Contravariant Tensor

A contravariant tensor is a Tensor having specific transformation properties (c.f., a Covariant Tensor). To examine the transformation properties of a contravariant tensor, first consider a Tensor of Rank 1 (a Vector)

$$d{\bf r} = dx_1 {\hat {\bf x}}_1 + dx_2 {\hat {\bf x}}_2 + dx_3 {\hat {\bf x}}_3,$$ (1)

for which

$$dx_i' = {\partial x_i'\over \partial x_j} dx_j.$$ (2)

Now let $A_i \equiv dx_i$, then any set of quantities $A_j$ which transform according to

$$A_i' = {\partial x_i'\over\partial x_j} A_j,$$ (3)

or, defining

$$a_{ij} \equiv {\partial x_i'\over\partial x_j},$$ (4)

according to

$$A_i' = a_{ij}A_j$$ (5)

is a contravariant tensor. Contravariant tensors are indicated with raised indices, i.e., $a^\mu$.

Covariant Tensors are a type of Tensor with differing transformation properties, denoted $a_\nu$. However, in 3-D Cartesian Coordinates,

$${\partial x_j\over\partial x_i'} = {\partial x_i'\over\partial x_j} \equiv a_{ij}$$ (6)

for $i,j = 1$, 2, 3, meaning that contravariant and covariant tensors are equivalent. The two types of tensors do differ in higher dimensions, however. Contravariant Four-Vectors satisfy

$$a^\mu =\Lambda^\mu_\nu a^\nu,$$ (7)

where $\Lambda$ is a Lorentz Tensor.

To turn a Covariant Tensor into a contravariant tensor, use the Metric Tensor $g^{\mu\nu}$ to write

$$a^\mu \equiv g^{\mu\nu}a_\nu.$$ (8)

Covariant and contravariant indices can be used simultaneously in a Mixed Tensor.

See also Covariant Tensor, Four-Vector, Lorentz Tensor, Metric Tensor, Mixed Tensor, Tensor

References

Arfken, G. ``Noncartesian Tensors, Covariant Differentiation.'' §3.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 158-164, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44-46, 1953.