"Signature" of the epsilon tensor

Question

Hi community.

It is (more or less) known that when defining a metric one can provide a parameter signature, e.g +1 for Riemannian metrics or -1 for Lorentzian metrics; and it is also known that one can parametrise the epsilon tensor with a metric.

I'm interested, however, in a generic epsilon tensor defined for a Lorentzian signature... without specifying a metric (because I want to consider spacetimes with diferent dimensionality). I tried the following, and it seems to work:

\delta{#}::KroneckerDelta.
\epsilon{#}::EpsilonTensor(delta=-\delta).

Question

Is the above proposal equivalent to assigning a signature = -1 to the epsilon?

Comments

I did the same the last days and it worked for me as well. However if you want to go the "canonical way", as far as I have seen from the documentation, you do not need to specify the metric, but only its signature, but I did not really tried it as I was happy with the workaround above.