Metric tensor
Basics
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:
Raising and lowering of metric tensor indices may involve Kronecker delta:
println ( '{_a -> ^a}' .mapping > > 'g_ab' .t )
|
The transformation that simplifies contractions with metric tensor is EliminateMetrics:
println EliminateMetrics > > 'g_am*F^ab*g_bn' .t
|
println EliminateMetrics > > 'g_am*g^an' .t
|
Details
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:
> 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:
CC .current () .setMetricName ( 'f' )
println EliminateMetrics > > 'f_am*F^ab*f_bn' .t
|
See also