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 $x^{\mu }=(x^0,x^1,x^2,x^3)$, 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 $p^{\mu }=(E/c,p_x,p_y,p_z)$ 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:

$F^{\mu \nu }=\left(\begin{array}{cccc}0& -E_x/c& -E_y/c& -E_z/c\\ E_x/c& 0& -B_z& B_y\\ E_y/c& B_z& 0& -B_x\\ E_z/c& -B_y& B_x& 0\end{array}\right)$

See also

• four-vector
• special relativity

References

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