Inphysics andmathematics, apseudotensor is usually a quantity that transforms like atensor under an orientation-preservingcoordinate transformation (e.g. aproper rotation) but additionally changes sign under an orientation-reversing coordinate transformation (e.g., animproper rotation), which is a transformation that can be expressed as a proper rotation followed byreflection. This is a generalization of apseudovector. To evaluate a tensor or pseudotensor sign, it has to becontracted with some vectors, as many as itsrank is, belonging to the space where the rotation is made while keeping the tensor coordinates unaffected (differently from what one does in the case of a base change). Under improper rotation a pseudotensor and a proper tensor of the same rank will have different sign which depends on the rank beingeven or odd. Sometimes inversion of the axes is used as an example of an improper rotation to see the behaviour of a pseudotensor, but it works only if vector space dimensions is odd otherwise inversion is a proper rotation without an additional reflection.

There is a second meaning forpseudotensor (and likewise forpseudovector), restricted togeneral relativity. Tensors obey strict transformation laws, but pseudotensors in this sense are not so constrained. Consequently, the form of a pseudotensor will, in general, change as theframe of reference is altered. An equation containing pseudotensors, such asstress–energy–momentum pseudotensors, which holds in one frame will not necessarily hold in a different frame. This makes pseudotensors of limited relevance because equations in which they appear are notinvariant in form.

Mathematical developments in the 1980s have allowed pseudotensors to be understood assections ofjet bundles.

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 pseudotensorP of the type${\displaystyle (p,q)}$ is a geometric object whose components in an arbitrary basis are enumerated by${\displaystyle (p+q)}$indices and obey the transformation rule$${\displaystyle {\hat{P}}_{j_1\dots j_p}^{i_1\dots i_q}=(-1{)}^A{A}^{i_1}{}_{k_1}\cdots A^{i_q}{}_{k_q}{B}^{l_1}{}_{j_1}\cdots B^{l_p}{}_{j_p}{P}_{l_1\dots l_p}^{k_1\dots k_q}}$$ under a change of basis.^[1][2][3]

Here${\displaystyle {\hat{P}}_{j_1\dots j_p}^{i_1\dots i_q},P_{l_1\dots l_p}^{k_1\dots k_q}}$ are the components of the pseudotensor in the new and old bases, respectively,${\displaystyle A^{i_q}{}_{k_q}}$ is the transition matrix for thecontravariant indices,${\displaystyle B^{l_p}{}_{j_p}}$ is the transition matrix for thecovariant indices, and${\displaystyle (-1{)}^A=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(det\left(A^{i_q}{}_{k_q}\right)\right)=\pm 1.}$ This transformation rule differs from the rule for an ordinary tensor only by the presence of the factor${\displaystyle (-1{)}^A.}$

The second context where the word "pseudotensor" is used isgeneral 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 theLandau–Lifshitz pseudotensor.

Onnon-orientable manifolds, one cannot define avolume form globally due to the non-orientability, but one can define avolume element, which is formally adensity, and may also be called apseudo-volume form, due to the additional sign twist (tensoring with the sign bundle). The volume element is a pseudotensor density according to the first definition.

Achange of variables in multi-dimensional integration may be achieved through the incorporation of a factor of the absolute value of thedeterminant of theJacobian matrix. The use of the absolute value introduces a sign change for improper coordinate transformations to compensate for the convention of keeping integration (volume) element positive; as such, anintegrand is an example of a pseudotensor density according to the first definition.

TheChristoffel symbols of anaffine connection on a manifold can be thought of as the correction terms to the partial derivatives of a coordinate expression of a vector field with respect to the coordinates to render it 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 according to the second definition.

• Action (physics) – Physical quantity of dimension energy × time
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• Noether's theorem – Statement relating differentiable symmetries to conserved quantities
• Pseudovector – Physical quantity that changes sign with improper rotation
• Variational principle – Scientific principles enabling the use of the calculus of variations

• Mathworld description for pseudotensor.

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