Conformal_Transformation.nb

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Cell[TextData[StyleBox["Cyril Pitrou (pitrou@iap.fr)\nMarch 2018",
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 " package.\n\nLimitations : \nWorks only for Levi-Civita connections.\nAny \
generalization or cleaning of definitions and implementations is welcome\nAn \
application to Brans-Dicke theory would be nice. TODO"
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Cell["Construction of the Conformal transformation.", "Section"],

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power ",
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derivative (Levi-Civita) CD of the original metric ",
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the conformally transformed version, that is multiplying by some power of the \
conformal factor ( -> ",
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first simplu replaced by the curvature tensors of the new metric \n   e.g. \
ConfHead[g1,g2][Ricci[cd1]] is replaced by Ricci[cd2] (a tautology).\n3) In a \
final stage, we express all quantities in terms of a base metric, with the \
CovD and curvature tensors associated with this base metric. \nIt can be the \
initial one, the final one, or just the default active metric. This is where \
we use \n   a) conformal rules to change metrics, \n   b) ChangeCovD to \
change covariant derivatives (having placed their indices down in the very \
beginning), \n   c) ChangeCurvature to express e.g. Ricci[cd2] in terms of \
Ricci[cd1] and terms involving the derivatives of the conformal factor.\nThe \
base metric, if not specified, is the active metric.\n\nThe syntax of the \
conformal transformation is   \n     Conformal[baseg][g1,g2][expr] \n     \n\
The base metric, if not specified, is the active metric, hence \
Conformal[g1,g2][expr]  is Conformal[activemetric][g1,g2][expr]. \n"
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ConfHead designated the conformally transformed version of a tensor. By \
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Then the package defines automatically all relevant conformal rules to relate \
two by two all metrics
In fact these conformal rules were actually built by xTensor thanks to an \
option \[OpenCurlyQuote]ConformalTo\[CloseCurlyQuote] in the definitions of \
the scondary metrics.\
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Finally we check that the behaviour makes sense for the Weyl tensor :
Indeed for Riemann we get many terms involving the CovD of the conformal \
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Changing the metric in which results are expressed (advanced)\
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We can express the result in function of the conformally transformed metric \
(and its associated connection). 
This does not change the ambient metric defined by DefMetric and used to \
raise and lower indices.
Indeed the first argument of Conformal (which is optional) specifies with \
which metric (and which associated covariant derivative and curvature \
tensors), the result should be expressed.

The syntax is :
Conformal[baseg][g1,g2][expr] , where baseg is the metric in which (with \
associated CovD and curvature tensors) the final results must be expressed. 
g1 -> g2 is the conformal transformation which is applied to the expression \
expr.\
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However if at each stage we express things in terms of the conformally \
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This just means we actively change a system (a tensor), but also actively \
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