In[]:=
<<xAct`xPert`;<<xAct`xTensor`;<<xAct`xTras`;
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Package xAct`xPerm` version 1.2.3, {2015,8,23}
CopyRight (C) 2003-2020, Jose M. Martin-Garcia, under the General Public License.
Connecting to external mac executable...
Connection established.
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Package xAct`xTensor` version 1.2.0, {2021,10,17}
CopyRight (C) 2002-2021, Jose M. Martin-Garcia, under the General Public License.
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Package xAct`xPert` version 1.0.6, {2018,2,28}
CopyRight (C) 2005-2020, David Brizuela, Jose M. Martin-Garcia and Guillermo A. Mena Marugan, under the General Public License.
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These packages come with ABSOLUTELY NO WARRANTY; for details type Disclaimer[]. This is free software, and you are welcome to redistribute it under certain conditions. See the General Public License for details.
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** Variable $PrePrint assigned value ScreenDollarIndices
** Variable $CovDFormat changed from Prefix to Postfix
** Option AllowUpperDerivatives of ContractMetric changed from False to True
** Option MetricOn of MakeRule changed from None to All
** Option ContractMetrics of MakeRule changed from False to True
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Package xAct`xTensor` version 1.2.0, {2021,10,17}
CopyRight (C) 2002-2021, Jose M. Martin-Garcia, under the General Public License.
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These packages come with ABSOLUTELY NO WARRANTY; for details type Disclaimer[]. This is free software, and you are welcome to redistribute it under certain conditions. See the General Public License for details.
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Package xAct`Invar` version 2.0.5, {2013,7,1}
CopyRight (C) 2006-2020, J. M. Martin-Garcia, D. Yllanes and R. Portugal, under the General Public License.
** DefConstantSymbol: Defining constant symbol sigma.
** DefConstantSymbol: Defining constant symbol dim.
** Option AllowUpperDerivatives of ContractMetric changed from False to True
** Variable $CovDFormat changed from Prefix to Postfix
** Option CurvatureRelations of DefCovD changed from True to False
** Variable $CommuteCovDsOnScalars changed from True to False
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Package xAct`xCoba` version 0.8.6, {2021,2,28}
CopyRight (C) 2005-2021, David Yllanes and Jose M. Martin-Garcia, under the General Public License.
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Package xAct`SymManipulator` version 0.9.5, {2021,9,14}
CopyRight (C) 2011-2021, Thomas Bäckdahl, under the General Public License.
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Package xAct`xTras` version 1.4.2, {2014,10,30}
CopyRight (C) 2012-2014, Teake Nutma, under the General Public License.
** Variable $CovDFormat changed from Postfix to Prefix
** Option CurvatureRelations of DefCovD changed from False to True
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These packages come with ABSOLUTELY NO WARRANTY; for details type Disclaimer[]. This is free software, and you are welcome to redistribute it under certain conditions. See the General Public License for details.
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In[]:=
DefManifold[M,4,{λ,μ,ν,ρ,σ,α,β,γ,δ}]DefMetric[-1,g[-μ,-ν],CD,PrintAs"g"]PrintAs[RiemannCD]^="R";PrintAs[RicciCD]^="R";PrintAs[RicciScalarCD]^="R";PrintAs[ChristoffelCD]^="Γ";PrintAs[EinsteinCD]^="G";$PrePrint=ScreenDollarIndices;
** DefManifold: Defining manifold M.
** DefVBundle: Defining vbundle TangentM.
** DefTensor: Defining symmetric metric tensor g[-μ,-ν].
** DefTensor: Defining antisymmetric tensor epsilong[-α,-β,-γ,-δ].
** DefTensor: Defining tetrametric Tetrag[-α,-β,-γ,-δ].
** DefTensor: Defining tetrametric Tetrag†[-α,-β,-γ,-δ].
** DefCovD: Defining covariant derivative CD[-μ].
** DefTensor: Defining vanishing torsion tensor TorsionCD[α,-β,-γ].
** DefTensor: Defining symmetric Christoffel tensor ChristoffelCD[α,-β,-γ].
** DefTensor: Defining Riemann tensor RiemannCD[-α,-β,-γ,-δ].
** DefTensor: Defining symmetric Ricci tensor RicciCD[-α,-β].
** DefCovD: Contractions of Riemann automatically replaced by Ricci.
** DefTensor: Defining Ricci scalar RicciScalarCD[].
** DefCovD: Contractions of Ricci automatically replaced by RicciScalar.
** DefTensor: Defining symmetric Einstein tensor EinsteinCD[-α,-β].
** DefTensor: Defining Weyl tensor WeylCD[-α,-β,-γ,-δ].
** DefTensor: Defining symmetric TFRicci tensor TFRicciCD[-α,-β].
** DefTensor: Defining Kretschmann scalar KretschmannCD[].
** DefCovD: Computing RiemannToWeylRules for dim 4
** DefCovD: Computing RicciToTFRicci for dim 4
** DefCovD: Computing RicciToEinsteinRules for dim 4
** DefTensor: Defining symmetrized Riemann tensor SymRiemannCD[-α,-β,-γ,-δ].
** DefTensor: Defining symmetric Schouten tensor SchoutenCD[-α,-β].
** DefTensor: Defining symmetric cosmological Schouten tensor SchoutenCCCD[LI[_],-α,-β].
** DefTensor: Defining symmetric cosmological Einstein tensor EinsteinCCCD[LI[_],-α,-β].
** DefTensor: Defining weight +2 density Detg[]. Determinant.
** DefParameter: Defining parameter PerturbationParameterg.
** DefTensor: Defining tensor Perturbationg[LI[order],-α,-β].
In[]:=
DefTensor[ϕ[],M];DefScalarFunction[V];DefConstantSymbol[n];DefScalarFunction[ω];
** DefTensor: Defining tensor ϕ[].
** DefScalarFunction: Defining scalar function V.
** DefConstantSymbol: Defining constant symbol n.
** DefScalarFunction: Defining scalar function ω.
** DefScalarFunction: Defining scalar function ω.
In[]:=
LEH=;Lq=ω[ϕ[]](CD[-μ][ϕ[]])(CD[μ][ϕ[]])-V[ϕ[]]L=LEH+Lq
RicciScalarCD[]
16π
-1
2
Out[]=
-V[ϕ]-ω[ϕ](ϕ)(ϕ)
1
2
▽
μ
μ
▽
Out[]=
R
16π
1
2
▽
μ
μ
▽
In[]:=
EE=(VarL[g[α,β],CD][L]//ContractMetric//RicciToEinstein//ToCanonical//Simplify)==0
Out[]=
G |
|
▽
α
▽
β
g |
|
▽
γ
γ
▽
16π
In[]:=
(*eqforscalarfield*)sfe=(VarL[g[α,β],CD][Lq]//ContractMetric//RicciToEinstein//ToCanonical//Simplify)==0
Out[]=
1
4
▽
α
▽
β
g |
|
▽
γ
γ
▽
In[]:=
<<xAct`xCoba`;
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Package xAct`xCoba` version 0.8.6, {2021,2,28}
CopyRight (C) 2005-2021, David Yllanes and Jose M. Martin-Garcia, under the General Public License.
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These packages come with ABSOLUTELY NO WARRANTY; for details type Disclaimer[]. This is free software, and you are welcome to redistribute it under certain conditions. See the General Public License for details.
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In[]:=
DefChart[qui,M,{0,1,2,3},{t[],r[],θ[],φ[]}];DefScalarFunction[f];
** DefChart: Defining chart qui.
** DefTensor: Defining coordinate scalar t[].
** DefTensor: Defining coordinate scalar r[].
** DefTensor: Defining coordinate scalar θ[].
** DefTensor: Defining coordinate scalar φ[].
** DefMapping: Defining mapping qui.