# 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

${\hat {P}}_{\,j_{1}\ldots j_{p}}^{i_{1}\ldots i_{q}}=(-1)^{A}A^{i_{1}}{}_{k_{1}}\cdots A^{i_{q}}{}_{k_{q}}B^{l_{1}}{}_{j_{1}}\cdots B^{l_{p}}{}_{j_{p}}P_{l_{1}\ldots l_{p}}^{k_{1}\ldots k_{q}}$ under a change of basis.

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.

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.