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This package is designed for the theory of perturbations in relativistic \
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It allows for the calculation of perturbations for all types of Bianchi \
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Version 0.4.3   (9/9/2015)\
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(* :Title: xPand *)

(* :Author: Cyril Pitrou, Xavier Roy and Obinna Umeh *)

(* :Summary: Computer algebra for cosmological perturbations around \
homogenous space-times *)

(* :Brief Discussion:
   - Background 3+1 splitting of space-time;
   - Spatial sections are homogeneous (Friedmann-Lemaitre or Bianchi of all \
types) *)
  
(* :Context: xAct`xPand` *)

(* :Package Version: 0.4.3 *)

(* :Copyright: Cyril Pitrou, Xavier Roy and Obinna Umeh (2012-2016) *)

(* :History: see xPand.History *)

(* :Keywords: *)

(* :Source: xPand.nb *)

(* :Warning: *)

(* :Mathematica Version: 7.0 and later *)

(* :Limitations: *)\
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Cell["1.4. Set up", "Subsubsection"],

Cell["\<\

Prefix format for derivatives:\
\>", "Text"],

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Cell["\<\

We do not modify the option MathLink of CanonicalPerm:\
\>", "Text"],

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Cell["1.5. Usage and error messages", "Subsubsection"],

Cell["\<\
Contents: 
\tVersions
\tBooleans
\tError messages
\tFunctions
\tLists of fields
\tPredefined fields
\tReserved words and protected names
\tTypes of gauge
\tTypes of manifold\
\>", "Text"],

Cell["\<\

Versions:\
\>", "Text"],

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     "\"\<$Version is a global variable giving the version of the package \
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     RowBox[{"$xTensorVersionExpected", "::", "usage"}], "=", 
     "\"\<$xTensorVersionExpected is a global variable giving the oldest \
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the package xPand in use.\>\""}], ";"}], "\[IndentingNewLine]", 
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     RowBox[{"$xPertVersionExpected", "::", "usage"}], "=", 
     "\"\<$xPertVersionExpected is a global variable giving the oldest \
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package xPand in use.\>\""}], ";"}]}]}]], "Input",
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Cell["\<\

Booleans:\
\>", "Text"],

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    RowBox[{
     RowBox[{"BackgroundFieldMethod", "::", "usage"}], " ", "=", " ", 
     "\"\<BackgroundFieldMethod is a boolean option, set by default to \
'False'. If set to 'True', then only the first order of the metric \
perturbation is kept, and it therefore stands for the total metric \
perturbation. \n\nThis option should be placed at the begining of the \
notebook, right after loading the xPand package.\>\""}], ";"}], 
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    RowBox[{
     RowBox[{"Recently", " ", 
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   RowBox[{
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     RowBox[{"$CacheRules", "::", "usage"}], " ", "=", " ", 
     "\"\<If set to 'True', the rules are remembered. This is potentially \
dangerous, so default is 'False'.\>\""}], ";"}], "\[IndentingNewLine]", 
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   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"$ConformalTime", "::", "usage"}], " ", "=", " ", 
     "\"\<If set to 'True' (default value), then the prime ' stands for the \
Lie derivative along the vector 'n' normal to the spatial slices. If the time \
coordinate is taken to be the proper time of the Eulerian observers \
(identified by the normal vector), then the Lie derivative along 'n' \
corresponds to the partial time derivative with respect to the conformal \
time, namely: \!\(\*SubscriptBox[\(\[ScriptCapitalL]\), \(n\)]\) .. = \
\[PartialD]/\[PartialD]\[Eta] .. . \n\nThe second Label-Index (LI) of \
projected tensors represents the number of Lie derivatives along 'n' that are \
applied to them. It also corresponds to the number of partial derivatives \
with respect to conformal time. \n\nIn the cases for which we prefer to use \
the cosmic time, we can choose to employ the second LI not for the number of \
\!\(\*SubscriptBox[\(\[ScriptCapitalL]\), \(n\)]\), but rather for the number \
of 1/a \!\(\*SubscriptBox[\(\[ScriptCapitalL]\), \(n\)]\), where 'a' is the \
scale factor. This is done by setting '$ConformalTime' to 'False'. The label \
index then indicates a derivative with respect to cosmic time.\n\nThe output \
format of the second LI is a prime for a conformal time derivative, and a dot \
for a cosmic time derivative.\>\""}], ";"}], "\[IndentingNewLine]", 
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     RowBox[{"$FirstOrderVectorPerturbations", "::", "usage"}], " ", "=", 
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   "\[IndentingNewLine]", 
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     RowBox[{"$FirstOrderTensorPerturbations", "::", "usage"}], " ", "=", 
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   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"$OpenConstantsOfStructure", "::", "usage"}], " ", "=", " ", 
     "\"\<Boolean value to decide if the constants of structure should be \
parameterized as in Ellis & MacCallum (1969). If 'False', then the constants \
of structure are kept general. If 'True', then the parameterization is: \
\!\(\*SubscriptBox[SuperscriptBox[\(C\), \(i\)], \(\(jk\)\(\\\ \)\)]\)= \
\!\(\*SubscriptBox[\(\[Epsilon]\), \(jkl\)]\)\!\(\*SuperscriptBox[\(N\), \(li\
\)]\) + \!\(\*SubscriptBox[\(A\), \([j\)]\)\!\(\*SubscriptBox[SuperscriptBox[\
\(\[Delta]\), \(i\)], \(\(k\)\(]\)\)]\).\>\""}], ";"}], "\[IndentingNewLine]",
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    RowBox[{
     RowBox[{"$SortCovDAutomatic", "::", "usage"}], " ", "=", " ", 
     "\"\<Boolean value to decide whether or not xTensor should commute \
automatically the induced derivatives. Default is 'True' in order to optimize \
the canonicalization of expression. Note that some adhoc commutation \
relations defined in '$CommutecdRules' are also applied in order to enforce \
the transverse properties of vector and tensor fields, on the one hand, and \
the appearance of Laplacians, on the other hand.\>\""}], ";"}]}]}]], "Input",
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Cell["\<\

Error messages:\
\>", "Text"],

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     "\"\<The dimension of the manifold is `1`; it is too small to perform a \
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   RowBox[{
    RowBox[{
     RowBox[{"SetSlicing", "::", "invalidnormalvector"}], " ", "=", " ", 
     "\"\<The normal vector `1` used to perform the background slicing should \
have only one index, belonging to the VBundle of the manifold. To avoid \
issues, it is possible to not predefine the normal vector and let the \
function SetSlicing define it automatically.\>\""}], ";"}], 
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     RowBox[{"SetSlicing", "::", "ninds"}], " ", "=", " ", 
     "\"\<The number of indices is insufficient to define the (N-1)+1 \
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   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"SetSlicing", "::", "noambientmetric"}], " ", "=", " ", 
     "\"\<The ambient metric `1` was not previously defined as a metric. \
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   RowBox[{
    RowBox[{
     RowBox[{"SetSlicing", "::", "signature"}], " ", "=", " ", 
     "\"\<The signature of the metric `1` should be -1 in order to perform a \
(N-1)+1 background splitting along a time-like vector. This comes from the \
fact that in the current version, the norm of the time-like vectors is fixed \
to -1.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
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    RowBox[{"*", " ", "DefProjectedTensor"}], " ", "**)"}], 
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   RowBox[{
    RowBox[{
     RowBox[{"DefProjectedTensor", "::", "invalidindices"}], " ", "=", " ", 
     "\"\<No label-index needs to be given for the definition of projected \
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   RowBox[{
    RowBox[{
     RowBox[{"DefProjectedTensor", "::", "notdownindices"}], " ", "=", " ", 
     "\"\<In order to be defined, the projected tensor `1` has to be written \
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   "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{"*", " ", "DefProjectedTensorProperties"}], " ", "**)"}], 
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   RowBox[{
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     RowBox[{"DefProjectedTensorProperties", "::", "symmetrictensors"}], "=", 
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   RowBox[{"(*", 
    RowBox[{"*", " ", "ExtractComponents"}], " ", "**)"}], 
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   RowBox[{
    RowBox[{
     RowBox[{"ExtractComponents", "::", "invalidprojector"}], " ", "=", " ", 
     "\"\<One of the projectors in `1` is not valid. The possible projectors \
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'*NameOfInducedMetric*'.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{"(*", 
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   RowBox[{
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   RowBox[{"(*", 
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   RowBox[{
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     RowBox[{"ToxPand", "::", "invalidconffactor"}], " ", "=", " ", 
     "\"\<A conformal factor should have two label indices, one for the \
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derivatives along u. And in the argument of SplitGaugeChange, you need only \
to put the name of the conformal factor (this is not a[LI[0],LI[0]] but just \
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     RowBox[{"xPand", "::", "makeboxes"}], " ", "=", " ", 
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Functions:\
\>", "Text"],

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    RowBox[{
     RowBox[{"SetSlicing", "::", "usage"}], " ", "=", " ", 
     "\"\<SetSlicing[g,n,nNorm,h,cd,{cdPost,cdPre},SpaceTimeType]. \n\nGiven \
a background manifold of dimension 'N' and an ambient metric 'g' on that \
manifold, this function performs a (N-1)+1 slicing. \n\nMore precisely, it \
builds the induced metric 'h' on the hypersurfaces, the associated covariant \
derivative 'cd', and the vector 'n' normal to these hypersurfaces. It also \
sets the type 'SpaceTimeType' for the hypersurfaces. The latter argument can \
take a value among {Anisotropic, BianchiB, BianchiA, BianchiI, FLCurved, \
FLFlat, Minkowski}. The norm 'nNorm' of the vector 'n' is set by default to \
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   RowBox[{
    RowBox[{
     RowBox[{"$CommutecdRules", "::", "usage"}], " ", "=", " ", 
     "\"\<Set of remembered rules for the commutation of the induced \
derivatives of the hypersurfaces. This is used to handle the successive \
operations of two, three or four induced derivatices.\>\""}], ";"}], 
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   RowBox[{
    RowBox[{
     RowBox[{"Conformal", "::", "usage"}], " ", "=", " ", 
     "\"\<Conformal[metricfinal][metric1,metric2][expr] performs a conformal \
transformation of 'expr' from 'metric1' to 'metric2' and expresses the result \
in terms of 'metricfinal' (along with its associated covariant derivative and \
curvature tensors). If needed, the function first reformulates 'expr' in \
terms of 'metric1'. The two metrics 'metric1' and 'metric2' have to be \
conformally related by DefConformalMetric. \n\n\
Conformal[metric1,metric2][expr] provides the final result with respect to \
the ambient metric, which is the first metric defined on the manifold.\>\""}],
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   RowBox[{
    RowBox[{
     RowBox[{"ConformalWeight", "::", "usage"}], " ", "=", " ", 
     "\"\<ConformalWeight[tensor[inds]] specifies the power of the conformal \
factor applied in the conformal transformation of the tensor 'tensor'. It \
corresponds to the number of down indices, minus the number of up indices, \
plus the quantity ConformalWeight[tensor] (without the indices). The latter \
is set by default to zero. \n\nFor instance, in the setting \
ConformalWeight[tensor] = 0, we have: \
ConformalWeight[tensor[\[Alpha],\[Beta]]] = -2, and ConformalWeight[tensor[-\
\[Alpha],-\[Beta]]] = 2.\>\""}], ";"}], "\[IndentingNewLine]", 
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   RowBox[{
    RowBox[{
     RowBox[{"DefConformalMetric", "::", "usage"}], " ", "=", " ", 
     "\"\<DefConformalMetric[g,S] defines a metric conformally related to \
'g', with the conformal factor 'S'. The new metric is named 'gS2', and its \
inverse is given by 'Inv[gS2]'. If other metrics are conformally related to \
'g', then 'gS2' is related to them by transitivity of the conformal \
transformation. \n\nWhen calling the function SetSlicing, the conformal \
factor is chosen to be the scale factor 'a[h]' (or, equivalently, 'ah'). The \
constructed metric 'gah2' then corresponds to the 'real' background metric \
(i.e. the one that would be obtained from the physical metric of the \
manifold).\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"DefMetricFields", "::", "usage"}], " ", "=", " ", 
     "\"\<DefMetricFields[g,gpert,h,parameter] defines the perturbed metric \
'gpert' from its background value 'g'. The argument 'h' is the induced metric \
on the spatial hypersurfaces, and the optional argument 'parameter' (set by \
default to \[Epsilon]) is the symbol used in the perturbative expansions. \
Several tensors that are used to decompose 'gpert' are defined when calling \
this function.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"DefMatterFields", "::", "usage"}], " ", "=", " ", 
     "\"\<DefMatterFields[uf,ufpert,h,parameter] defines the perturbed vector \
field 'ufpert' from its background value 'uf'. The argument 'h' is the \
induced metric on the spatial hypersurfaces, and the optional argument \
'parameter' (set by default to \[Epsilon]) is the symbol used in the \
perturbative expansions. Several tensors that are used to decompose 'ufpert' \
are defined when calling this function.\>\""}], ";"}], "\[IndentingNewLine]", 
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   RowBox[{
    RowBox[{
     RowBox[{"DefProjectedTensor", "::", "usage"}], " ", "=", " ", 
     "\"\<DefProjectedTensor[Name[inds],h,options] defines a projected tensor \
on the spatial hypersurfaces. \n\n'Name' refers to the name of the tensor, \
and 'inds' (which has to be with down indices) defines its rank. The argument \
'h' is the induced metric on the spatial hypersurfaces. \n\nThere exist 3 \
types of options. \n\nThe first type is: 'TensorProperties', by default set \
to 'Transverse', 'Traceless' and 'SymmetricTensor'. These can be overwritten \
with 'TensorProperties->ListOfProperties', where 'ListOfProperties' is a list \
whose elements can be: 'Transverse', 'Traceless' and 'SymmetricTensor'. \n\n\
The second option is: 'SpaceTimesOfDefinition', by default set to \
'Background' and 'Perturbed'. These can be also overwritten with \
'SpaceTimesOfDefinition->SpaceTimesList', where 'SpaceTimesList' is a list \
whose elements can be: 'Background' and 'Perturbed'. \n\nThe last option is: \
'PrintAs', by default set to 'Identity'. This can be overwritten with \
'PrintAs->String', in order to specify the output form of the tensor.\>\""}], 
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   RowBox[{
    RowBox[{
     RowBox[{"ExtractComponents", "::", "usage"}], " ", "=", " ", 
     "\"\<ExtractComponents[expr,h,ListOfProjectors,ListOfFreeIndices] \
projects 'expr' along its time component(s) (i.e. along the vector normal to \
the hypersurfaces) and along its space component(s) (i.e. onto the \
hypersurfaces), being defined by the background slicing associated with the \
induced metric 'h'. \n\n'ListOfProjectors' is a list whose elements are \
among: 'Time' (or, equivalently, '*NameOfNormalVector*') and 'Space' (or, \
equivalently, '*NameOfInducedMetric*'). The length of the list has to equal \
the rank of the tensors of 'expr'. The free indices (in canonical order) of \
'expr' are then projected according to the 'ListOfProjectors'. \n\n\
'ListOfFreeIndices' is the list of free indices on which the ListOfProjectors \
is to be applied. If this is omitted, then it is the list of free indices in \
alphabetic order. \n\nWhen the argument 'ListOfProjectors' is omitted, all \
the projections for rank-0 to rank-2 tensors are displayed in a table.\>\""}],
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   RowBox[{"(*", 
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   RowBox[{
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     "\"\<ExtractOrder[expr,order] displays the expression at order 'order' \
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   RowBox[{
    RowBox[{
     RowBox[{"IndicesUp", "::", "usage"}], " ", "=", " ", 
     "\"\<IndicesUp[expr] raises the indices of all the tensors of 'expr' \
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   RowBox[{
    RowBox[{
     RowBox[{"IndicesDown", "::", "usage"}], " ", "=", " ", 
     "\"\<IndicesDown[expr] lowers the indices of all the tensors of 'expr' \
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   RowBox[{"(*", 
    RowBox[{
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   RowBox[{
    RowBox[{
     RowBox[{"SplitMatter", "::", "usage"}], " ", "=", " ", 
     "\"\<SplitMatter[uf,ufpert,normuf,h,gauge,order,tiltspecification] \
builds the rules to split the perturbed four-velocity 'ufpert' (with index \
up) in a given 'gauge', and according to the background slicing associated \
with the induced metric 'h'. The rules are computed to the order 'order'. \n\n\
'normuf' is the squared norm of the four-velocity 'uf' (usually taken as -1), \
and the optional argument 'tiltspecification' (set by default to 'NotTilted') \
allows for a tilt between the fluid four-velocity and the vector normal to \
the hypersurfaces. \n\nThe perturbations of all the fields that depend on \
'uf' are also computed.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
     RowBox[{
      RowBox[{"**", "**", "**", "**"}], "*", " ", "The", " ", "following", 
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     RowBox[{
      RowBox[{"will", " ", "have", " ", "to", " ", "be", " ", "re"}], "-", 
      RowBox[{"edited", "."}]}]}], " ", "**********)"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"SplitMetric", "::", "usage"}], " ", "=", " ", 
     "\"\<SplitMetric[g,gpert,h,gauge] builds the rules to split the \
perturbed metric 'gpert' in a given gauge and according to the background \
slicing associated with the induced metric 'h'.\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
     RowBox[{"**", "**", "**", "**", " ", "This"}], " ", "command", " ", 
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     RowBox[{".2", ".", " ", "More"}], " ", "documentation", " ", "is", " ", 
     "needed", " ", "and", " ", "syntax", " ", "can", " ", "still", " ", 
     "evolve"}], " ", "********)"}], "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"SplitNormalVector", "::", "usage"}], " ", "=", " ", 
     "\"\<SplitNormalVector[h,order], where h is an induced metric, and order \
is the order up to which the perturbations of the normal vector should be \
computed in function of the perturbed metric. This outputs a list of rules, \
one for each order of perturbation of the normal vector. All rules are \
expressed in function of the perturbed metric\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"$MyRules", "::", "usage"}], " ", "=", " ", 
     "\"\<List of global rules that are remembered by xPand.\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
     RowBox[{
      RowBox[{"ConstantsDecompositionRules", "::", "usage"}], " ", "=", " ", 
      "\"\<ConstantsDecompositionRules[h] provides the rules to transform the \
constants of structure according to the Bianchi type of the homogeneous \
hypersurfaces associated with the induced metric 'h'. This option is only \
valid for three-dimensional hypersurfaces. \n\nThe parametrization follows \
Ellis & MacCallum (1969), that is, the constants of structure are decomposed \
as: \!\(\*SubscriptBox[SuperscriptBox[\(C\), \(i\)], \(\(jk\)\(\\\ \)\)]\)= \
\!\(\*SubscriptBox[\(\[Epsilon]\), \(jkl\)]\)\!\(\*SuperscriptBox[\(N\), \(li\
\)]\) + \!\(\*SubscriptBox[\(A\), \([j\)]\)\!\(\*SubscriptBox[SuperscriptBox[\
\(\[Delta]\), \(i\)], \(\(k\)\(]\)\)]\).\>\""}], ";"}], "*)"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"ToBianchiType", "::", "usage"}], " ", "=", " ", 
     "\"\<ToBianchiType[h][expr] applies to expr the necessary rules to \
express the constants of structure according to the Bianchi type of the \
homogeneous hypersurfaces associated with the induced metric 'h'. This option \
is only valid for three-dimensional hypersurfaces. \n\nThe parametrization \
follows Ellis & MacCallum (1969), that is, the constants of structure are \
decomposed as: \!\(\*SubscriptBox[SuperscriptBox[\(C\), \(i\)], \(\(jk\)\(\\\ \
\)\)]\)= \!\(\*SubscriptBox[\(\[Epsilon]\), \
\(jkl\)]\)\!\(\*SuperscriptBox[\(N\), \(li\)]\) + \!\(\*SubscriptBox[\(A\), \
\([j\)]\)\!\(\*SubscriptBox[SuperscriptBox[\(\[Delta]\), \(i\)], \(\(k\)\(]\)\
\)]\).\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
     RowBox[{
      RowBox[{"ConstantsOfStructureRules", "::", "usage"}], " ", "=", " ", 
      "\"\<ConstantsOfStructureRules[h]. \n\nGiven an induced metric h \
corresponding to a non trivial Bianchi type, ConstantsOfStructureRules[h] \
gives all the rules to express the Riemann, the Ricci and the Ricci scalar in \
function of the constants of structure of the spatial sections.\>\""}], ";"}],
     "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"ToConstantsOfStructure", "::", "usage"}], " ", "=", " ", 
     "\"\<ToConstantsOfStructure[h][expr]. \n\nGiven an induced metric h \
corresponding to a non trivial Bianchi type, ToConstantsOfStructure[h][expr] \
applies to expr all the rules to express the Riemann, the Ricci and the Ricci \
scalar in function of the constants of structure of the spatial \
sections.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{"(*", 
     RowBox[{
      RowBox[{
       RowBox[{"**", "**", "**", "**"}], "*", " ", "The", " ", "following", 
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      RowBox[{"is", " ", "not", " ", "yet", " ", 
       RowBox[{"defined", "."}]}]}], " ", "**********)"}], 
    "\[IndentingNewLine]", "\[IndentingNewLine]", 
    RowBox[{
     RowBox[{
      RowBox[{"RulesCovDsOfTensor", "::", "usage"}], " ", "=", " ", 
      "\"\<\>\""}], ";"}], "*)"}], "\[IndentingNewLine]", 
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   RowBox[{"(*", " ", 
    RowBox[{"**", "**", "**", "**", "**", "**", "**"}], " ", "*)"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"ToCosmicTime", "::", "usage"}], " ", "=", " ", 
     "\"\<Transforms an expression where the second label index of projected \
tensor means \!\(\*SubscriptBox[\(\[ScriptCapitalL]\), SuperscriptBox[\(n\), \
\(\[Alpha]\)]]\), that is a derivative with respect to conformal time, into \
an expression where the second index means '1/a *\!\(\*SubscriptBox[\(\
\[ScriptCapitalL]\), SuperscriptBox[\(n\), \(\[Alpha]\)]]\)', that is a \
derivative with respect to cosmic time. The global variable $ConformalTime is \
set to False in the process.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"ToConformalTime", "::", "usage"}], " ", "=", " ", 
     "\"\<Transforms an expression where the second label index of projected \
tensor means '1/a * \!\(\*SubscriptBox[\(\[ScriptCapitalL]\), SuperscriptBox[\
\(n\), \(\[Alpha]\)]]\)', that is a derivative with respect to cosmic time, \
into an expression where the second index means '\!\(\*SubscriptBox[\(\
\[ScriptCapitalL]\), SuperscriptBox[\(n\), \(\[Alpha]\)]]\)', that is a \
derivative with respect to conformal time. The global variable $ConformalTime \
is set to True in the process.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
     RowBox[{
      RowBox[{"**", "**", "**", "**"}], "*", " ", "The", " ", "following", 
      " ", "command"}], ",", " ", "RulesVelocitySpatial", ",", " ", 
     RowBox[{"is", " ", "not", " ", "yet", " ", 
      RowBox[{"edited", "."}]}]}], " ", "**********)"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"RulesVelocitySpatial", "::", "usage"}], " ", "=", " ", 
     "\"\<RulesVelocitySpatial[h,uf,duf,NormVectorSquare,gauge,order,\
tiltedvalue] gives the standard rules for the spatial velocity in a given \
gauge. h is the induced metric, uf and duf are the fluid velocity and its \
perturbation. NormVectorSquare is the norm squared of velocity, and gauge is \
the gauge choice. Order is the order up to which the perturbations are \
computed. The last argument, which can be '\\\"Tilted\\\"' or '\\\"NotTiled\\\
\"' is by default set to \\\"NotTilted\\\" and is optional.\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"SplitGaugeChange", "::", "usage"}], " ", "=", " ", 
     "\"\<SplitGaugeChange[expr,ListPairs,\[Xi],h,order] performs a gauge \
transformation of order 'order' on 'expr', with a conformal transformation of \
conformal factor 'a[h]', and splits the result according to the background \
slicing associated with the induced metric 'h'. \n\n'\[Xi]' is the general \
vector field used for the gauge transformation. It must have a Label-Index \
(LI) for the order of perturbation (i.e. it has to be of the form \
\[Xi][LI[order],index]). 'ListPairs' is the list of rules used for the \
background slicing. \n\nThe theory and formulas for the power expansion of \
gauge transformation was developed by Bruni et al. in: Class. Quantum Grav. \
14, 2585 (1997).\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   
   RowBox[{"(*", 
    RowBox[{
     RowBox[{
      RowBox[{"**", "**", "**", "**"}], "*", " ", "The", " ", "following", 
      " ", "command"}], ",", " ", "SplitFieldsAndGaugeChange", ",", " ", 
     RowBox[{"is", " ", "not", " ", "yet", " ", 
      RowBox[{"edited", "."}]}]}], " ", "**********)"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"SplitFieldsAndGaugeChange", "::", "usage"}], " ", "=", " ", 
     "\"\<SplitFieldsAndGaugeChange[expr,g,dg,uf,duf,h,order,\
tiltspecification] performs a gauge transformation on expr where expr must \
involve metric perturbations (or curvature tensor associated with this \
metric) or matter field perturbation such as the fluid velocity, and splits \
the results according to the most general gauge choice. \\nh is the induced \
metric, and refers to the background n+1 splitting. g is the metric and dg is \
the metric perturbation. uf is the fluid velocity and duf is the fluid \
velocity perturbation (when index is up). \\n\[Xi] is the general vector \
field used for the gauge transformation which must have a label index for the \
order of perturbation (it is of the form \[Xi][LI[order],Index]), and order \
is the order of the gauge transformation. 'tiltspecification' is optional and \
by default is \\\"NotTilted\\\". \\nThe theory and formulas for the power \
expansion of changes of gauge was developed by Bruni et al. in reference \
Class. Quantum Grav. 14, 2585 (1997).\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
     RowBox[{
      RowBox[{"SplitPerturbationsOld", "::", "usage"}], " ", "=", " ", 
      "\"\<This is the old version of SplitPerturbations. It is obsolete, and \
it will have to be removed.\>\""}], ";"}], "*)"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"SplitPerturbations", "::", "usage"}], "=", 
     "\"\<SplitPerturbations[expr,ListPairs,h] splits the expression 'expr', \
using the list of rules 'ListPairs', according to the background slicing \
associated with the induced metric 'h'.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
     RowBox[{
      RowBox[{"**", "**", "**", "**"}], "*", " ", "The", " ", "following", 
      " ", "command"}], ",", " ", "ToMetric", ",", " ", 
     RowBox[{
      RowBox[{"will", " ", "have", " ", "to", " ", "be", " ", "re"}], "-", 
      RowBox[{"edited", " ", "after", " ", "its", " ", 
       RowBox[{"modification", ".", " ", 
        RowBox[{"**", "**", "**", "**"}]}], "*"}]}]}], "*)"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"ToMetric", "::", "usage"}], " ", "=", " ", 
     "\"\<ToMetric[expr,metric] formulates 'expr' in terms of 'metric', and \
its associated covariant derivative and curvature tensors.\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"ToxPand", "::", "usage"}], " ", "=", " ", 
     "\"\<ToxPand[expr,gpert,uf,ufpert,h,gauge,order,tiltspecification]. \n\n\
This function first performs on 'expr' a conformal transformation with the \
scale factor 'a[h]' (except for a Minkowski 'SpaceType'); then it perturbs \
the result up to the order 'order' by means of xPert tools; and finally, it \
splits the result using the rules associated with the 'gauge', and according \
to the background slicing associated with the induced metric 'h'. \
'tiltspecification' is optional and by default is \\\"NotTilted\\\" \n\nAll \
the perturbed fields are derived from the perturbed metric 'gpert' and the \
perturbed fluid four-velocity 'ufpert'.\n\nThe syntax for multifluid is \
xPand[expr,gpert,{{uf1,uf1pert},{uf2,uf2pert},...},h,gauge,order,\
tiltspecification]. \n\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"ToxPandFromRules", "::", "usage"}], " ", "=", " ", 
     "\"\<ToxPandFromRules[expr,ListPairs,h,order]. \n\nThis function first \
performs on 'expr' a conformal transformation with the scale factor 'a[h]' \
(except for a Minkowski 'SpaceType'); then it perturbs the result up to the \
order 'order' by means of xPert tools; and finally, it splits the result \
using the list of rules 'ListPairs' and according to the background slicing \
associated with the induced metric 'h'. \n\nCompared to the function ToxPand, \
the present function requires to give the rules for the splitting of the \
perturbations.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"VisualizeTensor", "::", "usage"}], " ", "=", " ", 
     "\"\<VisualizeTensor[expr,h] displays in a table the projections of \
'expr' along its time components (i.e. along the vector normal to the \
hypersurfaces) and along its space components (i.e. onto the hypersurfaces), \
being defined by the background slicing associated with the induced metric \
'h'. \n\n'expr' has to be an expression involving tensors of rank 2 \
only.\>\""}], ";"}], "\[IndentingNewLine]", 
   RowBox[{"(*", " ", 
    RowBox[{
     RowBox[{
     "VisualizeTensor", " ", "will", " ", "be", " ", "put", " ", "as", " ", 
      "private"}], ",", " ", 
     RowBox[{
     "since", " ", "the", " ", "function", " ", "ExtractComponents", " ", 
      "already", " ", "handle", " ", 
      RowBox[{"this", "."}]}]}], " ", "*)"}]}]}]], "Input",
 InitializationCell->True],

Cell["\<\

Lists of fields:\
\>", "Text"],

Cell[BoxData[
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  RowBox[{"(*", 
   RowBox[{
    RowBox[{"**", " ", "LISTS"}], " ", "OF", " ", "FIELDS"}], " ", "***)"}], 
  "\[IndentingNewLine]", "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"$ListFieldsBackgroundOnly", "::", "usage"}], "=", 
     "\"\<$ListFieldsBackgroundOnly[h], with 'h' being the induced metric on \
the hypersurfaces, is the list of fields that live on the background only. It \
is automatically defined when calling the function SetSlicing.\>\""}], ";"}], 
   "\[IndentingNewLine]", "\n", 
   RowBox[{
    RowBox[{
     RowBox[{"$ListFieldsPerturbedOnly", "::", "usage"}], "=", 
     "\"\<$ListFieldsPerturbedOnly[h], with 'h' being the induced metric on \
the hypersurfaces, is the list of fields that live on the perturbed manifold \
only. It contains the metric perturbations. It is automatically defined when \
calling the function DefMetricFields.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"$ListFieldsBackgroundAndPerturbed", "::", "usage"}], "=", 
     "\"\<$ListFieldsBackgroundAndPerturbed[h,uf], with 'h' and 'uf' being \
respectively the induced metric on the hypersurfaces and the fluid \
four-velocity, is the list of fields that live on the background manifold and \
that can be perturbed. It is automatically defined when calling the function \
DefMatterFields.\>\""}], ";"}]}]}]], "Input",
 InitializationCell->True],

Cell["\<\

Predefined fields:

The energy density and the pressure of the fluid only depend on the fluid \
four-velocity. 
The spatial and time components of the fluid four-velocity depend on (i) the \
fluid four-velocity, and (ii) the induced metric (as it defines the spatial \
projection and is uniquely related to the time projection). 
All the other fields depend (solely) on the choice of the background slicing; \
hence they depend on the associated induced metric. \
\>", "Text"],

Cell[BoxData[
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  RowBox[{"(*", 
   RowBox[{
    RowBox[{"**", " ", "PREDEFINED"}], " ", 
    RowBox[{"FIELDS", ":", " ", 
     RowBox[{"Geometrical", " ", "quantities"}]}]}], " ", "***)"}], 
  "\[IndentingNewLine]", "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"a", "::", "usage"}], " ", "=", " ", 
     "\"\<a[h] is the scale factor of the spatial hypersurfaces whose induced \
metric is given by 'h'. It is automatically defined when calling the function \
SetSlicing. \n\nThe command a[h] also builds the symbol 'ah', so one can use \
a[h] or, equivalently, ah. \n\nThe default printed form of a[h][] is: \
'a'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"H", "::", "usage"}], " ", "=", " ", 
     "\"\<H[h] is the Hubble factor of the spatial hypersurfaces whose \
induced metric is given by 'h'. It is automatically defined when calling the \
function SetSlicing. \n\nThe command H[h] also builds the symbol 'Hh', so one \
can use H[h] or, equivalently, Hh. \n\nThe default printed form of H[h][] is: \
'\[ScriptCapitalH]'.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Connection", "::", "usage"}], " ", "=", " ", 
     "\"\<Connection[h][index,index,index] are the coefficients of the \
connection associated with the induced metric 'h'. They are automatically \
defined when calling the function SetSlicing. \n\nThe command Connection[h] \
builds the symbol 'Connectionh', so one can use \
Connection[h][index,index,index] or, equivalently, \
Connectionh[index,index,index].\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"CS", "::", "usage"}], " ", "=", " ", 
     "\"\<CS[h][index,index,index] are the constants of structure on the \
hypersurfaces associated with the induced metric 'h'. They are automatically \
defined when calling the function SetSlicing. \n\nThe command CS[h] builds \
the symbol 'CSh', so one can use CS[h][index,index,index] or, equivalently, \
CSh[index,index,index]. \n\nThe default printed form of \
CS[h][-index,-index,-index] is: '\!\(\*SubscriptBox[\(C\), \(index\\\ index\\\
\ index\)]\)'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"nt", "::", "usage"}], " ", "=", " ", 
     "\"\<nt[h][index,index] is the symmetric tensor part involved in the \
decomposition of the constants of structure: \
\!\(\*SubscriptBox[SuperscriptBox[\(C\), \(i\)], \(\(jk\)\(\\\ \)\)]\)= \
\!\(\*SubscriptBox[\(\[Epsilon]\), \(jkl\)]\)\!\(\*SuperscriptBox[\(N\), \(li\
\)]\) + \!\(\*SubscriptBox[\(A\), \([j\)]\)\!\(\*SubscriptBox[SuperscriptBox[\
\(\[Delta]\), \(i\)], \(\(k\)\(]\)\)]\), in Ellis and MacCallum (1969). It is \
automatically defined when calling the function SetSlicing for Bianchi \
space-times of type other then I. The presence as an argument of the induced \
metric 'h' is reminiscent of the choice of the background slicing. \n\nThe \
command nt[h] builds the symbol 'nth', so one can use nt[h][index,index] or, \
equivalently, nth[index,index]. \n\nThe default printed form of \
nt[h][-index,-index] is: '\!\(\*SubscriptBox[\(N\), \(index\\\ index\)]\)'.\>\
\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"av", "::", "usage"}], " ", "=", " ", 
     "\"\<av[h][index] is the 'vector' part involved in the decomposition of \
the constants of structure: \!\(\*SubscriptBox[SuperscriptBox[\(C\), \(i\)], \
\(\(jk\)\(\\\ \)\)]\)= \!\(\*SubscriptBox[\(\[Epsilon]\), \
\(jkl\)]\)\!\(\*SuperscriptBox[\(N\), \(li\)]\) + \!\(\*SubscriptBox[\(A\), \
\([j\)]\)\!\(\*SubscriptBox[SuperscriptBox[\(\[Delta]\), \(i\)], \(\(k\)\(]\)\
\)]\), in Ellis & MacCallum (1969). It is automatically defined when calling \
the function SetSlicing for Bianchi space-times of type other than I. The \
presence as an argument of the induced metric 'h' is reminiscent of the \
choice of the background slicing.\n\nThe command av[h] builds the symbol \
'avh', so one can use av[h][index] or, equivalently, avh[index]. \n\nThe \
default printed form of av[h][-index] is: '\!\(\*SubscriptBox[\(A\), \
\(index\)]\)'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"\[ScriptK]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[ScriptK][h] is the (constant) curvature parameter of the \
isotropic and homogeneous hypersurfaces associated with the induced metric \
'h'. It is automatically defined when calling the function SetSlicing for \
Friedmann-Lemaitre cosmologies. \n\nThe command \[ScriptK][h] also builds the \
symbol '\[ScriptK]h', so one can use \[ScriptK][h] or, equivalently, \
\[ScriptK]h. \n\nThe default printed form of \[ScriptK][h][] is: \
\[ScriptK].\>\""}], ";"}]}]}]], "Input",
 InitializationCell->True],

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   RowBox[{
    RowBox[{"**", " ", "PREDEFINED"}], " ", 
    RowBox[{"FIELDS", ":", " ", 
     RowBox[{
     "Scalar", " ", "field", " ", "and", " ", "fluid", " ", 
      "quantities"}]}]}], " ", "***)"}], "\[IndentingNewLine]", 
  "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"\[CurlyPhi]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[CurlyPhi][LI[p],LI[q]] is a general scalar field, automatically \
defined when calling the function DefMatterFields. \n\nThe first Label-Index \
(LI) gives the order 'p' of '\[CurlyPhi]' (hence, for p = 0, '\[CurlyPhi]' \
corresponds to the background value of the scalar field). The second LI \
represents the number 'q' of Lie derivatives along the vector normal to the \
hypersurfaces acting on the 'p'-order of '\[CurlyPhi]'. \n\nThe commands \
\[CurlyPhi][] and \[CurlyPhi][LI[p]] are defined to be equal to \
\[CurlyPhi][LI[0],LI[0]] and \[CurlyPhi][LI[p],LI[0]], respectively.\>\""}], 
    ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"\[Rho]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[Rho][uf][LI[p],LI[q]] is the energy density of the fluid, \
automatically defined when calling the function DefMatterFields. It depends \
on the fluid four-velocity 'uf'. \n\nThe first Label-Index (LI) gives the \
order 'p' of '\[Rho]' (hence, for p = 0, '\[Rho]' corresponds to the \
background value of the energy density). The second LI represents the number \
'q' of Lie derivatives along the vector normal to the hypersurfaces acting on \
the 'p'-order of '\[Rho]'. \n\nThe commands \[Rho][uf][] and \
\[Rho][uf][LI[p]] are defined to be equal to \[Rho][uf][LI[0],LI[0]] and \
\[Rho][uf][LI[p],LI[0]], respectively. \n\nThe command \[Rho][uf] builds the \
symbol '\[Rho]uf', so one can use \[Rho][uf][LI[p],LI[q]] or, equivalently, \
\[Rho]uf[LI[p],LI[q]].\>\""}], ";"}], "\[IndentingNewLine]", "\t", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"P", "::", "usage"}], " ", "=", " ", 
     "\"\<P[uf][LI[p],LI[q]] is the pressure of the fluid, automatically \
defined when calling the function DefMatterFields. It depends on the fluid \
four-velocity 'uf'. \n\nThe first Label-Index (LI) gives the order 'p' of 'P' \
(hence, for p = 0, 'P' corresponds to the background value of the pressure). \
The second LI represents the number 'q' of Lie derivatives along the vector \
normal to the hypersurfaces acting on the 'p'-order of 'P'. \n\nThe commands \
P[uf][] and P[uf][LI[p]] are defined to be equal to P[uf][LI[0],LI[0]] and \
P[uf][LI[p],LI[0]], respectively. \n\nThe command P[uf] builds the symbol \
'Puf', so one can use P[uf][LI[p],LI[q]] or, equivalently, \
Puf[LI[p],LI[q]].\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{"(*", " ", 
    RowBox[{
     RowBox[{
      RowBox[{"\[CapitalPi]", "::", "usage"}], " ", "=", " ", 
      "\"\<\[CapitalPi][uf][LI[p],LI[q],index,index] is the anisotropic \
stress tensor of the fluid, automatically defined when calling the function \
DefMatterFields. It depends on the fluid four-velocity 'uf'. \n\nThe first \
Label-Index (LI) gives the order 'p' of '\[CapitalPi]' (hence, for p = 0, '\
\[CapitalPi]' corresponds to the (vanishing) background value of the \
anisotropic stress). The second LI represents the number 'q' of Lie \
derivatives along the vector normal to the hypersurfaces acting on the \
'p'-order of '\[CapitalPi]'. \n\nThe commands \[CapitalPi][uf][index,index] \
and \[CapitalPi][uf][LI[p],index,index] are defined to be equal to \
\[CapitalPi][uf][LI[0],LI[0],index,index] and \
\[CapitalPi][uf][LI[p],LI[0],index,index], respectively. \n\nThe command \
\[CapitalPi][uf] builds the symbol '\[CapitalPi]uf', so one can use \
\[CapitalPi][uf][LI[p],LI[q],index,index] or, equivalently, \
\[CapitalPi]uf[LI[p],LI[q],index,index].\>\""}], ";"}], " ", 
    "*)"}]}]}]], "Input",
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  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"V0", "::", "usage"}], " ", "=", " ", 
     "\"\<V0[h,uf][LI[p],LI[q]] is the time part of the perturbed fluid \
four-velocity 'ufpert' (with index up): ufpert = V0 \[Times] \
'*NameOfNormalVector*' + Vspat. It is automatically defined when calling the \
function DefMatterFields. It depends on the choice of the background slicing, \
and hence on the induced metric 'h'. \n\nThe first Label-Index (LI) gives the \
order 'p' of 'V0' (hence, if the fluid is not tilted in the background, 'V0' \
equals 1 for p = 0). The second LI represents the number 'q' of Lie \
derivatives along the vector normal to the hypersurfaces acting on the \
'p'-order of 'V0'. \n\nThe commands V0[h,uf][] and V0[h,uf][LI[p]] are \
defined to be equal to V0[h,uf][LI[0],LI[0]] and V0[h,uf][LI[p],LI[0]], \
respectively. \n\nThe command V0[h,uf] builds the symbol 'V0huf', so one can \
use V0[h,uf][LI[p],LI[q]] or, equivalently, V0huf[LI[p],LI[q]]. \n\nThe \
default printed form of V0[h,uf][] is: 'V0'.\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Vspat", "::", "usage"}], " ", "=", " ", 
     "\"\<Vspat[h,uf][LI[p],LI[q],index] is the spatial part of the perturbed \
fluid four-velocity 'ufpert' (with index up): ufpert = V0 \[Times] \
'*NameOfNormalVector*' + Vspat. It is automatically defined when calling the \
function DefMatterFields. It depends on the choice of the background slicing, \
and hence on the induced metric 'h'. \n\nThe first Label-Index (LI) gives the \
order 'p' of 'Vspat' (hence, if the fluid is not tilted in the background, \
'Vspat' is null for p = 0). The second LI represents the number 'q' of Lie \
derivatives along the vector normal to the hypersurfaces acting on the \
'p'-order of 'Vspat'. \n\nThe commands Vspat[h,uf][index] and \
Vspat[h,uf][LI[p],index] are defined to be equal to \
Vspat[h,uf][LI[0],LI[0],index] and Vspat[h,uf][LI[p],LI[0],index], \
respectively. \n\nThe command Vspat[h,uf] builds the symbol 'Vspathuf', so \
one can use Vspat[h,uf][LI[p],LI[q],index] or, equivalently, \
Vspathuf[LI[p],LI[q],index]. \n\nThe default printed form of \
Vspat[h,uf][-index] is: '\!\(\*SubscriptBox[\(\[ScriptCapitalV]\), \(index\)]\
\)'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Vs", "::", "usage"}], " ", "=", " ", 
     "\"\<Vs[h,uf][LI[p],LI[q]] is the scalar part of the scalar-vector \
decomposition of 'Vspat', where 'Vspat' is the spatial part of the perturbed \
fluid four-velocity 'ufpert' (with index up). It is automatically defined \
when calling the function DefMatterFields. It depends on the choice of the \
background slicing, and hence on the induced metric 'h'. \n\nThe first \
Label-Index (LI) gives the order 'p' of 'Vs' (hence, if the fluid is not \
tilted in the background, 'Vs' is null for p = 0). The second LI represents \
the number 'q' of Lie derivatives along the vector normal to the \
hypersurfaces acting on the 'p'-order of 'Vs'. \n\nThe commands Vs[h,uf][] \
and Vs[h,uf][LI[p]] are defined to be equal to Vs[h,uf][LI[0],LI[0]] and \
Vs[h,uf][LI[p],LI[0]], respectively. \n\nThe command Vs[h,uf] builds the \
symbol 'Vshuf', so one can use Vs[h,uf][LI[p],LI[q]] or, equivalently, \
Vshuf[LI[p],LI[q]]. \n\nThe default printed form of Vs[h,uf][] is: \
'V'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Vv", "::", "usage"}], " ", "=", " ", 
     "\"\<Vv[h,uf][LI[p],LI[q],index] is the vector part of the scalar-vector \
decomposition of 'Vspat', where 'Vspat' is the spatial part of the perturbed \
fluid four-velocity 'ufpert' (with index up). It is automatically defined \
when calling the function DefMatterFields. It depends on the choice of the \
background slicing, and hence on the induced metric 'h'. \n\nThe first \
Label-Index (LI) gives the order 'p' of 'Vv' (hence, if the fluid is not \
tilted in the background, 'Vv' is null for p = 0). The second LI represents \
the number 'q' of Lie derivatives along the vector normal to the \
hypersurfaces acting on the 'p'-order of 'Vv'. \n\nThe commands \
Vv[h,uf][index] and Vv[h,uf][LI[p],index] are defined to be equal to \
Vv[h,uf][LI[0],LI[0],index] and Vv[h,uf][LI[p],LI[0],index], respectively. \n\
\nThe command Vv[h,uf] builds the symbol 'Vvhuf', so one can use \
Vv[h,uf][LI[p],LI[q],index] or, equivalently, Vvhuf[LI[p],LI[q],index]. \n\n\
The default printed form of Vv[h,uf][-index] is: '\!\(\*SubscriptBox[\(V\), \
\(index\)]\)'.\>\""}], ";"}]}]}]], "Input",
 InitializationCell->True],

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    RowBox[{"**", " ", "PREDEFINED"}], " ", 
    RowBox[{"FIELDS", ":", " ", 
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  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"\[Phi]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[Phi][h][LI[p],LI[q]] is the first Bardeen potential, \
corresponding to the 00 component of the metric perturbation. It is \
automatically defined when calling the function DefMetricFields. It depends \
on the choice of the background slicing, and hence on the induced metric 'h'. \
\n\nThe first Label-Index (LI) gives the order 'p' of the perturbation \
'\[Phi]' (hence '\[Phi]' is null for p = 0). The second LI represents the \
number 'q' of Lie derivatives along the vector normal to the hypersurfaces \
acting on the 'p'-order of '\[Phi]'. \n\nThe commands \[Phi][h][] and \
\[Phi][h][LI[p]] are defined to be equal to \[Phi][h][LI[0],LI[0]] and \
\[Phi][h][LI[p],LI[0]], respectively. \n\nThe command \[Phi][h] builds the \
symbol '\[Phi]h', so one can use \[Phi][h][LI[p],LI[q]] or, equivalently, \
\[Phi]h[LI[p],LI[q]].\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
    "*", " ", "0", "i", " ", "components", " ", "of", " ", "the", " ", 
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   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Bs", "::", "usage"}], " ", "=", " ", 
     "\"\<Bs[h][LI[p],LI[q]] is the scalar part of the scalar-vector \
decomposition of the 0i components of the metric perturbation. It is \
automatically defined when calling the function DefMetricFields. It depends \
on the choice of the background slicing, and hence on the induced metric 'h'. \
\n\nThe first Label-Index (LI) gives the order 'p' of the perturbation 'Bs' \
(hence 'Bs' is null for p = 0). The second LI represents the number 'q' of \
Lie derivatives along the vector normal to the hypersurfaces acting on the \
'p'-order of 'Bs'. \n\nThe commands Bs[h][] and Bs[h][LI[p]] are defined to \
be equal to Bs[h][LI[0],LI[0]] and Bs[h][LI[p],LI[0]], respectively. \n\nThe \
command Bs[h] builds the symbol 'Bsh', so one can use Bs[h][LI[p],LI[q]] or, \
equivalently, Bsh[LI[p],LI[q]]. \n\nThe default printed form of Bs[h][] is: \
'B'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Bv", "::", "usage"}], " ", "=", " ", 
     "\"\<Bv[h][LI[p],LI[q],index] is the vector part of the scalar-vector \
decomposition of the 0i components of the metric perturbation. It is \
automatically defined when calling the function DefMetricFields. It depends \
on the choice of the background slicing, and hence on the induced metric 'h'. \
\n\nThe first Label-Index (LI) gives the order 'p' of the perturbation 'Bv' \
(hence 'Bv' is null for p = 0). The second LI represents the number 'q' of \
Lie derivatives along the vector normal to the hypersurfaces acting on the \
'p'-order of 'Bv'. \n\nThe commands Bv[h][index] and Bv[h][LI[p],index] are \
defined to be equal to Bv[h][LI[0],LI[0],index] and Bv[h][LI[p],LI[0],index], \
respectively. \n\nThe command Bv[h] builds the symbol 'Bvh', so one can use \
Bv[h][LI[p],LI[q],index] or, equivalently, Bvh[LI[p],LI[q],index]. \n\nThe \
default printed form of Bv[h][-index] is: '\!\(\*SubscriptBox[\(B\), \
\(index\)]\)'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{"(*", 
    RowBox[{
    "*", " ", "Spatial", " ", "components", " ", "of", " ", "the", " ", 
     "metric", " ", "perturbation"}], " ", "**)"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"\[Psi]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[Psi][h][LI[p],LI[q]] is the second Bardeen potential, present in \
the spatial components of the metric perturbation. It is automatically \
defined when calling the function DefMetricFields. It depends on the choice \
of the background slicing, and hence on the induced metric 'h'. \n\nThe first \
Label-Index (LI) gives the order 'p' of the perturbation '\[Psi]' (hence '\
\[Psi]' is null for p = 0). The second LI represents the number 'q' of Lie \
derivatives along the vector normal to the hypersurfaces acting on the \
'p'-order of '\[Psi]'. \n\nThe commands \[Psi][h][] and \[Psi][h][LI[p]] are \
defined to be equal to \[Psi][h][LI[0],LI[0]] and \[Psi][h][LI[p],LI[0]], \
respectively. \n\nThe command \[Psi][h] builds the symbol '\[Psi]h', so one \
can use \[Psi][h][LI[p],LI[q]] or, equivalently, \[Psi]h[LI[p],LI[q]].\>\""}],
     ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Es", "::", "usage"}], " ", "=", " ", 
     "\"\<Es[h][LI[p],LI[q]] is the scalar part of the scalar-vector-tensor \
decomposition of the spatial components of the metric perturbation. It is \
automatically defined when calling the function DefMetricFields. It depends \
on the choice of the background slicing, and hence on the induced metric 'h'. \
\n\nThe first Label-Index (LI) gives the order 'p' of the perturbation 'Es' \
(hence 'Es' is null for p = 0). The second LI represents the number 'q' of \
Lie derivatives along the vector normal to the hypersurfaces acting on the \
'p'-order of 'Es'. \n\nThe commands Es[h][] and Es[h][LI[p]] are defined to \
be equal to Es[h][LI[0],LI[0]] and Es[h][LI[p],LI[0]], respectively. \n\nThe \
command Es[h] builds the symbol 'Esh', so one can use Es[h][LI[p],LI[q]] or, \
equivalently, Esh[LI[p],LI[q]]. \n\nThe default printed form of Es[h][] is: \
'E'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Ev", "::", "usage"}], " ", "=", " ", 
     "\"\<Ev[h][LI[p],LI[q],index] is the vector part of the \
scalar-vector-tensor decomposition of the spatial components of the metric \
perturbation. It is automatically defined when calling the function \
DefMetricFields. It depends on the choice of the background slicing, and \
hence on the induced metric 'h'. \n\nThe first Label-Index (LI) gives the \
order 'p' of the perturbation 'Ev' (hence 'Ev' is null for p = 0). The second \
LI represents the number 'q' of Lie derivatives along the vector normal to \
the hypersurfaces acting on the 'p'-order of 'Ev'. \n\nThe commands \
Ev[h][index] and Ev[h][LI[p],index] are defined to be equal to \
Ev[h][LI[0],LI[0],index] and Ev[h][LI[p],LI[0],index], respectively. \n\nThe \
command Ev[h] builds the symbol 'Evh', so one can use \
Ev[h][LI[p],LI[q],index] or, equivalently, Evh[LI[p],LI[q],index]. \n\nThe \
default printed form of Ev[h][-index] is: '\!\(\*SubscriptBox[\(E\), \
\(index\)]\)'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Et", "::", "usage"}], " ", "=", " ", 
     "\"\<Et[h][LI[p],LI[q],index,index] is the tensor part of the \
scalar-vector-tensor decomposition of the spatial components of the metric \
perturbation. It is automatically defined when calling the function \
DefMetricFields. It depends on the choice of the background slicing, and \
hence on the induced metric 'h'. \n\nThe first Label-Index (LI) gives the \
order 'p' of the perturbation 'Et' (hence 'Et' is null for p = 0). The second \
LI represents the number 'q' of Lie derivatives along the vector normal to \
the hypersurfaces acting on the 'p'-order of 'Et'. \n\nThe commands \
Et[h][index,index] and Et[h][LI[p],index,index] are defined to be equal to \
Et[h][LI[0],LI[0],index,index] and Et[h][LI[p],LI[0],index,index], \
respectively. \n\nThe command Et[h] builds the symbol 'Eth', so one can use \
Et[h][LI[p],LI[q],index,index] or, equivalently, \
Eth[LI[p],LI[q],index,index]. \n\nThe default printed form of \
Et[h][-index,-index] is: '\!\(\*SubscriptBox[\(E\), \(index\\\ index\)]\)'.\>\
\""}], ";"}]}]}]], "Input",
 InitializationCell->True],

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    RowBox[{
     RowBox[{"**", " ", "PREDEFINED"}], " ", 
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      RowBox[{"Gauge", " ", "vector"}]}]}], ",", " ", 
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  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"\[Xi]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[Xi][h][LI[order],index] is the general vector field used in a \
gauge transformation. It is automatically defined when calling the function \
SplitFieldsAndGaugeChange. It depends on the choice of the background \
slicing, and hence on the induced metric 'h'. \n\nThe Label-Index (LI) gives \
the order 'order' of the vector field '\[Xi]' (hence '\[Xi]' is null for p = \
0). \n\nThe command \[Xi][h] builds the symbol '\[Xi]h', so one can use \
\[Xi][h][LI[order],index] or, equivalently, \[Xi]h[LI[order],index].\>\""}], 
    ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"T", "::", "usage"}], " ", "=", " ", 
     "\"\<T[h][LI[p],LI[q]] is the factor of the time component of the vector \
field '\[Xi]' used in a gauge transformation. It is automatically defined \
when calling the function DefMetricFields. It depends on the choice of the \
background slicing, and hence on the induced metric 'h'. \n\nThe first \
Label-Index (LI) gives the order 'p' of 'T' (hence 'T' is null for p = 0). \
The second LI represents the number 'q' of Lie derivatives along the vector \
normal to the hypersurfaces acting on the 'p'-order of 'T'. \n\nThe commands \
T[h][] and T[h][LI[p]] are defined to be equal to T[h][LI[0],LI[0]] and \
T[h][LI[p],LI[0]], respectively. \n\nThe command T[h] builds the symbol 'Th', \
so one can use T[h][LI[p],LI[q]] or, equivalently, Th[LI[p],LI[q]]. \n\nThe \
default printed form of T[h][] is: 'T'.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Ls", "::", "usage"}], " ", "=", " ", 
     "\"\<Ls[h][LI[p],LI[q]] is the scalar part of the scalar-vector \
decomposition of the spatial component of the vector field '\[Xi]' used in a \
gauge transformation. It is automatically defined when calling the function \
DefMetricFields. It depends on the choice of the background slicing, and \
hence on the induced metric 'h'. \n\nThe first Label-Index (LI) gives the \
order 'p' of 'Ls' (hence 'Ls' is null for p = 0). The second LI represents \
the number 'q' of Lie derivatives along the vector normal to the \
hypersurfaces acting on the 'p'-order of 'Ls'. \n\nThe commands Ls[h][] and \
Ls[h][LI[p]] are defined to be equal to Ls[h][LI[0],LI[0]] and \
Ls[h][LI[p],LI[0]], respectively. \n\nThe command Ls[h] builds the symbol \
'Lsh', so one can use Ls[h][LI[p],LI[q]] or, equivalently, Lsh[LI[p],LI[q]]. \
\n\nThe default printed form of Ls[h][] is: 'L'.\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"Lv", "::", "usage"}], " ", "=", " ", 
     "\"\<Lv[h][LI[p],LI[q],index] is the vector part of the scalar-vector \
decomposition of the spatial component of the vector field '\[Xi]' used in a \
gauge transformation. It is automatically defined when calling the function \
DefMetricFields. It depends on the choice of the background slicing, and \
hence on the induced metric 'h'. \n\nThe first Label-Index (LI) gives the \
order 'p' of 'Lv' (hence 'Lv' is null for p = 0). The second LI represents \
the number 'q' of Lie derivatives along the vector normal to the \
hypersurfaces acting on the 'p'-order of 'Lv'. \n\nThe commands Lv[h][index] \
and Lv[h][LI[p],index] are defined to be equal to Lv[h][LI[0],LI[0],index] \
and Lv[h][LI[p],LI[0],index], respectively. \n\nThe command Lv[h] builds the \
symbol 'Lvh', so one can use Lv[h][LI[p],LI[q],index] or, equivalently, \
Lvh[LI[p],LI[q],index]. \n\nThe default printed form of Lv[h][-index] is: '\!\
\(\*SubscriptBox[\(L\), \(index\)]\)'.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"\[Epsilon]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[Epsilon] is the default perturbation parameter. It is \
automatically defined when calling either the function DefMetricFields or the \
function DefMatterFields.\>\""}], ";"}], "\[IndentingNewLine]", 
   "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"\[ScriptCapitalN]0", "::", "usage"}], " ", "=", " ", 
     "\"\<\[ScriptCapitalN]0[h][LI[p],LI[q]] is the time part of the \
perturbation of the normal vector to constant time hypersurfaces (whith index \
down). It is automatically defined when calling SetSlicing.\nIn the ADM \
formalism, it is equal to the lapse function.\n\nThe first Label-Index (LI) \
gives the order 'p' of the perturbation '\[ScriptCapitalN]0'. The second LI \
represents the number 'q' of Lie derivatives along the vector normal to the \
hypersurfaces acting on the 'p'-order of '\[ScriptCapitalN]'. Hence \
\[ScriptCapitalN]0[h][LI[0],LI[0]] is the background lapse and is equal to 1.\
\n\nThe commands \[ScriptCapitalN]0[h][index] and \
\[ScriptCapitalN]0[h][LI[p],index] are defined to be equal to \
\[ScriptCapitalN]0[h][LI[0],LI[0],index] and \
\[ScriptCapitalN]0[h][LI[p],LI[0],index], respectively. \n\nThe command \
\[ScriptCapitalN]0[h] builds the symbol '\[ScriptCapitalN]0h', so one can use \
\[ScriptCapitalN]0[h][LI[p],LI[q],index] or, equivalently, \
\[ScriptCapitalN]0h[LI[p],LI[q],index]. \n\nThe default printed form of \
\[ScriptCapitalN]0[h][-index] is: '\!\(\*SubscriptBox[\(\[ScriptCapitalN]0\), \
\(index\)]\)'.\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"\[ScriptCapitalN]", "::", "usage"}], " ", "=", " ", 
     "\"\<\[ScriptCapitalN][h][index] is the normal vector to constant time \
hypersurfaces. It is automatically defined when calling SetSlicing. Though on \
the background spacetime it is equal to \!\(\*SuperscriptBox[\(n\), \
\(\[Mu]\)]\), where \!\(\*SuperscriptBox[\(n\), \(\[Mu]\)]\) is the vector \
normal to h on the background space-time, this definition allows manipulation \
of the normal vector, like perturbation, conformal transformations, which \
would be impossible directly with \!\(\*SuperscriptBox[\(n\), \(\[Mu]\)]\) \
since many definitions are already automatized for it (e.g. normalization to \
-1).\>\""}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", 
   "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"d\[ScriptCapitalN]", "::", "usage"}], " ", "=", " ", 
     "\"\<d\[ScriptCapitalN][h][LI[p],index] is the perturbation of the \
normal vector to constant time hypersurfaces. 'p' is the order of \
perturbation.\n\nThe command d\[ScriptCapitalN][h] builds the symbol 'd\
\[ScriptCapitalN]h', so one can use d\[ScriptCapitalN][h][LI[p],index] or, \
equivalently, d\[ScriptCapitalN]h[LI[p]index].\>\""}], ";"}], 
   "\[IndentingNewLine]", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{
     RowBox[{"\[ScriptCapitalN]i", "::", "usage"}], " ", "=", " ", 
     "\"\<\[ScriptCapitalN]i[h][index] is the shift vector (in the ADM \
formalism) of the normal vector to constant time hypersurfaces. It is defined \
as \[ScriptCapitalN]i[h][upindex] = -Projector[h][ \[ScriptCapitalN]0[h] \
\[ScriptCapitalN][h][upindex] ].\nIt is only meaningful when used with and \
upper index. Its main purpose is the computation of perturbations for the \
shift vector in the ADM formalism.\n\nThe command \[ScriptCapitalN]i[h] \
builds the symbol '\[ScriptCapitalN]ih', so one can use \
\[ScriptCapitalN]i[h][upindex] or, equivalently, \
\[ScriptCapitalN]ih[upindex]. \n\>\""}], ";"}]}]}]], "Input",
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Types of gauge:\
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Types of manifold:\
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InertHead used to replace the scalars. This is to avoid the application of \
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\tColoration of the perturbation orders
\tConvenient functions to build rules
\tHandling expressions
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Coloration of the perturbation orders:\
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Definition of the ambient metric g, the normal vector u to the hypersurfaces \
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My Gauge transformation rule, which takes into account the conformal \
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2.12 Automatic perturbation of the time components for normalized vectors\
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The function which builds the time component of the vector from normalization\
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We gather all the rules for matter below. This function is used in he \
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This part is in development. We intend to perturb the normal vector, so as to \
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