11.12 Tensor Metric Based Variance Conversion

Tensor Metric Based Variance Conversion is the general umbrella concept covering both the index raising operation and the index lowering operation, describing the entire class of metric-dependent procedures by which a tensor's variance type, meaning its distribution of upper and lower indices, can be freely converted while continuing to represent the same underlying geometric object.

Definition and Unifying Idea

What Is Being Unified

Metric based variance conversion refers collectively to any application of the metric tensor, whether covariant or contravariant, to change the position of one or more indices on a tensor, encompassing raising a lower index, lowering an upper index, and any combination of such operations applied across a tensor's several indices.

Why a Single Umbrella Concept Is Useful

Treating raising and lowering as two instances of one underlying conversion process highlights that both operations share the same essential character: each is a linear, invertible contraction with a metric tensor that changes variance type while preserving the total index count and the identity of the underlying geometric object.

The Two Directions of Conversion

Conversion From Covariant to Contravariant

One direction of the conversion uses the contravariant metric tensor to move an index from a lower position to an upper position, which is the operation known individually as index raising, applicable to any covariant index present on a tensor.

Conversion From Contravariant to Covariant

The opposite direction of the conversion uses the covariant metric tensor to move an index from an upper position to a lower position, which is the operation known individually as index lowering, applicable to any contravariant index present on a tensor.

Properties Shared by Both Directions

Metric Dependence Throughout

Both directions of the conversion require the presence of a non-degenerate metric tensor on the space, since neither raising nor lowering can be performed without a specific metric supplying the components used in the contraction, distinguishing variance conversion from coordinate transformation, which requires no metric at all.

Mutual Invertibility

The two directions of conversion are mutual inverses of one another, since the covariant and contravariant metric tensors used in each direction are themselves matrix inverses, so converting an index in one direction and then immediately converting it back in the other direction restores the original tensor exactly.

Role Within Tensor Algebras

Providing a Complete Picture of Index Manipulation

Metric based variance conversion, taken as a whole, provides the complete picture of how a tensor's variance type can be reshaped freely at any point where a metric is defined, complementing the coordinate transformation laws, which describe how components change without altering variance type.

Foundation for Associated Tensors

The general concept of variance conversion underlies the notion of associated tensors, meaning the family of tensors of differing type, related through raising and lowering, that all represent one and the same geometric object once a metric has been fixed on the space.