Indifferent Tensor Field An exercise in the continuum mechanics book by Gurtin says:
A spatial tensor field $A$ is indifferent if, during any change in observer, A transforms according to $A^{*}(x^{*},t) = Q(t)A(x,t)Q(t)^{T}$.
We have that $x^{*}(p,t) = Q(t)x(p,t)$.
The book does not give a definition for $A^*.$
What does "transforms according to" mean?
 A: Perhaps this will do for your wish for concreteness?
Let $\{\boldsymbol{e}_i, \boldsymbol{e}_j, \boldsymbol{e}_k\}$ be a basis for the space. Under a change of reference frame, this basis becomes
$$\{\boldsymbol{e}^*_i, \boldsymbol{e}^*_j, \boldsymbol{e}^*_k\}=\{\boldsymbol{Q}\boldsymbol{e}_i, \boldsymbol{Q}\boldsymbol{e}_j, \boldsymbol{Q}\boldsymbol{e}_k\}.$$
Suppose two different observers are attempting to study the same tensor field. In the reference frame defined by the basis $\{\boldsymbol{e}_i, \boldsymbol{e}_j, \boldsymbol{e}_k\}$ call this tensor field $\boldsymbol{A}$. In the other reference frame call this tensor field, $\boldsymbol{A}^*$. Then $\boldsymbol{A}$ and $\boldsymbol{A}^*$ can be written in components.
$$\boldsymbol{A} = A_{ij}\boldsymbol{e}_i\otimes\boldsymbol{e}_j,\\ \boldsymbol{A^*} = A^*_{ij}\boldsymbol{e}^*_i\otimes\boldsymbol{e}^*_j.$$
But using the relation for $\{\boldsymbol{e}^*_i, \boldsymbol{e}^*_j, \boldsymbol{e}^*_k\}$ in terms of the basis $\{\boldsymbol{e}_i, \boldsymbol{e}_j, \boldsymbol{e}_k\}$, then
\begin{align}
\boldsymbol{A}^* &= A^*_{ij}\boldsymbol{Q}\boldsymbol{e}_i\otimes\boldsymbol{Q}\boldsymbol{e}_j\\
&= \boldsymbol{Q}\left(A^*_{ij}\boldsymbol{e}_i\otimes\boldsymbol{e}_j\right)\boldsymbol{Q}^{T}. 
\end{align}
Now if the tensor field under study is indifferent, then by your definiton
\begin{align}
\boldsymbol{A}^* &= \boldsymbol{Q}\boldsymbol{A}\boldsymbol{Q}^{T}\\
&= \boldsymbol{Q}\left(A_{ij}\boldsymbol{e}_i\otimes\boldsymbol{e}_j\right)\boldsymbol{Q}^{T}.
\end{align}
Comparing the two ways of writing $\boldsymbol{A}^*$, it is seen that for indifferent tensor fields $A^*_{ij}=A_{ij}$.
A: It means the following:
Let $x^*$ be a new coordinate system related to $x$ by $x^*(p,t) = Q(t)x(p,t)$. We denote by $A^*$ the spacial vector field in the coordinate $x^*$. Then $A$ is called indifferent if for any change of coordinate system as above we have
$$A^*(x^*,t) = Q(t)A(x,t)Q(t)^T.$$
