Introduction to the mathematics of general relativity: Difference between revisions

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The vector is called [[Covariance and contravariance of vectors|''covariant'' or ''contravariant'']] depending on how the transformation of the vector's components is related to the transformation of coordinates.

* Contravariant vectors have units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration) and transform in the opposite way as the coordinate system. For example, in changing units from meters to millimeters the coordinate units get smaller, but the numbers in a vector become larger: 1 m becomes 1000 mm.

* Covariant vectors, on the other hand, have units of one-over-distance (such as in a [[gradient]]) and transform in the same way as the coordinate system. For example, in changing from meters to millimeters, the coordinate units become smaller and the number measuring a gradient will also become smaller: 1 Kelvin per m becomes 0.001 Kelvin per mm.

In [[Einstein notation]], contravariant vectors and components of tensors are shown with superscripts, e.g. {{math|''xⁱ''}}, and covariant vectors and components of tensors with subscripts, e.g. {{math|''x_i''}}. Indices are "raised" or "lowered" by multiplication by an appropriate matrix, often the identity matrix.