History of Topics in Special Relativity/Four-force (mechanics)

Overview
The four-force $$K^{\mu}$$ is not only applicable to electromagnetic phenomena (compare with ../Four-force (electromagnetism)), but also applies to mechanics in general, thus it can be used in relation to fluids, dust, Lorentz invariant gravity models etc.. It is defined as


 * (a) the rate of change in the four-momentum $$P^{\mu}=mU^{\mu}$$ with respect to proper time $$\tau$$,


 * (b) function of three-force $$\mathbf{f}$$


 * (c) assuming constant mass as the product of invariant mass m and four-acceleration $$A^{\mu}$$.


 * (d) by integrating the four-force density $$D^{\mu}$$ with respect to rest unit volume $$V_{0}=V\gamma$$

The corresponding four-force density $$D^{\mu}$$ is defined as


 * (a1) the rate of change of four-momentum density $$M^{\mu}=\mu_{0}U^{\mu}$$ with rest mass density $$\mu_{0}=\mu/\gamma$$


 * (b1) function of three-force density $$\mathbf{d}$$


 * (c1) assuming constant mass the product of rest mass density $$\mu_{0}$$ and four-acceleration $$A^{\mu}$$


 * (d1) the four-divergence of the energy-momentum tensor $$T^{\alpha\beta}$$ (such as for fluids or dust). In case $$\partial_{\alpha}T^{\alpha\beta}=0$$, the four corresponding equations represent the energy and momentum conservation laws.


 * $$\begin{matrix}\begin{matrix}K^{\mu} & =\frac{\mathrm{d}P^{\mu}}{\mathrm{d}\tau} & =\gamma\left(\frac{1}{c}\mathbf{f}\cdot\mathbf{v},\ \mathbf{f}\right) & =mA^{\mu} & =\int D^{\mu}\,dV_{0}\\

& (a) & (b) & (c) & (d)\\ D^{\mu} & =\frac{\mathrm{d}M^{\mu}}{\mathrm{d}\tau} & =\left(\frac{1}{c}\mathbf{d\cdot v},\ \mathbf{d}\right) & =\mu_{0}A^{\mu} & =\partial_{\alpha}T^{\alpha\beta}\\ & (a1) & (b1) & (c1) & (d1) \end{matrix}\\ \left(\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mathbf{f}=\int\mathbf{d}\,dV\right) \end{matrix}$$

Examples are the four-force density using the perfect fluid stress energy tensor (compare with ../Stress-energy tensor (matter)):


 * $$(e)\ D^{\mu}=\partial_{\alpha}T^{\alpha\beta}$$ with $$T^{\alpha\beta}=\left(\mu_{0}+\frac{p}{c^{2}}\right)u^{\alpha}u^{\beta}+pg^{\alpha\beta}$$

or using the dust solution in case of vanishing pressure:


 * $$(f)\ D^{\mu}=\partial_{\alpha}T^{\alpha\beta}$$ with $$T^{\alpha\beta}=\mu_{0}u^{\alpha}u^{\beta}$$

Killing (1884/5)
Wilhelm Killing discussed Newtonian mechanics in non-Euclidean spaces by expressing position, velocity, acceleration and force in terms of four components. He expressed the four components of force R=(P,X,Y,Z), its norm, its inner product with coordinates (p,x,y,z), and the equations of motion as the product of mass with acceleration as follows:


 * $$\begin{matrix}R\Rightarrow P,X,Y,Z\\

\hline R^{2}=\frac{P^{2}}{k^{2}}+X^{2}+Y^{2}+Z^{2}\\ pP+xX+yY+zZ=0\\ \hline mk^{2}\frac{d^{2}p}{dt^{2}}=P,\ m\frac{d^{2}x}{dt^{2}}=X,\ m\frac{d^{2}y}{dt^{2}}=Y,\ m\frac{d^{2}z}{dt^{2}}=Z \end{matrix}$$

If the Gaussian curvature $$1/k^{2}$$ (with k as radius of curvature) is negative the force becomes related to the hyperboloid model of hyperbolic space, which at first sight becomes similar to the relativistic four-force in Minkowski space by setting $$k^{2}=-c^{2}$$ with c as speed of light. However, Killing obtained his results by differentiation with respect to Newtonian time t, not relativistic proper time, so his expressions aren't relativistic four-vectors in the first place, in particular they don't involve a limiting speed.

Poincaré (1905/6)
Henri Poincaré (July 1905, published January 1906) argued, that the Lorentz transformation not only applies to electrodynamics, but to all other phenomena as well including mechanics. For instance, he explicitly defined gravitation as non-electromagnetic in origin and applied the following expression of four-force to his Lorentz invariant model of gravitation:


 * $$\left( k_{0}X_{1},\quad k_{0}Y_{1},\quad k_{0}Z_{1},\quad k_{0}T_{1}\right )$$ with $$T_{1}=\Sigma X_{1}\xi$$ and $$k_{0}=\tfrac{1}{\sqrt{1-\epsilon^{2}}}$$

equivalent to (b) because


 * $$\epsilon=\frac{v}{c},\ \left(X_{1},\ Y_{1},\ Z_{1}\right)=\mathbf{f},\ \Sigma X_{1}\xi=\mathbf{f}\cdot\mathbf{v}$$.

Minkowski (1907)
In an appendix to his lecture from December 1907 (published 1908), Hermann Minkowski extended the postulate of relativity to mechanics in general, defining four-force density $$K+(K\overline{w})w$$ with w as four-velocity and S as stress energy tensor


 * $$\begin{matrix}K+(K\overline{w})w\\

\hline K=\text{lor }S\Rightarrow K_{h}=\frac{\partial S_{1h}}{\partial x_{1}}+\frac{\partial S_{2h}}{\partial x_{2}}+\frac{\partial S_{3h}}{\partial x_{3}}+\frac{\partial S_{4h}}{\partial x_{4}}\\ \left[\text{lor }=\left|\frac{\partial}{\partial x_{1}},\ \frac{\partial}{\partial x_{2}},\ \frac{\partial}{\partial x_{3}},\ \frac{\partial}{x_{4}}\right|\right] \end{matrix}$$

equivalent to (d1). He went on to show that these relations can be also used to define the equations of motion of mechanics in terms of constant rest mass density $$\nu$$:


 * $$\begin{matrix}\frac{\partial\nu\,w_{h}w_{1}}{\partial x_{1}}+\frac{\partial\nu\,w_{h}w_{2}}{\partial x_{2}}+\frac{\partial\nu\,w_{h}w_{3}}{\partial x_{3}}+\frac{\partial\nu\,w_{h}w_{4}}{\partial x_{4}}\\

\nu\frac{dw_{h}}{d\tau}\Rightarrow\nu\frac{d}{d\tau}\left(\frac{dx_{h}}{d\tau}\right)\Rightarrow K+(K\,\overline{w})w=(X,Y,Z,iT)\\ \hline \nu\frac{d}{d\tau}\frac{dx}{d\tau}=X,\ \nu\frac{d}{d\tau}\frac{dy}{d\tau}=Y,\ \nu\frac{d}{d\tau}\frac{dz}{d\tau}=Z,\ \nu\frac{d}{d\tau}\frac{dt}{d\tau}=T \end{matrix}$$

equivalent to (b1, c1, d1) as well as (f) since the first line includes the dust stress-energy tensor. Eventually he defined a “moving force” as the product of constant rest mass and four-acceleration


 * $$m\frac{d}{d\tau}\frac{dx}{d\tau}=R_{x},\quad m\frac{d}{d\tau}\frac{dy}{d\tau}=R_{y},\quad m\frac{d}{d\tau}\frac{dz}{d\tau}=R_{z},\quad m\frac{d}{d\tau}\frac{dt}{d\tau}=R_{t}$$

equivalent to (c). Minkowski's assumption of constant rest mass was later challenged by Abraham (see next section).

Abraham (1909-12)
In 1909, Max Abraham pointed out that the relativity principle requires that the mechanical forces must transform like the electromagnetic ones, so there must be a four-dimensional tensor for mechanics (i.e. mechanical stress energy tensor) in analogy to the electromagnetic one, and that the relation $$c^{2}\mathfrak{g}=\mathfrak{S}$$ can alternatively be interpreted as relation between mechanical momentum and energy density:


 * $$\begin{matrix}\begin{align}\mathfrak{K}_{x} & =\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}+\frac{\partial X_{u}}{\partial u}\\

\mathfrak{K}_{y} & =\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}+\frac{\partial Y_{u}}{\partial u}\\ \mathfrak{K}_{z} & =\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}+\frac{\partial Z_{u}}{\partial u}\\ \mathfrak{K}_{u} & =\frac{\partial U_{x}}{\partial x}+\frac{\partial U_{y}}{\partial y}+\frac{\partial U_{z}}{\partial z}+\frac{\partial U_{u}}{\partial u} \end{align} \\ \left[u=ict\right] \end{matrix}$$

equivalent to (a) when interpreted as mechanical force. While Minkowski (1907) assumed constant rest mass density, Abraham (1909) held that mass-energy equivalence, according to which mass depends on its energy content, would suggest a variable rest mass $$m_{0}$$.

In 1912, Abraham introduced the expression “world tensor of motion” $$T^{\ast}$$ (equivalent to the dust tensor) while formulating his first theory of gravitation. It has ten components representing kinetic stresses, energy flux $$\mathfrak{S}^{\ast}$$ and momentum $$\mathfrak{g}^{\ast}$$ of matter in terms of rest mass density $$\nu$$, which he combined with the world tensor $$T$$ (representing the electromagnetic-, gravitational-, and stress field) in order to formulate the momentum and energy conservation theorems:


 * $$\begin{align}\frac{\partial\left(c\mathfrak{g}_{x}+c\mathfrak{g}_{x}^{\ast}\right)}{c\partial t} & =\frac{\partial\left(X_{x}+X_{x}^{\ast}\right)}{\partial x}+\frac{\partial\left(X_{y}+X_{y}^{\ast}\right)}{\partial y}+\frac{\partial\left(X_{z}+X_{z}^{\ast}\right)}{\partial z}\\

\frac{\partial\left(c\mathfrak{g}_{y}+c\mathfrak{g}_{y}^{\ast}\right)}{c\partial t} & =\frac{\partial\left(Y_{x}+Y_{x}^{\ast}\right)}{\partial x}+\frac{\partial\left(Y_{y}+Y_{y}^{\ast}\right)}{\partial y}+\frac{\partial\left(Y_{z}+Y_{z}^{\ast}\right)}{\partial z}\\ \frac{\partial\left(c\mathfrak{g}_{z}+c\mathfrak{g}_{z}^{\ast}\right)}{c\partial t} & =\frac{\partial\left(Z_{x}+Z_{x}^{\ast}\right)}{\partial x}+\frac{\partial\left(Z_{y}+Z_{y}^{\ast}\right)}{\partial y}+\frac{\partial\left(Z_{z}+Z_{z}^{\ast}\right)}{\partial z}\\ \frac{\partial\left(\varepsilon+\varepsilon^{\ast}\right)}{\partial t} & =-c{\rm div}\left\{ \frac{\mathfrak{S}+\mathfrak{S}^{\ast}}{c}\right\} \end{align} $$

equivalent to (f) when only the dust tensor $$T^{\ast}$$ is considered.

Nordström (1910–13)
The force definitions of both Abraham (variable rest mass) and Minkowski (constant rest mass) were elaborated by Gunnar Nordström (1910), who defined two variants of four-force density using a “four-dimensional tensor” (i.e. dust solution) $$\gamma\mathfrak{a}_{m}\mathfrak{a}_{n}$$ consisting of rest mass density $$\gamma$$ and four-velocity $$\mathfrak{a}$$. The first formulation was based on Abraham's assumption of variable rest mass density:


 * $$\begin{align}\mathfrak{K}_{x}^{\prime} & =\frac{\partial}{\partial x}\gamma\mathfrak{a}_{x}^{2}+\frac{\partial}{\partial y}\gamma\mathfrak{a}_{x}\mathfrak{a}_{y}+\frac{\partial}{\partial z}\gamma\mathfrak{a}_{x}\mathfrak{a}_{z}+\frac{\partial}{\partial u}\gamma\mathfrak{a}_{x}\mathfrak{a}_{u}\\

\mathfrak{K}_{y}^{\prime} & =\frac{\partial}{\partial x}\gamma\mathfrak{a}_{y}\mathfrak{a}_{x}+\frac{\partial}{\partial y}\gamma\mathfrak{a}_{y}^{2}+\frac{\partial}{\partial z}\gamma\mathfrak{a}_{y}\mathfrak{a}_{z}+\frac{\partial}{\partial u}\gamma\mathfrak{a}_{y}\mathfrak{a}_{u}\\ \mathfrak{K}_{z}^{\prime} & =\frac{\partial}{\partial x}\gamma\mathfrak{a}_{z}\mathfrak{a}_{x}+\frac{\partial}{\partial y}\gamma\mathfrak{a}_{z}\mathfrak{a}_{y}+\frac{\partial}{\partial z}\gamma\mathfrak{a}_{z}^{2}+\frac{\partial}{\partial u}\gamma\mathfrak{a}_{z}\mathfrak{a}_{u}\\ \mathfrak{K}_{u}^{\prime} & =\frac{\partial}{\partial x}\gamma\mathfrak{a}_{u}\mathfrak{a}_{x}+\frac{\partial}{\partial y}\gamma\mathfrak{a}_{u}\mathfrak{a}_{y}+\frac{\partial}{\partial z}\gamma\mathfrak{a}_{u}\mathfrak{a}_{z}+\frac{\partial}{\partial u}\gamma\mathfrak{a}_{u}^{2}\\ & (u=ict) \end{align} $$

equivalent to (f), and the second one on Minkowski's assumption of constant rest mass density:


 * $$\begin{align}\mathfrak{K}_{x}^{\prime\prime} & =\gamma\mathfrak{a}_{x}\frac{\partial\mathfrak{a}_{x}}{\partial x}+\gamma\mathfrak{a}_{y}\frac{\partial\mathfrak{a}_{x}}{\partial y}+\gamma\mathfrak{a}_{z}\frac{\partial\mathfrak{a}_{x}}{\partial z}+\gamma\mathfrak{a}_{u}\frac{\partial\mathfrak{a}_{x}}{\partial u}\\

\mathfrak{K}_{y}^{\prime\prime} & =\gamma\mathfrak{a}_{x}\frac{\partial\mathfrak{a}_{y}}{\partial x}+\gamma\mathfrak{a}_{y}\frac{\partial\mathfrak{a}_{y}}{\partial y}+\gamma\mathfrak{a}_{z}\frac{\partial\mathfrak{a}_{y}}{\partial z}+\gamma\mathfrak{a}_{u}\frac{\partial\mathfrak{a}_{y}}{\partial u}\\ \mathfrak{K}_{z}^{\prime\prime} & =\gamma\mathfrak{a}_{x}\frac{\partial\mathfrak{a}_{z}}{\partial x}+\gamma\mathfrak{a}_{y}\frac{\partial\mathfrak{a}_{z}}{\partial y}+\gamma\mathfrak{a}_{z}\frac{\partial\mathfrak{a}_{z}}{\partial z}+\gamma\mathfrak{a}_{u}\frac{\partial\mathfrak{a}_{z}}{\partial u}\\ \mathfrak{K}_{u}^{\prime\prime} & =\gamma\mathfrak{a}_{x}\frac{\partial\mathfrak{a}_{u}}{\partial x}+\gamma\mathfrak{a}_{y}\frac{\partial\mathfrak{a}_{u}}{\partial y}+\gamma\mathfrak{a}_{z}\frac{\partial\mathfrak{a}_{u}}{\partial z}+\gamma\mathfrak{a}_{u}\frac{\partial\mathfrak{a}_{u}}{\partial u} \end{align} $$

equivalent to (f). In 1911, Nordström only used the first variant with variable rest mass density and considered pressure in a “material fluid” as well.

In 1913, he added an “elastic stress tensor” p in order to reformulated Laue's symmetrical four dimensional tensor T representing spatial stresses and mechanical momentum and energy density, which he used to add an elastic component $$\mathfrak{K}^{e}$$ to the four-force-density $$\mathfrak{K}$$ to give the equation of motion:


 * $$\begin{align}\mathfrak{K}_{x}+\mathfrak{K}_{x}^{e} & =\frac{\partial}{\partial x}\nu\mathfrak{B}_{x}^{2}+\frac{\partial}{\partial y}\nu\mathfrak{B}_{x}\mathfrak{B}_{y}+\frac{\partial}{\partial z}\nu\mathfrak{B}_{x}\mathfrak{B}_{z}+\frac{\partial}{\partial u}\nu\mathfrak{B}_{x}\mathfrak{B}_{u}\\

\mathfrak{K}_{y}+\mathfrak{K}_{y}^{e} & =\frac{\partial}{\partial x}\nu\mathfrak{B}_{y}\mathfrak{B}_{x}+\frac{\partial}{\partial y}\nu\mathfrak{B}_{y}^{2}+\frac{\partial}{\partial z}\nu\mathfrak{B}_{y}\mathfrak{B}_{z}+\frac{\partial}{\partial u}\nu\mathfrak{B}_{y}\mathfrak{B}_{u}\\ \mathfrak{K}_{z}+\mathfrak{K}_{z}^{e} & =\frac{\partial}{\partial x}\nu\mathfrak{B}_{z}\mathfrak{B}_{x}+\frac{\partial}{\partial y}\nu\mathfrak{B}_{z}\mathfrak{B}_{y}+\frac{\partial}{\partial z}\nu\mathfrak{B}_{z}^{2}+\frac{\partial}{\partial u}\nu\mathfrak{B}_{z}\mathfrak{B}_{u}\\ \mathfrak{K}_{u}+\mathfrak{K}_{u}^{e} & =\frac{\partial}{\partial x}\nu\mathfrak{B}_{u}\mathfrak{B}_{x}+\frac{\partial}{\partial y}\nu\mathfrak{B}_{u}\mathfrak{B}_{y}+\frac{\partial}{\partial z}\nu\mathfrak{B}_{u}\mathfrak{B}_{z}+\frac{\partial}{\partial u}\nu\mathfrak{B}_{u}^{2} \end{align} $$

equivalent to (f). He went on to employ this notion in his theory of gravitation.

Ignatowski (1911)
Wladimir Ignatowski derived the hydrodynamic four-force density in terms of mass density $$\varrho$$ and pressure $$\pi$$ in case of perfect fluids:


 * $$\begin{matrix}\left(\mathfrak{K}_{1},\ n\mathfrak{v}\mathfrak{K}_{1}+\sqrt{1-n\mathfrak{v}^{2}}D\right)=\left(-\Delta\pi,\ n\frac{\partial\pi}{\partial t}\right)=\left(-\Delta\pi,\ n\mathfrak{v}\Delta\pi+n\frac{d\pi}{dt}\right)\\

\hline \left[\mathfrak{K}_{1}=\varrho-\frac{d\frac{n\mathfrak{v}}{\sqrt{1-n\mathfrak{v}^{2}}}}{dt}+\frac{\mathfrak{v}}{\sqrt{1-n\mathfrak{v}^{2}}}D=-\Delta\pi,\ D=\frac{n}{\sqrt{1-n\mathfrak{v}^{2}}}\frac{d\pi}{dt}=\frac{d\varrho}{dt}+\varrho\mathrm{div}\mathfrak{v},\ n=\frac{1}{c^{2}},\ \Delta=\mathrm{grad}\right] \end{matrix}$$

which corresponds to the four-force of perfect fluid (e).

Von Laue (1911-20)
In the first textbook on relativity (1911), Max von Laue defined the mechanical ponderomotive force F based on world tensor T (i.e. mechanical stress-energy tensor), implying the complete reduction of mechanical inertia to energy and stresses:


 * $$F=-\varDelta iv\,T$$

equivalent to (c).

In the second edition (1912, published 1913), he followed and  in defining the four-force K in order to produce the Lagrangian (1) and Eulerian (2) fundamental equations of hydrodynamics, as well as for the case of least compressibility (3), where $$\delta m_{n}$$ is the normal rest mass, p the pressure, Y the four-velocity:


 * $$\begin{matrix}(1) & \frac{d}{d\tau}\left\{ \delta m_{n}\left(1+\frac{P}{c^{2}}\right)Y\right\} +\frac{\delta V^{0}}{c}\Gamma\varrho\alpha\delta\ p=K\\

(2) & Y_{x}\frac{\partial}{\partial x}\left(\varkappa^{0}Y_{m}\right)+Y_{y}\frac{\partial}{\partial y}\left(\varkappa^{0}Y_{m}\right)+Y_{z}\frac{\partial}{\partial z}\left(\varkappa^{0}Y_{m}\right)+Y_{l}\frac{\partial}{\partial l}\left(\varkappa^{0}Y_{m}\right)+\frac{1}{c^{2}}\Gamma\varrho\alpha\delta_{m}P=\frac{K_{m}}{c\delta m_{n}}\\ (3) & k^{0}\left(\frac{1}{c}\cdot\frac{dY}{d\tau}-Y\ Div\ Y\right)+\Gamma\varrho\alpha\delta\ k^{0}=\frac{k_{n}^{0}K}{c\delta m_{n}}\\ & \left(l=ict,\ \varkappa^{0}=1+\frac{P}{c^{2}},\ P=\int_{p_{n}}^{p}\frac{dp}{k^{0}},\ k^{0}=\frac{\delta m_{n}}{\delta V^{0}},\ \Gamma\varrho\alpha\delta=\text{four-gradient}\right) \end{matrix}$$

equivalent to (c).

In the fourth edition (1921), he defined the four-force density using the kinetic stress energy tensor, with rest energy density $$W^{0}$$, rest energy $$E^{0}$$, rest volume $$V^{0}$$:


 * $$\begin{matrix}T_{jk}=W^{0}Y_{j}Y_{k}\\

\varDelta iv_{x}T=Y_{x}Div\left(W^{0}Y\right)+W^{0}\left(Y_{x}\frac{\partial Y_{x}}{\partial x}+Y_{y}\frac{\partial Y_{x}}{\partial x}+Y_{z}\frac{\partial Y_{x}}{\partial x}+Y_{l}\frac{\partial Y_{x}}{\partial x}\right)\\ \varDelta iv_{x}T=\frac{W^{0}}{c}\frac{dY}{d\tau}=-F\\ \int T_{ik}\delta V^{0}=E^{0}Y_{i}Y_{k}\\ \left(l=ict,\ \begin{matrix}div=\text{four-divergence of four-vector}\\ \varDelta iv=\text{four-divergence of six-vector} \end{matrix}\right) \end{matrix}$$

equivalent to (b,c) and the dust solution (f).

Herglotz (1911)
Gustav Herglotz gave a complete theory of elasticity in special relativity, including equations of motion in different forms, which he defined using coordinates $$x\dots$$ after deformation, $$\xi\dots$$ and $$\xi^{0}\dots$$ before deformation, from which he derived the deformation quantities $$a_{ij}$$ and $$A_{ij}$$, together with the kinetic potential $$\varPhi$$. He gave the Lagrangian equation of motion in terms of four-force density:


 * $$\begin{matrix}\frac{1}{a_{44}}\Xi,\ \frac{1}{a_{44}}\mathrm{H},\ \frac{1}{a_{44}}\mathrm{Z},\ \frac{1}{a_{44}}\mathrm{T}\\

\hline \begin{align}\begin{align}\Xi & =\frac{\partial\varPhi_{11}}{\partial\xi}+\frac{\partial\varPhi_{12}}{\partial\eta}+\frac{\partial\varPhi_{13}}{\partial\zeta}+\frac{\partial\varPhi_{14}}{\partial\tau}\\ \mathrm{H} & =\frac{\partial\varPhi_{21}}{\partial\xi}+\frac{\partial\varPhi_{22}}{\partial\eta}+\frac{\partial\varPhi_{23}}{\partial\zeta}+\frac{\partial\varPhi_{24}}{\partial\tau}\\ \mathrm{Z} & =\frac{\partial\varPhi_{31}}{\partial\xi}+\frac{\partial\varPhi_{32}}{\partial\eta}+\frac{\partial\varPhi_{33}}{\partial\zeta}+\frac{\partial\varPhi_{34}}{\partial\tau}\\ \mathrm{T} & =\frac{\partial\varPhi_{41}}{\partial\xi}+\frac{\partial\varPhi_{42}}{\partial\eta}+\frac{\partial\varPhi_{43}}{\partial\zeta}+\frac{\partial\varPhi_{44}}{\partial\tau}\\ & =-\left(u\Xi+v\mathrm{H}+w\mathrm{Z}+\vartheta\frac{\partial\varepsilon}{\partial\tau}\right)\\ \mathrm{E} & =-\frac{\partial\varPhi}{\partial\varepsilon}=a_{44}\vartheta \end{align} \end{align} \end{matrix}$$

and the Euler equations of motion by defining stress-energy tensor $$F_{ij}$$, whose components can be related to momentum density $$\mathfrak{X},\mathfrak{Y},\mathfrak{Z}$$, energy density $$\mathfrak{E}$$, velocity u,v,w:


 * $$\begin{align}\begin{align}X & =\frac{\partial F_{11}}{\partial x}+\frac{\partial F_{12}}{\partial y}+\frac{\partial F_{13}}{\partial z}+\frac{\partial F_{14}}{\partial t}\\

Y & =\frac{\partial F_{21}}{\partial x}+\frac{\partial F_{22}}{\partial y}+\frac{\partial F_{23}}{\partial z}+\frac{\partial F_{24}}{\partial t}\\ Z & =\frac{\partial F_{31}}{\partial x}+\frac{\partial F_{32}}{\partial y}+\frac{\partial F_{33}}{\partial z}+\frac{\partial F_{34}}{\partial t}\\ T & =\frac{\partial F_{41}}{\partial x}+\frac{\partial F_{42}}{\partial y}+\frac{\partial F_{43}}{\partial z}+\frac{\partial F_{44}}{\partial t}\\ & =-\left(uX+vY+wZ+Q\right)\\ \vartheta & =\varDelta\sqrt{1-s^{2}}\frac{\partial F}{\partial\varepsilon}=a_{44}\vartheta \end{align} \end{align} $$

Then he formulated a third kind of equations of motion by introducing “relative” stresses $$S_{ij}$$ into the Euler equations:


 * $$\begin{matrix}\begin{align}\begin{align}X & =D_{t}\mathfrak{X}+\frac{\partial S_{11}}{\partial x}+\frac{\partial S_{12}}{\partial y}+\frac{\partial S_{13}}{\partial z}\\

Y & =D_{t}\mathfrak{Y}+\frac{\partial S_{21}}{\partial x}+\frac{\partial S_{22}}{\partial y}+\frac{\partial S_{23}}{\partial z}\\ Z & =D_{t}\mathfrak{Z}+\frac{\partial S_{31}}{\partial x}+\frac{\partial S_{32}}{\partial y}+\frac{\partial S_{33}}{\partial z}\\ -T & =D_{t}\mathfrak{E}+\frac{\partial\left(\mathfrak{X}-u\mathfrak{E}\right)}{\partial x}+\frac{\partial\left(\mathfrak{Y}-v\mathfrak{E}\right)}{\partial y}+\frac{\partial\left(\mathfrak{Z}-w\mathfrak{E}\right)}{\partial z} \end{align} \end{align} \\ \left[D_{t}(f)=\frac{\partial f}{\partial t}+\frac{\partial(uf)}{\partial x}+\frac{\partial(vf)}{\partial y}+\frac{\partial(wf)}{\partial z}\right] \end{matrix}$$

He finally showed how to modify $$\mathfrak{X,Y,Z,E},S_{ij}$$ using mass density m and pressure p, so that previous equations become the equations of motion of a perfect fluid:


 * $$\begin{matrix}D_{t}(mu)+\frac{\partial p}{\partial x}=X\\

D_{t}(mv)+\frac{\partial p}{\partial y}=Y\\ D_{t}(mw)+\frac{\partial p}{\partial z}=Z\\ -D_{t}(m)+\frac{\partial p}{\partial t}=T\\ \left[m=\frac{F+p}{1-s^{2}},\ s=\sqrt{u^{2}+v^{2}+w^{2}}\right] \end{matrix}$$

which corresponds to the four-force of perfect fluid (e).

Lamla (1911/12)
Ernst Lamla (1911, published 1912) derived the equation of motion of hydrodynamics independently of Herglotz. Using pressure p and rest density g, he gave the Lagrangian form:


 * $$\begin{matrix}c\cdot\frac{d}{d\tau}\left\{ w\left(1+\frac{P}{c^{2}}\right)\right\} +\mathrm{lor}\,P-R=0\\

\left[R_{1}=\frac{X}{\sqrt{1-\frac{q^{2}}{c^{2}}}},\dots R_{4}=\frac{i}{c\sqrt{1-\frac{q^{2}}{c^{2}}}}\left(X\dot{x}+Y\dot{y}+Z\dot{z}\right)\right]\\ \hline \begin{align}\begin{align}\frac{d}{dt}\left\{ \frac{\dot{x}}{\sqrt{1-\frac{q^{2}}{c^{2}}}}\left(1+\frac{P}{c^{2}}\right)\right\} +\sqrt{1-\frac{q^{2}}{c^{2}}}\cdot\frac{\delta P}{\delta x}-X & =0\\ \frac{d}{dt}\left\{ \frac{\dot{y}}{\sqrt{1-\frac{q^{2}}{c^{2}}}}\left(1+\frac{P}{c^{2}}\right)\right\} +\sqrt{1-\frac{q^{2}}{c^{2}}}\cdot\frac{\delta P}{\delta y}-Y & =0\\ \frac{d}{dt}\left\{ \frac{\dot{z}}{\sqrt{1-\frac{q^{2}}{c^{2}}}}\left(1+\frac{P}{c^{2}}\right)\right\} +\sqrt{1-\frac{q^{2}}{c^{2}}}\cdot\frac{\delta P}{\delta z}-Z & =0\\ \frac{d}{dt}\left\{ \frac{c^{2}}{\sqrt{1-\frac{q^{2}}{c^{2}}}}\left(1+\frac{P}{c^{2}}\right)\right\} -\sqrt{1-\frac{q^{2}}{c^{2}}}\frac{\delta P}{\delta z}-\left(X\dot{x}+Y\dot{y}+Z\dot{z}\right) & =0 \end{align} \end{align} \\ \hline \left[g=k\sqrt{1-\frac{q^{2}}{c^{2}}},\ P(g)=\int_{p=0}\frac{dp}{g},\ \mathrm{lor}\,P=\left|\frac{\partial P}{\partial x_{1}},\ \frac{\partial P}{\partial x_{2}},\ \frac{\partial P}{\partial x_{3}},\ \frac{\partial P}{\partial x_{4}}\right|\right] \end{matrix}$$

which corresponds to the four-force of perfect fluid (e), as well as Euler's equation of motion using four-velocity w:


 * $$\begin{matrix}\left(w\overline{\mathrm{lor}}\right)\left[w\left(c^{2}+P\right)\right]+\mathrm{lor}\,P-R=0\\

\hline \begin{align}\frac{\delta(u\varepsilon)}{\delta t}+u\frac{\delta(u\varepsilon)}{\delta x}+v\frac{\delta(u\varepsilon)}{\delta y}+w\frac{\delta(u\varepsilon)}{\delta z}+\sqrt{1-\frac{q^{2}}{c^{2}}}\frac{\delta P}{\delta x}-X & =0\\ \frac{\delta(v\varepsilon)}{\delta t}+u\frac{\delta(v\varepsilon)}{\delta x}+v\frac{\delta(v\varepsilon)}{\delta y}+w\frac{\delta(v\varepsilon)}{\delta z}+\sqrt{1-\frac{q^{2}}{c^{2}}}\frac{\delta P}{\delta y}-Y & =0\\ \frac{\delta(w\varepsilon)}{\delta t}+u\frac{\delta(w\varepsilon)}{\delta x}+v\frac{\delta(w\varepsilon)}{\delta y}+w\frac{\delta(w\varepsilon)}{\delta z}+\sqrt{1-\frac{q^{2}}{c^{2}}}\frac{\delta P}{\delta z}-Z & =0\\ c^{2}\left(\frac{\delta\varepsilon}{\delta t}+u\frac{\delta\varepsilon}{\delta x}+v\frac{\delta\varepsilon}{\delta y}+w\frac{\delta\varepsilon}{\delta z}\right)-\frac{1}{k}\frac{\delta p}{\delta t}-\left(Xu+Yv+Zw\right) & =0 \end{align} \\ \hline \left(\varepsilon=\frac{1+\frac{P}{c^{2}}}{\sqrt{1-\frac{q^{2}}{c^{2}}}},\ \left(w\overline{\mathrm{lor}}\right)=w_{1}\frac{\delta}{\delta x_{1}}+w_{2}\frac{\delta}{\delta x_{2}}+w_{3}\frac{\delta}{\delta x_{3}}+w_{4}\frac{\delta}{\delta x_{4}}\right) \end{matrix}$$

and also gave the Euler equations of motion in the case of substances of least compressibility


 * $$\begin{matrix}\mathrm{lor}\,P=g\left(w\left(w\overline{\mathrm{lor}}\right)-\frac{1}{c}\frac{dw}{d\tau}\right)+\frac{g_{0}}{c^{2}}R\\

\hline \begin{align}\left(1-\frac{q^{2}}{c^{2}}\right)\frac{\delta g}{\delta x} & =-\frac{g}{c^{2}}\left\{ \frac{\delta u}{\delta t}+u\frac{\delta u}{\delta x}+v\frac{\delta u}{\delta y}+w\frac{\delta u}{\delta z}-u\,\mathrm{div}\,q\right\} +X\frac{g_{0}}{c^{2}}\sqrt{1-\frac{q^{2}}{c^{2}}}\\ \left(1-\frac{q^{2}}{c^{2}}\right)\frac{\delta g}{\delta y} & =-\frac{g}{c^{2}}\left\{ \frac{\delta v}{\delta t}+u\frac{\delta v}{\delta x}+v\frac{\delta v}{\delta y}+w\frac{\delta v}{\delta z}-v\,\mathrm{div}\,q\right\} +Y\frac{g_{0}}{c^{2}}\sqrt{1-\frac{q^{2}}{c^{2}}}\\ \left(1-\frac{q^{2}}{c^{2}}\right)\frac{\delta g}{\delta z} & =-\frac{g}{c^{2}}\left\{ \frac{\delta w}{\delta t}+u\frac{\delta w}{\delta x}+v\frac{\delta w}{\delta y}+w\frac{\delta w}{\delta z}-w\,\mathrm{div}\,q\right\} +Z\frac{g_{0}}{c^{2}}\sqrt{1-\frac{q^{2}}{c^{2}}}\\ \left(1-\frac{q^{2}}{c^{2}}\right)\frac{\delta g}{\delta t} & =-g\,\mathrm{div}\,q-\frac{g_{0}}{c^{2}}\sqrt{1-\frac{q^{2}}{c^{2}}}\left(Xu+Yv+Zw\right) \end{align} \\ \hline \left(g=g_{0}\sqrt{1+\frac{2p}{g_{0}c^{2}}}\right) \end{matrix}$$

Lewis & Wilson (1912)
Edwin Bidwell Wilson and Gilbert Newton Lewis devised an alternative 4D vector calculus based on Dyadics which, however, never gained widespread support. They defined the dyadic $$\mu_{0}\mathbf{ww}$$ using four-velocity $$\mathbf{w}$$ and rest mass density $$\mu_{0}$$ in order to formulate the fundamental equation of hydrodynamics:


 * $$\begin{matrix}\lozenge\cdot(\mu_{0}\mathbf{ww})=0\\

\left[\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}}\right] \end{matrix}$$

equivalent to the (f).

Einstein (1913-16)
In 1913, in the context of his Entwurf theory (a precursor of general relativity), Albert Einstein defined the equation for incoherent matter using the dust stress energy tensor $$\Theta_{\mu\nu}$$:


 * $$\begin{matrix}\sum_{\nu}\frac{\partial\mathfrak{T}_{\sigma\nu}}{\partial x_{\nu}}=\frac{1}{2}\sum_{\mu\nu\tau}\frac{\partial g_{\mu\nu}}{x_{\sigma}}\gamma_{\mu\tau}\mathfrak{T}_{\tau\nu}\\

\left[\mathfrak{T}_{\sigma\nu}=\sum_{\mu}\sqrt{-g}g_{\sigma\mu}\Theta_{\mu\nu}\right] \end{matrix}$$

equivalent to (f), and in 1916 he used the perfect fluid tensor $$\left.T_{\sigma}\right.^{\alpha}$$:


 * $$\frac{\partial\left.T_{\sigma}\right.^{\alpha}}{\partial x_{\alpha}}+\frac{1}{2}\frac{\partial g^{\mu\nu}}{x_{\sigma}}T_{\mu\nu}=0$$

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

Kottler (1914)
Friedrich Kottler derived the equations of motion from the “Nordström tensor” (i.e. dust tensor) in terms of rest mass density $$\nu_{0}$$:


 * $$\sum_{k=1}^{4}\frac{\partial}{\partial x^{(k)}}\left(\nu\frac{dx^{(h)}}{dt}\frac{dx^{(k)}}{dt}\right)=\nu\frac{d^{2}x^{(h)}}{dt^{2}}$$

equivalent to (f), which he then related to the action of a constant external electromagnetic field.