A.2. The Levi-Civita Tensor Density - ckw

A.2. The Levi-Civita Tensor Density
The Levi-Civita symbol [pic] is defined as
[pic] for [pic]
in every 4-D coordinate system. Consider now the Levi-Civita tensor [pic]
which equals to [pic] in one coordinate system [pic]. In another
coordinate system [pic], we have
[pic][pic] [pic] (A.12)
[pic]
( [pic] (A.14)
Thus, the Levi-Civita symbol is a tensor density of weight (1. The metric
determinant g can be written as
[pic] [pic] [pic] (A.15)
Thus, g is a scalar density of weight (2. [ A tensor density of weight n
is an object that transforms like a tensor but with an extra factor [pic] ]
For convenience, we define the covariant symbol by [pic]. Note that using
the metric tensor to lower indices gives
[pic] [pic] (A.16)
Since the left-hand side is a tensor density of weight (1, [pic] must be a
tensor density of weight +1. This can also be seen from
[pic]
( [pic] (A.17)
[pic]
Finally, it is easy to see that [pic] and [pic] transform like tensors.