4.2. PermToRiemann

Any permutation of N indices can be converted into a Riemann monomial of the appropiate case with the function PermToRiemann. There is a second argument to specify whether we want the contracted Riemann tensors to be automatically transformed into Ricci or not.

The default value of the argument for conversion into Ricci is stored in the global variable

In[81]:=

$CurvatureRelations

Out[81]=

False

In[82]:=

rinv2 = RInv[metric][{0, 0}, 1]

Out[82]=

I_00,1

In[83]:=

rperm2 = InvToPerm[rinv2]

Out[83]=

RPerm[metric][{{0, 0}, 0}, Cycles[{2, 3}, {4, 5}, {6, 7}]]

If we set that variable to True, we get a product of Ricci tensors:

In[84]:=

PermToRiemann[rperm2, True]

Out[84]=

Scalar[R_ab^   R_  ^ba]

If we set that argument to False or use the default, no Riccis appear

In[85]:=

PermToRiemann[rperm2, False]

Out[85]=

Scalar[R_ (  a )^(ab c) R_ (b cd)^( d  )]

In[86]:=

PermToRiemann[rperm2]

Out[86]=

Scalar[R_ (  a )^(ab c) R_ (b cd)^( d  )]


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