LEVI!-CIVITA switch

 

It is possible to convert the product of Levi-Civita tensors (or symbols) to expressions containing metric tensors only. The effect is caused by the LEVI!-CIVITA switch: if the metric tensor and its signature have been declared then all products of Levi-Civita tensors (symbols) are substituted by the appropriate expression of metric tensors during the evaluation:

$^1532a$₁&ldots;a_n^₁₅₃₂b₁&ldots;b_n = (-1)^qn!g[a₁\>\> b₁ga₂\>\> b₂&cdots;ga_n]\>\> a_n ¹⁵³²a₁&ldots;a_n1532b₁&ldots;b_n = n!g[a₁\>\> b₁ga₂\>\> b₂&cdots;ga_n]\>\> a_n

where $^$a₁&ldots;a_n is the Levi-Civita tensor, $$a₁&ldots;a_n is the Levi-Civita symbol, $g$_ij the metric tensor. $q=\{n-s2\}$ where $n$ is the dimension of space-time and $s$ is the signature of the metric tensor.

Default setting of the LEVI!-CIVITA switch is OFF.