Pseudotensor

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}^{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}

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

Here \hat{P}^{i_1\ldots i_q}_{\,j_1\ldots j_p}, P^{k_1\ldots k_q}_{l_1\ldots l_p} 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.

The Christoffel symbols of an affine connection on a manifold can be thought of as the correction term to the total derivative of a coordinate expression of a vector field that renders the vector field's covariant derivative. While the affine connection itself doesn't depend on the choice of coordinates, its Christoffel symbols do, making them a pseudotensor quantity.

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