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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>
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| ==Syntax==
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| 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/>
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| ==Examples==
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| 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/>
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| 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/>
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| The [[electromagnetic field tensor]] is an example of a rank two contravariant tensor:<ref name=Lambourne_2010/>
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| <math>F^{\mu\nu} = \begin{pmatrix}
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| 0 & -E_x/c & -E_y/c & -E_z/c\\
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| E_x/c & 0 & -B_z & B_y\\
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| E_y/c & B_z & 0 & -B_x\\
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| E_z/c & -B_y & B_x & 0
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| \end{pmatrix}</math>
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| ==See also==
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| * [[four-vector]]
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| * [[special relativity]]
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Four-Tensor}}
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| [[Category:Tensors]]
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| [[Category:Theory of relativity]]
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| {{Relativity-stub}}
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Latest revision as of 07:12, 28 November 2014
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