GR Calculations in Specific Bases Using Mathematica

Transcription

1 GR Calculations in Specific Bases Using Mathematica Geoge E. Habovsky MAST Midwest Relativity Meeting, 2015 CIERA, Nowesten Univesity

2 2 GR Calculations in Specific Bases Using Mathematica.nb What I Will Cove Intoduction How to Establish a Manifold How to Establish a Coodinate Chat How to Define a Metic How to Define a Tenso Computing the Chistoffel Symbols The Riemann Tenso, The Ricci Tenso, The Ricci Scala, and The Einstein Tenso The Stess-Enegy Tenso Einstein s Field Equations

3 GR Calculations in Specific Bases Using Mathematica.nb 3 Intoduction This is the thid of an appaently endless seies of talks on how to use Mathematica in geneal elativity. Two yeas ago I talked about the built-in capabilities fo handling tensos. Last yea I talked about the xact package in geneal and how to apply it to petubative geneal elativity, deiving the scala and tenso field equations fo a gavitational petubation given a Lagangian. This yea I am talking about pefoming calculations in specific coodinate bases. Past talks can be found at the website: http : // This talk will appea thee also.

4 4 GR Calculations in Specific Bases Using Mathematica.nb Establishing You Manifold The fist thing too do is activate xact. << xact`xtenso` Package xact`xpem` vesion 1.2.2, {2014, 9, 28} CopyRight (C) , Jose M. Matin-Gacia, unde the Geneal Public License. Connecting to extenal MinGW executable... Connection established Package xact`xtenso` vesion 1.1.1, {2014, 9, 28} CopyRight (C) , Jose M. Matin-Gacia, unde the Geneal Public License These packages come with ABSOLUTELY NO WARRANTY; fo details type Disclaime[]. This is fee softwae, and you ae welcome to edistibute it unde cetain conditions. See the Geneal Public License fo details Then you define you manifold.

5 GR Calculations in Specific Bases Using Mathematica.nb 5 DefManifold[M4, 4, {α, β, γ, μ, ν, λ, σ, η}] ** DefManifold: Defining manifold M4. ** DefVBundle: Defining vbundle TangentM4.

6 6 GR Calculations in Specific Bases Using Mathematica.nb DefMetic[-1, metic[-α, -β], CD, {";", " "}, PintAs "g"] ** DefTenso: Defining symmetic metic tenso metic[- α, - β]. ** DefTenso: Defining antisymmetic tenso epsilonmetic[- α, - β, - γ, - η]. ** DefTenso: Defining tetametic Tetametic[- α, - β, - γ, - η]. ** DefTenso: Defining tetametic Tetametic [- α, - β, - γ, - η]. ** DefCovD: Defining covaiant deivative CD[- α]. ** DefTenso: Defining vanishing tosion tenso TosionCD[α, - β, - γ]. ** DefTenso: Defining symmetic Chistoffel tenso ChistoffelCD[α, - β, - γ]. ** DefTenso: Defining Riemann tenso RiemannCD[- α, - β, - γ, - η]. ** DefTenso: Defining symmetic Ricci tenso RicciCD[- α, - β]. ** DefCovD: Contactions of Riemann automatically eplaced by Ricci. ** DefTenso: Defining Ricci scala RicciScalaCD[]. ** DefCovD: Contactions of Ricci automatically eplaced by RicciScala. ** DefTenso: Defining symmetic Einstein tenso EinsteinCD[- α, - β]. ** DefTenso: Defining Weyl tenso WeylCD[-α, -β, -γ, -η]. ** DefTenso: Defining symmetic TFRicci tenso TFRicciCD[- α, - β]. ** DefTenso: Defining Ketschmann scala KetschmannCD[]. ** DefCovD: Computing RiemannToWeylRules fo dim 4 ** DefCovD: Computing RicciToTFRicci fo dim 4 ** DefCovD: Computing RicciToEinsteinRules fo dim 4 ** DefTenso: Defining weight +2 density Detmetic[]. Deteminant.

7 GR Calculations in Specific Bases Using Mathematica.nb 7 Establishing You Chat << xact`xcoba` Package xact`xcoba` vesion 0.8.2, {2014, 9, 28} CopyRight (C) , David Yllanes and Jose M. Matin-Gacia, unde the Geneal Public License These packages come with ABSOLUTELY NO WARRANTY; fo details type Disclaime[]. This is fee softwae, and you ae welcome to edistibute it unde cetain conditions. See the Geneal Public License fo details

8 8 GR Calculations in Specific Bases Using Mathematica.nb $DefInfoQ = False; $PePint = SceenDollaIndices; $CVSimplify = Simplify;

9 GR Calculations in Specific Bases Using Mathematica.nb 9 DefChat[cb, M4, {0, 1, 2, 3}, {t[], [], θ[], ϕ[]}] cb /: CIndexFom[0, cb] := "t"; cb /: CIndexFom[1, cb] := ""; cb /: CIndexFom[2, cb] := "θ"; cb /: CIndexFom[3, cb] := "ϕ"; You should then define any scala functions and constants you will need fo you metic. DefConstantSymbol[M] DefConstantSymbol[G]

10 10 GR Calculations in Specific Bases Using Mathematica.nb Two Ways to Define You Metic MatixFom met = DiagonalMatix 1-2 M [], 1-2 M M Sin[θ] M [] -1, -[] 2, 2 [] Sin[θ[]] 2

11 GR Calculations in Specific Bases Using Mathematica.nb 11 MeticInBasis[metic, - cb, met] // TableFom Added independent ule g tt 1-2 M fo tenso metic Added independent ule Added independent ule Added independent ule g t 0 fo tenso metic g tθ 0 fo tenso metic g tϕ 0 fo tenso metic Added dependent ule g t g t fo tenso metic Added independent ule g M fo tenso metic Added independent ule Added independent ule g θ 0 fo tenso metic g ϕ 0 fo tenso metic Added dependent ule Added dependent ule g θt g tθ fo tenso metic g θ g θ fo tenso metic Added independent ule g θθ - 2 fo tenso metic Added independent ule g θϕ 0 fo tenso metic Added dependent ule Added dependent ule Added dependent ule g ϕt g tϕ fo tenso metic g ϕ g ϕ fo tenso metic g ϕθ g θϕ fo tenso metic Added independent ule g ϕϕ 2 Sin[θ] 2 fo tenso metic g tt 1-2 M g t 0 g tθ 0 g tϕ 0 g t 0 g M g θ 0 g ϕ 0 g θt 0 g θ 0 g θθ - 2 g θϕ 0 g ϕt 0 g ϕ 0 g ϕθ 0 g ϕϕ 2 Sin[θ] 2

12 12 GR Calculations in Specific Bases Using Mathematica.nb metic FoldedRule g t g t, g θt g tθ, g θ g θ, g ϕt g tϕ, g ϕ g ϕ, g ϕθ g θϕ, g tt 1-2 M, g t 0, g tθ 0, g tϕ 0, g M g θ 0, g ϕ 0, g θθ - 2, g θϕ 0, g ϕϕ 2 Sin[θ] 2, MeticCompute[metic, cb, "Weyl"[-1, -1, -1, -1]]

13 GR Calculations in Specific Bases Using Mathematica.nb 13 Now we can exploe the second method of defining the metic. g = CTenso[met, {-cb, -cb}]; SetCMetic[g, -cb]; Hee was can specify the g tt component, g[{0, -cb}, {0, -cb}] 1-2 M MeticCompute[g, cb, "Weyl"[-1, -1, -1, -1]];

14 14 GR Calculations in Specific Bases Using Mathematica.nb Hee we define the covaiant deivative, cd = CovDOfMetic[g] CCovD PDcb, M CTenso 0, - 2 M -, 0, 0, - M, 0, 0, 0, {0, 0, 0, 0}, {0, 0, 0, 0}, 2 2 M - 2 M (2 M - ) M, 0, 0, 0, 0,, 0, 0, 3 2 M - 2 (2 M - ) Sin[θ]2 {0, 0, -2 M +, 0}, 0, 0, 0,, {0, 0, 0, 0}, 0, 0, 1, 0, 0, 1 Sin[2 θ], 0, 0, 0, 0, 0,, {0, 0, 0, 0}, 0, 0, 0, 1 1, {0, 0, 0, Cot[θ]}, 0,, Cot[θ], 0, 2 2 {cb, -cb, -cb}, 0, CTenso 1-2 M, 0, 0, 0, 0, 1, 0, 0, 1-2 M 0, 0, - 2, 0, 0, 0, 0, 2 Sin[θ] 2, {-cb, -cb}, 0

15 GR Calculations in Specific Bases Using Mathematica.nb 15 Chistoffel Symbols in a Coodinate Basis In geneal we can wite the Chistoffel symbols Chistoffel[CD, PDcb][α, -β, -γ] Γ[, ] α βγ We can make a table of these in ou coodinate basis. Pat[TensoValues@ ChistoffelCDPDcb, 2] // TableFom Γ[, ] t tt 0 Γ[, ] t t - Γ[, ] t tθ 0 Γ[, ] t tϕ 0 Γ[, ] t 0 Γ[, ] t θ 0 Γ[, ] t ϕ 0 Γ[, ] t θθ 0 Γ[, ] t θϕ 0 Γ[, ] t ϕϕ 0 Γ[, ] tt Γ[, ] t 0 Γ[, ] tθ 0 Γ[, ] tϕ 0 Γ[, ] Γ[, ] θ 0 Γ[, ] ϕ 0 M 2 M - 2 M (2 M-) 3 M 2 M - 2 Γ[, ] θθ -2 M + Γ[, ] θϕ 0 Γ[, ] ϕϕ Γ[, ] θ tt 0 Γ[, ] θ t 0 Γ[, ] θ tθ 0 Γ[, ] θ tϕ 0 (2 M-) Sin[θ]2

16 16 GR Calculations in Specific Bases Using Mathematica.nb Γ[, ] θ 0 Γ[, ] θ θ 1 Γ[, ] θ ϕ 0 Γ[, ] θ θθ 0 Γ[, ] θ θϕ 0 Γ[, ] θ ϕϕ Γ[, ] ϕ tt 0 Γ[, ] ϕ t 0 Γ[, ] ϕ tθ 0 Γ[, ] ϕ tϕ 0 Γ[, ] ϕ 0 Γ[, ] ϕ θ 0 Sin[2 θ] Γ[, ] ϕ ϕ 1 2 Γ[, ] ϕ θθ 0 Γ[, ] ϕ θϕ Cot[θ] Γ[, ] ϕ ϕϕ 0

17 GR Calculations in Specific Bases Using Mathematica.nb 17 The Riemann Tenso iemann = Riemann[cd] CTenso {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 2 M (-2 M + ) 2 M 0,, 0, 0,, 0, 0, 0, {0, 0, 0, 0}, {0, 0, 0, 0}, 4 (2 M - ) 2 M (2 M - ) 0, 0,, 0, {0, 0, 0, 0}, - M, 0, 0, 0, {0, 0, 0, 0}, 4 M (2 M - ) 0, 0, 0,, {0, 0, 0, 0}, {0, 0, 0, 0}, M Sin[θ]2, 0, 0, 0, M (-2 M + ) 2 M 0, -, 0, 0, -, 0, 0, 0, {0, 0, 0, 0}, {0, 0, 0, 0}, 4 (2 M - ) 2 {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, M {0, 0, 0, 0}, 0, 0, (2 M - ), 0, 0, - M, 0, 0, {0, 0, 0, 0}, 2-4 M + Cot[θ] {0, 0, 0, 0}, 0, 0, 0,, 0, 0, 0,, 4 2 (-2 M + ) 2 (4 M - ) Sin[θ]2 Cos[θ] Sin[θ] 0,,, 0, M (2 M - ) 0, 0, -, 0, {0, 0, 0, 0}, M, 0, 0, 0, {0, 0, 0, 0}, 4 M {0, 0, 0, 0}, 0, 0, - (2 M - ), 0, 0, M, 0, 0, {0, 0, 0, 0}, 2 {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {0, 0, 0, 0}, 0, 0, 0, Cot[θ], 0, 0, 0, M, Cos[θ] (2 M - ) Sin[θ] (-2 M + 3 ) Sin[θ]2 0,,, 0, 2 M (2 M - ) 0, 0, 0, -, {0, 0, 0, 0}, {0, 0, 0, 0}, - M Sin[θ]2, 0, 0, 0, M + Cot[θ] {0, 0, 0, 0}, 0, 0, 0, -, 0, 0, 0, -, 4 2 (-2 M + ) 2 (4 M - ) Sin[θ]2 Cos[θ] Sin[θ] 0, -, -, 0, {0, 0, 0, 0}, 0, 0, 0, - Cot[θ], , 0, 0, M Cos[θ] (2 M - ) Sin[θ] (-2 M + 3 ) Sin[θ]2, 0, -, -, 0, 2 {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {-cb, -cb, -cb, cb}, 0 iemann[{0, -cb}, {1, -cb}, {0, -cb}, {1, cb}] 2 M (-2 M + ) 4 iemann[{3, -cb}, {2, -cb}, {3, -cb}, {2, cb}] (-2 M + 3 ) Sin[θ]2-2

18 18 GR Calculations in Specific Bases Using Mathematica.nb The Ricci Tenso and Ricci Scala Ricci[cd][-α, -β] M (-2 M+) Cot[θ] 0-8 M Cot[θ] (2 M+5 ) Sin[θ]2-2 2 αβ Ricci[cd][{0, -cb}, {0, -cb}] M (-2 M + ) 2 4 s = RicciScala[cd][[1]] - 2 M

19 GR Calculations in Specific Bases Using Mathematica.nb 19 The Einstein Tenso Einstein[cd][-α, -β] (2 M-) (4 M+5 ) M+3 Cot[θ] 0-4 M Cot[θ] 2 M αβ Einstein[cd][{0, -cb}, {0, -cb}] (2 M - ) (4 M + 5 ) - 4 4

20 20 GR Calculations in Specific Bases Using Mathematica.nb The Stess Enegy Tenso We next need to calculate the stess-enegy tenso. we begin by defining the density field. DefTenso[ρ, M4] DefTenso[ρ, M4, GenSet[]] Hee we define the 4-velocity. U = CTenso[{1, 0, 0, 0}, {-cb}] CTenso[{1, 0, 0, 0}, {-cb}, 0]

21 GR Calculations in Specific Bases Using Mathematica.nb 21 Hee is the stess-enegy tenso fo a pessue-less dust Td[α_, β_] := ρ[] U[-α] U[-β] Td[α, β] ρ[] βα Hee we have the T tt component. Td[{0, cb}, {0, cb}] ρ[]

22 22 GR Calculations in Specific Bases Using Mathematica.nb The pessue-less dust is vey simple. A little moe complicated is the pefect fluid. This equies us to define a pessue field. DefTenso[p, M4] DefTenso[p, M4, GenSet[]] The stess-enegy tenso fo this situation is, Tf[α_, β_] := (ρ[] + p[]) U[-α] U[-β] + p[] g[-α, -β] Tf[α, β] p[]+p[] 1-2 M p[] M p[] Sin[θ] 2 αβ 0 0 -p[] 2 0 Tf[{-0, cb}, {-0, cb}] p[] + p[] 1-2 M + ρ[] Tf[{-1, cb}, {-1, cb}] p[] 1-2 M

23 GR Calculations in Specific Bases Using Mathematica.nb 23 Einstein s Field Equations We will now ty to wite Einstein s equation fo the tt components of the Einstein and stess-enegy tensos. We begin with this fomulation, R tt R g tt = 8 π G T tt

24 24 GR Calculations in Specific Bases Using Mathematica.nb tteq = Ricci[cd][{0, -cb}, {0, -cb}] - 1 s g[{0, -cb}, {0, -cb}] 2 8 π G Tf[{-0, cb}, {-0, cb}] M (-2 M + ) M (2 M + 5 ) 8 G π p[] + p[] 1-2 M ρ[] tteq // FullSimplify - (2 M - ) (4 M + 5 ) 8 G π p[] 2-2 M ρ[] We can also wite the equation, G tt = 8 π G T tt tteq2 = Einstein[cd][{0, -cb}, {0, -cb}] 8 π G Tf[{-0, cb}, {-0, cb}] // FullSimplify - (2 M - ) (4 M + 5 ) 8 G π p[] 2-2 M ρ[]

25 GR Calculations in Specific Bases Using Mathematica.nb 25 We can genalize this eineq[a_, b_] := Einstein[cd][{a, -cb}, {b, -cb}] - 8 π G Tf[{-a, cb}, {-b, cb}] // FullSimplify eineq[0, 0] - (2 M - ) (4 M + 5 ) - 8 G π p[] 2-2 M ρ[] eineq[1, 1] M G π p[] (-2 M + ) eineq[0, 1] 0

26 26 GR Calculations in Specific Bases Using Mathematica.nb Table[eineq[a, b], {a, 0, 3}, {b, 0, 3}] // TableFom (2 M-) (4 M+5 ) G π p[] 2-2 M + ρ[] M+3-16 G π p[] (-2 M+) Cot[θ] M + 8 G π p[] G π p[] Sin[θ] 2 Cot[θ] 2 0

27 Thank You! GR Calculations in Specific Bases Using Mathematica.nb 27