3.4 KiB
Tensor transformations
We have a n \in \mathbb{N}
finite dimensional vector space V
such that \dim V = n
, with a basis \{\mathbf{e}_i\}_{i=1}^n,
a corresponding dual space V^*
with a basis \{\mathbf{\hat e}^i\}_{i=1}^n
and a pseudo inner product \bm{g}
on V.
Let us introduce a different basis \{\mathbf{f}_i\}_{i=1}^n
of V
with a corresponding dual basis \{\mathbf{\hat f}^i\}_{i=1}^n
of V^*
which are related to the former basis \{\mathbf{e}_i\}_{i=1}^n
by
\mathbf{f}_j = A^i_j \mathbf{e}_i,
so that \mathbf{\hat e}^i = A^i_j \mathbf{\hat f}^j
.
Transformation of tensors
Recall from the section of tensor-formalism that a holor depends on the chosen basis, but the corresponding tensor itself does not. This implies that holors transform in a particular way under a change of basis, which is characteristic for tensors.
Theorem 1: let
\mathbf{T} \in \mathscr{T}^p_q(V)
be a tensor withp=q=1
without loss of generality andB = A^{-1}
. Then\mathbf{T}
may be decomposed into
\begin{align*} \mathbf{T} &= T^i_j \mathbf{e}_i \otimes \mathbf{\hat e}^j, \ &= \overline T^i_j \mathbf{f}_i \otimes \mathbf{\hat f}^j, \end{align*}
with the holors related by
\overline T^i_j = B^i_k A^j_l T^k_l.
??? note "Proof:"
Will be added later.
The homogeneous nature of the tensor transformation implies that a holor equation of the form T^i_j = 0
holds relative to any basis if it holds relative to a particular one.
Transformation of volume forms
Lemma 1: let
(V, \bm{\mu})
be a vector space with an oriented volume form with
\begin{align*} \bm{\mu} &= \mu_{i_1 \dots i_n} \mathbf{\hat e}^{i_1} \otimes \cdots \otimes \mathbf{\hat e}^{i_n}, \ &= \overline \mu_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \cdots \otimes \mathbf{\hat f}^{i_n}, \end{align*}
then we have
\overline \mu_{j_1 \dots j_n} = A^{i_1}{j_1} \cdots A^{i_n}{j_n} \mu_{i_1 \dots i_n} = \mu_{j_1 \dots j_n} \det (A).
??? note "Proof:"
Will be added later.
Then \det(A)
is the volume scaling factor of the transformation with A
. So that if \bm{\mu}(\mathbf{e}_1, \dots, \mathbf{e}_n) = 1
, then \bm{\mu}(\mathbf{f}_1, \dots, \mathbf{f}_n) = \det(A).
Theorem 2: let
(V, \bm{\mu})
be a vector space with an oriented volume form with
\begin{align*} \bm{\mu} &= \mu_{i_1 \dots i_n} \mathbf{\hat e}^{i_1} \otimes \cdots \otimes \mathbf{\hat e}^{i_n}, \ &= \overline \mu_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \cdots \otimes \mathbf{\hat f}^{i_n}, \end{align*}
and if we define
\overline \mu_{i_1 \dots i_n} \overset{\text{def}}{=} \frac{1}{\det (A)} A^{j_1}{i_1} \cdots A^{j_n}{i_n} \mu_{j_1 \dots j_n},
then
\mu_{i_1 \dots i_n} = \overline \mu_{i_1 \dots i_n} = [i_1, \dots, i_n]
is an invariant holor.
??? note "Proof:"
Will be added later.
Transformation of Levi-Civita form
Theorem 3: let
\bm{\epsilon} \in \bigwedge_n(V)
be the Levi-Civita tensor with
\begin{align*} \bm{\epsilon} &= \epsilon_{i_1 \dots i_n} \mathbf{\hat e}^{i_1} \otimes \cdots \otimes \mathbf{\hat e}^{i_n}, \ &= \overline \epsilon_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \cdots \otimes \mathbf{\hat f}^{i_n}, \end{align*}
then
\epsilon_{i_1 \dots i_n} = \overline \epsilon_{i_1 \dots i_n}
is an invariant holor.
??? note "Proof:"
Will be added later.