# History of Topics in Special Relativity/Four-momentum

History of 4-Vectors (edit)

## Overview

The w:four-momentum $P^{\mu }$  is defined as the product of mass and w:four-velocity $U^{\mu }$  or alternatively can be obtained by integrating the four-momentum density $P^{\mu }/V$  with respect to volume V (the four-momentum density corresponds to components $T^{\alpha 0}$  of the stress energy tensor combining energy density W and momentum density $\mathbf {g}$ ). In addition, replacing rest mass with rest mass density $\mu _{0}$  in terms of rest volume $V_{0}$  produces the mass four-current $J^{\mu }$  in analogy to the electric four-current:

${\begin{matrix}{\begin{matrix}P^{\mu }&\underbrace {=mU^{\mu }=m\gamma \left(c,\mathbf {v} \right)=\left({\frac {E}{c}},\mathbf {p} \right)} &=\underbrace {{\frac {1}{c}}\int \int \int T^{\alpha 0}dV} \\&(a)&(b)\\P^{\mu }/V&\underbrace {={\frac {1}{c}}T^{\alpha 0}=\left({\frac {W}{c}},\mathbf {g} \right)} \\&(c)\\J^{\mu }&\underbrace {=\mu _{0}U^{\mu }=\mu \left(c,\mathbf {v} \right)} \quad \left[\partial \cdot J^{\mu }=0\right]\\&(d)\end{matrix}}\\\left[\gamma ={\frac {\mu }{\mu _{0}}}={\frac {V_{0}}{V}}={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ m=\mu V=\mu _{0}V_{0}\right]\end{matrix}}$

Without explicitly defining the four-momentum vector, the Lorentz transformation of all components of (a) was given by #Planck (1907), while the Lorentz transformation of all components of (c) were given by #Laue (1911-13). The first explicit definition of (a) was given by #Minkowski (1908), followed by #Lewis and Wilson (1912), #Einstein (1912-14), #Cunningham (1914), #Weyl (1918-19). Four-momentum density (c) played a role in the papers of #Einstein (1912-14) and #Lewis and Wilson (1912). The material four-current (d) was given by #Laue (1913) and #Weyl (1918-19).

## Historical notation

### Planck (1907)

After w:Albert Einstein gave the energy transformation into the rest frame in 1905 and the general energy transformation in May 1907, w:Max Planck in June 1907 defined the transformation of both momentum ${\mathfrak {G}}$  and energy E as follows[R 1]

${\mathfrak {G}}_{x'}^{'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {G}}_{x}-{\frac {v(E+pV)}{c^{2}}}\right),\ {\mathfrak {G}}_{y'}^{'}={\mathfrak {G}}_{y},\ {\mathfrak {G}}_{z'}^{'}={\mathfrak {G}}_{z},\ E'={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left(E-v{\mathfrak {G}}_{x}-{\frac {v({\dot {x}}-v)}{c^{2}-v{\dot {x}}}}pV\right)$

or simplifying in terms of enthalpy R=E+pV:[R 2]

${\mathfrak {G}}_{x'}^{'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {G}}_{x}-{\frac {vR}{c^{2}}}\right),\ {\mathfrak {G}}_{y'}^{'}={\mathfrak {G}}_{y},\ {\mathfrak {G}}_{z'}^{'}={\mathfrak {G}}_{z},\ R'={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left(R-v{\mathfrak {G}}_{x}\right)$

and the transformations into the rest frame[R 3]

${\begin{matrix}E={\frac {c}{\sqrt {c^{2}-v^{2}}}}E_{0}^{\prime }+{\frac {q^{2}}{\sqrt {c^{2}-v^{2}}}}Vp_{0}^{\prime },\quad R={\frac {c}{\sqrt {c^{2}-v^{2}}}}R_{0}^{\prime },\quad G={\frac {q}{c^{2}}}R={\frac {q}{c{\sqrt {c^{2}-v^{2}}}}}R_{0}^{\prime }\\\left[{\mathfrak {G}}_{x}=G{\frac {\dot {x}}{q}},\ {\mathfrak {G}}_{x}=G{\frac {\dot {y}}{q}},\ {\mathfrak {G}}_{z}=G{\frac {\dot {z}}{q}}\right]\end{matrix}}$

Even though Planck wasn't using four-vectors, his formulas correspond to the Lorentz transformation of four-vector $\left[c{\mathfrak {G}}_{x},c{\mathfrak {G}}_{y},c{\mathfrak {G}}_{z},R=E+pV\right]$ , becoming the ordinary four-momentum $\left[c{\mathfrak {G}}_{x},c{\mathfrak {G}}_{y},c{\mathfrak {G}}_{z},E\right]$  by setting the pressure p=0.

### Minkowski (1907-09)

In 1907 (published 1908) w:Hermann Minkowski defined the following continuity equation with $\nu$  as rest mass density and w as four-velocity:[R 4]

${\begin{matrix}{\text{lor }}\nu {\overline {w}}={\frac {\partial \nu w_{1}}{\partial x_{1}}}+{\frac {\partial \nu w_{2}}{\partial x_{2}}}+{\frac {\partial \nu w_{3}}{\partial x_{3}}}+{\frac {\partial \nu w_{4}}{\partial x_{4}}}=0\\\left[{\text{lor }}=\left|{\frac {\partial }{\partial x_{1}}},\ {\frac {\partial }{\partial x_{2}}},\ {\frac {\partial }{\partial x_{3}}},\ {\frac {\partial }{\partial x_{4}}}\right|\right]\end{matrix}}$

which implies the mass four-current equivalent to (d).

The first mention of four-momentum (a) was given by Minkowski in his lecture “space and time” from 1908 (published 1909), calling it "momentum-vector" (“Impulsvektor”) as the product of mass m with the motion-vector (i.e. four-velocity) at a point P[R 5]. He further noted that if the time component of four-momentum is multiplied by $c^{2}$  it becomes the kinetic energy:

$m\,c^{2}{\frac {dt}{d\tau }}=m\,c^{2}\left/{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right.$

### Laue (1911-13)

w:Max von Laue (1911) in his influential first textbook on relativity, gave the Lorentz transformation of the components of the symmetric “world tensor” T (i.e. stress energy tensor), with the l=ict components being energy flux ${\mathfrak {S}}$ , momentum density ${\mathfrak {g}}$ , energy density W, and pointed out that the divergence of those l-components represents the energy conservation theorem (with A as power of the force density):[R 6]

{\begin{matrix}\left(T_{lx},T_{ly},T_{lz},T_{ll}\right)\Rightarrow \left({\frac {i}{c}}{\mathfrak {S}}_{x},\ {\frac {i}{c}}{\mathfrak {S}}_{y},\ {\frac {i}{c}}{\mathfrak {S}}_{z},\ -W\right)\\A+div{\mathfrak {S}}+{\frac {\partial W}{\partial t}}=0\\\hline {\begin{aligned}{\mathfrak {S}}_{x}&={\frac {\left(1+\beta ^{2}\right){\mathfrak {S}}_{x}^{\prime }+v\left(\mathbf {p} _{xx}^{\prime }+W^{\prime }\right)}{1-\beta ^{2}}}\\&={\frac {qc^{2}}{c^{2}-q^{2}}}\left(\mathbf {p} _{xx}^{0}+W^{0}\right)\end{aligned}},\ {\begin{aligned}{\mathfrak {S}}_{y}&={\frac {{\mathfrak {S}}_{y}^{\prime }+v\mathbf {p} _{xy}^{\prime }}{\sqrt {1-\beta ^{2}}}}\\&={\frac {qc}{c^{2}-q^{2}}}\mathbf {p} _{xy}^{0}\end{aligned}},\ {\begin{aligned}{\mathfrak {S}}_{z}&={\frac {{\mathfrak {S}}_{z}^{\prime }+v\mathbf {p} _{xz}^{\prime }}{\sqrt {1-\beta ^{2}}}}\\&={\frac {qc}{c^{2}-q^{2}}}\mathbf {p} _{xz}^{0}\end{aligned}},\ {\begin{aligned}W&={\frac {W'+\beta ^{2}\mathbf {p} _{xx}^{\prime }+2{\frac {v}{c^{2}}}{\mathfrak {S}}_{x}^{\prime }}{1-\beta ^{2}}}\\&={\frac {c^{2}W^{0}+q^{2}\mathbf {p} _{xx}^{0}}{c^{2}-q^{2}}}\end{aligned}}\\\left[{\mathfrak {g}}={\frac {\mathfrak {S}}{c^{2}}}\right]\end{matrix}}

which components correspond to four-momentum density (c) in case of vanishing pressure p, even though Laue didn't directly denoted it as a four-vector.

In the second edition (1912, published 1913), Laue discussed hydrodynamics in special relativity, defining the four-current of a material volume element in terms of rest mass density $k^{0}$  and four-velocity Y, and its continuity equation:[R 7]

${\begin{matrix}M=k^{0}Y\\M_{x}={\frac {k{\mathfrak {q}}_{x}}{c}},\ M_{y}={\frac {k{\mathfrak {q}}_{y}}{c}},\ M_{z}={\frac {k{\mathfrak {q}}_{z}}{c}},\ M_{l}=ik\\Div\,M=k^{0}Div\,Y+\left(Y,\Gamma \varrho \alpha \delta \,k^{0}\right)=0\\\left[k^{0}=k{\frac {\sqrt {c^{2}-q^{2}}}{c}},\ Div={\text{four-divergence}},\ \Gamma \varrho \alpha \delta ={\text{four-gradient}},\ l=ict\right]\end{matrix}}$

equivalent to material four-current (d).

### Lewis and Wilson (1912)

w:Edwin Bidwell Wilson and w:Gilbert Newton Lewis (1912) devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They explicitly defined “extended momentum” (i.e. four-momentum) $m_{0}\mathbf {w}$  and used it to derive the “extended force” (i.e. four force) together with $\mathbf {c}$  as four-acceleration:[R 8]

${\begin{matrix}m_{0}\mathbf {w} ={\frac {m_{0}v}{\sqrt {1-v^{2}}}}\mathbf {k} _{1}+{\frac {m_{0}}{\sqrt {1-v^{2}}}}\mathbf {k} _{4}=mv\mathbf {k} _{1}+m\mathbf {k} _{4}\\m_{0}\mathbf {w} =m\mathbf {v} +m\mathbf {k} _{4}\\\hline m_{0}\mathbf {c} ={\frac {dm_{0}\mathbf {w} }{ds}}={\frac {dmv}{ds}}\mathbf {k} _{1}+{\frac {dm}{ds}}\mathbf {k} _{4}={\frac {1}{\sqrt {1-v^{2}}}}\left({\frac {dmv}{dt}}\mathbf {k} _{1}+{\frac {dm}{dt}}\mathbf {k} _{4}\right)\\\left(m={\frac {m_{0}}{\sqrt {1-v^{2}}}},\ \mathbf {v} =\mathbf {k} _{1}{\frac {dx_{1}}{dx_{4}}}+\mathbf {k} _{2}{\frac {dx_{2}}{dx_{4}}}+\mathbf {k} _{3}{\frac {dx_{3}}{dx_{4}}}\right)\end{matrix}}$

equivalent to (a). Using rest mass density $\mu _{0}$ , they also defined the extended vector[R 9]

$\mu _{0}\mathbf {w} ={\frac {\mu _{0}}{\sqrt {1-v^{2}}}}\left(\mathbf {v} +\mathbf {k} _{4}\right)$

equivalent to the material four-current (d). Then they defined the four-momentum of radiant energy representing total momentum and energy per volume $d{\mathfrak {S}}$  by integrating electromagnetic energy density $e^{\prime 2}$  and the Poynting vector $e^{\prime 2}{\tfrac {\mathbf {l} _{s}}{l_{4}}}$ :[R 10]

$d\mathbf {g} =\left(e^{\prime 2}{\frac {\mathbf {l} _{s}}{l_{4}}}+e^{\prime 2}\mathbf {k} _{4}\right)d{\mathfrak {S}}$

equivalent to (b). They added, however, that the corresponding energy density vector $d\mathbf {g} /d{\mathfrak {S}}$  is not a four-vector because it is not independent of the chose axis.

### Einstein (1912-14)

In an unpublished manuscript on special relativity (written around 1912/14), w:Albert Einstein showed how to derive the components of the momentum-energy four-vector from the components $T_{\mu 4}$  (four-momentum density) of the stress-energy tensor (overline indicates integration over volume, $G_{\mu }$  is four-velocity):[R 11]

${\begin{matrix}{\overline {T}}_{14}=ic{\overline {\mathfrak {g}}}_{1}=ic\int {\mathfrak {g}}_{x}dxdydz\\{\overline {T}}_{44}=-{\overline {\eta }}=-\int \eta \,dxdydz\\\left({\overline {\mathfrak {g}}}_{1},{\overline {\mathfrak {g}}}_{1},{\overline {\mathfrak {g}}}_{1},{\frac {i}{c}}{\overline {\eta }}\right)\Rightarrow {\frac {{\overline {\eta }}_{0}}{c}}\left(G_{\mu }\right)\\\hline {\overline {\mathfrak {g}}}={\frac {m{\mathfrak {q}}_{x}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}},\ {\overline {\eta }}={\frac {mc^{2}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}\\\left[{\frac {{\bar {\eta }}_{0}}{c^{2}}}=m\right]\end{matrix}}$

equivalent to (a,b,c).

In the context of his Entwurf theory (a precursor of general relativity), Einstein (1913) formulated the following equations for momentum J and energy E using the metric tensor $g_{\mu \nu }$ , from which he concluded that momentum and energy of a material point form a “covariant vector” (i.e. covariant four-momentum), and also showed that the corresponding volume densities are equal to certain components of the stress-energy tensor $\Theta _{\mu \nu }$  (i.e. w:dust solution):[R 12]

${\begin{matrix}J_{x}=m{\frac {\partial H}{\partial {\dot {x}}}}=m{\frac {\dot {x}}{\sqrt {c^{2}-q^{2}}}},\ {\text{etc.}}\\E={\frac {\partial H}{\partial {\dot {x}}}}{\dot {x}}+{\frac {\partial H}{\partial {\dot {y}}}}{\dot {y}}+{\frac {\partial H}{\partial {\dot {z}}}}{\dot {z}}-H=m{\frac {c^{2}}{\sqrt {c^{2}-q^{2}}}}\\\hline J_{x}=-m{\frac {g_{11}{\dot {x_{1}}}+g_{12}{\dot {x_{2}}}+g_{13}{\dot {x_{3}}}+g_{14}}{\frac {ds}{dt}}}=-m{\frac {g_{11}dx_{1}+g_{12}dx_{2}+g_{13}dx_{3}+g_{14}dx_{4}}{ds}},\\-E=-\left({\dot {x}}{\frac {\partial H}{\partial {\dot {x}}}}+\cdot +\cdot \right)+H=-m\left(g_{41}{\frac {dx_{1}}{ds}}+g_{42}{\frac {dx_{2}}{ds}}+g_{43}{\frac {dx_{3}}{ds}}+g_{44}{\frac {dx_{4}}{ds}}\right)\\\hline {\frac {J_{x}}{V}}=-\varrho _{0}{\sqrt {-g}}\cdot \sum _{\nu }g_{1\nu }{\frac {dx_{\nu }}{ds}}\cdot {\frac {dx_{4}}{ds}}\\-{\frac {E}{V}}=-\varrho _{0}{\sqrt {-g}}\cdot \sum _{\nu }g_{4\nu }{\frac {dx_{\nu }}{ds}}\cdot {\frac {dx_{4}}{ds}}\\\left[\Theta _{\mu \nu }=\varrho _{0}{\frac {dx_{\mu }}{ds}}\cdot {\frac {dx_{\nu }}{ds}},\ \varrho _{0}={\frac {m}{V_{0}}}\right]\end{matrix}}$

equivalent to (a,b,c) in the case of $g_{\mu \nu }$  being the Minkowski tensor.

In 1914 Einstein summarized his previous arguments using the covariant four-vector $\mathbf {I} _{\sigma }$  (i.e. covariant four-momentum) and explicitly showed that in the case of $g_{\mu \nu }$  being the Minkowski tensor it becomes the ordinary four-momentum of special relativity. He also argued in a footnote why (in terms of his theory of gravitation) this covariant four-momentum $\mathbf {I} _{\sigma }$  is preferable over the contravariant four-momentum $\mathbf {I} ^{\sigma }$ :[R 13]

{\begin{matrix}\mathbf {I} _{\sigma }=m\sum _{\mu }g_{\sigma \mu }{\frac {dx_{\mu }}{ds}}\\{\frac {d\mathbf {I} _{\sigma }}{dx_{4}}}=\sum _{\nu \tau }\Gamma _{\nu \sigma }^{\tau }{\frac {dx_{\nu }}{dx_{4}}}\mathbf {I} _{\tau }+\int {\mathfrak {K}}_{\sigma }dv\\g_{\mu \nu }={\begin{matrix}-1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&1\end{matrix}}\Rightarrow \left.{\begin{aligned}-\mathbf {I} _{1}&={\frac {m{\mathfrak {q}}_{x}}{\sqrt {1-q^{2}}}}\\&\dots \\\mathbf {I} _{4}&={\frac {m}{\sqrt {1-q^{2}}}}\end{aligned}}\right\}\\\hline \mathbf {I} ^{\sigma }=m{\frac {dx_{\sigma }}{ds}}\end{matrix}}

equivalent to (a).

### Cunningham (1914)

Like Wilson and Lewis, w:Ebenezer Cunningham used the expression “extended momentum” ${\mathfrak {g}}$  (i.e. four-momentum), and derived the four-force from it:[R 14]

${\begin{matrix}{\mathfrak {g}}=(\mathbf {g} ,iw/c)\\\delta {\mathfrak {g}}=(\delta \mathbf {g} ,i\delta w/c)\\{\frac {d{\mathfrak {g}}}{dt_{0}}}=\kappa \left({\frac {d\mathbf {g} }{dt}},\ ic{\frac {dw}{dt}}\right)\\\hline \mathbf {g} ={\frac {w_{0}\mathbf {v} }{c^{2}\left(1-v^{2}/c^{2}\right)^{\frac {1}{2}}}},\ w={\frac {w_{0}}{\left(1-v^{2}/c^{2}\right)^{\frac {1}{2}}}}\end{matrix}}$

equivalent to (a, b).

### Weyl (1918-19)

In the first edition of his book “space time matter”, w:Hermann Weyl (1918) defined the “material current” in terms of rest mass density and four-velocity, together with its continuity equation:[R 15]

${\begin{matrix}\mu _{0}u^{i}\\\sum _{i}{\frac {\partial \left(\mu _{0}u^{i}\right)}{\partial x_{i}}}=0\\\left[{\frac {dv}{dV}}=\mu ,\ {\frac {dm}{dV_{0}}}=\mu _{0},\ \mu _{0}=\mu {\sqrt {1-v^{2}}},\ dV=dV_{0}{\sqrt {1-v^{2}}}\right]\end{matrix}}$

equivalent to (d).

In 1919, in the framework of general relativity, he expressed the pseudotensor density of total energy as $\left.{\mathfrak {S}}_{i}\right.^{k}$ , with the integral $J_{i}$  (i.e. four-momentum) of $\left.{\mathfrak {S}}_{i}\right.^{0}$  (i.e. four-momentum density) in space $x_{0}$  = const. representing energy (i=0) and momentum (i=1,2,3). For an arbitrary coordinate system he defined is as the product of mass and four-velocity[R 16]

$\left.{\mathfrak {S}}_{i}\right.^{0}\Rightarrow J_{i}=mu_{i},\quad u_{i}={\frac {dx_{i}}{ds}}$

equivalent to (a,b,c).

In the third edition of his book (1919), the description of the material current remained the same as in the first edition,[R 17] but this time he also included a description of four-momentum $J_{i}$  in terms of four-momentum density $\left.{\mathfrak {S}}_{i}\right.^{0}$ :[R 18]

${\begin{matrix}J_{i}=\int {\mathfrak {U}}_{i}^{0}dx_{1}dx_{2}dx_{3};\quad {\sqrt {J_{0}^{2}-J_{1}^{2}-J_{2}^{2}-J_{3}^{2}}}={\text{mass}}\\J_{i}=\int _{\Omega }{\mathfrak {S}}_{i}^{0}dx_{1}dx_{2}dx_{3}\\J_{i}=mu_{i}\quad \left(u^{i}={\frac {dx_{i}}{ds}}\right)\\{\frac {dJ_{i}}{dt}}=K_{i}\end{matrix}}$

equivalent to (a,b,c).