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 gains an additional sign flip under an orientation reversing coordinate transformation (e.g., an improper rotation, which is a transformation that can be expressed as an inversion followed by a proper rotation).

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.

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.

See also

• Tensor
• Tensor density
• General relativity
• Noether's theorem
• Pseudovector
• Variational principle
• Conservation law
• Action (physics)

External links

• Mathworld description for pseudotensor.