Berezinian

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

Syntax

General four-tensors are usually written as $A_{\;\nu _{1},\nu _{2},...,\nu _{m}}^{\mu _{1},\mu _{2},...,\mu _{n}}$, with the indices taking integral values from 0 to 3. Such a tensor is said to have contravariant rank n and covariant rank m.

Examples

One of the simplest non-trivial examples of a four-tensor is the four-displacement $x^{\mu }=\left(x^{0},x^{1},x^{2},x^{3}\right)$, 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 $x^{0}=ct$ gives the displacement of a body in time (time is multiplied by the speed of light $c$ so that $x^{0}$ has dimensions of length). The remaining components of the four-displacement form the spatial displacement vector $\mathbf {x}$.

Similarly, the four-momentum $p^{\mu }=\left(E/c,p_{x},p_{y},p_{z}\right)$ of a body is equivalent to the energy-momentum tensor of said body. The element $p^{0}=E/c$ 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 $p^{1}$, $p^{2}$ and $p^{3}$ correspond to the momentum of the body as a result of it travelling through space, written in vector notation as $\mathbf {p}$.

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

$F^{\mu \nu }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}$

See also

• four-vector
• special relativity

References

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