Editing Pseudotensor (section)

==Definition==
Two quite different mathematical objects are called a pseudotensor in different contexts.

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 <math>(p, q)</math> is a geometric object whose components in an arbitrary basis are enumerated by <math>(p + q)</math>indices and obey the transformation rule
<math display=block>\hat{P}^{i_1\ldots i_q}_{\,j_1\ldots j_p} =
(-1)^A A^{i_1} {}_{k_1}\cdots A^{i_q} {}_{k_q}
B^{l_1} {}_{j_1}\cdots B^{l_p} {}_{j_p}
P^{k_1\ldots k_q}_{l_1\ldots l_p}</math> 
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>

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 [[Covariance and contravariance of vectors|contravariant]] indices, <math>B^{l_p} {}_{j_p}</math> is the transition matrix for the [[Covariance|covariant]] indices, and <math>(-1)^A = \mathrm{sign}\left(\det\left(A^{i_q} {}_{k_q}\right)\right) = \pm{1}.</math> 
This transformation rule differs from the rule for an ordinary tensor only by the presence of the factor <math>(-1)^A.</math>

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]].