Editing Tensor contraction (section)

== Metric contraction ==
{{see also|Raising and lowering indices#An example from Minkowski spacetime}}
As in the previous example, contraction on a pair of indices that are either both contravariant or both covariant is not possible in general. However, in the presence of an [[inner product]] (also known as a [[Metric tensor|metric]]) ''g'', such contractions are possible. One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction. The combined operation is known as ''[[metric contraction]]''.<ref name="o'neill">{{cite book |first=Barrett |last=O'Neill |title=Semi-Riemannian Geometry with Applications to Relativity |publisher=Academic Press |year=1983 |page=86 |isbn=0-12-526740-1 }}</ref>