Metric volume tensor (${ϵ}_{a_1\dots a_d}$)

On an oriented $d$-dimensional pseudo-Riemannian manifold, the metric volume tensor is the totally antisymmetric tensor with ${ϵ}_{1\dots d}=\sqrt{|\det g|}$ in positively oriented coordinates. It supplies the components of the volume form. It differs from the coordinate Levi-Civita symbol, whose entries are just $0,1,-1$. Raising indices with a Lorentzian metric can change signs; the metric, dimension and orientation must be specified.