Berezinian
Four-tensor is a frequent abbreviation for a tensor in a four-dimensional spacetime.[1]
Syntax
General four-tensors are usually written as , with the indices taking integral values from 0 to 3. Such a tensor is said to have contravariant rank n and covariant rank m.[1]
Examples
One of the simplest non-trivial examples of a four-tensor is the four-displacement , a four-tensor with contravariant rank 1 and covariant rank 0. Four-tensors of this kind are usually known as four-vectors. Here the component gives the displacement of a body in time (time is multiplied by the speed of light so that has dimensions of length). The remaining components of the four-displacement form the spatial displacement vector .[1]
Similarly, the four-momentum of a body is equivalent to the energy-momentum tensor of said body. The element represents the momentum of the body as a result of it travelling through time (directly comparable to the internal energy of the body). The elements , and correspond to the momentum of the body as a result of it travelling through space, written in vector notation as .[1]
The electromagnetic field tensor is an example of a rank two contravariant tensor:[1]
See also
References
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