# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## Appendix A Transformation Properties of the Projector in Galilei/Newton Spacetime  
In Galilei/Newton spacetime, consider the manner in which the tensors \(\gamma\), \(\overrightarrow{\gamma}\), \(\overleftarrow{\gamma}\), and \(\overrightarrow{\overrightarrow{\gamma}}\)--discussed in the paragraph containing Equation (45)--behave under a Galilei transformation \(G\) between a frame \(\Theta\) and a frame \(\Theta^{\prime}\). Given components \(\gamma_{\mu\
u}\) of \(\gamma\) in \(\Theta\), the components in \(\Theta^{\prime}\) are  
\[\gamma_{\mu^{\prime}\
u^{\prime}}=G^{\alpha}_{\ \mu^{\prime}}G^{\beta}_{\ \
u^{ \prime}}\gamma_{\alpha\beta}. \tag{49}\]  
Expressing Equation (45) in block matrix form as  
\[\left[\gamma_{\mu\
u}\right]= \begin{bmatrix}0&0\\ 0&1\end{bmatrix} , \tag{50}\]  
and given a Galilei transformation matrix  
\[\left[G^{\mu}_{\ \mu^{\prime}}\right]= \begin{bmatrix}1&0\\ v&R\end{bmatrix}  \tag{51}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Description** & **References** \\ \hline \(N\) & Particle number density & Equation (87) on \(\mathbb{M}\), \\  & & Equation (88) on \(\mathbb{G}\) \\ \(\mathbf{S}\) & Three-momentum density & Equation (169), \\  & & Equations (170) and (172) on \(\mathbb{M}\), \\  & & Equations (171) and (173) on \(\mathbb{G}\) \\ \(E\) & Internal/kinetic energy density & Equation (179), \\  & & Equations (180) and (182) on \(\mathbb{M}\), \\  & & Equations (181) and (183) on \(\mathbb{G}\) \\ \(\mathbf{F}_{N}\) & Particle number flux & Equation (159) \\ \(\mathbf{F}_{\mathbf{S}}\) & Three-momentum flux & Equation (174), \\  & & Equations (175) on \(\mathbb{M}\), \\  & & Equations (176) on \(\mathbb{G}\) \\ \(\mathbf{F}_{E}\) & Internal/kinetic energy flux & Equation (184), \\  & & Equation (185) on \(\mathbb{M}\), \\  & & Equation (186) on \(\mathbb{G}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Densities and fluxes measured in the fiducial frame \(\Theta\) in terms of the fluid three-velocity \(V\) measured by \(\Theta\) and quantities determined by thermodynamics and constitutive relations in the comoving frame \(\Theta\). in which \(v\) is a \(3\times 1\) matrix of Galilei boost parameters and \(R\) is a \(3\times 3\) orthogonal rotation matrix (\(R^{T}=R^{-1}\)), Equation (A1) can be expressed in block matrix form as  
\[\left[\gamma_{\mu^{\prime}v^{\prime}}\right] = \begin{bmatrix}1&v^{T}\\ 0&R^{T}\end{bmatrix}\begin{bmatrix}0&0\\ 0&1\end{bmatrix}\begin{bmatrix}1&0\\ v&R\end{bmatrix} \] (A4) \[= \begin{bmatrix}v^{T}v&v^{T}R\\ R^{T}v&1\end{bmatrix} .\]  
The inverse Galilei transformation matrix is  
\[\left[G^{\mu^{\prime}}{}_{\mu}\right]= \begin{bmatrix}1&0\\ -R^{-1}v&R^{-1}\end{bmatrix} ,\] (A5)  
used to find \(\left(\overrightarrow{\gamma}\right)^{\mu^{\prime}}{}_{v^{\prime}}=G^{\mu^{ \prime}}{}_{\alpha}G^{\beta}{}_{v^{\prime}}\left(\overrightarrow{\gamma} \right)^{\alpha}{}_{\beta}\):  
\[\left[\left(\overrightarrow{\gamma}\right)^{\mu^{\prime}}{}_{v^ {\prime}}\right] = \begin{bmatrix}1&0\\ -R^{-1}v&R^{-1}\end{bmatrix}\begin{bmatrix}0&0\\ 0&1\end{bmatrix}\begin{bmatrix}1&0\\ v&R\end{bmatrix} \] (A6) \[= \begin{bmatrix}0&0\\ R^{-1}v&1\end{bmatrix} .\]  
A similar calculation for \(\left(\overleftarrow{\gamma}\right)_{\mu^{\prime}}{}^{v^{\prime}}=G^{\alpha}{}_ {\mu^{\prime}}G^{\
u^{\prime}}{}_{\beta}\left(\overleftarrow{\gamma}\right) _{\alpha}{}^{\beta}\) yields  
\[\left[\left(\overleftarrow{\gamma}\right)_{\mu^{\prime}}{}^{v^{ \prime}}\right] = \begin{bmatrix}1&v^{T}\\ 0&R^{T}\end{bmatrix}\begin{bmatrix}0&0\\ 0&1\end{bmatrix}\begin{bmatrix}1&-v^{T}R\\ 0&R\end{bmatrix} \] (A7) \[= \begin{bmatrix}0&v^{T}R\\ 0&1\end{bmatrix}