1.6.1 Tensorial Invariance

Tensorial Invariance ensures mathematical consistency under coordinate transformations, preserving physical laws across different reference frames.

Tensorial Invariance is the property of a tensor whose component array is numerically identical in every basis admitted by a specified group of transformations, distinguishing such a tensor from the general case in which components change value from basis to basis while the underlying abstract object stays fixed. Where the foundations of tensor invariance explain why an abstract tensor is basis-independent even though its components are not, tensorial invariance singles out the special tensors for which the components themselves also fail to change, making them canonical fixed points of the transformation law.

Defining an Invariant Tensor

The Fixed-Component Condition

A tensor T is called invariant under a group G of basis changes if, for every transformation in G, applying the ordinary tensor transformation law to its components returns exactly the same numbers it started with.

T_{' j_{1} \dots}^{' i^{1} \dots} = T_{j_{1} \dots}^{i^{1} \dots} for every transformation in G

Invariance Relative to a Group

Whether a given tensor counts as invariant depends on which group G is specified. A tensor invariant under the full general linear group is automatically invariant under every subgroup of it, but a tensor invariant only under a restricted subgroup, such as the rotation group, need not be invariant under the larger group of all linear changes of basis.

Canonical Examples

The Kronecker Delta

The Kronecker delta, the type (1, 1) tensor equal to 1 when its two indices coincide and 0 otherwise, is invariant under the full general linear group, since it represents the identity map on the vector space, and the identity map commutes with every change of basis.

δ_{j}^{i}' = A_{k}^{i} B_{j}^{l} δ_{l}^{k} = A_{k}^{i} B_{j}^{k} = δ_{j}^{i}

The Metric Tensor Under Isometries

A metric tensor is not, in general, invariant under arbitrary changes of basis, since a generic change of basis alters its components. It is invariant, however, under the subgroup of transformations that preserve it by definition, the isometries of the geometry it defines, such as orthogonal transformations for a Euclidean metric.

The Levi-Civita Symbol

The Levi-Civita symbol, totally antisymmetric with entries +1, -1, or 0, is invariant under the group of transformations with determinant equal to 1, since a general change of basis multiplies it by the determinant of the change-of-basis matrix, leaving it unchanged only when that determinant equals one.

ε_{'} i_{1} \dots i_{n} = det (A) ε_{i_{1} \dots i_{n}}

Isotropic Tensors

Definition

A tensor invariant under the full orthogonal group, meaning under every rotation and reflection of the vector space, is called isotropic. Isotropic tensors describe quantities with no preferred direction, and they play a distinguished role wherever a physical or geometric relation must not depend on the orientation of the coordinate axes used to express it.

Classification by Rank

Isotropic tensors are severely constrained: the only isotropic tensor of rank one is the zero vector, the only isotropic tensors of rank two are scalar multiples of the Kronecker delta, and the isotropic tensors of rank three, in three dimensions, are scalar multiples of the Levi-Civita symbol. Higher-rank isotropic tensors are built from sums of products of these elementary invariant tensors.

The Role of Invariant Tensors in Constructing Invariant Quantities

Contracting with Invariant Tensors Preserves Invariance

Contracting an arbitrary tensor with an invariant tensor, such as raising or lowering an index with an invariant metric, or forming a determinant using the invariant Levi-Civita symbol, produces a new quantity whose invariance properties are inherited directly from those of the invariant tensor used in the contraction.

s = δ_{i j} u^{i} v^{j}

Building Group-Compatible Physical Laws

Physical and geometric laws that must hold regardless of orientation are constructed by combining tensors exclusively through operations involving isotropic tensors, guaranteeing that the resulting expressions are themselves isotropic and therefore valid in every rotated frame.

Diagrammatic Summary

The diagram sets the ordinary case of a tensor, whose components change under transformation while the object stays fixed, against tensorial invariance proper, in which the components of tensors like the Kronecker delta or Levi-Civita symbol remain numerically identical in every basis reachable by the relevant group.