Hodge Dual - Index Notation For The Star Operator

Index Notation For The Star Operator

Using index notation, the Hodge dual is obtained by contracting the indices of a k-form with the n-dimensional completely antisymmetric Levi-Civita tensor. This differs from the Levi-Civita symbol by a factor of |det g|½, where g is an inner product (the metric tensor). The absolute value of the determinant is necessary if g is not positive-definite, e.g. for tangent spaces to Lorentzian manifolds.

Thus one writes

where η is an arbitrary antisymmetric tensor in k indices. It is understood that indices are raised and lowered using the same inner product g as in the definition of the Levi-Civita tensor. Although one can take the star of any tensor, the result is antisymmetric, since the symmetric components of the tensor completely cancel out when contracted with the completely anti-symmetric Levi-Civita symbol.