# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 4 Conclusions
See also Table 3. Two types of reference frames must be invoked. First, a fiducial frame \(\Theta\) is necessary to cover an extended region of spacetime with a single coordinate system. (On \(\mathbb{M}\) and \(\mathbb{G}\), as addressed in this paper, the fiducial frame is a global inertial one in which the paradigmatic forces of electromagnetism and gravity can be defined respectively.) Second, a comoving frame \(\Theta\) is also necessary, in order to define quantities related by thermodynamics, so that a closed system of equations can be obtained. Defining and working with quantities related to these two frames does not require use of components and changes of coordinates; use of coordinate-free geometric notation is fully viable, and even conceptually simplifying, for this purpose. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Description** & **References** \\ \hline \(\mathcal{M}\) & Generic spacetime & Section 2.1 \\  & (four-dimensional manifold) & \\ \(\mathbb{M}\) & Minkowski spacetime & Sections 2.2 and 2.3 \\  & (affine; pseudo-Riemann metric embodying & \\  & absolute light cones) & \\ \(\mathbb{G}\) & Galilei/Newton spacetime & Sections 2.2 and 2.4 \\  & (affine; fiber bundle embodying absolute time) & \\ \(\Theta\) & Fiducial global inertial reference frame & Section 2.1 \\ \((x^{\mu})\) & Coordinates on \(\mathbb{M}\), \(\mathbb{G}\) according to \(\Theta\) & Section 2.1 \\  & (global inertial spacetime coordinates) & \\ \(t\) & Time coordinate \(x^{0}\) according to \(\Theta\) & Section 2.1 \\  & (relative on \(\mathbb{M}\), absolute on \(\mathbb{G}\)) & \\ \(\mathcal{S}_{t}\) & Hypersurfaces of simultaneity & Section 2.1 \\  & (level surfaces of \(t\)) & \\  & (Euclid ’position space’ on \(\mathbb{M}\) and \(\mathbb{G}\)) & \\  & (relative on \(\mathbb{M}\), absolute on \(\mathbb{G}\)) & \\ \((x^{i})\) & Coordinates on \(\mathcal{S}_{t}\) according to \(\Theta\) & Section 2.1 \\  & (rectangular position space coordinates) & \\ \(t\) & Time one-form & \\  & (relative on \(\mathbb{M}\), absolute on \(\mathbb{G}\)) & \\  & (Invariant under Galilei transformations on \(\mathbb{G}\)) & \\ \(\boldsymbol{w}\) & Four-velocity vector of \(\Theta\) observers & Section 2.1; \\ \(\gamma\), \(\overrightarrow{\gamma}\) & Metric and inverse on \(\mathcal{S}_{t}\) & \\  & (flat on \(\mathbb{M}\) and \(\mathbb{G}\)) & \\ \(g\), \(\overrightarrow{\mathfrak{X}}\) & Metric and inverse on \(\mathbb{M}\) & \\  & (Invariant under Lorentz transformations) & \\ \(\delta\) & Identity tensor on \(\mathbb{M}\) and \(\mathbb{G}\) & \\  & ( \((1,1)\); Kronecker delta) & \\ \(\gamma\), \(\overrightarrow{\gamma}\), \(\overrightarrow{\gamma}\), \(\overrightarrow{\gamma}\) & Projection tensors to \(\mathcal{S}_{t}\) on \(\mathbb{M}\) and \(\mathbb{G}\) & \\  & ( \((0,2)\), \((1,1)\), \((1,1)\), \((2,0)\) ) & \\  & (\(\overrightarrow{\gamma}\) invariant under Galilei transformations on \(\mathbb{G}\)) & \\ \(\ddot{\Theta}\) & Accelerated reference frame & Section 3.1 \\  & (Often associated with the fluid) & Section 3.2 \\ \(\boldsymbol{U}\) & Four-velocity vector of \(\ddot{\Theta}\) observers & Section 3.1 \\  & (Often the four-velocity of the fluid) & Section 3.2 \\ \(\boldsymbol{h}\) & Projector orthogonal to \(\boldsymbol{U}\) on \(\mathbb{M}\) & Equation (110) on \(\mathbb{M}\) \\  & ( \(\overrightarrow{\gamma}\) on \(\mathbb{G}\) plays a role analogous to \(\overrightarrow{\boldsymbol{h}}\) on \(\mathbb{M}\)) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric entities characterizing \(3+1\) Minkowski and Galilei/Newton spacetimes.