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 .

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 units of length). The remaining components of the four-displacement form the spatial displacement vector .

Similarly, the four-momentum

Four-momentum

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...

  of a body is equivalent to the energy-momentum tensor of said body. The element represents the momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 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

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 of the body as a result of it travelling through space, written in vector notation as .

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