TOPICS
Search

Covariant Tensor


A covariant tensor, denoted with a lowered index (e.g., a_mu) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor.

To examine the transformation properties of a covariant tensor, first consider the gradient

 del phi=(partialphi)/(partialx_1)x_1^^+(partialphi)/(partialx_2)x_2^^+(partialphi)/(partialx_3)x_3^^,
(1)

for which

 (partialphi^')/(partialx_i^')=(partialphi)/(partialx_j)(partialx_j)/(partialx_i^'),
(2)

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

 A_i=(partialphi)/(partialx_i),
(3)

then any set of quantities A_j which transform according to

 A_i^'=(partialx_j)/(partialx_i^')A_j
(4)

or, defining

 a_i^j=(partialx_j)/(partialx_i^'),
(5)

according to

 A_i^'=a_i^jA_j
(6)

is a covariant tensor.

Contravariant tensors are a type of tensor with differing transformation properties, denoted a^nu. To turn a contravariant tensor a^nu into a covariant tensor a_mu (index lowering), use the metric tensor g_(munu) to write

 g_(munu)a^nu=a_mu.
(7)

Covariant and contravariant indices can be used simultaneously in a mixed tensor.

In Euclidean spaces, and more generally in flat Riemannian manifolds, a coordinate system can be found where the metric tensor is constant, equal to Kronecker delta

 g_(munu)=delta_(munu).
(8)

Therefore, raising and lowering indices is trivial, hence covariant and contravariant tensors have the same coordinates, and can be identified. Such tensors are known as Cartesian tensors.

A similar result holds for flat pseudo-Riemannian manifolds, such as Minkowski space, for which covariant and contravariant tensors can be identified. However, raising and lowering indices changes the sign of the temporal components of tensors, because of the negative eigenvalue in the Minkowski metric.


See also

Cartesian Tensor, Contravariant Tensor, Four-Vector, Index Lowering, Lorentz Tensor, Metric Tensor, Mixed Tensor, Tensor

Portions of this entry contributed by Manuel F. González Lázaro

Explore with Wolfram|Alpha

References

Arfken, G. "Noncartesian Tensors, Covariant Differentiation." §3.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 158-164, 1985.Lichnerowicz, A. Elements of Tensor Calculus. New York: Wiley, 1962.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44-46, 1953.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.

Referenced on Wolfram|Alpha

Covariant Tensor

Cite this as:

Lázaro, Manuel F. González and Weisstein, Eric W. "Covariant Tensor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CovariantTensor.html

Subject classifications