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| In [[physics]] and [[mathematics]], a '''pseudotensor''' is usually a quantity that transforms like a [[tensor]] under an orientation-preserving coordinate transformation (''e.g.'', a [[proper rotation]]), but additionally changes sign under an orientation reversing coordinate transformation (''e.g.'', an [[improper rotation]], which is a transformation that can be expressed as a proper rotation followed by reflection).
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| There is a second meaning for '''pseudotensor''', restricted to [[general relativity]]; tensors obey strict transformation laws, whilst pseudotensors are not so constrained. Consequently the form of a pseudotensor will, in general, change as the frame of reference is altered. An equation which holds in a frame containing pseudotensors will not necessarily hold in a different frame; this makes pseudotensors of limited relevance because equations in which they appear are not [[Covariance and contravariance|invariant]] in form.
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| ==Definition==
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| Two quite different mathematical objects are called a pseudotensor in different contexts.
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| The first context is essentially a tensor multiplied by an extra sign factor, such that the pseudotensor changes sign under reflections when a normal tensor does not. According to one definition, a pseudotensor '''P''' of the type (''p'',''q'') is a geometric object whose components in an arbitrary basis are enumerated by (''p'' + ''q'') indices and obey the transformation rule
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| :<math>\hat{P}^{i_1\ldots i_q}_{\,j_1\ldots j_p} =
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| (-1)^A A^{i_1} {}_{k_1}\cdots A^{i_q} {}_{k_q}
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| B^{l_1} {}_{j_1}\cdots B^{l_p} {}_{j_p}
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| P^{k_1\ldots k_q}_{l_1\ldots l_p}</math>
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| under a change of basis.<ref>Sharipov, R.A. (1996). Course of Differential Geometry, Ufa:Bashkir State University, Russia, p. 34, eq. 6.15. ISBN 5-7477-0129-0 [arXiv:math/0412421v1]</ref><ref>Lawden, Derek F. (1982). An Introduction to Tensor Calculus, Relativity and Cosmology. Chichester:John Wiley & Sons Ltd., p. 29, eq. 13.1. ISBN 0-471-10082-X</ref><ref>Borisenko, A. I. and Tarapov, I. E. (1968). Vector and Tensor Analysis with Applications, New York:Dover Publications, Inc. , p. 124, eq. 3.34. ISBN 0-486-63833-2</ref>
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| Here <math>\hat{P}^{i_1\ldots i_q}_{\,j_1\ldots j_p}, P^{k_1\ldots k_q}_{l_1\ldots l_p}</math> are the components of the pseudotensor in the new and old bases, respectively, <math>A^{i_q} {}_{k_q}</math> is the transition matrix for the [[contravariant]] indices, <math>B^{l_p} {}_{j_p}</math> is the transition matrix for the [[Covariance|covariant]] indices, and <math> (-1)^A = \mathrm{sign}(\det(A^{i_q} {}_{k_q})) = \pm{1}</math>.
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| This transformation rule differs from the rule for an ordinary tensor in the [[Intermediate treatment of tensors#Transformation rules|intermediate treatment]] only by the presence of the factor (−1)<sup>''A''</sup>.
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| The second context where the word "pseudotensor" is used is [[general relativity]]. In that theory, one cannot describe the energy and momentum of the gravitational field by an energy-momentum tensor. Instead, one introduces objects that behave as tensors only with respect to restricted coordinate transformations. Strictly speaking, such objects are not tensors at all. A famous example of such a pseudotensor is the [[Landau-Lifshitz pseudotensor]].
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| ==Examples==
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| On [[orientable manifold|non-orientable manifolds]], one cannot define a [[volume form]] due to the non-orientability, but one can define a [[volume element]], which is formally a [[density on a manifold|density]], and may also be called a ''pseudo-volume form,'' due to the additional sign twist (tensoring with the sign bundle).
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| A [[Integration by substitution|change of variable]]s in multi-dimensional integration is achieved by incorporation of a factor of the absolute value of the [[determinant]] of the [[Jacobian matrix and determinant|Jacobian matrix]]. The use of the absolute value introduces a sign-flip for improper coordinate transformations; as such, an [[integrand]] is an example of a pseudotensor density according to the first definition.
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| ==References==
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| {{reflist}}
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| ==See also==
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| *[[Tensor]]
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| *[[Tensor density]]
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| *[[General relativity]]
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| *[[Noether's theorem]]
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| *[[Pseudovector]]
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| *[[Variational principle]]
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| *[[Conservation law]]
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| *[[Action (physics)]]
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| ==External links==
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| *[http://mathworld.wolfram.com/Pseudotensor.html Mathworld description for pseudotensor].
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| [[Category:Tensors]]
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| [[Category:Tensors in general relativity]]
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Hi there, I am Andrew Berryhill. Her family life in Ohio but her husband desires them to transfer. I am presently a travel agent. Doing ballet is some thing she would never give up.
my web blog; clairvoyance (www.Dabuckchannel.com)