Lowerindex

Lowerindex refers to the covariant components of a tensor or vector obtained by lowering an index through a metric or inner product. In tensor notation, indices indicate variance: contravariant indices are upper, covariant indices are lower. The term is commonly used in differential geometry and physics, especially in general relativity, where tensors carry both upper and lower indices.

The lowering process uses the metric tensor g. If a vector v has contravariant components v^i, its covariant components are v_i = g_{ij} v^j, with summation over j. More generally, a tensor T^{i j ...} can be converted to T_{k l ...} by contracting each contravariant index with the corresponding metric: T_{kl...} = g_{km} g_{ln} T^{mn...}. Conversely, indices can be raised with the inverse metric g^{ij}: v^i = g^{ij} v_j.

Lower indices naturally live in the dual space: covariant components describe linear functionals acting on vectors. This duality is formalized by the metric, which provides a canonical isomorphism between a vector space and its dual. The transformation properties of covariant components under coordinate changes—transforming covariantly—differ from those of contravariant components, which transform contravariantly.

Notationally, a lower index denotes a covariant component, while an upper index denotes a contravariant component. The operation of lowering or raising an index depends on a chosen metric; in spaces with a natural inner product, lowering is achieved via the inner product, and in flat Euclidean space with the standard metric, numerical values may coincide under the usual identification of vectors and covectors. In relativity, the choice of metric (for example, the Minkowski metric) affects signs and interpretation.