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'''Four-tensor''' is a frequent abbreviation for a [[tensor]] in a four-dimensional [[spacetime]].<ref name=Lambourne_2010>Lambourne, Robert J A. Relativity, Gravitation and Cosmology. Cambridge University Press. 2010.</ref> | |||
==Syntax== | |||
General four-tensors are usually written as <math>A^{\mu_1,\mu_2,...,\mu_n}_{\;\nu_1,\nu_2,...,\nu_m}</math>, with the indices taking integral values from 0 to 3. Such a tensor is said to have [[Covariance and contravariance of vectors|contravariant]] rank n and [[Covariance and contravariance of vectors|covariant]] rank m.<ref name=Lambourne_2010/> | |||
==Examples== | |||
One of the simplest non-trivial examples of a four-tensor is the four-displacement <math>x^\mu=\left(x^0, x^1, x^2, x^3\right)</math>, 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 <math>x^0=ct</math> gives the displacement of a body in time (time is multiplied by the speed of light <math>c</math> so that <math>x^0</math> has dimensions of length). The remaining components of the four-displacement form the spatial displacement vector <math>\mathbf{x}</math>.<ref name=Lambourne_2010/> | |||
Similarly, the [[four-momentum]] <math>p^{\mu}=\left(E/c,p_x,p_y,p_z\right)</math> of a body is equivalent to the energy-momentum tensor of said body. The element <math>p^0=E/c</math> 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 <math>p^1</math>, <math>p^2</math> and <math>p^3</math> correspond to the [[momentum]] of the body as a result of it travelling through space, written in vector notation as <math>\mathbf{p}</math>.<ref name=Lambourne_2010/> | |||
The [[electromagnetic field tensor]] is an example of a rank two contravariant tensor:<ref name=Lambourne_2010/> | |||
<math>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}</math> | |||
==See also== | |||
* [[four-vector]] | |||
* [[special relativity]] | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Four-Tensor}} | |||
[[Category:Tensors]] | |||
[[Category:Theory of relativity]] | |||
{{Relativity-stub}} | |||
Revision as of 22:03, 14 March 2013
Four-tensor is a frequent abbreviation for a tensor in a four-dimensional spacetime.[1]
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.[1]
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 dimensions of length). The remaining components of the four-displacement form the spatial displacement vector .[1]
Similarly, the four-momentum of a body is equivalent to the energy-momentum tensor of said body. The element 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 , and correspond to the momentum of the body as a result of it travelling through space, written in vector notation as .[1]
The electromagnetic field tensor is an example of a rank two contravariant tensor:[1]
See also
References
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