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^^,

for which


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


then any set of quantities A_j which transform according to


or, defining


according to


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


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


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

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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.

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Covariant Tensor

Cite this as:

Lázaro, Manuel F. González and Weisstein, Eric W. "Covariant Tensor." From MathWorld--A Wolfram Web Resource.

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