Four-tensor

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

Syntax

General four-tensors are usually written as $A^{\mu_1,\mu_2,...,\mu_n}_{\;\nu_1,\nu_2,...,\nu_m}$, 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 $x^\mu=\left(x^0, x^1, x^2, x^3\right)$, 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 $\mathbf{x}$.^[1]

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 $\mathbf{p}$.^[1]

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

$F^{\mu\nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c\\ E_x/c & 0 & -B_z & B_y\\ E_y/c & B_z & 0 & -B_x\\ E_z/c & -B_y & B_x & 0 \end{pmatrix}$

See also

• four-vector
• special relativity

References

1. Lambourne, Robert J A. Relativity, Gravitation and Cosmology. Cambridge University Press. 2010.