A covariant tensor, denoted with a lowered index (e.g.,
) 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
then any set of quantities
which transform according to
is a covariant tensor.
Contravariant tensors are a type of tensor with differing transformation properties, denoted . To turn a contravariant
into a covariant tensor ( index lowering), use
the metric tensor to write
Covariant and contravariant indices can be used simultaneously in a
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
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
Portions of this entry contributed by
F. González Lázaro Explore with Wolfram|Alpha
References Arfken, G. "Noncartesian Tensors, Covariant Differentiation." §3.8 in Orlando, FL: Academic Press, pp. 158-164,
Methods for Physicists, 3rd ed. Lichnerowicz, A. New York: Wiley, 1962. Elements
of Tensor Calculus. Morse, P. M.
and Feshbach, H. New York: McGraw-Hill, pp. 44-46, 1953. Methods
of Theoretical Physics, Part I. Weinberg,
New York: Wiley, 1972. Gravitation
and Cosmology: Principles and Applications of the General Theory of Relativity. Referenced on Wolfram|Alpha Covariant Tensor
Cite this as:
Lázaro, Manuel F. González and Weisstein, Eric W. "Covariant Tensor."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/CovariantTensor.html