Covariant Tensor

A covariant tensor is a Tensor having specific transformation properties (c.f., a Contravariant Tensor). To examine the transformation properties of a covariant tensor, first consider the Gradient

$$\nabla\phi \equiv {\partial\phi\over\partial x_1} {\hat {\bf... ...{\bf x}}_2 + {\partial\phi\over\partial x_3} {\hat{\bf x}}_3,$$ (1)

for which

$${\partial\phi'\over\partial x_i'} = {\partial\phi\over\partial x_j}{\partial x_j\over \partial x_i'},$$ (2)

where $\phi (x_1,x_2,x_3) = \phi'(x_1',x_2',x_3')$. Now let

$$A_i \equiv {\partial\phi\over\partial x_i},$$ (3)

then any set of quantities $A_j$ which transform according to

$$A_i' = {\partial x_j\over\partial x_i'} A_j'$$ (4)

or, defining

$$a_{ij} \equiv {\partial x_j\over\partial x_i'},$$ (5)

according to

$$A_i = a_{ij}A_j'$$ (6)

is a covariant tensor. Covariant tensors are indicated with lowered indices, i.e., $a_\mu$.

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

$${\partial x_j\over\partial x_i'} = {\partial x_i'\over\partial x_j} \equiv a_{ij}$$ (7)

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. Covariant Four-Vectors satisfy

$$a_\mu = \Lambda^\nu_\mu a_\nu,$$ (8)

where $\Lambda$ is a Lorentz Tensor.

To turn a Contravariant Tensor into a covariant tensor, use the Metric Tensor $g_{\mu\nu}$ to write

$$a_\mu \equiv g_{\mu\nu}a^\nu.$$ (9)

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

See also Contravariant 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.