====== 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:
<sxh groovy; gutter: false>
println 'g_ab - g_ba'.t
</sxh>
<sxh plain; gutter: false>
   > 0
</sxh>
<sxh groovy; gutter: false>
println 'g^AB + g^BA'.t
</sxh>
<sxh plain; gutter: false>
   > 2*g^AB
</sxh>

Raising and lowering of metric tensor indices may involve [[documentation:ref:kronecker_delta]]:
<sxh groovy; gutter: false>
println ('{_a -> ^a}'.mapping >> 'g_ab'.t)
</sxh>
<sxh plain; gutter: false>
   > d^{a}_{b}
</sxh>

The transformation that simplifies contractions with metric tensor is [[documentation:ref:eliminatemetrics]]:
<sxh groovy; gutter: false>
println EliminateMetrics >> 'g_am*F^ab*g_bn'.t
</sxh>
<sxh plain; gutter: false>
   > F_mn
</sxh>
<sxh groovy; gutter: false>
println EliminateMetrics >> 'g_am*g^an'.t
</sxh>
<sxh plain; gutter: false>
   > d^n_m
</sxh>

====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 [[IndexType|index types]]. So, if one will enter tensor with name ''g'' and non-metric indices (e.g. ''Matrix1''), one will cause exception:
<sxh groovy; gutter: false>
''' g_{a' b'} '''.t
</sxh>
<sxh plain; gutter: false>
   > ParserException: Metric is not specified for non metric index type.
</sxh>

One can specify different name for metric tensor by putting the following line in the beginning of the code:
<sxh groovy; gutter: false>
//change default metric name
CC.current().setMetricName('f')
println EliminateMetrics >> 'f_am*F^ab*f_bn'.t
</sxh>
<sxh plain; gutter: false>
   > F_mn
</sxh>

=====See also=====
  * Related guides: [[documentation:guide:types_of_indices_and_metric]]
  * Reference material: [[documentation:ref:kronecker_delta]], [[documentation:ref:eliminatemetrics]], [[documentation:ref:indextype]]