Four-dimensional, ``space-time'' structured
deformations

Let $\mathcal{E}$ and $\mathcal{V}$ denote a three-dimensional Euclidean point space and its translation space, respectively, and put $\mathcal{E}% ^{(4)}:=\mathcal{E}\times \Bbb{R}$ and $\mathcal{V}^{(4)}:=\mathcal{V}\times \Bbb{R}$. The objects $\mathcal{E}^{(4)}$ and $\mathcal{V}^{(4)}$ then define a four-dimensional Euclidean point space that we view as describing non-relativistic, ``space-time''. Because our study of structured deformations is applicable in any finite-dimensional Euclidean point space, we may consider for each piecewise fit region $\mathcal{A}\subset \mathcal{E}$ and each T>0 the piecewise fit region $\mathcal{A}\times (0,T)\subset \mathcal{E}^{(4)}$ and let $(\kappa ^{(4)},g^{(4)},G^{(4)})\in Std(\mathcal{A% }\times (0,T))$ be given. We call $(\kappa ^{(4)},g^{(4)},G^{(4)})$ a space-time structured deformation, and we note that the pair $(\kappa ^{(4)},g^{(4)})\in Sid(\mathcal{A}\times (0,T))$ is a simple deformation. Moreover, $G^{(4)}:\mathcal{A}\times (0,T)\backslash \kappa ^{(4)}\rightarrow Lin\mathcal{V}^{(4)}$ is a tensor field, so that each of its values G⁽⁴⁾(x,t) for $(x,t)\in \mathcal{A}\times (0,T)$ has the block form

$$G^{(4)}(x,t)=\left[ \begin{array}{cc} G^{11}(x,t) & G^{12}(x... ...x,t) & G^{22}(x,t) \end{array}\right] \in Lin\mathcal{V}^{(4)}$$ (1)

with

 

$$\in Lin\mathcal{V},\,\,\quad G^{12}(x,t)\in Lin(\Bbb{R},% \mathcal{V})\approx \mathcal{V}$$ G¹¹(x,t)

$$\,G^{21}(x,t) \in Lin(\mathcal{V},\Bbb{R})\approx \mathcal{V},\,\,\quad G^{22}(x,t)\in Lin(\Bbb{R},\Bbb{R})\approx \Bbb{R}.$$ (2)

Similarly, $\nabla ^{(4)}g^{(4)}$, the gradient of the transplacement $% g^{(4)}=(g,\tau ):\mathcal{A}\times (0,T)\rightarrow \mathcal{E}^{(4)}$, gives rise to the block form

$$\nabla ^{(4)}g^{(4)}(x,t)=\left[ \begin{array}{cc} \nabla g(... ... & \dot{\tau}(x,t) \end{array}\right] \in Lin\mathcal{V}^{(4)}$$ (3)

with

 

$$\nabla g(x,t) \in Lin\mathcal{V},\,\,\quad \dot{g}(x,t)\in Lin(\Bbb{R},% \mathcal{V})\approx \mathcal{V}$$

$$\nabla \tau (x,t) \in Lin(\mathcal{V},\Bbb{R})\approx \mathcal{V}% ,\,\,\quad \dot{\tau}(x,t)\in Lin(\Bbb{R},\Bbb{R})\approx \Bbb{R}.$$ (4)

Of course, the Approximation Theorem can be applied to obtain a sequence of ``space-time'' simple deformations that converges to the space-time structured deformation $(\kappa ^{(4)},g^{(4)},G^{(4)}).$ Henceforth, the symbol `` $\,\nabla \,$'' denotes differentiation with respect to the spatial variable ``x'', and the superimposed `` $\,\cdot \,$'' denotes differentiation with respect to the time variable ``t''.

We do not pursue the details of the theory of ``space-time'' structured deformations, because of one shortcoming of this setting: in general, the time-coordinate $\tau (x,t)$ of the transplacement value g<sup>(4)</sup>(x,t) does not equal t. In other words, ``time-travel'' can occur through a space-time structured deformation. This is a shortcoming, because it makes it possible for a body occupying the reference configuration $\mathcal{A}$to disappear during an interval of time and then reappear. Therefore, we shall develop our kinematical ideas by assuming the relation

$$\tau (x,t)=t\ \ {\rm for\ all }\ (x,t) \in \mathcal{A}\times (0,T).$$ (5)

This assumption leads us to notions of ``classical motion'', ``simple motion'', and ``structured motion'' as described in the next subsections.