# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Minkowski Spacetime
For later comparison with Galilei/Newton spacetime, it is worth remembering that the component \(p^{0}\) is not the energy itself, but the inertia \(\epsilon_{\mathbf{q}}/c^{2}\), which in Minkowski spacetime varies with the particle momentum. A spacetime diagram of \(\mathbb{M}\) as charted by the inertial frame \(\Theta\) (red), with \(c=1\), is shown in Figure 1. By convention, the worldlines of the fiducial observers--relative to whose reference frame, in this case \(\Theta\), measurements are made--are vertical, and the hyperplanes of simultaneity of \(\Theta\) are horizontal. According to this convention, a speed measured by \(\Theta\) corresponds to the slope of a worldline relative to vertical. With \(c=1\), light rays are at \(45^{\circ}\), midway between the \(\Theta\) observer worldlines and their hyperplanes of simultaneity. The future null cone of an event \(\mathcal{A}\) consists of all null vectors \(\mathbf{a}\) with time component \(a^{0}>0\); it bounds (from 'below') all events that can be affected by \(\mathcal{A}\). Similarly, a past null cone bounds (from 'above') all events that can influence \(\mathcal{A}\). The future and past of \(\mathcal{A}\) are separated by a region of spacelike displacements from \(\mathcal{A}\), which can neither influence nor be influenced by \(\mathcal{A}\). A particle with mass \(m>0\) has a timelike worldline; that \(c\) is a maximum speed corresponds geometrically to the requirement that a worldline passing through \(\mathcal{A}\) cannot pass outside the past and future null cones of \(\mathcal{A}\). The future and past null cones of the origin are also shown in Figure 1 (black). Minkowski spacetime is invariant under Poincare transformations. These leave Equation (14) in the same mathematical form, including the metric components as given in Equations (14)-(16), and therefore preserve the causal structure. The Poincare group includes 10 continuous symmetries: four spacetime translations, three position space rotations, and three pseudo-rotations--the familiar Lorentz boosts that mix time and space. The discrete symmetries are spatial inversion (mirror reflection or parity reversal) and time reversal. Lorentz boosts are transformations between inertial frames in uniform relative motion, with different families of inertial observers. An inertial frame \(\Theta^{\prime}\) (blue) in relative motion to \(\Theta\) is also shown in Figure 1, in particular the worldlines of a few of its observers and a few of its surfaces of simultaneity. Some of the remarkable aspects of Minkowski spacetime are discernible. For instance, the relativity of simultaneity is obvious in the non-coincidence of the hyperplanes of simultaneity \(\mathcal{S}\) and \(\mathcal{S}^{\prime}\). Length contraction can be seen, for example, in the fact that in the hyperplane \(\mathcal{S}_{t=0}\) of \(\Theta\), the event labeled \(x^{\prime}=2\) by \(\Theta^{\prime}\) has a coordinate \(x<2\) according to \(\Theta\). An example of time dilation can be seen by noting that the time interval experienced by an \(\Theta^{\prime}\) observer traveling along \(x^{\prime}=0\) from the origin to \(t^{\prime}=2\) corresponds to a time interval \(t>2\) as reckoned in \(\Theta\). The principle of special relativity is that the laws of physics must take the same mathematical form in any global inertial frame. As applied to \(\mathbb{M}\), this means invariance under Poincare transformations. For instance, the laws of physics as expressed in \(\Theta^{\prime}\), whose charting of \(\mathbb{M}\) is shown in Figure 2--a Lorentz boost or pseudo-rotation of Figure 1--must have the same form as in \(\Theta\). (That the null cones remain unchanged under this transformation reflects the invariance of the Maxwell equations under Lorentz boosts, so that the speed of light is the same for all inertial observers.) This requirement is satisfied mathematically by expressing the laws of physics in terms of geometric objects, e.g., tensors on \(\mathbb{M}\).