History of Topics in Special Relativity/Four-current

w:Henri Poincaré (June 1905^{[R 1]}; July 1905, published 1906^{[R 2]}) showed that the four quantities related to charge density ${\displaystyle \rho }$ are connected by a Lorentz transformation:

${\displaystyle {\begin{matrix}\rho ,\ \rho \xi ,\ \rho \eta ,\ \rho \zeta \\\hline \rho ^{\prime }={\frac {k}{l^{3}}}\rho (1+\epsilon \xi ),\quad \rho ^{\prime }\xi ^{\prime }={\frac {k}{l^{3}}}\rho (\xi +\epsilon ),\quad \rho ^{\prime }\eta ^{\prime }={\frac {\rho \eta }{l^{3}}},\ \quad \rho ^{\prime }\zeta ^{\prime }={\frac {\rho \zeta }{l^{3}}}&({\text{June}})\\\rho ^{\prime }={\frac {k}{l^{3}}}(\rho +\epsilon \rho \xi ),\quad \rho '\xi ^{\prime }={\frac {k}{l^{3}}}(\rho \xi +\epsilon \rho ),\quad \rho ^{\prime }\eta ^{\prime }={\frac {1}{l^{3}}}\rho \eta ,\quad \rho ^{\prime }\zeta ^{\prime }={\frac {1}{l^{3}}}\rho \zeta '&({\text{July}})\\\left(k={\frac {1}{\sqrt {1-\epsilon ^{2}}}},\ l=1\right)\end{matrix}}}$

and in his July paper he further stated the continuity equation and the invariance of Jacobian D:^{[R 3]}

${\displaystyle {\begin{matrix}{\frac {d\rho ^{\prime }}{dt^{\prime }}}+\sum {\frac {d\rho ^{\prime }\xi ^{\prime }}{dx^{\prime }}}=0\\D_{1}^{'}={\frac {d\rho ^{\prime }}{dt^{\prime }}}+\sum {\frac {d\rho ^{\prime }\xi ^{\prime }}{dx^{\prime }}}=0,\ D_{1}={\frac {d\rho }{dt}}+\sum {\frac {d\rho \xi }{dx}}=0\end{matrix}}}$

Even though Poincaré didn't directly use four-vector notation in those cases, his quantities are the components of four-current (a).

Following Poincaré, w:Roberto Marcolongo defined the general Lorentz transformation ${\displaystyle \alpha ,\beta ,\gamma ,\delta }$ of the components of the four independent variables ${\displaystyle \mathbf {V} ,\varrho }$ and its continuity equation:^{[R 4]}

${\displaystyle {\begin{matrix}(\xi ,\eta ,\zeta )=\mathbf {V} ,\ (\xi ',\eta ',\zeta ')=\mathbf {V} '\\\hline \varrho '\xi '=\varrho \left(\alpha _{1}\xi +\beta _{1}\eta +\gamma _{1}\zeta -i\delta _{1}\right)\\\dots \\\varrho '=\varrho \left(\alpha _{4}\xi +\beta _{4}\eta +\gamma _{4}\zeta -i\delta _{4}\right)\\\hline {\frac {\partial \varrho '}{\partial t'}}+{\frac {\partial \varrho '\xi '}{\partial x'}}+{\frac {\partial \varrho '\eta '}{\partial y'}}+{\frac {\partial \varrho '\zeta '}{\partial z'}}=0\\(t=iu)\end{matrix}}}$

equivalent to the components of four-current (a), and pointed out its relation to the components ${\displaystyle \mathbf {J} ,\varphi }$ of the four-potential

${\displaystyle {\begin{matrix}\Box \mathbf {J} '_{x}=-4\pi \varrho '\xi '=-4\pi \varrho \left(\alpha _{1}\xi +\beta _{1}\eta +\gamma _{1}\zeta -i\delta _{1}\right),\dots \\\Box \mathbf {J} =-4\pi \varrho \mathbf {V,\ \Box \varphi } =-4\pi \rho ,\ \Box \varphi '=-4\pi \varrho '\end{matrix}}}$

equivalent to the components of Maxwell's equations (b).

w:Hermann Minkowski from the outset employed vector and matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product. In a lecture held in November 1907, published 1915, Minkowski defined the four-current in vacuum with ${\displaystyle \varrho }$ as charge density and ${\displaystyle {\mathfrak {v}}}$ as velocity:^{[R 5]}

${\displaystyle \left(\varrho _{1},\varrho _{2},\varrho _{3},\varrho _{4}\right)=(\varrho {\mathfrak {v}},\ i\varrho )}$

equivalent to (a), and the electric four-current in matter with ${\displaystyle \mathbf {i} }$ as current and ${\displaystyle \sigma }$ as charge density:^{[R 6]}

${\displaystyle (\sigma )=\left(\sigma _{1},\ \sigma _{2},\ \sigma _{3},\ \sigma _{4}\right)=(i_{x},i_{y},i_{z},\ i\sigma )}$

In another lecture from December 1907, Minkowski defined the “space-time vector current” and its Lorentz transformation^{[R 7]}

${\displaystyle {\begin{matrix}\left(\varrho \,{\mathfrak {w}}_{x},\ \varrho \,{\mathfrak {w}}_{y},\ \varrho \,{\mathfrak {w}}_{z},\ i\varrho \right)\Rightarrow \left(\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}\right)\\\hline \varrho '_{3}=x_{3}\cos \ i\psi +\varrho _{4}\sin \ i\psi ,\quad \varrho '_{4}=-\varrho _{3}\sin \ i\psi +\varrho _{4}\cos \ i\psi ,\quad \varrho '_{1}=\varrho _{1},\quad \varrho '_{2}=\varrho _{2}\\\varrho '{\mathfrak {w}}'_{z'}=\varrho \left({\frac {{\mathfrak {w}}_{z}-q}{\sqrt {1-q^{2}}}}\right),\quad \varrho '=\varrho \left({\frac {-q\,{\mathfrak {w}}_{z}+1}{\sqrt {1-q^{2}}}}\right),\quad \varrho '{\mathfrak {w}}'_{x'}=\varrho \,{\mathfrak {w}}_{x},\quad \varrho '{\mathfrak {w}}'_{y'}=\varrho \,{\mathfrak {w}}_{y},\\\hline -(\varrho _{1}^{2}+\varrho _{2}^{2}+\varrho _{3}^{2}+\varrho _{4}^{2})=\varrho ^{2}(1-{\mathfrak {w}}_{x}^{2}-{\mathfrak {w}}_{y}^{2}-{\mathfrak {w}}_{z}^{2})=\varrho ^{2}(1-{\mathfrak {w}}^{2})\end{matrix}}}$

equivalent to (a). In moving media and dielectrics, Minkowski more generally used the current density vector “electric current” ${\displaystyle {\mathfrak {s}}}$ which becomes ${\displaystyle {\mathfrak {s}}=\sigma {\mathfrak {E}}}$ in isotropic media:^{[R 8]}

${\displaystyle {\begin{matrix}\left({\mathfrak {s}}_{x},\ {\mathfrak {s}}_{y},\ {\mathfrak {s}}_{z},\ i\varrho \right)\Rightarrow \left(s_{1},\ s_{2},\ s_{3},\ s_{4}\right)\end{matrix}}}$

Following Minkowski, w:Max Born (1909) defined the “space-time vector of first kind” (four-vector) and its continuity equation^{[R 9]}

${\displaystyle {\begin{matrix}\left({\frac {\varrho w_{x}}{c}},\ {\frac {\varrho w_{y}}{c}},\ {\frac {\varrho w_{z}}{c}},\ i\varrho \right)\Rightarrow \left(\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}\right)\\{\frac {\partial \varrho w_{x}}{\partial x}}+{\frac {\partial \varrho w_{y}}{\partial y}}+{\frac {\partial \varrho w_{z}}{\partial z}}+{\frac {\partial \varrho }{\partial t}}=0\end{matrix}}}$

equivalent to (a), and pointed out its relation to Maxwell's equations as the product of the D'Alembert operator with the electromagnetic potential ${\displaystyle \Phi _{\alpha }}$:

${\displaystyle {\frac {\partial }{\partial x_{\alpha }}}\sum _{\beta =1}^{4}{\frac {\partial \Phi _{\beta }}{\partial x_{\beta }}}-\sum _{\beta =1}^{4}{\frac {\partial ^{2}\Phi _{\alpha }}{\partial x_{\beta }^{2}}}=\varrho _{\alpha }\quad \left(\sum _{\beta =1}^{4}{\frac {\partial \Phi _{\beta }}{\partial x_{\beta }}}=0\right)}$

equivalent to (c). He also expressed the four-current in terms of rest charge density and four-velocity

${\displaystyle {\begin{matrix}\varrho _{1}={\frac {\varrho ^{\ast }}{c}}\cdot {\frac {\partial x}{\partial \tau }},\ \varrho _{2}={\frac {\varrho ^{\ast }}{c}}\cdot {\frac {\partial y}{\partial \tau }},\ \varrho _{3}={\frac {\varrho ^{\ast }}{c}}\cdot {\frac {\partial z}{\partial \tau }},\ \varrho _{4}=i\varrho ^{\ast }{\frac {\partial t}{\partial \tau }}\\\left(\varrho ^{\ast }=\varrho {\sqrt {1-{\frac {w^{2}}{c^{2}}}}}={\sqrt {-\left(\varrho _{1}^{2}+\varrho _{2}^{2}+\varrho _{3}^{2}+\varrho _{4}^{2}\right)}},\ d\tau =dt{\sqrt {1-{\frac {w^{2}}{c^{2}}}}}\right)\\\varrho _{\alpha }=i\varrho ^{\ast }{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}\quad \left(\xi _{4}=ic\tau \right)\end{matrix}}}$

equivalent to (b).

A discussion of four-current in terms of integral forms (even though in the broader context of w:spherical wave transformations), was given by w:Harry Bateman in a paper read 1909 and published 1910, who defined the Lorentz transformations of its components ${\displaystyle \left(\rho w_{x},\rho w_{y},\rho w_{z},\rho \right)}$^{[R 10]}

${\displaystyle \rho w_{x}=\beta (\rho 'w'-v\rho '),\ \rho w_{y}=\rho 'w'_{y},\ \rho w_{z}=\rho 'w'_{z},\ -\rho =\beta (v\rho 'w'_{x}-\rho '),\ \left[\beta ={\frac {1}{\sqrt {1-v^{2}}}}\right]}$

forming the following invariant relations together with the differential four-position and four-potential:^{[R 11]}

${\displaystyle {\begin{matrix}{\frac {1}{\lambda ^{2}}}\left[\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt\right]\\{\frac {\rho ^{2}}{\lambda ^{2}}}\left(1-w^{2}\right)dx\ dy\ dz\ dt\\\rho \left[A_{x}w_{x}+A_{y}w_{y}+A_{z}w_{z}-\Phi \right]dx\ dy\ dz\ dt\end{matrix}}}$

with ${\displaystyle \lambda ^{2}=1}$ in relativity.

w:Wladimir Ignatowski (1910) defined the “vector of first kind” using charge density ${\displaystyle \varrho }$ and three-velocity ${\displaystyle {\mathfrak {v}}}$:^{[R 12]}

${\displaystyle {\begin{matrix}\left(\varrho {\mathfrak {v}},\ \varrho \right)\\\hline \left[\varrho {\sqrt {1-n{\mathfrak {v}}^{2}}}=\varrho '{\sqrt {1-n{\mathfrak {v}}^{\prime 2}}}=\varrho _{0}\right]\end{matrix}}}$

equivalent to four-current (a).

In influential papers on 4D vector calculus in relativity, w:Arnold Sommerfeld defined the four-current P, which he called four-density (Viererdichte):^{[R 13]}

${\displaystyle {\begin{matrix}P_{x}=\varrho {\frac {{\mathfrak {v}}_{x}}{c}},\ P_{y}=\varrho {\frac {{\mathfrak {v}}_{y}}{c}},\ P_{z}=\varrho {\frac {{\mathfrak {v}}_{z}}{c}},\ P_{l}=i\varrho \\\hline \beta ^{2}={\frac {1}{c^{2}}}\left({\mathfrak {v}}_{x}^{2}+{\mathfrak {v}}_{y}^{2}+{\mathfrak {v}}_{z}^{2}\right)\quad \Rightarrow \quad \left|P\right|=i\varrho {\sqrt {1-\beta ^{2}}}\\{}[l=ict]\end{matrix}}}$

equivalent to (a). In the second paper he pointed out its relation to four-potential ${\displaystyle \Phi }$ and the electromagnetic tensor (six-vector) f together with the continuity condition:^{[R 14]}

${\displaystyle {\begin{matrix}{\begin{aligned}P&={\mathfrak {Div}}\mathrm {Rot} \ \Phi ={\mathfrak {Div}}\ f\\-P&=\square \Phi ,\ (\mathrm {Div} \ \Phi =0)\\\mathrm {Div} \ P&=0\end{aligned}}\\\left[{\begin{aligned}\mathrm {Rot} &={\text{exterior product}}\\\mathrm {Div} &={\text{divergence four-vector}}\\{\mathfrak {Div}}&={\text{divergence six-vector}}\\\square &={\text{D'Alembert operator}}\end{aligned}}\right]\end{matrix}}}$

equivalent to Maxwell's equations (c). The scalar product with the four-potential^{[R 15]}

${\displaystyle (P\Phi )}$

he called “electro-kinetic potential” whereas the vector product with the electromagnetic tensor^{[R 16]}

${\displaystyle (Pf)={\mathfrak {F}}}$

he called the electrodynamic force (four-force density).

w:Gilbert Newton Lewis (1910) devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. The four-current is a “1-vector”:^{[R 17]}

${\displaystyle {\begin{aligned}\mathbf {q} &={\frac {\varrho }{c}}\mathbf {v} +i\varrho \mathbf {k} _{4}\\&={\frac {\varrho }{c}}v_{1}\mathbf {k} _{1}+{\frac {\varrho }{c}}v_{2}\mathbf {k} _{2}+{\frac {\varrho }{c}}v_{3}\mathbf {k} _{3}+i\varrho \mathbf {k} _{4}\end{aligned}}}$

equivalent to (a) and its relation to the four-potential ${\displaystyle \mathbf {m} }$ and electromagnetic tensor ${\displaystyle \mathbf {M} }$:

${\displaystyle {\begin{matrix}{\begin{aligned}\lozenge \lozenge \times \mathbf {m} &=\mathbf {q} \\\lozenge \mathbf {M} &=\mathbf {q} \\\lozenge ^{2}\mathbf {m} &=-\mathbf {q} \end{aligned}}\\{\begin{aligned}\left({\frac {\partial H_{12}}{\partial x_{2}}}+{\frac {\partial H_{13}}{\partial x_{3}}}+{\frac {\partial E_{14}}{\partial x_{4}}}\right)\mathbf {k} _{1}&={\frac {\varrho }{c}}v_{1}\mathbf {k} _{1}\\\left({\frac {\partial H_{21}}{\partial x_{1}}}+{\frac {\partial H_{23}}{\partial x_{3}}}+{\frac {\partial E_{24}}{\partial x_{4}}}\right)\mathbf {k} _{2}&={\frac {\varrho }{c}}v_{2}\mathbf {k} _{2}\\\left({\frac {\partial H_{31}}{\partial x_{1}}}+{\frac {\partial H_{32}}{\partial x_{2}}}+{\frac {\partial E_{34}}{\partial x_{4}}}\right)\mathbf {k} _{3}&={\frac {\varrho }{c}}v_{3}\mathbf {k} _{3}\\\left({\frac {\partial H_{41}}{\partial x_{1}}}+{\frac {\partial H_{42}}{\partial x_{2}}}+{\frac {\partial E_{43}}{\partial x_{4}}}\right)\mathbf {k} _{4}&={\frac {\varrho }{c}}i\mathbf {k} _{4}\end{aligned}}\\\left[{\begin{matrix}\lozenge =\mathbf {k} _{1}{\frac {\partial }{\partial x_{1}}}+\mathbf {k} _{2}{\frac {\partial }{\partial x_{2}}}+\mathbf {k} _{3}{\frac {\partial }{\partial x_{3}}}+\mathbf {k} _{4}{\frac {\partial }{\partial x_{4}}}\\\lozenge ^{2}={\frac {\partial ^{2}}{\partial x_{1}}}+{\frac {\partial ^{2}}{\partial x_{2}}}+{\frac {\partial ^{2}}{\partial x_{3}}}+{\frac {\partial ^{2}}{\partial x_{4}}}\end{matrix}}\right]\end{matrix}}}$

equivalent to (c,d).

In 1912, Lewis and w:Edwin Bidwell Wilson used only real coordinates, writing the above expressions as^{[R 18]}

${\displaystyle {\begin{matrix}{\begin{aligned}\lozenge \cdot \mathbf {M} &=4\pi \mathbf {q} \\\lozenge ^{2}\mathbf {m} &=-4\pi \mathbf {q} \end{aligned}}\\\left[{\begin{matrix}\lozenge =\mathbf {k} _{1}{\frac {\partial }{\partial x_{1}}}+\mathbf {k} _{2}{\frac {\partial }{\partial x_{2}}}+\mathbf {k} _{3}{\frac {\partial }{\partial x_{3}}}-\mathbf {k} _{4}{\frac {\partial }{\partial x_{4}}}\\\lozenge ^{2}={\frac {\partial ^{2}}{\partial x_{1}}}+{\frac {\partial ^{2}}{\partial x_{2}}}+{\frac {\partial ^{2}}{\partial x_{3}}}-{\frac {\partial ^{2}}{\partial x_{4}}}\end{matrix}}\right]\end{matrix}}}$

equivalent to (c,d).

In the first textbook on relativity in 1911, w:Max von Laue elaborated on Sommerfeld's methods and explicitly used the term “four-current” (Viererstrom) of density ${\displaystyle \varrho }$ in relation to four-potential ${\displaystyle \Phi }$ and electromagnetic tensor ${\displaystyle {\mathfrak {M}}}$:^{[R 19]}

${\displaystyle {\begin{matrix}P\Rightarrow \left(P_{x}={\frac {\varrho {\mathfrak {q}}_{x}}{c}},\ P_{y}={\frac {\varrho {\mathfrak {q}}_{y}}{c}},\ P_{z}={\frac {\varrho {\mathfrak {q}}_{z}}{c}},\ P_{l}=i\varrho \right)\\\hline {\begin{aligned}P&=\varDelta iv\ ({\mathfrak {M}})\\-P&=\square \Phi \ (Div\ \Phi =0)\\Div\ (P)&=0\end{aligned}}\\\left[{\begin{aligned}{\mathfrak {Rot}}&={\text{exterior product}}\\Div&={\text{divergence four-vector}}\\\varDelta iv&={\text{divergence six-vector}}\\\square &={\text{D'Alembert operator}}\end{aligned}}\right]\end{matrix}}}$

equivalent to (a,c,d). He went on to define four-force density F as vector-product with ${\displaystyle {\mathfrak {M}}}$, four-convection K and four-conduction ${\displaystyle \Lambda }$ using four-velocity Y:^{[R 20]}

${\displaystyle {\begin{matrix}F=[P{\mathfrak {M}}],\ (PF)=(P[P{\mathfrak {M}}])=0\\K=-(YP)Y\\\Lambda =P+(YP)Y\end{matrix}}}$,

w:Ludwik Silberstein devised an alternative 4D calculus based on w:Biquaternions which, however, never gained widespread support. He defined the “current-quaternion” (i.e. four-current) C and its relation to the “electromagnetic bivector” (i.e. field tensor) ${\displaystyle \mathbf {F} }$ and “potential-quaternion” (i.e. four-potential) ${\displaystyle \Phi }$^{[R 21]}

${\displaystyle {\begin{matrix}{\begin{aligned}\mathrm {C} &=\rho \left(\iota +{\frac {1}{c}}\mathbf {p} \right)\\&=\iota \rho {\frac {dq}{dl}}\\\mathrm {C} &=\mathrm {D} \mathbf {F} =-\Box \Phi \\\mathrm {S} \mathrm {D} _{c}\mathrm {C} &=0\end{aligned}}\\\left[\mathrm {D} ={\frac {\partial }{\partial l}}-\nabla ,\ \mathrm {D} \mathrm {D} _{c}=\Box ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}+{\frac {\partial ^{2}}{\partial l^{2}}}\right]\end{matrix}}}$

w:Friedrich Kottler defined the four-current ${\displaystyle \mathbf {P} ^{(\alpha )}}$ and its relation to four-velocity ${\displaystyle V^{(\alpha )}}$, four-potential ${\displaystyle \Phi _{\alpha }}$, four-force ${\displaystyle F_{\alpha }}$, electromagnetic field-tensor ${\displaystyle F_{\alpha \beta }}$, stress-energy tensor ${\displaystyle S_{\alpha \beta }}$:^{[R 22]}

${\displaystyle {\begin{matrix}P^{(1)}=\rho {\frac {{\mathfrak {v}}_{x}}{c}}=i\rho _{0}V^{(1)},\ P^{(2)}=\rho {\frac {{\mathfrak {v}}_{y}}{c}}=i\rho _{0}V^{(2)},\ P^{(3)}=\rho {\frac {{\mathfrak {v}}_{z}}{c}}=i\rho _{0}V^{(3)},\ P^{(4)}=i\rho =i\rho _{0}V^{(4)}\\\hline \sum _{h=1}^{4}{\frac {\partial F_{gh}}{\partial x^{(h)}}}=\mathbf {P} ^{(g)},\ \Box \Phi _{\alpha }=-\mathbf {P} ^{(\alpha )}\\F_{\alpha }(y)=\sum _{\beta }{\frac {F_{\alpha \beta }(y)\mathbf {P} ^{(\beta )}(y)}{\sqrt {1-{\mathfrak {w}}^{2}/c^{2}}}}\\\left[{\underset {\beta }{\sum }}F_{\alpha \beta }(y)\mathbf {P} ^{(\beta )}(y)={\underset {\beta }{\sum }}F_{\alpha \beta }(y){\underset {\gamma }{\sum }}{\frac {\partial }{\partial y^{(\gamma )}}}F_{\beta \gamma }(y)={\underset {\beta }{\sum }}{\frac {\partial }{\partial y^{(\beta )}}}S_{\alpha \beta },\ \rho _{0}=\rho {\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right]\end{matrix}}}$

equivalent to (a,b,c,d) and subsequently was the first to give the generally covariant formulation of Maxwell's equations using metric tensor ${\displaystyle c_{\alpha \beta }}$^{[R 23]}

${\displaystyle {\begin{matrix}\sum c^{(1\alpha )}\sum _{\beta ,\gamma }c^{(\beta \gamma )}\Phi _{\alpha /\beta \gamma }=-\mathbf {P} ^{(\alpha )}\ {\text{etc}}.\\\left[\sum _{\beta ,\gamma }c^{(\beta \gamma )}\Phi _{\beta /\gamma }=0\right]\end{matrix}}}$

equivalent to (e).

Independently of Kottler (1912), w:Albert Einstein defined the general covariant four-current in the context of his Entwurf theory (a precursor of general relativity):^{[R 24]}

${\displaystyle \varrho _{0}{\frac {dx_{\nu }}{ds}}={\frac {1}{\sqrt {-g}}}\varrho _{0}{\frac {dx_{\nu }}{dt}}}$

equivalent to (a), and the generally covariant formulation of Maxwell's equations

${\displaystyle {\begin{matrix}\sum _{\nu }{\frac {\partial }{\partial x_{\nu }}}\left({\sqrt {-g}}\cdot \varphi _{\mu \nu }\right)=\varrho _{0}{\frac {dx_{\mu }}{dt}}\\\hline {\begin{aligned}{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}-{\frac {\partial {\mathfrak {E}}_{x}}{\partial t}}&=u_{x}\\\dots \\\dots \\{\frac {\partial {\mathfrak {E}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {E}}_{y}}{\partial z}}+{\frac {\partial {\mathfrak {E}}_{x}}{\partial z}}&=\varrho \end{aligned}}\\\left[\varrho _{0}{\frac {dx_{\mu }}{dt}}=u_{\mu }\right]\end{matrix}}}$

equivalent to (e) in the case of ${\displaystyle g_{\mu \nu }}$ being the Minkowski tensor.