Raising and Lowering Indices - Vectors (order-1 Tensors)

Vectors (order-1 Tensors)

Multiplying by the contravariant metric tensor (and contracting) raises an index:

and multiplying by the covariant metric tensor (and contracting) lowers an index:

The form g_ij need not be nonsingular to lower an index, but to get the inverse (and thus raise an index) it must be nonsingular.

Raising and then lowering the same index (or conversely) are inverse operations, which is reflected in the covariant and contravariant metric tensors being inverse to each other:

where δi_k is the Kronecker delta or identity matrix. Since there are different choices of metric with different metric signatures (signs along the diagonal elements, i.e. tensor components with equal indices), the name and signature is usually indicated to prevent confusion. Different authors use different metrics and signatures for different reasons.

Mnemonically (though incorrectly), one could think of indices "cancelling" between a metric and another tensor, and the metric stepping up or down the index. In the above examples, such "cancellations" and "steps" are like

Again, while a helpful guide, this is only mnemonical and not a property of tensors since the indices do not cancel like in equations, it is only a concept of the notation. The results are continued below, for tensors higher orders (i.e. more indices).

When raising indices of quantities in spacetime, it helps to decopose summations into "timelike components" (where indices are zero) and "spacelike components" (where indices are 1, 2, 3, represented conventionally by Latin indices).

Example from Minkowski spacetime

The covariant 4-position is given by

in components:

(where x_j are the usual Cartesian coordinates) and the Minkowski metric tensor with signature (+−−−) is defined as

in components:

To raise the index, multiply by the tensor and contract:

then for λ = 0:

and for λ = j = 1, 2, 3:

So the index-raised contravariant 4-position is: