# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Global Inertial Frames  
Spacetime \(\mathcal{M}\) is a four-dimensional differentiable manifold whose points are called 'events'. Any four-dimensional differentiable manifold can be locally charted by coordinates \((x^{\mu})\in\mathbb{R}^{4}\), with Greek indices running over \(0,1,2,3\); call this a reference frame \(\Theta\). Thus, \(\Theta\) labels each event by four real coordinate values. Call \(t=x^{0}\) the time coordinate. It can be regarded as a scalar field on \(\mathcal{M}\), whose non-intersecting level surfaces \(\mathcal{S}_{t}\) are three-dimensional hypersurfaces. Let the remainingcoordinates \(\left(x^{i}\right)\)--with Latin indices running over \(1,2,3\)--chart the hypersurfaces \(\mathcal{S}_{t}\), and call them position space coordinates. The hypersurfaces \(\mathcal{S}_{t}\) constitute a foliation of \(\mathcal{M}\), and the coordinates as described are said to be adapted to the foliation. Define a time one-form  
\[\mathbf{t}=\mathbf{\
abla}t=\mathbf{\mathrm{d}}t \tag{1}\]  
as the spacetime gradient of the time coordinate \(t\) (which coincides with the natural basis one-form \(\mathbf{\mathrm{d}}t\) associated with \(t\)), exploiting the fact that no metric is needed to take the gradient of a scalar field. (Tensors are denoted with a boldface symbol; without indices, their type--e.g., vector \((1,0)\), bilinear form \((0,2)\), etc.--should be stated when the symbol is introduced.) This one-form \(\mathbf{t}\) has components  
\[\left[t_{\mu}\right]_{\varTheta}=\left[\frac{\partial t}{\partial x^{\mu}} \right]_{\varTheta}=\left[1\qquad 0\qquad 0\qquad 0\right]_{\varTheta} \tag{2}\]  
with respect to \(\varTheta\), i.e., the above-discussed coordinates adapted to the foliation. A metric is also not needed to define curves on \(\mathcal{M}\), and tangent vectors in terms of them; exploit this to represent particles in spacetime by their worldlines, which are curves on \(\mathcal{M}\) that intersect each \(\mathcal{S}_{t}\) once. Parametrize a particle worldline \(\mathcal{X}(\lambda)\) by an affine parameter \(\lambda\), and consider the four-vector  
\[\mathbf{p}=\frac{\mathrm{d}\mathcal{X}(\lambda)}{\mathrm{d}\lambda} \tag{3}\]  
tangent to \(\mathcal{X}(\lambda)\). Denote by the four functions \(X^{\mu}(\lambda)\) the coordinates assigned by \(\varTheta\) to the event on \(\mathcal{X}(\lambda)\) specified by a particular value of \(\lambda\); then,  
\[p^{\mu}=\frac{\mathrm{d}X^{\mu}}{\mathrm{d}\lambda} \tag{4}\]  
are the components of \(\mathbf{p}\) according to \(\varTheta\). Assuming the coordinate \(t\) has units of time, scale \(\lambda\) such that it has units of time/mass; then, \(\mathbf{t}(\mathbf{p})=\mathbf{t}\cdot\mathbf{p}=t_{\alpha}\,p^{\alpha}=p^{0}\) has units of mass, and is interpreted as the particle inertia as measured by \(\varTheta\). (In this paper, the dot operator \((\cdot)\) only represents a contraction of indices in which one is already up and the other already down. Note that it does not represent a scalar product, which is not available on Galilei/Newton spacetime; a scalar product will always be expressed explicitly in terms of a metric tensor. Summation over repeated indices is implied where one is up and one is down.) Call \(\mathbf{p}\) the particle four-momentum. For a particle of mass \(m>0\), define also its four-velocity \(\mathbf{u}\) in terms of its four-momentum \(\mathbf{p}\) by  
\[\mathbf{u}=\frac{\mathbf{p}}{m}. \tag{5}\]  
A particle has mass \(m>0\) if at any point along its worldline there exist reference frames in which it is at rest; in such a frame, the particle's inertia is its mass \(m\). Since \(\mathbf{p}\) is a four-vector and \(m\) is a scalar, \(\mathbf{u}\) is also a four-vector.