History of Topics in Special Relativity/Four-momentum

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

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

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).

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 ${\displaystyle {\mathfrak {G}}}$ and energy E as follows^{[R 1]}

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

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

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

and the transformations into the rest frame^{[R 3]}

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

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

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

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

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 ${\displaystyle c^{2}}$ it becomes the kinetic energy:

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

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 ${\displaystyle {\mathfrak {S}}}$, momentum density ${\displaystyle {\mathfrak {g}}}$, 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]}

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

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 ${\displaystyle k^{0}}$ and four-velocity Y, and its continuity equation:^{[R 7]}

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

equivalent to material four-current (d).

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) ${\displaystyle m_{0}\mathbf {w} }$ and used it to derive the “extended force” (i.e. four force) together with ${\displaystyle \mathbf {c} }$ as four-acceleration:^{[R 8]}

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

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

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

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

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

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

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 ${\displaystyle T_{\mu 4}}$ (four-momentum density) of the stress-energy tensor (overline indicates integration over volume, ${\displaystyle G_{\mu }}$ is four-velocity):^{[R 11]}

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

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 ${\displaystyle g_{\mu \nu }}$, 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 ${\displaystyle \Theta _{\mu \nu }}$ (i.e. w:dust solution):^{[R 12]}

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

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

In 1914 Einstein summarized his previous arguments using the covariant four-vector ${\displaystyle \mathbf {I} _{\sigma }}$ (i.e. covariant four-momentum) and explicitly showed that in the case of ${\displaystyle g_{\mu \nu }}$ 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 ${\displaystyle \mathbf {I} _{\sigma }}$ is preferable over the contravariant four-momentum ${\displaystyle \mathbf {I} ^{\sigma }}$:^{[R 13]}

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

equivalent to (a).

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

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

equivalent to (a, b).

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]}

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

equivalent to (d).

In 1919, in the framework of general relativity, he expressed the pseudotensor density of total energy as ${\displaystyle \left.{\mathfrak {S}}_{i}\right.^{k}}$, with the integral ${\displaystyle J_{i}}$ (i.e. four-momentum) of ${\displaystyle \left.{\mathfrak {S}}_{i}\right.^{0}}$ (i.e. four-momentum density) in space ${\displaystyle x_{0}}$ = 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]}

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

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 ${\displaystyle J_{i}}$ in terms of four-momentum density ${\displaystyle \left.{\mathfrak {S}}_{i}\right.^{0}}$:^{[R 18]}

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

equivalent to (a,b,c).

• Cunningham, E. (1914), The principle of relativity, Cambridge: University Press
• Einstein, A. (1912–14), "Einstein's manuscript on the special theory of relativity", The collected papers of Albert Einstein, vol. 4, pp. 3–108{{citation}}: CS1 maint: date format (link)
• Einstein, A. & Grossmann, M. (1913), "Entwurf einer verallgemeinerten Relativitätstheorie und eine Theorie der Gravitation", Zeitschrift für Mathematik und Physik, 62: 225–261{{citation}}: CS1 maint: multiple names: authors list (link)
• Einstein, A. (1914), "Die Formale Grundlage der allgemeinen Relativitätstheorie", Berliner Sitzungsberichte, 1914 (2): 1030–1085
• Laue, M. v. (1911), Das Relativitätsprinzip, Braunschweig: Vieweg
• Laue, M. v. (December 1912), Das Relativitätsprinzip (2. Edition), Braunschweig: Vieweg (published 1913)
• Lewis, G. N. & Wilson, E. B. (1912), "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences, 48: 387–507{{citation}}: CS1 maint: multiple names: authors list (link)
• Minkowski, H. (1908) [1907], "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111

See also the transcription Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern on German Wikisource

See also the transcription The Fundamental Equations for Electromagnetic Processes in Moving Bodies on English Wikisource

• Minkowski, H. (1909) [1908], "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88

See also the transcription Raum und Zeit on German Wikisource

See also the transcription Space and Time on English Wikisource

• Planck, M. (1907), "Zur Dynamik bewegter Systeme", Berliner Sitzungsberichte, Erster Halbband, 29: 542–570
  Reprinted in: Planck, M. (1908) [1907], "Zur Dynamik bewegter Systeme", Annalen der Physik, 331 (6): 1–34

See also the transcription Zur Dynamik bewegter Systeme on German Wikisource

See also the transcription On the Dynamics of Moving Systems on English Wikisource

• Weyl, H. (March 1918), Raum-Zeit-Relativität, Berlin: Springer
• Weyl, H. (1919), "Eine neue Erweiterung der Relativitätstheorie", Annalen der Physik, 364 (10): 101–133
• Weyl, H. (1919), Raum-Zeit-Relativität (3. Auflage), Berlin: Springer