# Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3 + 1 Spacetime Perspective
## 2 Spacetime
### Minkowski Spacetime  
As just described in Section 2.2, Minkowski spacetime \(\mathbb{M}\) is a flat pseudo-Riemann manifold, adding to its affine structure an indefinite non-degenerate bilinear form--the Minkowski metric \(\boldsymbol{g}\). The metric defines a scalar product \(\boldsymbol{g}\left(\boldsymbol{a},\boldsymbol{b}\right)=\boldsymbol{g}\left( \boldsymbol{b},\boldsymbol{a}\right)\) of two vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\). The metric \(\boldsymbol{g}\) is called indefinite because the scalar product of a vector with itself need not be positive: a vector \(\boldsymbol{a}\) is spacelike if \(\boldsymbol{g}\left(\boldsymbol{a},\boldsymbol{a}\right)>0\), timelike if \(\boldsymbol{g}\left(\boldsymbol{a},\boldsymbol{a}\right)<0\), and null if \(\boldsymbol{g}\left(\boldsymbol{a},\boldsymbol{a}\right)=0\). This nomenclature arises by comparing the relative contributions of the time and position space components of a vector to its scalar product with itself. The canonical form of the Minkowski metric is displayed with respect to the Minkowski coordinates of a global inertial reference frame. The scalar product of an infinitesimal spacetime displacement \(\mathrm{d}\boldsymbol{\ell}=\mathrm{d}x^{\alpha}\,\partial_{\alpha}\) with itself is called the line element. In particular, let \(\left(x^{\mu}\right)=\left(t,x,y,z\right)\in\mathbb{R}^{4}\) be the coordinates of a global inertial frame \(\Theta\) on \(\mathbb{M}\); these are Minkowski coordinates if at every event the line element is expressed as  
\[\mathrm{d}\ell^{2} = \boldsymbol{g}\left(\mathrm{d}\boldsymbol{\ell},\mathrm{d} \boldsymbol{\ell}\right)\] \[= g_{\alpha\beta}\,\mathrm{d}x^{\alpha}\mathrm{d}x^{\beta}\] \[= -c^{2}\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z ^{2},\]  
where \(c\) is the speed of light, so that the matrix of components of \(\boldsymbol{g}\) has the canonical form  
\[\left[g_{\mu\
u}\right]_{\Theta}= \begin{bmatrix}-c^{2}&0_{j}\\ 0_{i}&\gamma_{ij}\end{bmatrix} _{\Theta}=\left[\eta_{\mu\
u}\right], \tag{14}\]  
where  
\[\left[\eta_{\mu\
u}\right]= \begin{bmatrix}-c^{2}&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{15}\]  
is the Minkowski matrix. Here, \(t\) is the time coordinate, corresponding to physical time according to \(\Theta\). The position space coordinates \(\left(x^{i}\right)=\left(x,y,z\right)\) chart the parallel hyperplanes \(\mathcal{S}_{t}\); these are hypersurfaces of simultaneity in inertial frame \(\Theta\). The Minkowski metric \(g\) is manifestly compatible with the three-metric \(\gamma\), previewed in Equation (12), that provides Euclid geometry on \(\mathcal{S}_{t}\); in geometric terms, \(\gamma\) is the pull-back of \(g\) onto \(\mathcal{S}_{t}\). The metric provides a duality between vectors and linear forms, which allows the raising and lowering of indices of tensors on \(\mathbb{M}\). This is made possible by the fact that \(g\) is non-degenerate, implying it has an inverse, whose components in Minkowski coordinates are  
\[\left[g^{\mu\
u}\right]_{\Theta}= \begin{bmatrix}-1/c^{2}&0^{j}\\ 0^{i}&\gamma^{ij}\end{bmatrix} _{\Theta}= \begin{bmatrix}-1/c^{2}&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} _{\Theta}. \tag{16}\]  
In coordinate-free notation, denote by an underbar the linear form \(\underline{a}\) that is associated by metric duality with a vector \(\boldsymbol{a}\). However, this is not necessary when indices are used, as the index position in \(a^{\mu}\) or \(a_{\mu}=g_{\mu\alpha}a^{\alpha}\) indicates whether the components are those of a vector or linear form, respectively. In a similar spirit, denote by \(\overrightarrow{\boldsymbol{A}}\) the tensor with first index raised associated by metric duality with a multilinear form \(\boldsymbol{A}\), and by \(\overrightarrow{\boldsymbol{A}}\) the tensor with the first and second indices raised.