Section II.6.2

Tensor contraction, symmetrization, and alternation

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Problem II.6.2.1

Raising and lowering of indices. Let (V,g)(V, g) again be a Euclidean vector space, TT a tensor of type (p,q)(p, q) with coordinates Ti1i2…ipj1j2…jqT_{i_{1} i_{2} \ldots i_{p}}^{j_{1} j_{2} \ldots j_{q}}. Let us construct a tensor with coordinates ∑kgikTki2…ipj1j2…jq\sum_{k} g^{i k} T_{k i_{2} \ldots i_{p}}^{j_{1} j_{2} \ldots j_{q}}, and denote the index ii of the coordinates of the resulting tensor by i1i_{1}, and then set

Ti2…ipi1j1j2…jq=∑kgi1kTki2…ipj1j2…jq T_{i_{2} \ldots i_{p}}^{i_{1} j_{1} j_{2} \ldots j_{q}}=\sum _{k} g^{i_{1} k} T_{k i_{2} \ldots i_{p}}^{j_{1} j_{2} \ldots j_{q}}

The tensor on the left-hand side of this equality is said to be the result of applying to TT the operation of raising the index i1i_{1}. The operation of lowering an index by means of the metric tensor GG is defined analogously. For example,

Ti1…ipj2j1j3…jq=∑kgj2kTi1…ipj1kj3…jq T_{i_{1} \ldots i_{p} j_{2}}^{j_{1} j_{3} \ldots j_{q}}=\sum _{k} g_{j_{2} k} T_{i_{1} \ldots i_{p}}^{j_{1} k j_{3} \ldots j_{q}}

are the coordinates of the tensor obtained by lowering the index j2j_{2} of the tensor TT to the last place in the sequence of lower indices.

In general, raising the ss-th index and lowering the tt-th index are linear maps

Tpq(V)→Tp−1q+1,Tpq→Tp+1q−1 \mathbb {T}_{p}^{q}(V) \rightarrow \mathbb {T}_{p-1}^{q+1}, \quad \mathbb {T}_{p}^{q} \rightarrow \mathbb {T}_{p+1}^{q-1}

applying which yields tensors with components

Ti1…is−1 is+1…ipis j1…jq+1,Ti1…ip itj1…jt−1 jt+1…jq. T_{i_1\ldots i_{s-1}\, i_{s+1}\ldots i_p}^{i_s\, j_1\ldots j_{q+1}}, \qquad T_{i_1\ldots i_p\, i_t}^{j_1\ldots j_{t-1}\, j_{t+1}\ldots j_q}.

Here we have used a "block arrangement" of the compound indices of the tensor coordinates. For example, a tensor T∈V⊗V∗T \in V \otimes V^{*} can be given by components denoted TijT_{i}^{j}, while T∈V⊗V∗⊗V⊗VT \in V \otimes V^{*} \otimes V \otimes V is given by components Ti1j1i2i3T_{i_{1}}{ }^{j_{1}}{ }_{i_{2} i_{3}}.

The operations of raising and lowering indices can be applied several times, including to the metric tensor itself.

Show that

gikgjlgkl=gij,gikgjlgkl=gij g^{i k} g^{j l} g_{k l}=g^{i j}, \quad g_{i k} g_{j l} g^{k l}=g_{i j}

Verify the validity of the equality gikxk=xig^{i k} x_{k}=x^{i}, meaning that the contravariant coordinates xix^{i} of a vector x\mathbf{x} are obtained by applying the index-raising operation to the covariant coordinates xix_{i} of the same vector.

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Problem II.6.2.2

Let dim⁡V>1\operatorname {dim} V>1. Show that T(V)\mathbb {T}(V) is a noncommutative ring without zero divisors. The only invertible elements in T(V)\mathbb {T}(V) are the nonzero scalars.

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Problem II.6.2.3

Compute explicitly the tensor invariants J2,J3J_{2}, J_{3} in Example 2.

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Remark.
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Example 2: the moment-of-inertia tensor J=(Jij)J = (J_{ij}) of a rigid body rotating about a fixed point o˙\dot{o} is the order-3 symmetric matrix with JiiJ_{ii} the axial moment of inertia about the axis ei\mathbf{e}_i and JijJ_{ij}, i≠ji \neq j, the centrifugal moments of inertia taken with the opposite sign. If the body consists of point masses mkm_k at positions [xk,yk,zk][x_k, y_k, z_k], 1⩽k⩽n1 \leqslant k \leqslant n (or a continuous mass distribution, with the sums replaced by integrals), then

J=∥∑kmk(yk2+zk2)−∑kmkxkyk−∑kmkxkzk−∑kmkxkyk∑kmk(xk2+zk2)−∑kmkykzk−∑kmkxkzk−∑kmkykzk∑kmk(xk2+yk2)∥. J=\left\Vert \begin{smallmatrix} \sum _{k} m_{k}(y_{k}^{2}+z_{k}^{2}) & -\sum _{k} m_{k} x_{k} y_{k} & -\sum _{k} m_{k} x_{k} z_{k} \\ -\sum _{k} m_{k} x_{k} y_{k} & \sum _{k} m_{k}(x_{k}^{2}+z_{k}^{2}) & -\sum _{k} m_{k} y_{k} z_{k} \\ -\sum _{k} m_{k} x_{k} z_{k} & -\sum _{k} m_{k} y_{k} z_{k} & \sum _{k} m_{k}(x_{k}^{2}+y_{k}^{2}) \end{smallmatrix}\right\Vert .

Three tensor invariants are associated with JJ (unchanged under orthogonal change of frame):

J1=tr⁡J=2∑kmk(xk2+yk2+zk2),J3=det⁡J, J_{1}=\operatorname {tr} J=2\sum _{k} m_{k}(x_{k}^{2}+y_{k}^{2}+z_{k}^{2}), \qquad J_{3}=\operatorname {det} J,

and J2J_{2} is the sum of the three principal 2×22 \times 2 minors of JJ,

J2=∣J11J12J21J22∣+∣J11J13J31J33∣+∣J22J23J32J33∣. J_{2}=\begin{vmatrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{vmatrix}+\begin{vmatrix} J_{11} & J_{13} \\ J_{31} & J_{33} \end{vmatrix}+\begin{vmatrix} J_{22} & J_{23} \\ J_{32} & J_{33} \end{vmatrix}.