# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Minkowski Spacetime
Again, the arrow is not necessary when indices are used, as the index position is sufficient to distinguish, for example, \(A_{\mu\
u}\) from \(A^{\mu\
u}=g^{\mu\mu}A_{\alpha\
u}\) and from \(A^{\mu\
u}=g^{\mu\alpha}g^{\
u\beta}A_{\alpha\beta}\). Note that, in this notation, the inverse metric is denoted \(\overrightarrow{\boldsymbol{g}}\). Moreover, the metric with a single index raised is \(\overrightarrow{\boldsymbol{g}}=\delta\), that is, the \((1,1)\) identity tensor \(\delta\) with components \(\delta^{\mu}{}_{\
u}\) of the Kronecker delta in any basis. Finally, denote by \(\overleftarrow{\boldsymbol{A}}\) the tensor with the last index raised associated by metric duality with a multilinear form \(\boldsymbol{A}\), for instance \(A_{\mu}{}^{\
u}=g^{\
u\alpha}A_{\mu\alpha}\). The Minkowski metric allows an interpretation of the affine parameter \(\tau\) of the particle four-velocity discussed in Section 2.1. Consider an infinitesimal displacement tangent to a timelike particle worldline (i.e., one whose tangent vectors are timelike at each point on the curve). On the one hand, Equation (14) yields  
\[\sqrt{-\mathrm{d}\ell^{2}}=\frac{c\,\mathrm{d}t}{\Lambda_{\boldsymbol{v}}}, \tag{17}\]  
where  
\[\frac{1}{\Lambda_{\boldsymbol{v}}}=\sqrt{1-\frac{\gamma\left(\boldsymbol{v}, \boldsymbol{v}\right)}{c^{2}}}=\sqrt{1-\frac{\gamma_{ij}v^{i}v^{j}}{c^{2}}} \tag{18}\]  
defines the Lorentz factor \(\Lambda_{\boldsymbol{v}}\) in terms of the instantaneous particle three-velocity. On the other hand, in a (possibly accelerated) 'comoving' reference frame carried along with the particle, this displacement involves no change in the spatial coordinates; it is purely an interval in the comoving time coordinate. Let the \(\tau\) affinely parametrizing the particle worldline be this 'proper time' coordinate; then, since \(\mathrm{d}\ell^{2}\) is a scalar, \(\sqrt{-\mathrm{d}\ell^{2}}=c\,\mathrm{d}\tau\), and  
\[\mathrm{d}t=\Lambda_{\boldsymbol{v}}\,\mathrm{d}\tau. \tag{19}\]  
Thus, the particle four-velocity of Equation (6) becomes  
\[\left[u^{\mu}\right]_{\Theta}=\Lambda_{\boldsymbol{v}} \begin{bmatrix}1\\ v^{i}\end{bmatrix} _{\Theta} \tag{20}\]  
as reckoned by \(\Theta\). The scalar product of a four-velocity with itself is readily seen to be \(g\left(\boldsymbol{u},\boldsymbol{u}\right)=-c^{2}\), in units with \(c=1\), it is a timelike unit vector. That \(c\) is a maximum speed is indicated by the expression for the Lorentz factor in Equation (18). Turn next to the particle four-momentum \(\boldsymbol{p}\). For a particle of mass \(m>0\), \(\boldsymbol{p}=mu\) satisfies \(g\left(\boldsymbol{p},\boldsymbol{p}\right)=-m^{2}c^{2}\) and has components  
\[\left[p^{\mu}\right]_{\Theta}= \begin{bmatrix}\epsilon_{\boldsymbol{q}}/c^{2} \\ q^{i}\end{bmatrix} _{\Theta}, \tag{21}\]with particle energy \(\epsilon_{\mathbf{q}}=\Lambda_{\mathbf{v}}\,mc^{2}\) and three-momentum \(q^{i}=\Lambda_{\mathbf{v}}\,mv^{i}\) measured by \(\Theta\). Note that \(\epsilon_{\mathbf{q}}=c\sqrt{m^{2}c^{2}+\gamma\left(\mathbf{q},\mathbf{q}\right)}\), and that \(\mathbf{q}\) is a three-vector on \(\mathcal{S}_{t}\), whose push-forward onto \(\mathbb{M}\) (denoted by the same symbol \(\mathbf{q}\)) has components  
\[\left[q^{\mu}\right]_{\Theta}= \begin{bmatrix}0\\ q^{i}\end{bmatrix} _{\Theta} \tag{22}\]  
according to \(\Theta\). The four-momentum of a massless particle is null, i.e., \(\mathbf{g}\left(\mathbf{p},\mathbf{p}\right)=0\). Its worldline cannot be parametrized by a proper time; instead, its affine parameter \(\lambda\) must be scaled such that  
\[\left[p^{\mu}\right]_{\Theta}=\left[\frac{\mathrm{d}X^{\mu}}{\mathrm{d} \lambda}\right]_{\Theta}= \begin{bmatrix}\epsilon_{\mathbf{q}}/c^{2}\\ q^{i}\end{bmatrix} _{\Theta}, \tag{23}\]  
with particle energy \(\epsilon_{\mathbf{q}}=c\sqrt{\gamma\left(\mathbf{q},\mathbf{q}\right)}\).