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| + | ====== 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]] | ||