The notation for metric tensor used in Redberry is ordinary g_ab and similarly for other index types (g_AB, g_\\alpha\\beta etc.). Metric tensors are automatically symmetric:
println 'g_ab - g_ba'.t
> 0
println 'g^AB + g^BA'.t
> 2*g^AB
Raising and lowering of metric tensor indices may involve Kronecker delta:
println ('{_a -> ^a}'.mapping >> 'g_ab'.t)
> d^{a}_{b}
The transformation that simplifies contractions with metric tensor is EliminateMetrics:
println EliminateMetrics >> 'g_am*F^ab*g_bn'.t
> F_mn
println EliminateMetrics >> 'g_am*g^an'.t
> d^n_m
In addition to g_ab notation Redberry also uses d_ab, which is the notation for Kronecker delta with raised or lowered indices.
By default, metric tensor defined only for metric index types. So, if one will enter tensor with name g and non-metric indices (e.g. Matrix1), one will cause exception:
''' g_{a' b'} '''.t
> ParserException: Metric is not specified for non metric index type.
One can specify different name for metric tensor by putting the following line in the beginning of the code:
//change default metric name
CC.current().setMetricName('f')
println EliminateMetrics >> 'f_am*F^ab*f_bn'.t
> F_mn