Pseudotensor

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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).

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 invariant in form.

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 (p,q) is a geometric object whose components in an arbitrary basis are enumerated by (p + q) indices and obey the transformation rule

{\hat {P}}_{{\,j_{1}\ldots j_{p}}}^{{i_{1}\ldots i_{q}}}=(-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_{{l_{1}\ldots l_{p}}}^{{k_{1}\ldots k_{q}}}

under a change of basis.[1][2][3]

Here {\hat {P}}_{{\,j_{1}\ldots j_{p}}}^{{i_{1}\ldots i_{q}}},P_{{l_{1}\ldots l_{p}}}^{{k_{1}\ldots k_{q}}} are the components of the pseudotensor in the new and old bases, respectively, A^{{i_{q}}}{}_{{k_{q}}} is the transition matrix for the contravariant indices, B^{{l_{p}}}{}_{{j_{p}}} is the transition matrix for the covariant indices, and (-1)^{A}={\mathrm {sign}}(\det(A^{{i_{q}}}{}_{{k_{q}}}))=\pm {1}. This transformation rule differs from the rule for an ordinary tensor in the intermediate treatment only by the presence of the factor (−1)A.

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.

Examples

On 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, and may also be called a pseudo-volume form, due to the additional sign twist (tensoring with the sign bundle).

A change of variables in multi-dimensional integration is achieved by incorporation of a factor of the absolute value of the determinant of the 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.

References

  1. ↑ 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]
  2. ↑ 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
  3. ↑ 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

See also

External links

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