chapter 5. \ We see below that the matrix H is not symmetric and thus the K system is not \ symmetric hyperbolic. As symmetric hyperbolicity is such a useful property, \ we choose instead to work with the five dimensional system, which was \ determined in Section 5 above. We will therefore go no further with the four \ dimensional system. 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