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Four-tensor is a frequent abbreviation for a tensor in a four-dimensional spacetime.[1]

Syntax

General four-tensors are usually written as Aν1,ν2,...,νmμ1,μ2,...,μn, 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 xμ=(x0,x1,x2,x3), 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 x0=ct gives the displacement of a body in time (time is multiplied by the speed of light c so that x0 has dimensions of length). The remaining components of the four-displacement form the spatial displacement vector x.[1]

Similarly, the four-momentum pμ=(E/c,px,py,pz) of a body is equivalent to the energy-momentum tensor of said body. The element p0=E/c 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 p1, p2 and p3 correspond to the momentum of the body as a result of it travelling through space, written in vector notation as p.[1]

The electromagnetic field tensor is an example of a rank two contravariant tensor:[1]

Fμν=(0Ex/cEy/cEz/cEx/c0BzByEy/cBz0BxEz/cByBx0)

See also

References

  1. 1.0 1.1 1.2 1.3 1.4 Lambourne, Robert J A. Relativity, Gravitation and Cosmology. Cambridge University Press. 2010.


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