5.1. RiemannToPerm

The function RiemannToPerm converts all Riemann scalars of a given metric (or list of metrics) into their canonical permutations:

Suppose we start from this simple expression:

In[94]:=

rexpr = RiemannCD[a, b, c, d] RiemannCD[-a, -b, -c, -d] + 6RiemannCD[e, f, -c, -d] RiemannCD[-a, -b, -e, -f] epsilonmetric[a, b, c, d]

Out[94]=

R_abcd^     R_    ^abcd + 6 ε_    ^abcd R_abef^     R_ (  cd)^ef  

Then we can transform all terms into their canonical permutations,

In[95]:=

rperm = RiemannToPerm[rexpr]

Out[95]=

RPerm[metric][{{0, 0}, 0}, Cycles[{2, 3, 5}, {4, 7, 6}]] + 6 RPerm[metric][{{0, 0}, 1}, Cycles[{2, 3, 5}, {4, 7, 9, 6}, {8, 11, 10}]]

that is, not the permutations which would regenerate the previous objects:

In[96]:=

PermToRiemann[rperm]

Out[96]=

Scalar[R_abcd^     R_    ^abcd] + 6 Scalar[ε_cdef^     R_ab  ^(  ef) R_    ^abcd]

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