History of Topics in Special Relativity/Electromagnetic tensor

History of relativistic Tensors ()
History of Electromagnetic field tensor History of Electromagnetic stress-energy tensor History of Material stress-energy tensor
History of Topics in Special Relativity () Multiple chapter topics History of Lorentz transformations History of relativistic four vectors History of relativistic tensors Single page topics History of twin paradox History of time dilation History of three-acceleration transformation Müller's history of relativity

Overview

The w:Electromagnetic tensor is an antisymmetric tensor that describes the electromagnetic field in spacetime. Its six independent components are composed of three electric ($\mathbf {E}$) plus three magnetic ($\mathbf {B}$) components. Those six components are analogous to six homogeneous line coordinates (w:Plücker coordinates), whose conditional equation corresponds to the invariant scalar product $\mathbf {E} \cdot \mathbf {B}$. It can be expressed as the exterior product of the w:four-gradient and the w:electromagnetic four-potential, producing a contravariant matrix as follows:

${\begin{matrix}F^{\mu \nu }={\underset {(a)}{\underbrace {\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}} }}={\underset {(b)}{\underbrace {\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }} }}\\\left[F^{\mu \nu }=-F^{\nu \mu },\ {\frac {1}{2}}F_{\mu \nu }F^{\mu \nu }=B^{2}-{\frac {E^{2}}{c^{2}}}\right]\end{matrix}}$

or in covariant form

$F_{\mu \nu }=\eta _{\mu \alpha }F^{\alpha \beta }\eta _{\beta \nu }={\underset {(c)}{\underbrace {\begin{bmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}} }}$

and the dual

${\begin{matrix}G^{\alpha \beta }={\frac {1}{2}}\epsilon ^{\alpha \beta \gamma \delta }F_{\gamma \delta }={\underset {(d)}{\underbrace {\begin{bmatrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\end{bmatrix}} }}\\\left[{\frac {1}{4}}G_{\gamma \delta }F^{\gamma \delta }=-{\frac {1}{c}}\mathbf {B} \cdot \mathbf {E} \right]\end{matrix}}$

The divergence of $F^{\mu \nu }$ can be related to the product of the four-potential $A^{\alpha }$ with the w:D'Alembert operator (in terms of the w:Lorenz gauge condition) and the four-current $J^{\alpha }$ representing the inhomogeneous w:Maxwell equations, while the divergence of $G^{\alpha \beta }$ represents the homogeneous equations:

$\partial _{\alpha }F^{\alpha \beta }={\underset {(e)}{\underbrace {\mu _{0}J^{\beta }} }}={\underset {(f)}{\underbrace {\square A^{\beta }} }};\quad {\underset {(g)}{\underbrace {\partial _{\alpha }G^{\alpha \beta }=0} }}$

It produces the four-force density using four-velocity $U^{\alpha }$ and rest charge density $\rho _{0}$:

$(h)\quad f_{\alpha }=F_{\alpha \beta }J^{\beta }=\rho _{0}F_{\alpha \beta }U^{\beta }$

and it also forms the basis of the electromagnetic stress-energy tensor:

$(i)\quad T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]$

The six independent components (corresponding to Plücker coordinates mentioned above) of the tensor can be used to formulate a "six-vector" and its dual:

$(j)\quad {\begin{matrix}\mathbf {F} =\left(F_{32},F_{13,}F_{21},F_{10},F_{20},F_{30}\right)=(\mathbf {B} ,\ -\mathbf {E} /c)\\\mathbf {F} ^{\ast }=\left(F_{32}^{\ast },F_{13}^{\ast },F_{21}^{\ast },F_{10}^{\ast },F_{20}^{\ast },F_{30}^{\ast }\right)=(-\mathbf {E} /c,\ \mathbf {B} )\end{matrix}}$

Alternatively, all those components appear in the Weber vector (also known as w:Riemann-Silberstein vector or electromagnetic bivector) and its conjugate, used in w:geometric algebra

$(k)\quad {\begin{matrix}\mathbf {F} =\mathbf {E} +i\mathbf {B} \\\mathbf {F} _{c}=\mathbf {E} -i\mathbf {B} \end{matrix}}\Rightarrow \left({\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }}\right)\mathbf {F} ={\frac {1}{\epsilon _{0}}}\rho -{\frac {1}{c}}\mathbf {J}$

Plücker coordinates were given by Grassmann (1844), Cayley (1859, 1867), Plücker (1865), Gordan (1868) and others. The Weber vector was given by #Weber (1901), Silberstein (1907). The invariants of the field tensor were known to #Poincaré (1905/6). The tensor itself was first given by #Minkowski (1907/8) in matrix notation, while #Born (1909), #Bateman (1909/10), #Abraham (1910), #Sommerfeld (1910), #Ignatowski (1910), #Lewis/Wilson (1910-12), #Laue (1911) devised alternative vector formulations. Quaternions were used by #Conway (1911) and #Silberstein (1911-12). Finally, the tensor was used in a generally covariant framework by #Kottler (1912) and #Einstein (1913).

Historical notation

Line coordinates: Grassmann (1844), Cayley (1859, 1867), Plücker (1865), Gordan (1868)

w:Hermann Grassmann (1844) introduced concepts that can be found in the formulation of the electromagnetic tensor: the w:exterior product, the Grassmann complement or dual, and essential parts of w:Plücker coordinates including its conditional equation. However, Grassmann's methods were essentially ignored by his contemporaries, while Plücker coordinates were independently rediscovered and further developed by others.

w:Arthur Cayley (1859, published 1860) defined a matrix of two points having four coordinates $x,y,z,w$ and $\alpha ,\beta ,\gamma ,\delta$, writing them in terms of six homogeneous line coordinates and their conditional equation in order to represent cones:

${\begin{matrix}\left|{\begin{matrix}x,&y,&z,&w\\\alpha ,&\beta ,&\gamma ,&\delta \end{matrix}}\right|\\\hline {\begin{aligned}p&=\gamma y-\beta z&s&=\delta x-\alpha w\\q&=\alpha z-\gamma x&t&=\delta y-\beta w\\r&=\beta x-\alpha y&u&=\delta z-\gamma w\end{aligned}}\\ps+qt+ru=0\end{matrix}}$

The full implications of these relations in terms of line geometry were first pointed out by w:Julius Plücker (1865), who independently derived them as the six homogeneous coordinates of the right line:

${\begin{matrix}\pm (uv'-u'v),\ \pm (tv'-t'v),\ \pm (tu'-t'u),\ \pm (tw'-t'w),\ \pm (uw'-u'w),\ \pm (vw'-v'w)\\(tu'-t'u)(vw'-v'w)-(tv'-t'v)(vw'-v'w)+(uv'-u'v)(tw'-t'w)=0\end{matrix}}$

After recounting his previous paper and the one of Plücker, Cayley (1867, published 1869) started with the definition of two points $(\alpha ,\beta ,\gamma ,\delta )$ and $(\alpha ',\beta ',\gamma ',\delta ')$, as well as two planes (A,B,C,D) and (A',B',C',D'), which he expressed in terms of six homogeneous line coordinates:

${\begin{matrix}{\begin{aligned}(A,B,C,D)\cdot (\alpha ,\beta ,\gamma ,\delta )\end{aligned}}\\\hline {\begin{matrix}&\beta \gamma '-\beta \gamma '&:&\gamma \alpha '-\gamma '\alpha &:&\alpha \beta '-\alpha '\beta &:&\alpha \delta '-\alpha '\delta &:&\beta \delta '-\beta '\delta &:&\gamma \delta '-\gamma '\delta \\=&AD'-A'D&:&BD'-B'D&:&CD'-C'D&:&BC'-B'C&:&CA'-C'A&:&AB'-A'B\\=&a&:&b&:&c&:&f&:&g&:&h\end{matrix}}\\\hline af+bg+ch=0\end{matrix}}$

He represented those relations by four matrices (w:Plücker matrix), of which two were discussed in more detail by Cayley: The first one he identified with the condition that a line (a,b,c,f,g,h) may be in a given plane (A,B,C,D)

$(A){\begin{matrix}\left|{\begin{matrix}0,&\alpha \beta '-\alpha '\beta ,&-(\gamma \alpha '-\gamma '\alpha ),&\alpha \delta '-\alpha '\delta \\-(\alpha \beta '-\alpha '\beta ),&0,&\beta \gamma '-\beta \gamma ',&\beta \delta '-\beta '\delta \\\gamma \alpha '-\gamma '\alpha ,&-(\beta \gamma '-\beta \gamma '),&0,&\gamma \delta '-\gamma '\delta \\-(\alpha \delta '-\alpha '\delta ),&-(\beta \delta '-\beta '\delta ),&-(\gamma \delta '-\gamma '\delta ),&0\end{matrix}}\right|(A,B,C,D)=0\\\Rightarrow \left|{\begin{matrix}0,&c,&-b,&f\\-c,&0,&a,&g\\b,&-a,&0,&h\\-f,&-g,&-h,&0\end{matrix}}\right|(A,B,C,D)=0\end{matrix}}$

and the second one with the condition that a line (a,b,c,f,g,h) may pass through a given point $(\alpha ,\beta ,\gamma ,\delta )$:

$(B){\begin{matrix}\left|{\begin{matrix}0,&AB'-A'B,&-(CA'-C'A),&AD'-A'D\\-(AB'-A'B),&0,&BC'-B'C,&BD'-B'D\\CA'-C'A,&-(BC'-B'C),&0,&CD'-C'D\\-(AD'-A'D),&-(BD'-B'D),&-(CD'-C'D),&0\end{matrix}}\right|(\alpha ,\beta ,\gamma ,\delta )=0\\\Rightarrow \left|{\begin{matrix}0,&h,&-g,&a\\-h,&0,&g,&b\\g,&-f,&0,&c\\-a,&-b,&-c,&0\end{matrix}}\right|(\alpha ,\beta ,\gamma ,\delta )=0\end{matrix}}$

He further introduced the "tractor" as a line that meets any given line.

w:Paul Gordan (1868) expressed these relations in compact index notation

${\begin{matrix}p_{ik}=x_{i}y_{k}-y_{i}x_{k},\ \left(p_{ik}=-p_{ki}\right)\\P=p_{12}p_{34}+p_{13}p_{42}+p_{14}p_{23}\end{matrix}}$

Weber (1901), Silberstein (1907)

w:Heinrich Martin Weber published a completely rewritten fourth edition of what he called "The partial differential equations of mathematical physics according to Riemann's lectures" in two volumes (1900, 1901). As he pointed out in the preface of the first volume (1900), the fourth edition represented Weber's own work on differential equations (unlike the first three editions (1882) which were based on Riemann's actual lectures), though he was still using the name of Riemann in the title because it preserved the overall conception of the original edition and that he tried to continue the work in Riemann's sense and spirit In the second volume (1901), Weber combined the electric and magnetic field components of the inhomogeneous Maxwell equations into a single complex vector having the same components as the electromagnetic tensor:

$c\ \mathrm {curl} \,({\mathfrak {E}}+i{\mathfrak {M}})=i{\frac {\partial ({\mathfrak {E}}+i{\mathfrak {M}})}{\partial t}}$

w:Ludwik Silberstein (1906/07) independently derived this relation for all four Maxwell equations using an "electromagnetic bivector" $\eta$ and its conjugate $\eta '$. While the term "bivector" was originally developed by Hamilton in terms of w:biquaternions, Silberstein represented it in terms of Heaviside's vector calculus, and also gave the expressions for energy density $e$ and w:Poynting vector $F$:

${\begin{matrix}{\begin{matrix}{\frac {\partial \eta }{\partial t}}=-i\,\mathrm {curl} \,\eta &&&{\frac {\partial \eta '}{\partial t}}=i\,\mathrm {curl} \,\eta '\\\mathrm {div} \eta =0&&&\mathrm {div} \eta '=0\\\left[\eta =E_{1}+iE_{2}\right]&&&\left[\eta '=E_{1}-iE_{2}\right]\end{matrix}}\\\hline {\frac {1}{2}}\left(\eta +\eta '\right)=E_{1},\ {\frac {1}{2i}}\left(\eta -\eta '\right)=E_{2}\\e={\frac {1}{2}}\eta \eta ',\ F={\frac {i}{2}}V\eta \eta '\end{matrix}}$

In a subsequent paper he gave credit to Weber as well.

Poincaré (1905/6)

Using the Lorentz transformation of the electromagnetic field first derived by w:Hendrik Lorentz (1904), it was shown by w:Henri Poincaré in July 1905 (published 1906) that the six electromagnetic quantities $f,g,h,\alpha ,\beta ,\gamma$ (= $\mathbf {E} ,\mathbf {B}$) can be combined to form the following Lorentz invariant relations (Poincaré signifies a vector by using the symbol $\Sigma$ followed by the first vector component):

${\begin{matrix}l^{4}\left(\sum f^{\prime 2}-\sum \alpha ^{\prime 2}\right)=\sum f^{2}-\sum \alpha ^{2}\\l^{4}\sum f^{\prime }\alpha ^{\prime }=\sum f\alpha \\\left[l=1\right]\end{matrix}}$

which are proportional to the invariants in (a) and (d), even though Poincaré wasn't in possession of the concept of the electromagnetic tensor.

Minkowski (1907/8)

w:Hermann Minkowski employed matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product from the outset. In a lecture held in November 1907, published 1915, Minkowski defined the electromagnetic tensor $\psi _{jk}$ (which he called "Traktor") in terms of four-potential $\psi _{j}$ using the field quantities ${\mathfrak {E}},{\mathfrak {H}}$ (= $\mathbf {E} ,\mathbf {B}$):

${\begin{matrix}\psi _{jk}={\frac {\partial \psi _{k}}{\partial x_{j}}}-{\frac {\partial \psi _{j}}{\partial x_{k}}},\ \left(\psi _{jk}=-\psi _{kj},\ \psi _{jj}=0\right)\\\left(\psi _{23},\ \psi _{31},\ \psi _{12};\ \psi _{14},\ \psi _{24},\ \psi _{34}\right)=\left({\mathfrak {H}}_{x},{\mathfrak {H}}_{y},{\mathfrak {H}}_{z};\ -i{\mathfrak {E}}_{x},-i{\mathfrak {E}}_{y},-i{\mathfrak {E}}_{z}\right)\end{matrix}}$:

equivalent to (a,b).

In another lecture from December 1907, he represented the six field quantities ${\mathfrak {e}},{\mathfrak {m}}$ (= $\mathbf {E} ,\mathbf {B}$) together as a "vector of second kind" $f$ (= the electromagnetic tensor), its dual $f^{*}$, and its Lorentz transformation using transformation matrix $A$, and its two invariants:

${\begin{matrix}\left(f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}\right)=\left({\mathfrak {m}}_{x},\ {\mathfrak {m}}_{y},\ {\mathfrak {m}}_{z},\ -i{\mathfrak {e}}_{x},\ -i{\mathfrak {e}}_{y},\ -i{\mathfrak {e}}_{z}\right)\\\left[f_{kh}=-f_{hk}\right]\\\hline f={\begin{vmatrix}0,&f_{12},&f_{13},&f_{14}\\f_{21},&0,&f_{23},&f_{24}\\f_{31},&f_{32},&0,&f_{34}\\f_{41},&f_{42},&f_{43},&0\end{vmatrix}},\ f^{*}={\begin{vmatrix}0,&f_{34},&f_{42},&f_{23}\\f_{43},&0,&f_{14},&f_{31}\\f_{24},&f_{41},&0,&f_{12}\\f_{32},&f_{13},&f_{21},&0\end{vmatrix}}\\\hline f'\Rightarrow {\bar {\rm {A}}}f{\rm {A}}={\rm {A}}^{-1}f{\rm {A}}\\f'_{41}=f_{41}\cos i\psi +f_{13}\sin i\psi ,\quad f'_{13}=-f_{41}\sin i\psi +f_{13}\cos i\psi ,\quad f'_{34}=f_{34},\\f'_{32}=f_{32}\cos i\psi +f_{42}\sin i\psi ,\quad f'_{42}=-f_{32}\sin i\psi +f_{42}\cos i\psi ,\quad f'_{12}=f_{12},\\\left[f'_{kh}=-f'_{hk}\right]\\\hline {\mathfrak {m}}^{2}-{\mathfrak {e}}^{2}=f_{23}^{2}+f_{31}^{2}+f_{12}^{2}+f_{14}^{2}+f_{24}^{2}+f_{34}^{2},\\{\mathfrak {me}}=i\left(f_{23}f_{14}+f_{31}f_{24}+f_{12}f_{34}\right)\end{matrix}}$

equivalent to (a,b,c,d), which he used to express the microscopic Maxwell equations in terms of four-current $\varrho$:

$(A)\ {\begin{array}{cccccc}&{\frac {\partial f_{12}}{\partial x_{2}}}&+{\frac {\partial f_{13}}{\partial x_{3}}}&+{\frac {\partial f_{14}}{\partial x_{4}}}&=&\varrho _{1},\\{\frac {\partial f_{21}}{\partial x_{1}}}&&+{\frac {\partial f_{23}}{\partial x_{3}}}&+{\frac {\partial f_{24}}{\partial x_{4}}}&=&\varrho _{2},\\{\frac {\partial f_{31}}{\partial x_{1}}}&+{\frac {\partial f_{32}}{\partial x_{2}}}&&+{\frac {\partial f_{34}}{\partial x_{4}}}&=&\varrho _{3},\\{\frac {\partial f_{41}}{\partial x_{1}}}&+{\frac {\partial f_{42}}{\partial x_{2}}}&+{\frac {\partial f_{43}}{\partial x_{3}}}&&=&\varrho _{4}.\end{array}}\quad (B)\ {\begin{array}{cccccc}&{\frac {\partial f_{34}}{\partial x_{2}}}&+{\frac {\partial f_{42}}{\partial x_{3}}}&+{\frac {\partial f_{23}}{\partial x_{4}}}&=&0,\\{\frac {\partial f_{43}}{\partial x_{1}}}&&+{\frac {\partial f_{14}}{\partial x_{3}}}&+{\frac {\partial f_{31}}{\partial x_{4}}}&=&0,\\{\frac {\partial f_{24}}{\partial x_{1}}}&+{\frac {\partial f_{41}}{\partial x_{2}}}&&+{\frac {\partial f_{12}}{\partial x_{4}}}&=&0,\\{\frac {\partial f_{32}}{\partial x_{1}}}&+{\frac {\partial f_{13}}{\partial x_{2}}}&+{\frac {\partial f_{21}}{\partial x_{3}}}&&=&0.\end{array}}$

equivalent to (e,f,g), and implicitly used the Weber vector in order to simplify the Lorentz transformations:

${\begin{matrix}{\mathfrak {e}}'_{x'}+i{\mathfrak {m}}'_{x'}=\left({\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x}\right)\cos i\psi +\left({\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y}\right)\sin i\psi ,\\{\mathfrak {e}}'_{y'}+i{\mathfrak {m}}'_{y'}=-\left({\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x}\right)\sin i\psi +\left({\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y}\right)\cos i\psi ,\\{\mathfrak {e}}'_{z'}+i{\mathfrak {m}}'_{z'}={\mathfrak {e}}_{z}+i{\mathfrak {m}}_{z}\end{matrix}}$

equivalent to (k). In addition he defined $F$ (with its Hodge dual $F^{*}$) as the electromagnetic tensor in the presence of matter using the field quantities ${\mathfrak {E}},{\mathfrak {M}}$ (= $\mathbf {D} ,\mathbf {H}$ in modern notation), by which he expressed the macroscopic Maxwell equations more generally in terms of "electric current" ${\mathfrak {s}}$ which becomes ${\mathfrak {s}}=\sigma {\mathfrak {E}}$ in isotropic media, which he further simplified using differential operator "lor"

${\begin{matrix}{\mathfrak {e}}=\varepsilon {\mathfrak {E}},\quad {\mathfrak {M}}=\mu {\mathfrak {m}},\quad {\mathfrak {s}}=\sigma {\mathfrak {E}},\quad \left({\mathfrak {s}}_{x},{\mathfrak {s}}_{y},{\mathfrak {s}}_{z},i\varrho \right)\Rightarrow \left(s_{1},s_{2},s_{3},s_{4}\right)\\\left(f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}\right)=\left({\mathfrak {m}}_{x},\ {\mathfrak {m}}_{y},\ {\mathfrak {m}}_{z},\ -i{\mathfrak {e}}_{x},\ -i{\mathfrak {e}}_{y},\ -i{\mathfrak {e}}_{z}\right)\\\left(F_{23},\ F_{31},\ F_{12},\ F_{14},\ F_{24},\ F_{34}\right)=\left({\mathfrak {M}}_{x},\ {\mathfrak {M}}_{y},\ {\mathfrak {M}}_{z},\ -i{\mathfrak {E}}_{x},\ i{\mathfrak {E}}_{y},\ i{\mathfrak {E}}_{z}\right)\\\left[f_{kh}=-f_{hk},\ F_{kh}=-F_{hk}\right]\\\hline {\begin{matrix}{\text{lor }}f=-s&{\text{lor }}F^{*}=0.\\\hline (A)\ {\begin{array}{cccccc}&{\frac {\partial f_{12}}{\partial x_{2}}}&+{\frac {\partial f_{13}}{\partial x_{3}}}&+{\frac {\partial f_{14}}{\partial x_{4}}}&=&s_{1},\\{\frac {\partial f_{21}}{\partial x_{1}}}&&+{\frac {\partial f_{23}}{\partial x_{3}}}&+{\frac {\partial f_{24}}{\partial x_{4}}}&=&s_{2},\\{\frac {\partial f_{31}}{\partial x_{1}}}&+{\frac {\partial f_{32}}{\partial x_{2}}}&&+{\frac {\partial f_{34}}{\partial x_{4}}}&=&s_{3},\\{\frac {\partial f_{41}}{\partial x_{1}}}&+{\frac {\partial f_{42}}{\partial x_{2}}}&+{\frac {\partial f_{43}}{\partial x_{3}}}&&=&s_{4},\end{array}}&\quad (B)\ {\begin{array}{cccccc}&{\frac {\partial F_{34}}{\partial x_{2}}}&+{\frac {\partial F_{42}}{\partial x_{3}}}&+{\frac {\partial F_{23}}{\partial x_{4}}}&=&0,\\{\frac {\partial F_{43}}{\partial x_{1}}}&&+{\frac {\partial F_{14}}{\partial x_{3}}}&+{\frac {\partial F_{31}}{\partial x_{4}}}&=&0,\\{\frac {\partial F_{24}}{\partial x_{1}}}&+{\frac {\partial F_{41}}{\partial x_{2}}}&&+{\frac {\partial F_{12}}{\partial x_{4}}}&=&0,\\{\frac {\partial F_{32}}{\partial x_{1}}}&+{\frac {\partial F_{13}}{\partial x_{2}}}&+{\frac {\partial F_{21}}{\partial x_{3}}}&&=&0.\end{array}}\end{matrix}}\\\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}}$

He used his tensors to define the electric rest force $\Phi$ and magnetic rest force $\Psi$ as the product with four-velocity $w$, which in turn can be used to express $F$ and $f$ and the four-conductivity

${\begin{matrix}{\begin{matrix}\Phi =-wF&\Psi =iwf^{*},\\\hline \left.{\begin{array}{ccccccccr}\Phi _{1}&=&&&w_{2}F_{12}&+&w_{3}F_{13}&+&w_{4}F_{14},\\\Phi _{2}&=&w_{1}F_{21}&&&+&w_{3}F_{23}&+&w_{4}F_{24},\\\Phi _{3}&=&w_{1}F_{31}&+&w_{2}F_{32}&&&+&w_{4}F_{34},\\\Phi _{4}&=&w_{1}F_{41}&+&w_{2}F_{42}&+&w_{3}F_{43}&&.\end{array}}\right|&{\begin{array}{cclcccccr}\Psi _{1}&=&-i(&&w_{2}f_{34}&+&w_{3}f_{42}&+&w_{4}f_{23}),\\\Psi _{2}&=&-i(w_{1}f_{43}&&&+&w_{3}f_{14}&+&w_{4}f_{31}),\\\Psi _{3}&=&-i(w_{1}f_{24}&+&w_{2}f_{41}&&&+&w_{4}f_{12}),\\\Psi _{4}&=&-i(w_{1}f_{32}&+&w_{2}f_{13}&+&w_{3}f_{21}&&).\end{array}}\end{matrix}}\\\hline wF=-\Phi ,\quad wF^{*}=-i\mu \Psi ,\quad wf=-\varepsilon \Phi ,\quad wf^{*}=-i\Psi \\F=[w,\Phi ]+i\mu [w,\Psi ]^{*},\\f=\varepsilon [w,\Phi ]+i[w,\Psi ]^{*},\end{matrix}}$

equivalent to (h). He finally used it to define the stress-energy tensor $S$ and Lagrangian $L$:

${\begin{matrix}\hline fF=S-L=\left|{\begin{array}{llll}S_{11}-L,&S_{12},&S_{13},&S_{14}\\S_{21},&S_{22}-L,&S_{23},&S_{23}\\S_{31},&S_{32},&S_{33}-L,&S_{34}\\S_{41},&S_{42},&S_{43},&S_{44}-L\end{array}}\right|,\quad F^{*}f^{*}=-S-L=\left|{\begin{array}{llll}-S_{11}-L,&-S_{12},&-S_{13},&-S_{14}\\-S_{21},&-S_{22}-L,&-S_{23},&-S_{23}\\-S_{31},&-S_{32},&-S_{33}-L,&-S_{34}\\-S_{41},&-S_{42},&-S_{43},&-S_{44}-L\end{array}}\right|\\\hline L={\frac {1}{2}}(f_{23}F_{23}+f_{31}F_{31}+f_{12}F_{12}+f_{14}F_{14}+f_{24}F_{24}+f_{34}F_{34}),\\S_{11}={\frac {1}{2}}(f_{23}F_{23}+f_{34}F_{34}+f_{42}F_{42}-f_{12}F_{12}-f_{13}F_{13}-f_{14}F_{14})\\S_{12}=f_{13}F_{32}+f_{14}F_{42},etc.\\\hline {\text{lor }}S={\text{lor }}L+{\text{lor }}fF=-sF+N\\{\scriptstyle N_{h}={\frac {1}{2}}\left({\frac {\partial f_{23}}{\partial x_{h}}}F_{23}+{\frac {\partial f_{31}}{\partial x_{h}}}F_{31}+{\frac {\partial f_{12}}{\partial x_{h}}}F_{12}+{\frac {\partial f_{14}}{\partial x_{h}}}F_{14}+{\frac {\partial f_{24}}{\partial x_{h}}}F_{24}+{\frac {\partial f_{34}}{\partial x_{h}}}F_{34}-f_{23}{\frac {\partial F_{23}}{\partial x_{h}}}-f_{31}{\frac {\partial F_{31}}{\partial x_{h}}}-f_{12}{\frac {\partial F_{12}}{\partial x_{h}}}-f_{14}{\frac {\partial F_{14}}{\partial x_{h}}}-f_{24}{\frac {\partial F_{24}}{\partial x_{h}}}-{\frac {\partial f_{34}}{\partial x_{h}}}F_{34}\right)}\end{matrix}}$

equivalent to (i).

Born (1909)

w:Max Born (1909) summarized Minkowski's work, defining the electromagnetic field and its Lagrangian as:

${\begin{matrix}f_{\alpha \beta }={\frac {\partial \Phi _{\beta }}{\partial x_{\alpha }}}-{\frac {\partial \Phi _{\alpha }}{\partial x_{\beta }}}=-f_{\beta \alpha }\\\left(f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}\right)=\left({\mathfrak {M}}_{x},\ {\mathfrak {M}}_{y},\ {\mathfrak {M}}_{z},\ -i{\mathfrak {E}}_{x},-i{\mathfrak {E}}_{y},-i{\mathfrak {E}}_{z}\right)\\L={\frac {1}{2}}\left({\mathfrak {M}}^{2}-{\mathfrak {E}}^{2}\right)={\frac {1}{2}}\sum _{\alpha \neq \beta }f_{\alpha \beta }^{2}\end{matrix}}$

equivalent to (a,b), and used it to express Maxwell's equations:

${\begin{array}{cccccc}&{\frac {\partial f_{12}}{\partial x_{2}}}&+{\frac {\partial f_{13}}{\partial x_{3}}}&+{\frac {\partial f_{14}}{\partial x_{4}}}&=&\varrho _{1},\\{\frac {\partial f_{21}}{\partial x_{1}}}&&+{\frac {\partial f_{23}}{\partial x_{3}}}&+{\frac {\partial f_{24}}{\partial x_{4}}}&=&\varrho _{2},\\{\frac {\partial f_{31}}{\partial x_{1}}}&+{\frac {\partial f_{32}}{\partial x_{2}}}&&+{\frac {\partial f_{34}}{\partial x_{4}}}&=&\varrho _{3},\\{\frac {\partial f_{41}}{\partial x_{1}}}&+{\frac {\partial f_{42}}{\partial x_{2}}}&+{\frac {\partial f_{43}}{\partial x_{3}}}&&=&\varrho _{4}.\end{array}}\quad {\begin{array}{cccccc}&{\frac {\partial f_{34}}{\partial x_{2}}}&+{\frac {\partial f_{42}}{\partial x_{3}}}&+{\frac {\partial f_{23}}{\partial x_{4}}}&=&0,\\{\frac {\partial f_{43}}{\partial x_{1}}}&&+{\frac {\partial f_{14}}{\partial x_{3}}}&+{\frac {\partial f_{31}}{\partial x_{4}}}&=&0,\\{\frac {\partial f_{24}}{\partial x_{1}}}&+{\frac {\partial f_{41}}{\partial x_{2}}}&&+{\frac {\partial f_{12}}{\partial x_{4}}}&=&0,\\{\frac {\partial f_{32}}{\partial x_{1}}}&+{\frac {\partial f_{13}}{\partial x_{2}}}&+{\frac {\partial f_{21}}{\partial x_{3}}}&&=&0.\end{array}}$

equivalent to (e,f,g), and defined the four-force density

$ci\mu {\frac {\partial ^{2}x_{\alpha }}{\partial \xi _{4}^{2}}}={\frac {\varrho _{0}^{\ast }}{c}}\sum _{\beta =1}^{4}f_{\alpha \beta }{\frac {\partial x_{\beta }}{\partial \xi _{4}}}$

equivalent to (h).

Bateman (1909/10)

In a paper read 1909 and published 1910, w:Harry Bateman discussed the electrodynamic equations in terms of four-dimensional integral forms (even though in the broader context of w:spherical wave transformations, with $\lambda ^{2}=1$ in relativity):

${\begin{matrix}H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\\E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\\\left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt\\\left(E_{x}H_{x}+E_{y}H_{y}+E_{z}H_{z}\right)dx\ dy\ dz\ dt\end{matrix}}$

which are the invariants of the electromagnetic field.

Abraham (1910)

w:Max Abraham expressed Minkowski's vectors of first kind (four-vector) as $V_{I}^{4}$ and vectors of second kind (six-vector) as $V_{II}^{4}$. Corresponding to Minkowski's $f$ and $F$, Abraham defined an electrodynamic $V_{II}^{4}$ in terms or ${\mathfrak {E}},{\mathfrak {B}}$ (= $\mathbf {E} ,\mathbf {B}$ modern notation) and another one in terms of ${\mathfrak {H}},{\mathfrak {D}}$ (= $\mathbf {H} ,\mathbf {D}$ in modern notation) as follows

${\begin{matrix}\left\{{\mathfrak {B}},\ -i{\mathfrak {E}}\right\},\quad \left\{{\mathfrak {H}},\ -i{\mathfrak {D}}\right\}\\\hline V_{II}^{4}\left\{{\mathfrak {a,b}}\right\},\ V_{II}^{4}\left\{{\mathfrak {a',b'}}\right\}\\{\mathfrak {a}}=\left[{\mathfrak {r}}_{1}{\mathfrak {r}}_{2}\right],\ {\mathfrak {b}}={\mathfrak {r_{1}}}u_{2}-{\mathfrak {r}}_{2}u_{1}\\\left\{{\begin{array}{ccccc}{\mathfrak {a}}_{x}=\left|{\begin{array}{cc}y_{1}&z_{1}\\y_{2}&z_{2}\end{array}}\right|,&&{\mathfrak {a}}_{y}=\left|{\begin{array}{cc}z_{1}&x_{1}\\z_{2}&x_{2}\end{array}}\right|,&&{\mathfrak {a}}_{z}=\left|{\begin{array}{cc}x_{1}&y_{1}\\x_{2}&y_{2}\end{array}}\right|\\{\mathfrak {b}}_{x}=\left|{\begin{array}{cc}x_{1}&u_{1}\\x_{2}&u_{2}\end{array}}\right|,&&{\mathfrak {b}}_{y}=\left|{\begin{array}{cc}y_{1}&u_{1}\\y_{2}&u_{2}\end{array}}\right|,&&{\mathfrak {b}}_{z}=\left|{\begin{array}{cc}z_{1}&u_{1}\\z_{2}&u_{2}\end{array}}\right|\end{array}}\right.\\{\begin{cases}{\mathfrak {a}}={\mathfrak {H}},&{\mathfrak {b}}=-i{\mathfrak {D}}\\{\mathfrak {a}}'={\mathfrak {B}},&{\mathfrak {b}}'=-i{\mathfrak {E}}.\end{cases}}\end{matrix}}$

equivalent to (a). The product of those $V_{II}^{4}$ with $V_{I}^{4}$-velocity (= four-velocity) gives Minkowski's electric and magnetic rest forces:

${\begin{matrix}\left\{{\begin{matrix}\left\{{\mathfrak {B}},\ -i{\mathfrak {E}}\right\}\\V_{I}^{4}-\mathrm {velocity} \end{matrix}}\right\}&\Rightarrow \left\{{\mathfrak {R}}^{e},\ U^{e}\right\}\\\left\{{\begin{matrix}\left\{{\mathfrak {H}},\ -i{\mathfrak {D}}\right\}\\\left\{V_{I}^{4}-\mathrm {velocity} \right\}\end{matrix}}\right\}&\Rightarrow \left\{{\mathfrak {R}}^{m},\ U^{m}\right\}\end{matrix}}$

equivalent to (h), and used it to determine some components of the stress energy tensor, namely momentum density ${\mathfrak {f}}$ and energy density $\psi$:

${\begin{matrix}V_{II}^{4}\left\{{\mathfrak {a,b}}\right\},\ V_{II}^{4}\left\{{\mathfrak {a',b'}}\right\}\\2{\mathfrak {f}}=i[{\mathfrak {b'a}}]+i[{\mathfrak {b'a}}]\\2\psi =({\mathfrak {aa}}')-({\mathfrak {bb}}')\\\hline {\begin{cases}{\mathfrak {a}}={\mathfrak {H}},&{\mathfrak {b}}=-i{\mathfrak {D}}\\{\mathfrak {a}}'={\mathfrak {B}},&{\mathfrak {b}}'=-i{\mathfrak {E}}.\end{cases}}\\2{\mathfrak {f}}=[{\mathfrak {EH}}]+[{\mathfrak {DB}}]\\2\psi ={\mathfrak {ED}}+{\mathfrak {HB}}.\end{matrix}}$

equivalent to the corresponding components of (i).

Sommerfeld (1910)

w:Arnold Sommerfeld translated Minkowski's matrix formalism into a four-dimensional vector formalism involving four-vectors and six-vectors in two papers. With reference to Grassmann, in the first paper he defined the general six-vector $f$ in terms of the special six-vector $\varphi$ and its supplement $\varphi ^{\ast }$

${\begin{matrix}\varphi _{xy}=u_{x}v_{y}-v_{x}u_{y}\\\varphi _{yz}\varphi _{xl}+\varphi _{zx}\varphi _{yl}+\varphi _{xy}\varphi _{zl}=0\\\left|\varphi \right|^{2}=\varphi _{yz}^{2}+\varphi _{zx}^{2}+\varphi _{xy}^{2}+\varphi _{xl}^{2}+\varphi _{yl}^{2}+\varphi _{zl}^{2}=1\\\varphi _{yz}^{*}=\varphi _{xl},\ \varphi _{zx}^{*}=\varphi _{yl},\ \varphi _{xy}^{*}=\varphi _{zl},\ \varphi _{xl}^{*}=\varphi _{yz}\dots \\\hline {\begin{array}{ll}u_{x}^{*}u_{x}+u_{y}^{*}u_{y}+u_{z}^{*}u_{z}+u_{l}^{*}u_{l}&=0,\\u_{x}^{*}v_{x}+u_{y}^{*}v_{y}+u_{z}^{*}v_{z}+u_{l}^{*}v_{l}&=0,\\v_{x}^{*}u_{x}+v_{y}^{*}u_{y}+v_{z}^{*}u_{z}+v_{l}^{*}u_{l}&=0,\\v_{x}^{*}v_{x}+v_{y}^{*}v_{y}+v_{z}^{*}v_{z}+v_{l}^{*}v_{l}&=0.\end{array}}\\\hline f=\varrho \varphi +\varrho ^{*}\varphi ^{*}\\f^{*}=\varrho ^{*}\varphi +\varrho \varphi ^{*}\end{matrix}}$

The general six-vector $f$ was then related to the electromagnetic field, together with the definition of its supplement or dual, its invariants, and its Lorentz transformation as:

${\begin{matrix}f=({\mathfrak {H}},-i{\mathfrak {E}})\\\left\{{\begin{array}{lrlrlr}f_{yz}=&{\mathfrak {H}}_{x},&f_{zx}=&{\mathfrak {H}}_{y},&f_{xy}=&{\mathfrak {H}}_{z},\\f_{xl}=&-i{\mathfrak {E}}_{x},&f_{yl}=&-i{\mathfrak {E}}_{y},&f_{zl}=&-i{\mathfrak {E}}_{z},\end{array}}\right.\\f_{ik}=-f_{ki},\ f_{ii}=0\\\hline f_{zx}^{*}=f_{yl},\ f_{xy}^{*}=f_{zl},\ f_{xl}^{*}=f_{yz},\ f_{yl}^{*}=f_{zx},\ f_{zl}^{*}=f_{xy}\\f_{ik}^{*}=f_{(ik)},\ f_{ki}^{*}=-f_{ik}^{*},\ f_{ii}^{*}=0\\\hline \left|f\right|^{2}={\mathfrak {H}}^{2}-{\mathfrak {E}}^{2}=2L,\ \left(ff^{*}\right)=-2i({\mathfrak {EH}})\\\hline {\begin{array}{ll}f_{x'y'}=f_{xy}\cos \varphi -f_{yl}\sin \varphi ,&f_{z'x'}=f_{zx}\cos \varphi +f_{zl}\sin \varphi ,\\f_{y'z'}=f_{yz},&f_{x'l'}=f_{xl},\\f_{y'l'}=f_{yl}\cos \varphi +f_{xy}\sin \varphi ,&f_{z'l'}=f_{zl}\cos \varphi -f_{zx}\sin \varphi ,\end{array}}\end{matrix}}$

equivalent to (a, j), from which he derived the stress-energy tensor $T=(ff)$:

${\begin{matrix}\left(f_{j}f_{h}\right)=f_{jx}f_{hx}+f_{jy}f_{hy}+f_{jz}f_{hz}+f_{jl}f_{hl}\\\left(f_{j}f_{h}\right)=\left(f_{h}f_{j}\right)\\\hline T={\frac {1}{2}}\left((f\cdot f)-\left(f^{*}f^{*}\right)\right)\end{matrix}}$

equivalent to (i), and the four-force density

${\begin{matrix}[Pf]\\\hline {\mathfrak {F}}_{j}=\varrho \left({\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {vH}}]\right),\ {\mathfrak {F}}_{l}={\frac {i\varrho }{c}}({\mathfrak {Ev}})\\(j=x,y,z;\ l=ict)\end{matrix}}$

equivalent to (h). In the second paper he defined Maxwell's equations and the relation to the four-potential

${\begin{matrix}{\mathfrak {Div}}\ f=P\\{\mathfrak {Div}}\ f^{*}=0\\\hline f=\mathrm {Rot} \ \Phi \\\left[{\begin{aligned}\mathrm {Rot} &={\text{exterior product}}\\{\mathfrak {Div}}&={\text{divergence six-vector}}\end{aligned}}\right]\end{matrix}}$

equivalent to (f,g,b).

Ignatowski (1910)

w:Wladimir Ignatowski (1910) formulated Minkowski's vector of second kind $f$ and its vectorial Lorentz transformation using unit velocity vector ${\mathfrak {c}}_{0}$ as follows:

${\begin{matrix}({\mathfrak {H,E}})\\\hline \left\{{\begin{aligned}{\mathfrak {H}}'=&p{\mathfrak {H}}-(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {H}}+pqn\left[{\mathfrak {Ec}}_{0}\right]\\{\mathfrak {E}}'=&p{\mathfrak {E}}-(p-1){\mathfrak {c}}_{0}\cdot {\mathfrak {c}}_{0}{\mathfrak {E}}-pq\left[{\mathfrak {Hc}}_{0}\right]\end{aligned}}\right.\\{\mathfrak {EH=E'H'}}=R\\{\mathfrak {E^{2}-H^{2}=E'^{2}-H'^{2}}}=S\end{matrix}}{\scriptstyle \left[n={\frac {1}{c^{2}}}\right]}$

equivalent to (a). The product of $f$ with four-velocity gives Minkowski's electric and magnetic rest forces:

${\begin{matrix}{\frac {\mathfrak {v'H'}}{\sqrt {1-n{\mathfrak {v}}^{\prime 2}}}}={\frac {p}{\sqrt {1-n{\mathfrak {v}}^{2}}}}\left\{{\mathfrak {Hv}}-q{\mathfrak {Hc}}_{0}+qn{\mathfrak {v}}\left[{\mathfrak {Ec}}_{0}\right]\right\}\\{\frac {\mathfrak {v'E'}}{\sqrt {1-n{\mathfrak {v}}^{\prime 2}}}}={\frac {p}{\sqrt {1-n{\mathfrak {v}}^{2}}}}\left\{{\mathfrak {Ev}}-q{\mathfrak {Ec}}_{0}-q{\mathfrak {v}}\left[{\mathfrak {Hc}}_{0}\right]\right\}\end{matrix}}$

equivalent to (h).

Lewis/Wilson (1910-12)

w:Gilbert Newton Lewis (1910) devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. He defined the electromagnetic 2-vector $\mathbf {M}$ (= electromagnetic tensor) in relation to the four-potential $\mathbf {m}$

${\begin{matrix}{\begin{aligned}\mathbf {M} &=\lozenge \times \mathbf {m} \\&=\nabla \times \mathbf {a} +i\left(\nabla \phi +{\frac {1}{c}}{\frac {\partial \mathbf {a} }{\partial t}}\right)\times \\&=\mathbf {E} +\mathbf {H} \end{aligned}}\mathbf {k} _{4}\\\hline \mathbf {E} =-i\mathbf {e} \times \mathbf {k} _{4},\ \mathbf {H} =\mathbf {h} \mathbf {k} _{123}\\{\begin{aligned}\mathbf {M} &=\mathbf {k} _{12}\left({\frac {\partial a_{2}}{\partial x_{1}}}-{\frac {\partial a_{1}}{\partial x_{2}}}\right)+\mathbf {k} _{13}\left({\frac {\partial a_{3}}{\partial x_{1}}}-{\frac {\partial a_{1}}{\partial x_{3}}}\right)+\mathbf {k} _{23}\left({\frac {\partial a_{3}}{\partial x_{2}}}-{\frac {\partial a_{2}}{\partial x_{2}}}\right)\\&\quad +\mathbf {k} _{14}\left({\frac {\partial i\phi }{\partial x_{1}}}-{\frac {\partial a_{1}}{\partial x_{4}}}\right)+\mathbf {k} _{23}\left({\frac {\partial i\phi }{\partial x_{1}}}-{\frac {\partial a_{1}}{\partial x_{4}}}\right)+\mathbf {k} _{34}\left({\frac {\partial i\phi }{\partial x_{3}}}-{\frac {\partial a_{3}}{\partial x_{4}}}\right)\\&=M_{12}\mathbf {k} _{12}+M_{13}\mathbf {k} _{13}+M_{23}\mathbf {k} _{23}+M_{14}\mathbf {k} _{14}+M_{24}\mathbf {k} _{24}+M_{34}\mathbf {k} _{34}\\&=H_{12}\mathbf {k} _{12}+H_{13}\mathbf {k} _{13}+H_{23}\mathbf {k} _{23}+E_{14}\mathbf {k} _{14}+E_{24}\mathbf {k} _{24}+E_{34}\mathbf {k} _{34}\end{aligned}}\\\left[\lozenge =\mathbf {k} _{1}{\frac {\partial }{\partial x_{1}}}+\mathbf {k} _{2}{\frac {\partial }{\partial x_{2}}}+\mathbf {k} _{3}{\frac {\partial }{\partial x_{3}}}+\mathbf {k} _{4}{\frac {\partial }{\partial x_{4}}}\right]\end{matrix}}$

equivalent to (a, b), from which he derived Maxwell's equations

${\begin{matrix}\lozenge \mathbf {M} =\mathbf {q} \\\lozenge \times \mathbf {M} =0\\{\begin{aligned}\left({\frac {\partial H_{12}}{\partial x_{2}}}+{\frac {\partial H_{13}}{\partial x_{3}}}+{\frac {\partial E_{14}}{\partial x_{4}}}\right)\mathbf {k} _{1}&={\frac {\varrho }{c}}v_{1}\mathbf {k} _{1}\\\left({\frac {\partial H_{21}}{\partial x_{1}}}+{\frac {\partial H_{23}}{\partial x_{3}}}+{\frac {\partial E_{24}}{\partial x_{4}}}\right)\mathbf {k} _{2}&={\frac {\varrho }{c}}v_{2}\mathbf {k} _{2}\\\left({\frac {\partial H_{31}}{\partial x_{1}}}+{\frac {\partial H_{32}}{\partial x_{2}}}+{\frac {\partial E_{34}}{\partial x_{4}}}\right)\mathbf {k} _{3}&={\frac {\varrho }{c}}v_{3}\mathbf {k} _{3}\\\left({\frac {\partial H_{41}}{\partial x_{1}}}+{\frac {\partial H_{42}}{\partial x_{2}}}+{\frac {\partial E_{43}}{\partial x_{4}}}\right)\mathbf {k} _{4}&={\frac {\varrho }{c}}i\mathbf {k} _{4}\end{aligned}}\end{matrix}}$

equivalent to (e,f).

In 1912, Lewis and w:Edwin Bidwell Wilson used real coordinates and also introduced the dual $\mathbf {M} ^{\ast }$, writing the above expressions as

${\begin{matrix}{\begin{aligned}\mathbf {M} &=\lozenge \times \mathbf {m} \\&=\nabla \times \mathbf {a} +\left(\nabla \phi +{\frac {1}{c}}{\frac {\partial \mathbf {a} }{\partial t}}\right)\times \\&=\mathbf {E} +\mathbf {H} \end{aligned}}\mathbf {k} _{4}\\\hline \mathbf {M} =h_{1}\mathbf {k} _{23}+h_{2}\mathbf {k} _{31}+h_{3}\mathbf {k} _{12}-e_{1}\mathbf {k} _{14}-h_{2}\mathbf {k} _{24}-e_{3}\mathbf {k} _{34}\\\mathbf {M} ^{\ast }=e_{1}\mathbf {k} _{23}+e_{2}\mathbf {k} _{31}+e_{3}\mathbf {k} _{12}+h_{1}\mathbf {k} _{14}+h_{2}\mathbf {k} _{24}+h_{3}\mathbf {k} _{34}\\\mathbf {M} \cdot \mathbf {M} =h^{2}-e^{2}=2L,\ \mathbf {M} \cdot \mathbf {M} ^{\ast }=2\mathbf {e} \cdot \mathbf {h} \\\left[\lozenge =\mathbf {k} _{1}{\frac {\partial }{\partial x_{1}}}+\mathbf {k} _{2}{\frac {\partial }{\partial x_{2}}}+\mathbf {k} _{3}{\frac {\partial }{\partial x_{3}}}-\mathbf {k} _{4}{\frac {\partial }{\partial x_{4}}}\right]\end{matrix}}$

equivalent to (a, b), and finally defined the dyadic $\Psi$ (= the stress-energy tensor) using unit dyadic $\mathbf {I}$

${\begin{matrix}\Psi ={\frac {1}{2}}\left(\Phi +\Phi ^{\ast }\right)\\\hline \Phi =(\mathbf {I} \cdot \mathbf {M} )\cdot (\mathbf {I} \cdot \mathbf {M} ),\ \Phi ^{\ast }=(\mathbf {I} \cdot \mathbf {M} ^{\ast })\cdot (\mathbf {I} \cdot \mathbf {M} ^{\ast })\end{matrix}}$

equivalent to (i).

Laue (1911)

In the influential first textbook on relativity, w:Max von Laue elaborated on the work of Minkowski and Sommerfeld. He defined the "field vector" ${\mathfrak {M}}$ (= electromagnetic tensor):

${\begin{matrix}{\begin{aligned}{\mathfrak {M}}_{xl}&=-i{\mathfrak {E}}_{x}&{\mathfrak {M}}_{yl}&=-i{\mathfrak {E}}_{y}&{\mathfrak {M}}_{zl}&=-i{\mathfrak {E}}_{z}\\{\mathfrak {M}}_{yz}&={\mathfrak {H}}_{x}&{\mathfrak {M}}_{zx}&={\mathfrak {H}}_{y}&{\mathfrak {M}}_{xy}&=i{\mathfrak {H}}_{z}\end{aligned}}\\\hline {\mathfrak {M}}^{2}={\mathfrak {H}}^{2}-{\mathfrak {E}}^{2},\ \left({\mathfrak {M}}{\mathfrak {M}}^{\ast }\right)=-2i({\mathfrak {EH}})\end{matrix}}$

equivalent to (a), the Maxwell equations and the relation to the four-potential

${\begin{matrix}{\begin{matrix}\varDelta iv\,{\mathfrak {M}}=P\\\varDelta iv\,{\mathfrak {M}}^{\ast }=0\\\hline {\mathfrak {Rot}}\Phi ={\mathfrak {M}}\\{\mathfrak {Rot}}^{\ast }\Phi ={\mathfrak {M}}^{\ast }\end{matrix}}\\\left[{\begin{aligned}{\mathfrak {Rot}}&={\text{exterior product}}\\\varDelta iv&={\text{divergence six-vector}}\end{aligned}}\right]\end{matrix}}$

equivalent to (g, b), the four-force density

$F=[P{\mathfrak {M}}]$

equivalent to (h), and the stress-energy tensor

$T=\left[[{\mathfrak {MM}}]\right]$

equivalent to (i).

Conway (1911)

Instead of Minkowski's matrix formulation or Sommerfeld's six-vector, w:Arthur Conway used the Weber vector $\sigma$ in his w:biquaternion representation of Maxwell's equation and the relativity principle, denoting $\mathrm {e}$ as the four-density and $\mathrm {p}$ as four-potential, and defined its Lorentz transformation using rotor $a$ as well (where the subscript $0$ means conjugate)

${\begin{matrix}\sigma =\varepsilon +hc\eta =-{\mathfrak {D}}_{0}\mathrm {p} \\{\mathfrak {D}}\sigma =\sigma _{0}D=-{\mathfrak {D}}{\mathcal {\mathfrak {D}}}_{0}\mathrm {p} =0\ {\text{or}}\ =-4\pi \mathrm {e} \\{\mathfrak {D}}_{0}\sigma _{0}=\sigma D_{0}=-4\pi \mathrm {e} _{0}\\\left[{\begin{matrix}{\mathfrak {D}}=\nabla -hc^{-1}\partial /\partial t,\ D=\Delta -hc^{-1}\partial /\partial t\\\mathrm {e} =e-hc^{-1}\iota ,\ \mathrm {p} =p-hc\omega \end{matrix}}\right]\\\hline \sigma =a\sigma 'a^{-1}\end{matrix}}$

equivalent to k.

Silberstein (1911-12)

w:Ludwik Silberstein (1911, published 1912), independently of Conway discussed the relativistic properties of the electromagnetic bivector $\mathbf {F}$ and its conjugate $\mathbf {G}$ (that he previously defined in 1907), used it to combine Maxwell's equations into a single one, and establishing the relation to the current-quaternion $C$, potential-quaternion $\Phi$, and force-quaternion $P_{e}$ (where the subscript $c$ means conjugate):

${\begin{matrix}{\begin{matrix}\mathbf {F} =\mathbf {M} -\iota \mathbf {E} =-D\Phi &&&\mathbf {G} =\mathbf {M} +\iota \mathbf {E} \\D\mathbf {F} =\mathrm {C} &&&D_{c}\mathbf {F} =C_{c}\end{matrix}}\\P=C\mathbf {F} =D[\mathbf {F} \cdot \mathbf {F} ]\Rightarrow P_{e}={\frac {1}{2}}\left\{C\mathbf {F} -\mathbf {G} C\right\}\\\left[D={\frac {\partial }{\partial l}}-\nabla \right]\\\hline {\begin{aligned}\mathbf {F} '&=Q_{c}\mathbf {F} Q\\&=(1-\gamma )(\mathbf {F} u)u+\gamma \mathbf {E} +\beta \gamma V\mathbf {F} u\\\mathbf {G} '&=Q\mathbf {G} Q_{c}\end{aligned}}\end{matrix}}$

equivalent to k. In a subsequent paper (1912) he went on to derive the Poynting vector and energy density analogous to his expressions in 1907, as well as the Maxwell stresses:

${\begin{aligned}{\mathfrak {P}}&=-{\frac {\iota c}{2}}V\mathbf {GF} =cV\mathbf {EM} \\u&={\frac {1}{2}}(\mathbf {GF} )={\frac {1}{2}}\left(E^{2}+M^{2}\right)\\f_{n}&=fn={\frac {1}{2}}(\mathbf {GF} )\mathbf {n} -{\frac {1}{2}}\mathbf {F} (\mathbf {Gn} )-{\frac {1}{2}}\mathbf {G} (\mathbf {Fn} )\end{aligned}}$

Silberstein used equivalent expressions also in his textbook on quaternionic special relativity in 1914. There he also discussed Minkowski's electrodynamics of media, obtaining the bivectors and Maxwell's macroscopic equations, their Lorentz transformation and invariants:

${\begin{matrix}{\mathfrak {E}}=K\mathbf {E} ',\ {\mathfrak {M}}=\mu \mathbf {M} ',\ \mathbf {I} =\sigma \mathbf {E} ',\ C'=\iota \rho '+{\frac {\mathbf {I} '}{c}}\\\hline \mathbf {L} '={\mathfrak {M}}'-i\mathbf {E} ',\ \mathbf {R} '={\mathfrak {M}}'+i\mathbf {E} '\\{\mathfrak {L}}'=\mathbf {M} '-i{\mathfrak {E}}',\ {\mathfrak {R}}'=\mathbf {M} '+i{\mathfrak {E}}'\\\hline D'{\mathfrak {L}}'-{\mathfrak {R}}'D'=2C',\ D'\mathbf {L} '+\mathbf {R} 'D'=0\\\hline {\mathfrak {L}}=Q{\mathfrak {L}}'Q_{c},\ {\mathfrak {R}}=Q_{c}{\mathfrak {R}}'Q,\ C=Q_{c}C'Q\\{\mathfrak {M}}^{2}-E^{2},\ ({\mathfrak {M}}\mathbf {E} ),\ M^{2}-{\mathfrak {E}}^{2},\ (\mathbf {M} {\mathfrak {E}})\end{matrix}}$

and Minkowski's electric and magnetic rest force:

${\begin{aligned}\eta &={\frac {1}{2c}}[Y\mathbf {L} -\mathbf {R} Y]\\\zeta &={\frac {1}{2c\iota }}[Y{\mathfrak {L}}-{\mathfrak {R}}Y]\end{aligned}}$

as well as the expressions of stress, momentum, energy.

Kottler (1912)

w:Friedrich Kottler formulated the electromagnetic field-tensor $F_{\alpha \beta }$ in order to express Maxwell's equations in terms of four-current $\mathbf {P} ^{(\beta )}$ and four-potential $\Phi _{\alpha }$:

${\begin{matrix}{\begin{aligned}{\mathfrak {H}}_{x}&=F_{23},&{\mathfrak {H}}_{y}&=F_{31},&{\mathfrak {H}}_{z}&=F_{12},\\-i{\mathfrak {E}}_{x}&=F_{14},&-i{\mathfrak {E}}_{y}&=F_{24},&-i{\mathfrak {E}}_{z}&=F_{34},\end{aligned}}\\{\begin{aligned}F_{23}^{*}&=-i{\mathfrak {E}}_{x},&F_{31}^{*}&=-i{\mathfrak {E}}_{y},&F_{12}^{*}&=-i{\mathfrak {E}}_{z},\\F_{14}^{*}&={\mathfrak {H}}_{x},&F_{24}^{*}&={\mathfrak {H}}_{y},&F_{34}^{*}&={\mathfrak {H}}_{z}.\end{aligned}}\\\hline \sum _{h=1}^{4}{\frac {\partial F_{gh}}{\partial x^{(h)}}}=\mathrm {P} ^{(g)}\\\sum _{h=1}^{4}{\frac {\partial F_{gh}^{\ast }}{\partial x^{(h)}}}=0\\F_{\alpha \beta }={\frac {\partial }{\partial x^{(\alpha )}}}\Phi _{\beta }-{\frac {\partial }{\partial x^{(\beta )}}}\Phi _{\alpha },\ \left(\Phi _{\alpha }=-F_{\alpha }\right)\end{matrix}}$

equivalent to (a,b,c,d,e,f,g). He was the first to define Maxwell's equations in a generally covariant way using metric tensor $c_{\alpha \beta }$:

${\begin{matrix}{\frac {1}{\sqrt {c}}}\mathrm {F} _{234}^{*}={\mathfrak {Div}}^{(1)}\left(F_{12}\dots F_{34}\right)={\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}F_{\alpha \beta /\gamma }={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta ,\gamma }{\sum }}c^{(1\beta )}c^{(\alpha \gamma )}F_{\beta \gamma }\right)=\mathrm {P} ^{(1)}\ \mathrm {etc} .\\{\frac {1}{\sqrt {c}}}\mathrm {F} _{234}={\mathfrak {Div}}^{(1)}\left(F_{12}^{\ast }\dots F_{34}^{\ast }\right)={\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}F_{\alpha \beta /\gamma }^{\ast }={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta ,\gamma }{\sum }}c^{(1\beta )}c^{(\alpha \gamma )}F_{\beta \gamma }^{\ast }\right)=0\ \mathrm {etc.} \\\hline F_{\alpha _{1}\alpha _{2}}^{\ast }={\frac {1}{\sqrt {c}}}{\underset {\beta _{1},\beta _{2}}{\sum }}c_{\alpha _{1}\beta _{1}}c_{\alpha _{2}\beta _{2}}F_{\alpha _{3}\beta _{3}}\\F_{\alpha \beta }=-\mathrm {F} _{\alpha \beta }={\frac {\partial \Phi _{\beta }}{\partial x^{(\alpha )}}}-{\frac {\partial \Phi _{\alpha }}{\partial x^{(\beta )}}}=\Phi _{\beta /\alpha }-\Phi _{\alpha /\beta }\end{matrix}}$

He went on to define the Minkowski four force

$F_{\alpha }(y)={\underset {\beta }{\sum }}{\frac {F_{\alpha \beta }(y)\mathrm {P} ^{(\alpha )}(y)}{\sqrt {1-{\mathfrak {w}}^{2}/c^{2}}}}$

equivalent to (h) and the stress-energy tensor $S$:

${\underset {\beta }{\sum }}F_{\alpha \beta }(y)\mathrm {P} ^{(\beta )}(y)={\underset {\beta }{\sum }}F_{\alpha \beta }(y){\underset {\gamma }{\sum }}{\frac {\partial }{\partial y^{(\gamma )}}}F_{\beta \gamma }(y)={\underset {\beta }{\sum }}{\frac {\partial }{\partial y^{(\beta )}}}S_{\alpha \beta }$

equivalent to (i).

Einstein (1913)

In the context of his Entwurf theory (a precursor of general relativity), w:Albert Einstein discussed the generally covariant formulation of Maxwell's equations using covariant second rank tensor or six-vector $\varphi _{\mu \nu }$ (= electromagnetic tensor), its dual $\varphi _{\mu \nu }^{\ast }$, and its complement $f_{\mu \nu }$, as well as rest charge density $\varrho _{0}$:

${\begin{matrix}\varphi _{\mu \nu }\quad \left[{\mathfrak {H}}_{x},{\mathfrak {H}}_{y},{\mathfrak {H}}_{z},-{\mathfrak {E}}_{x},-{\mathfrak {E}}_{y},-{\mathfrak {E}}_{z}\right]\\f_{\mu \nu }\quad \left[-{\mathfrak {E}}_{x},-{\mathfrak {E}}_{y},-{\mathfrak {E}}_{z},{\mathfrak {H}}_{x},{\mathfrak {H}}_{y},{\mathfrak {H}}_{z}\right]\\{\sqrt {-g}}\cdot \varphi _{23}={\mathfrak {H}}_{x},\ {\sqrt {-g}}\cdot \varphi _{31}={\mathfrak {H}}_{y},\ {\sqrt {-g}}\cdot \varphi _{12}={\mathfrak {H}}_{z}\\{\sqrt {-g}}\cdot \varphi _{14}=-{\mathfrak {E}}_{x},\ {\sqrt {-g}}\cdot \varphi _{24}=-{\mathfrak {E}}_{y},\ {\sqrt {-g}}\cdot \varphi _{34}=-{\mathfrak {E}}_{z}\\\hline {\begin{matrix}\sum _{\nu }{\frac {\partial }{\partial x_{\nu }}}\left({\sqrt {-g}}\cdot \varphi _{\mu \nu }\right)=\varrho {\frac {dx_{\mu }}{dt}}&&&\sum _{\nu }{\frac {\partial }{\partial x_{\nu }}}\left({\sqrt {-g}}\cdot \varphi _{\mu \nu }^{\ast }\right)=0\\\hline {\begin{aligned}{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}-{\frac {\partial {\mathfrak {E}}_{x}}{\partial t}}\end{aligned}}=u_{x}&&&{\begin{aligned}-{\frac {\partial {\mathfrak {E}}_{x}}{\partial y}}+{\frac {\partial {\mathfrak {E}}_{y}}{\partial z}}-{\frac {1}{c^{2}}}{\frac {\partial {\mathfrak {H}}_{x}}{\partial t}}\end{aligned}}=0\\\dots &&&\dots \\\dots &&&\dots \\{\begin{aligned}{\frac {\partial {\mathfrak {E}}_{x}}{\partial y}}+{\frac {\partial {\mathfrak {E}}_{y}}{\partial z}}+{\frac {\partial {\mathfrak {E}}_{z}}{\partial t}}\end{aligned}}=\varrho &&&{\begin{aligned}-{\frac {1}{c^{2}}}{\frac {\partial {\mathfrak {H}}_{x}}{\partial x}}-{\frac {1}{c^{2}}}{\frac {\partial {\mathfrak {H}}_{y}}{\partial t}}-{\frac {1}{c^{2}}}{\frac {\partial {\mathfrak {H}}_{z}}{\partial z}}\end{aligned}}=0\\\left[\varrho {\frac {dx_{\mu }}{dt}}=u_{\mu }\right]\end{matrix}}\end{matrix}}$

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

References

Historical mathematical sources

• Cayley, A. (1860) [1859], "On a new analytical representation of curves in space", Quarterly Journal of Pure and Applied Mathematics, 3: 225–236
• Cayley, A. (1869) [1867], "On the Six Coordinates of a Line", Transactions of the Cambridge Philosophical Society, XI: 290–323
• Gordan, Paul (1868), "Ueber eine das Hyperboloid betreffende Aufgabe", Zeitschr. Math. Phys., 13: 59–63
• Grassmann, H. (1844), Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, Leipzig: Wiegand
• Plücker, Julius (1865), "On a new geometry of space", Phil. Trans. R. Soc., 155: 725–791, doi:10.1098/rstl.1865.0017

Historical relativity sources

• Abraham, M. (1910), "Sull'Elletrodinamica di Minkowski", Rendiconti del Circolo Matematico di Palermo, 30: 33–46

• On the Electrodynamics of Minkowski on English Wikisource

• Bateman, H. (1910) [1909], "The Transformation of the Electrodynamical Equations", Proceedings of the London Mathematical Society, 8: 223–264

• The Transformation of the Electrodynamical Equations on English Wikisource

• Born, M. (1909), "Die träge Masse und das Relativitätsprinzip", Annalen der Physik, 333 (3): 571–584
• Conway, A. W. (1911), "On the application of quaternions to some recent developments of electrical theory", Proceedings of the Royal Irish Academy, Section A, 29: 1–9
• 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)
• Ignatowsky, W. v. (1910), "Das Relativitätsprinzip", Archiv der Mathematik und Physik 17: 1-24, 18: 17-40

• Das Relativitätsprinzip on German Wikisource

• Kottler, F. (1912), "Über die Raumzeitlinien der Minkowski'schen Welt", Wiener Sitzungsberichte 2a, 121: 1659–1759

• On the spacetime lines of a Minkowski world on English Wikisource

• Laue, M. v. (1911), Das Relativitätsprinzip, Braunschweig: Vieweg
• Lewis, G. N. (1910), "On Four-Dimensional Vector Analysis, and Its Application in Electrical Theory", Proceedings of the American Academy of Arts and Sciences, 43 (7): 165–181
• 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. (1915) [1907], "Das Relativitätsprinzip", Annalen der Physik, 352 (15): 927–938

• Das Relativitätsprinzip on German Wikisource

• 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

• Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern on German Wikisource
• The Fundamental Equations for Electromagnetic Processes in Moving Bodies on English Wikisource

• Poincaré, H. (1906) [1905], "Sur la dynamique de l'électron", Rendiconti del Circolo Matematico di Palermo, 21: 129–176

• Sur la dynamique de l’électron on French Wikisource
• On the Dynamics of the Electron on English Wikisource

• Silberstein, L. (1907) [1906], "Elektromagnetische Grundgleichungen in bivectorieller Behandlung" (PDF), Annalen der Physik, 327 (3): 579–586, Bibcode:1907AnP...327..579S, doi:10.1002/andp.19073270313
• Silberstein, L. (1907), "Nachtrag zur Abhandlung über 'Elektromagnetische Grundgleichungen in bivectorieller Behandlung'" (PDF), Annalen der Physik, 329 (14): 783–784, Bibcode:1907AnP...329..783S, doi:10.1002/andp.19073291409
• Silberstein, L. (1912) [1911], "Quaternionic form of relativity", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23 (137): 790–809, doi:10.1080/14786440508637276
• Silberstein, L. (1913) [1912], "Second memoir on quaternionic relativity", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25 (CXLV): 135–144
• Silberstein, L. (1914), The Theory of Relativity, London: Macmillan
• Sommerfeld, A. (1910a), "Zur Relativitätstheorie I: Vierdimensionale Vektoralgebra", Annalen der Physik, 337 (9): 749–776

• On the Theory of Relativity I: Four-dimensional Vector Algebra on English Wikisource

• Sommerfeld, A. (1910b), "Zur Relativitätstheorie II: Vierdimensionale Vector Analysis", Annalen der Physik, 338 (14): 649–689

• On the Theory of Relativity II: Four-dimensional Vector Analysis on English Wikisource

• Weber, H. M. (1900), Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen (4. edition, volume I), Braunschweig: Vieweg
• Weber, H. M. (1901), Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen (4. edition, volume II), Braunschweig: Vieweg