# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Global Inertial Frames
Reparametrize the worldline in terms of a new affine parameter such that \(\mathrm{d}\tau=m\,\mathrm{d}\lambda\), where the new parameter \(\tau\) has units of time; then  
\[\left[u^{\mu}\right]_{\varTheta}=\left[\frac{\mathrm{d}X^{\mu}}{\mathrm{d} \tau}\right]_{\varTheta}=\frac{\mathrm{d}t}{\mathrm{d}\tau}\left[\frac{1}{v^{i }}\right]_{\varTheta} \tag{6}\]  
are the components of \(\mathbf{u}\) reckoned by \(\varTheta\), where  
\[v^{i}=\frac{\mathrm{d}X^{i}}{\mathrm{d}t} \tag{7}\]are the components of a velocity three-vector \(\mathbf{v}\) on \(\mathcal{S}_{t}\). The push-forward of \(\mathbf{v}\) onto \(\mathcal{M}\), denoted with the same symbol, has components  
\[\left[\mathbf{v}^{\mu}\right]_{\Theta}= \begin{bmatrix}0\\ \mathbf{v}^{i}\end{bmatrix} _{\Theta} \tag{8}\]  
in coordinates (such as \(\Theta\)) adapted to the foliation; it is a four-vector on \(\mathcal{M}\) that is tangent to \(\mathcal{S}_{t}\). The leading factor \(\mathrm{d}t/\mathrm{d}\tau\) in Equation (6) cannot be specified further until additional details are given about a particular spacetime. The coordinate curves of a reference frame \(\Theta\) can be used to define the worldlines of a family of observers, with four-velocity field \(\mathbf{w}\), associated with \(\Theta\). The coordinate curves for coordinate \(x^{\mu}\) are those curves parametrized by \(x^{\mu}\), with the other coordinates \((x^{\
u},\,\
u\
eq\mu)\) held fixed. Given the coordinates \(\left(t,x^{i}\right)\) of reference frame \(\Theta\), the \(t\) coordinate curves--those of fixed \(\left(x^{i}\right)\)--are the worldlines of the \(\Theta\) observers. The four-velocity \(\mathbf{w}\) of these observers is proportional, by some factor \(a\), to the natural basis vector \(\mathbf{\partial}_{t}\) tangent to the \(t\) coordinate curves, so that \(\mathbf{w}\) has components  
\[\left[\mathbf{w}^{\mu}\right]_{\Theta}=a \begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix} _{\Theta} \tag{9}\]  
according to \(\Theta\). Compared with Equation (6), it is apparent that the \(\Theta\) observers are--rather obviously--at rest with respect to themselves. While what has been said so far pertains to spacetimes based on any four-dimensional differentiable manifold, Minkowski and Galilei/Newton spacetimes are based on a special type of differentiable manifold--an affine space--that provides some additional features. Prior to being endowed with additional structure, an affine space is invariant under general linear transformations and translations, admitting infinitely extended parallel lines. With additional structure to be discussed in Sections 2.2-2.4, the classes of linear transformations under which Minkowski and Galilei/Newton spacetimes are invariant are more restricted, but they retain invariance under spacetime translations, and admit infinitely extended parallel lines in the four dimensions of spacetime. This means that coordinates \(\left(t,x^{i}\right)\) of a reference frame \(\Theta\) can be chosen such that the entire spacetime is covered by a global coordinate chart, with the worldlines of \(\Theta\) observers being infinitely extended parallel lines. If the coordinates \(\left(x^{i}\right)\) are adapted to the foliation defined by level surfaces of \(t\), the fact that their coordinate curves are both tangent to the hypersurfaces \(\mathcal{S}_{t}\) and can be chosen to be infinitely extended parallel lines implies that the \(\mathcal{S}_{t}\) can be chosen to be infinite parallel affine hyperplanes. Begin the construction of what shall be called a 'global inertial' reference frame \(\Theta\) in Minkowski or Galilei/Newton spacetime by choosing its coordinate curves to be infinitely extended parallel lines, and the hypersurfaces \(\mathcal{S}_{t}\) to be parallel hyperplanes. The worldlines of the \(\Theta\) observers are'straight', and are therefore geodesics. Physically, their straightness corresponds to an absence of acceleration; hence, the designation of \(\Theta\) as a global inertial reference frame.