History of Topics in Special Relativity/Stress-energy tensor (electromagnetic)

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Overview

The w:electromagnetic stress–energy tensor is the contribution to the w:stress–energy tensor due to the w:electromagnetic field in terms of the electromagnetic field tensor $F^{\mu \nu }$, Poynting vector $𝐒$, Maxwell stress tensor ${\sigma }_{ij}$:

$\begin{array}{c}{T}^{\mu \nu }=\left[\begin{array}{cccc}\frac{1}{2}({ϵ}_0{E}^2+\frac{1}{{\mu }_0}{B}^2)& S_{\text{x}}/c& S_{\text{y}}/c& S_{\text{z}}/c\\ S_{\text{x}}/c& -{\sigma }_{\text{xx}}& -{\sigma }_{\text{xy}}& -{\sigma }_{\text{xz}}\\ S_{\text{y}}/c& -{\sigma }_{\text{yx}}& -{\sigma }_{\text{yy}}& -{\sigma }_{\text{yz}}\\ S_{\text{z}}/c& -{\sigma }_{\text{zx}}& -{\sigma }_{\text{zy}}& -{\sigma }_{\text{zz}}\end{array}\right]=\frac{1}{{\mu }_0}[F^{\mu \alpha }{F}^{\nu }{}_{\alpha }-\frac{1}{4}{\eta }^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }]\\ [𝐒=\frac{1}{{\mu }_0}𝐄\times 𝐁,{\sigma }_{ij}={ϵ}_0{E}_i{E}_j+\frac{1}{{\mu }_0}{B}_i{B}_j-\frac{1}{2}({ϵ}_0{E}^2+\frac{1}{{\mu }_0}{B}^2){\delta }_{ij}]\end{array}$

This tensor in the form written above is universally accepted when it comes to electromagnetic processes in vacuum. It was introduced by #Minkowski (1907/8) in matrix notation and used in different notation by #Sommerfeld (1910), #Laue (1910-11), #Einstein (1912-14). Alternative representations have been used by #Lewis/Wilson (1912) (dyadics) and #Silberstein (1912-14) (quaternions). However, several competing tensors have been proposed when it comes to electromagnetic processes in physical media, most notably the asymmetric one by #Minkowski (1907/8) and the symmetric one by #Abraham (1909-10) (w:Abraham-Minkowski controversy). Abraham's form was used by #Laue (1910-11) and #Henschke (1912/13), while Minkowski's was used by #Ishiwara (1910-11), with #Grammel (1913) summarizing and combining both approaches.

Historical notation

Minkowski (1907/8)

In his lecture from December 1907 (published 1908), w:Hermann Minkowski used the electromagnetic tensor in vacuum $f$ and in matter $F$, as well as their duals $f^{*},F^{*}$, in order to define the (dielectric) electromagnetic energy-momentum tensor $S$ using $𝔢,𝔪$ (i.e. $𝐄,𝐁$) and $𝔈,𝔐$ (= $𝐃,𝐇$):

$\begin{array}{c}S=\left|\begin{array}{cccc}{S}_{11},& S_{12},& S_{13},& S_{14}\\ S_{21},& S_{22},& S_{23},& S_{23}\\ S_{31},& S_{32},& S_{33},& S_{34}\\ S_{41},& S_{42},& S_{43},& S_{44}\end{array}\right|=\left|\begin{array}{cccc}{X}_x,& Y_x,& Z_x,& -i{T}_x\\ X_y,& Y_y,& Z_y,& -i{T}_y\\ X_z,& Y_z,& Z_z,& -i{T}_z\\ -i{X}_t,& -i{Y}_t,& -i{Z}_t,& T_t\end{array}\right|\\ fF=S-L=\left|\begin{array}{cccc}{S}_{11}-L,& S_{12},& S_{13},& S_{14}\\ S_{21},& S_{22}-L,& S_{23},& S_{23}\\ S_{31},& S_{32},& S_{33}-L,& S_{34}\\ S_{41},& S_{42},& S_{43},& S_{44}-L\end{array}\right|,F^{*}{f}^{*}=-S-L=\left|\begin{array}{cccc}-S_{11}-L,& -S_{12},& -S_{13},& -S_{14}\\ -S_{21},& -S_{22}-L,& -S_{23},& -S_{23}\\ -S_{31},& -S_{32},& -S_{33}-L,& -S_{34}\\ -S_{41},& -S_{42},& -S_{43},& -S_{44}-L\end{array}\right|\\ L=\frac{1}{2}(f_{23}{F}_{23}+f_{31}{F}_{31}+f_{12}{F}_{12}+f_{14}{F}_{14}+f_{24}{F}_{24}+f_{34}{F}_{34})\\ S_{11}=\frac{1}{2}(f_{23}{F}_{23}+f_{34}{F}_{34}+f_{42}{F}_{42}-f_{12}{F}_{12}-f_{13}{F}_{13}-f_{14}{F}_{14})\\ S_{12}=f_{13}{F}_{32}+f_{14}{F}_{42},etc.\\ X_y={𝔪}_x{𝔐}_y+{𝔢}_y{𝔈}_x,Y_x={𝔪}_y{𝔐}_x+{𝔢}_x{𝔈}_y,\mathrm{e}\mathrm{t}\mathrm{c}.\\ X_t={𝔢}_y{𝔐}_z-{𝔢}_z{𝔐}_y,\\ T_x={𝔪}_z{𝔈}_y-{𝔪}_y{𝔈}_z,\mathrm{e}\mathrm{t}\mathrm{c}.\\ T_t=\frac{1}{2}({𝔪}_x{𝔐}_x+{𝔪}_y{𝔐}_y+{𝔪}_z{𝔐}_z+{𝔢}_x{𝔈}_x+{𝔢}_y{𝔈}_y+{𝔢}_z{𝔈}_z)\\ L=\frac{1}{2}({𝔪}_x{𝔐}_x+{𝔪}_y{𝔐}_y+{𝔪}_z{𝔐}_z-{𝔢}_x{𝔈}_x-{𝔢}_y{𝔈}_y-{𝔢}_z{𝔈}_z)\end{array}$

where $X_x,Y_x,Z_x,\dots$ are the Maxwell stresses, $T_x,T_y,T_z$ the Poynting vector, $T_t$ the energy density and $L$ the Lagrangian. While Minkowski's tensor is symmetrical in vacuum, it is evidently asymmetric in a dielectric because $(X_t,Y_t,Z_t)=𝐄\times 𝐇$ while their counterparts have the form $(T_x,T_y,T_z)=𝐁\times 𝐃$. Alternatively, #Abraham (1909-10) symmetrically defined the tensor even in a dielectric, which started the w:Abraham-Minkowski controversy.

Using that tensor, Minkowski went on to derive the four-force density using four-velocity $w$:

$\begin{array}{c}K=\text{lor }S\Rightarrow K+(w\bar{K})w\\ [\text{lor }=|\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4}|]\end{array}$

Abraham (1909-10)

w:Max Abraham (1909) expressed the stress energy tensor components in terms of arbitrary dielectric constant and permeability. He defined the fields $𝔈,ℌ$ (= $𝐄,𝐇$) and $𝔇,𝔅$ (i.e. $𝐃,𝐁$) for resting media, and ${𝔈}^{\prime },{ℌ}^{\prime }$ as the fields acting on moving electric and magnetic unit poles. He started by relating the six components of the spatial part $X_x\dots$ of the stress-energy tensor to the “relative stresses” $X_x^{\prime }\dots$ (= the ordinary elastic stresses seen in the moving frame) using momentum density $𝔤$ and velocity $𝔴$:

$\begin{array}{c}{X}_x,Y_y,Z_z,X_y=Y_x,Y_z=Z_y,Z_x=X_z\\ \{\begin{array}{ccccc}X{\text{'}}_x=X_x+{𝔴}_x{𝔤}_x,& & X{\text{'}}_y=X_y+{𝔴}_y{𝔤}_x,& & X{\text{'}}_z=X_z+{𝔴}_z{𝔤}_x,\\ Y{\text{'}}_x=Y_x+{𝔴}_x{𝔤}_y,& & Y{\text{'}}_y=Y_y+{𝔴}_y{𝔤}_y,& & Y{\text{'}}_z=Y_z+{𝔴}_z{𝔤}_y,\\ Z{\text{'}}_x=Z_x+{𝔴}_x{𝔤}_z,& & Z{\text{'}}_y=Z_y+{𝔴}_y{𝔤}_z,& & Z{\text{'}}_z=Z_z+{𝔴}_z{𝔤}_z,\end{array}\\ \{\begin{array}{c}X{\text{'}}_x=𝔈{\text{'}}_x{𝔇}_x+ℌ{\text{'}}_x{𝔅}_x-\frac{1}{2}[{𝔈}^{\prime }𝔇\mathfrak{+}{ℌ}^{\prime }𝔅\}\\ X{\text{'}}_y=𝔈{\text{'}}_x{𝔇}_y+ℌ{\text{'}}_x{𝔅}_y,\\ X{\text{'}}_z=𝔈{\text{'}}_x{𝔇}_z+ℌ{\text{'}}_x{𝔅}_z;\\ Y{\text{'}}_x=𝔈{\text{'}}_y{𝔇}_x+ℌ{\text{'}}_y{𝔅}_x,\\ Y{\text{'}}_y=𝔈{\text{'}}_y{𝔇}_y+ℌ{\text{'}}_y{𝔅}_y-\frac{1}{2}[{𝔈}^{\prime }𝔇\mathfrak{+}{ℌ}^{\prime }𝔅\},\\ Y{\text{'}}_z=𝔈{\text{'}}_y{𝔇}_z+ℌ{\text{'}}_y{𝔅}_z;\\ Z{\text{'}}_x=𝔈{\text{'}}_z{𝔇}_x+ℌ{\text{'}}_z{𝔅}_x,\\ Z{\text{'}}_y=𝔈{\text{'}}_z{𝔇}_y+ℌ{\text{'}}_z{𝔅}_y,\\ Z{\text{'}}_z=𝔈{\text{'}}_z{𝔇}_z+ℌ{\text{'}}_z{𝔅}_z-\frac{1}{2}[{𝔈}^{\prime }𝔇\mathfrak{+}{ℌ}^{\prime }𝔅\}.\end{array}\end{array}$

and obtained the other components using energy density $\psi$ and Poynting vector $𝔖$, by which he analyzed all electrodynamic theories of that time, including the following adaptation of Minkowski's model

$\begin{array}{c}\{\begin{array}{c}𝔇=ϵ{𝔈}^{\prime }-[𝔮ℌ],\\ 𝔅=\mu {ℌ}^{\prime }+[𝔮𝔈];\end{array}\{\begin{array}{c}{𝔈}^{\prime }=𝔈+[𝔮𝔅],\\ {ℌ}^{\prime }=ℌ-[𝔮𝔇].\end{array}\\ {𝔖}^{\prime }=c[{𝔈}^{\prime }{ℌ}^{\prime }],𝔤\mathfrak{=}\frac{𝔖}{c^2}\\ \begin{array}{c}\begin{array}{cc}𝔖& =c[{𝔈}^{\prime }{ℌ}^{\prime }]+𝔴\{{𝔈}^{\prime }𝔇+{ℌ}^{\prime }𝔅\}-𝔇\mathfrak{(}𝔴{𝔈}^{\prime }\mathfrak{)}\mathfrak{-}𝔅\mathfrak{(}𝔴{ℌ}^{\prime }\mathfrak{)}\mathfrak{+}𝔴\mathfrak{(}𝔴𝔤\mathfrak{)}\\ c𝔤& =\mathfrak{[}{𝔈}^{\prime }{ℌ}^{\prime }\mathfrak{]}\mathfrak{+}𝔮\mathfrak{(}{𝔈}^{\prime }𝔇\mathfrak{)}\mathfrak{+}𝔮\mathfrak{(}{ℌ}^{\prime }𝔅\mathfrak{)}\mathfrak{-}𝔇\mathfrak{(}𝔮{𝔈}^{\prime }\mathfrak{)}\mathfrak{-}𝔅\mathfrak{(}𝔮{ℌ}^{\prime }\mathfrak{)}\mathfrak{+}𝔮\mathfrak{(}𝔮c𝔤\mathfrak{)}\\ & =[𝔈ℌ]-𝔮\mathfrak{(}𝔮𝔚\mathfrak{)}\\ & =[𝔈ℌ](𝔮=0)\\ \psi & =\frac{1}{2}𝔈𝔇+\frac{1}{2}ℌ𝔅+𝔮𝔚\end{array}\end{array}\\ \left[\begin{array}{cc}{𝔚}_x& =k^{-2}\{[𝔇𝔅{]}_x-[𝔈ℌ{]}_x\}\\ {𝔚}_y& =[𝔇𝔅{]}_y-[𝔈ℌ{]}_y\\ {𝔚}_z& =[𝔇𝔅{]}_z-[𝔈ℌ{]}_z\end{array}[k^2=1-|𝔮{|}^2,𝔮=\frac{𝔴}{c}]\right]\\ c{𝔤}_x=X_t,c{𝔤}_y=Y_t,c{𝔤}_z=Z_t\\ {𝔖}_x=c{T}_x,{𝔖}_y=c{T}_y,{𝔖}_z=c{T}_z,\psi =T_t\\ \begin{array}{cccccc}{X}_y& =Y_x& Y_z& =Z_y& Z_x& =X_z\\ X_t& =T_x& Y_t& =T_y& Z_t& =T_z\end{array}\end{array}$

He noted that the last relations exhibit a "remarkable" symmetry that doesn't occur in Minkowski's original formulation, that is, Abraham's tensor is symmetric because it satisfies Planck's (1907) relation $𝔤\mathfrak{=}\frac{𝔖}{c^2}$ even in a dielectric, which is the basis of the Abraham–Minkowski controversy. For instance, the above relations give in the case of a dielectric at rest $(X_t,Y_t,Z_t)=(T_x,T_y,T_z)=𝐄\times 𝐇$ representing momentum density times the speed of light.

In a subsequent paper (1909) he denoted this system as a "four-dimensional tensor" or "tensor quadruple":

$\begin{array}{c}{X}_x=\frac{1}{2}({𝔈}_x^2-{𝔈}_y^2-{𝔈}_z^2)+\frac{1}{2}({ℌ}_x^2-{ℌ}_y^2-{ℌ}_z^2)\\ Y_y=\frac{1}{2}({𝔈}_y^2-{𝔈}_z^2-{𝔈}_x^2)+\frac{1}{2}({ℌ}_y^2-{ℌ}_z^2-{ℌ}_x^2)\\ Z_z=\frac{1}{2}({𝔈}_z^2-{𝔈}_x^2-{𝔈}_y^2)+\frac{1}{2}({ℌ}_z^2-{ℌ}_x^2-{ℌ}_y^2)\\ X_y=Y_x={𝔈}_x{𝔈}_y+{ℌ}_x{ℌ}_y\\ Y_z=Z_y={𝔈}_y{𝔈}_z+{ℌ}_y{ℌ}_z\\ Z_x=X_z={𝔈}_z{𝔈}_x+{ℌ}_z{ℌ}_x\\ X_u=U_x=-i{𝔰}_x,Y_u=U_y=-i{𝔰}_y,Z_u=U_z=-i{𝔰}_z,U_u=\psi \\ [𝔰=𝔖/c=c𝔤=[𝔈ℌ],\psi =\frac{1}{2}{𝔈}^2+\frac{1}{2}{ℌ}^2,u=ict]\\ \begin{array}{cccccc}{X}_y& =Y_x& Y_z& =Z_y& Z_x& =X_z\\ {𝔖}_x& =c^2{𝔤}_x& {𝔖}_y& =c^2{𝔤}_y& {𝔖}_z& =c^2{𝔤}_z\end{array}\end{array}$

and defined the corresponding ponderomotive four-force density. He again referred to the Abraham–Minkowski controversy by alluding to the result of this previous paper, that his tensor remains symmetrical even in the general case of arbitrary dielectric constant and permeability (as opposed to Minkowski's asymmetrical one). He further pointed out another difference to Minkowski's formulation: While Minkowski uses constant rest mass, Abraham showed that mass-energy equivalence requires variable rest mass because of the contribution of heat.

In 1910 he brought Minkowski's matrix formulation into vector form, writing the ten components of the "four-dimensional tensor" in a dielectric as:

$\begin{array}{c}\{\begin{array}{ccccccc}{X}_x,Y_y,Z_z,& & Y_z=Z_y,& & Z_x=X_z,& & X_y=Y_x;\\ X_u=U_x,& & Y_u=U_y,& & Z_u=U_z,& & U_u,\end{array}\\ X_u=-ic{𝔤}_x,Y_u=-ic{𝔤}_y,Z_u=-ic{𝔤}_z\\ U_x=-\frac{i}{c}{𝔖}_x,U_y=\frac{-i}{c}{𝔖}_y,U_z=\frac{-i}{c}{𝔖}_z,U_u=\psi ;\\ 2𝔣=[𝔈ℌ]+[𝔇𝔅]-𝔚-𝔮(𝔮𝔚)\\ 2\psi =𝔈𝔇+ℌ𝔅-2(𝔮𝔚)\\ [𝔚=(ϵ\mu -1)\{k^{-2}{𝔣}^{\prime }+k^{-4}𝔮({𝔮𝔣}^{\prime })\},𝔣=c𝔤=\frac{1}{c}𝔖,k=\sqrt{1-{𝔮}^2}]\end{array}$

again pointing out that they satisfy symmetry conditions which are not satisfied by Minkowski's approach.

Sommerfeld (1910)

In an influential paper, w:Arnold Sommerfeld translated Minkowski's matrix formalism into a four-dimensional vector formalism involving four-vectors and six-vectors. Sommerfeld defined a tensor $T=(ff)$ as the product of six-vector $f$ (i.e. electromagnetic field tensor) and its dual $f^{*}$ using fields $𝔈,ℌ$ (i.e. $𝐄,𝐇$), with $T$ having 10 components:

$\begin{array}{c}{T}_{jh}=\left({𝔅}_j{𝔅}_h\right)={𝔅}_{jx}{𝔅}_{hx}+{𝔅}_{jy}{𝔅}_{hy}+{𝔅}_{jz}{𝔅}_{hz}\\ T_{jh}=T_{hj},T_{jj}\ne 0\\ \left({𝔅}_j{𝔅}_h\right)\Rightarrow \left(f_j{f}_h\right)\\ \left(f_j{f}_h\right)=f_{jx}{f}_{hx}+f_{jy}{f}_{hy}+f_{jz}{f}_{hz}+f_{jl}{f}_{hl}\\ \left(f_j{f}_h\right)=\left(f_h{f}_j\right)\\ j=h\Rightarrow \begin{array}{cc}\left(f_x{f}_x\right)=& {ℌ}_y^2+{ℌ}_z^2-{𝔈}_x^2\\ \left(f_y{f}_y\right)=& {ℌ}_z^2+{ℌ}_x^2-{𝔈}_y^2\\ \left(f_z{f}_z\right)=& {ℌ}_x^2+{ℌ}_y^2-{𝔈}_z^2\\ \left(f_l{f}_l\right)=-& {𝔈}_x^2-{𝔈}_y^2-{𝔈}_z^2\end{array}\\ j\ne h\Rightarrow \begin{array}{ccc}\left(f_x{f}_y\right)=& \left(f_y{f}_x\right)=& -({ℌ}_x{ℌ}_y+{𝔈}_x{𝔈}_y)\\ \left(f_y{f}_z\right)=& \left(f_z{f}_y\right)=& -({ℌ}_y{ℌ}_z+{𝔈}_y{𝔈}_u)\\ \left(f_z{f}_x\right)=& \left(f_x{f}_z\right)=& -({ℌ}_z{ℌ}_x+{𝔈}_z{𝔈}_x)\\ \left(f_x{f}_l\right)=& \left(f_l{f}_x\right)=& i({𝔈}_y{ℌ}_z-{𝔈}_z{ℌ}_y)\\ \left(f_y{f}_l\right)=& \left(f_l{f}_y\right)=& i({𝔈}_z{ℌ}_x-{𝔈}_x{ℌ}_z)\\ \left(f_z{f}_l\right)=& \left(f_l{f}_z\right)=& i({𝔈}_x{ℌ}_y-{𝔈}_y{ℌ}_x)\end{array}\\ \begin{array}{cc}\left(f_x^{*}{f}_x^{*}\right)=& -{𝔈}_y^2-{𝔈}_z^2+{ℌ}_x^2\\ \left(f_x^{*}{f}_y^{*}\right)=& \left(f_x^{*}{f}_y^{*}\right)=+({𝔈}_x{𝔈}_y+{𝔈}_x{𝔈}_y)\\ \left(f_x^{*}{f}_l^{*}\right)=& \left(f_l^{*}{f}_x^{*}\right)=i({ℌ}_y{𝔈}_z-{ℌ}_z{𝔈}_y)\end{array}\end{array}$

which he rewrote following a suggestion by Laue (see next section).

Laue (1910-11)

w:Max von Laue suggested to Sommerfeld in private conversation the following formulation of the stress energy tensor, published by Sommerfeld in 1910:

$\begin{array}{c}T=\frac{1}{2}((f\cdot f)-\left(f^{*}{f}^{*}\right))\\ \{\begin{array}{cc}{T}_{xx}& =\frac{1}{2}(-{ℌ}_x^2+{ℌ}_y^2+{ℌ}_z^2-{𝔈}_x^2+{𝔈}_y^2+{𝔈}_z^2)\\ T_{yy}& =\frac{1}{2}(-{ℌ}_y^2+{ℌ}_z^2+{ℌ}_x^2-{𝔈}_y^2+{𝔈}_z^2+{𝔈}_x^2)\\ T_{zz}& =\frac{1}{2}(-{ℌ}_z^2+{ℌ}_x^2+{ℌ}_y^2-{𝔈}_z^2+{𝔈}_x^2+{𝔈}_y^2)\\ T_{ll}& =\frac{1}{2}(-{ℌ}_x^2-{ℌ}_y^2-{ℌ}_z^2-{𝔈}_x^2-{𝔈}_y^2-{𝔈}_z^2)\end{array}\\ \{\begin{array}{cc}{T}_{xy}=T_{yx}& =-{ℌ}_x{ℌ}_y+{𝔈}_x{𝔈}_y\\ T_{yz}=T_{zx}& =-{ℌ}_y{ℌ}_z+{𝔈}_y{𝔈}_z\\ T_{zx}=T_{xz}& =-{ℌ}_z{ℌ}_x+{𝔈}_z{𝔈}_x\end{array}\\ \{\begin{array}{cc}{T}_{xl}=T_{lx}& =i({𝔈}_y{ℌ}_z-{𝔈}_z{ℌ}_y)\\ T_{yl}=T_{ly}& =i({𝔈}_z{ℌ}_x-{𝔈}_x{ℌ}_z)\\ T_{zl}=T_{lz}& =i({𝔈}_x{ℌ}_y-{𝔈}_y{ℌ}_x)\end{array}\\ T_{xx}+T_{yy}+T_{zz}+T_{ll}=0\end{array}$

In 1911, Laue himself published a paper on the dynamics of relativity, the contents of which he also published at almost the same time in his influential first textbook on relativity. He followed Abraham in defining a symmetric "world tensor" $T$ (i.e. electromagnetic stress energy tensor) as the tensor product of the electromagnetic field tensor $𝔐$, including Poynting vector $𝔖$, energy density $W$, momentum density ${𝔤}_e$, and its Lorentz transformation:

$\begin{array}{c}\begin{array}{c}T=[[𝔐𝔐]]=\begin{array}{cccc}{𝐩}_{xx}& {𝐩}_{xy}& {𝐩}_{xz}& \frac{i}{c}{𝔖}_x\\ {𝐩}_{yx}& {𝐩}_{yy}& {𝐩}_{yz}& \frac{i}{c}{𝔖}_y\\ {𝐩}_{zx}& {𝐩}_{zy}& {𝐩}_{zz}& \frac{i}{c}{𝔖}_z\\ \frac{i}{c}{𝔖}_x& \frac{i}{c}{𝔖}_y& \frac{i}{c}{𝔖}_z& -W\end{array}\\ \begin{array}{cccc}{T}_{xx}& =\frac{1}{2}({𝔈}^2+{ℌ}^2)-{𝔈}_x^2-{ℌ}_x^2& & ={𝐩}_{xx}\\ T_{xy}& =-({𝔈}_x{𝔈}_y+{ℌ}_x{ℌ}_x)& & ={𝐩}_{xy}\\ T_{ll}& =-\frac{1}{2}({𝔈}^2+{ℌ}^2)& & =-W\\ T_{xl}& =i({𝔈}_y{ℌ}_z-{𝔈}_z{ℌ}_y)& & =\frac{i}{c}{𝔖}_x\end{array}\\ {𝔤}_e=\frac{𝔖}{c^2}=\frac{1}{c}[𝔈ℌ]\end{array}\\ \\ \begin{array}{c}{𝔖}_x=\frac{(1+{\beta }^2){𝔖}_x^{\prime }+v({𝐩}_{xx}^{\prime }+W^{\prime })}{1-{\beta }^2},\\ {𝔖}_y=\frac{{𝔖}_y^{\prime }+v{𝐩}_{xy}^{\prime }}{\sqrt{1-{\beta }^2}},{𝔖}_z=\frac{{𝔖}_z^{\prime }+v{𝐩}_{xz}^{\prime }}{\sqrt{1-{\beta }^2}}\\ W=\frac{W^{\prime }+{\beta }^2{𝐩}_{xx}^{\prime }+2\frac{v}{c^2}{𝔖}_x^{\prime }}{1-{\beta }^2}\\ {𝐩}_{yy}={𝐩}_{yy}^{\prime },{𝐩}_{zz}={𝐩}_{zz}^{\prime },{𝐩}_{yz}={𝐩}_{yz}^{\prime }\\ {𝐩}_{xx}=\frac{{𝐩}_{xx}^{\prime }+2\frac{v}{c^2}{𝔖}_x^{\prime }+{\beta }^2{W}^{\prime }}{c^2-q^2},\\ {𝐩}_{xy}=\frac{{𝐩}_{xy}^{\prime }+\frac{v}{c^2}{𝔖}_y^{\prime }}{\sqrt{1-{\beta }^2}},{𝐩}_{xz}=\frac{{𝐩}_{xz}^{\prime }+\frac{v}{c^2}{𝔖}_z^{\prime }}{\sqrt{1-{\beta }^2}}\\ W=\frac{1}{1-{\beta }^2}\{(W^{\prime }+({𝔳𝔤}^{\prime }))+\frac{1}{c^2}(𝔳,{𝔖}^{\prime }+[𝔳{𝐩}^{\prime }])\}\\ 𝔖=\frac{1}{1-{\beta }^2}\{({𝔖}^{\prime }+[𝔳{𝐩}^{\prime }])+𝔳(W^{\prime }+({𝔳𝔤}^{\prime }))+\frac{1-\sqrt{1-{\beta }^2}}{v^2}[v,[𝔳,{𝔖}^{\prime }+[𝔳{𝐩}^{\prime }]]]\}\end{array}\end{array}$

From that he derived, among other things, the electromagnetic energy and momentum of a spherical electron

$\begin{array}{c}{𝔊}_e=\frac{4E^0𝔮}{3c\sqrt{c^2-q^2}},E=\frac{c^2+\frac{1}{3}{q}^2}{c\sqrt{c^2-q^2}}{E}^0\\ [{𝔊}_e=\frac{1}{c^2}\int 𝔖dS,E=\int WdS]\end{array}$

and its relation to four-force density $𝔉$ and four-current $P$

$𝔉=-𝞓ivT=-[P𝔐]$

Referring to Planck's (1907) "universal relation" relating momentum and energy density which he called "theorem of inertia of energy", he reproduced Abraham's expressions for energy and momentum in a dielectric at rest as well as in motion:

$\begin{array}{c}{𝔉}^0=-𝔡𝔦𝔳{𝐩}^0-\frac{1}{c}\frac{\partial }{\partial t}\left[{𝔈}^0{ℌ}^0\right]\\ {𝔖}^0=c\left[{𝔈}^0{ℌ}^0\right],{𝔤}_e^0=\frac{1}{c}\left[{𝔈}^0{ℌ}^0\right]\Rightarrow {𝔤}_e^0=\frac{{𝔖}^0}{c^2}\\ 𝔖=c^2{𝔤}_e=c[𝔈ℌ]+𝔮\frac{(𝔮,[𝔈ℌ]-[𝔇𝔅])}{c^2-q^2}\\ W=\frac{1}{2}((𝔈𝔇)+(ℌ𝔅))+\frac{c}{c^2-q^2}(𝔮,[𝔈ℌ]-[𝔇𝔅])\end{array}$

and wrote the relations also in terms of Abraham's "relative stresses" (i.e. the elastic stresses) as follows (Laue not only referred these equations to electrodynamics but to mechanics as well):

$\begin{array}{c}{𝐭}_{jk}={𝐩}_{jk}-{𝔤}_j{𝔮}_k\\ c^2𝔤=𝔖=W𝔮+[𝔮𝐭]-[𝔮[𝔮𝔤]]=W𝔮+[𝔮𝐭]-𝔮[𝔮𝔤]+𝔤q^2\end{array}$

Ishiwara (1910-11)

w:Jun Ishiwara (1910) chose Minkowski's asymmetric formulation of the stress-energy tensor in a dielectric over Abraham's symmetric one, and also gave its Lorentz transformation:

$\begin{array}{cccc}\begin{array}{cc}{X}_x& =\frac{1}{4\pi }\{{𝔈}_x{𝔇}_x+{ℌ}_x{𝔅}_x-\frac{1}{2}(𝔈𝔇+ℌ𝔅)\}\\ X_y& =\frac{1}{4\pi }\{{𝔈}_x{𝔇}_y+{ℌ}_x{𝔅}_y\}\\ X_z& =\frac{1}{4\pi }\{{𝔈}_x{𝔇}_z+{ℌ}_x{𝔅}_z\}\\ Y_x& =\frac{1}{4\pi }\{{𝔈}_y{𝔇}_x+{ℌ}_y{𝔅}_x\}\\ Y_y& =\frac{1}{4\pi }\{{𝔈}_y{𝔇}_y+{ℌ}_y{𝔅}_y-\frac{1}{2}(𝔈𝔇+ℌ𝔅)\}\\ Y_z& =\frac{1}{4\pi }\{{𝔈}_y{𝔇}_z+{ℌ}_y{𝔅}_z\}\\ Z_x& =\frac{1}{4\pi }\{{𝔈}_z{𝔇}_x+{ℌ}_z{𝔅}_x\}\\ Z_y& =\frac{1}{4\pi }\{{𝔈}_z{𝔇}_y+{ℌ}_z{𝔅}_y\}\\ Z_z& =\frac{1}{4\pi }\{{𝔈}_z{𝔇}_z+{ℌ}_z{𝔅}_z-\frac{1}{2}(𝔈𝔇+ℌ𝔅)\}\\ 𝔤& =\frac{1}{4\pi c}[𝔇𝔅]\\ 𝔖& =\frac{c}{4\pi }[𝔈ℌ]\\ u& =\frac{1}{8\pi }\{𝔈𝔇+ℌ𝔅\}\end{array}& & & |\begin{array}{cc}{\varkappa }^2\frac{{𝔖}_x^{\prime }}{c}& =\frac{{𝔖}_x}{c}+\beta X_x-\beta u+{\beta }^{2}c{𝔤}_x\\ \varkappa \frac{{𝔖}_y^{\prime }}{c}& =\frac{{𝔖}_y}{c}+\beta X_y\\ \varkappa \frac{{𝔖}_z^{\prime }}{c}& =\frac{{𝔖}_z}{c}+\beta X_z\\ {\varkappa }^2{u}^{\prime }& =u-\beta \frac{{𝔖}_x}{c}-\beta c{𝔤}_x-{\beta }^2{X}_x\\ {\varkappa }^2{X}_x^{\prime }& =X_x+\beta \frac{{𝔖}_x}{c}+\beta c{𝔤}_x-{\beta }^{2}u\\ \varkappa X_y^{\prime }& =X_y+\beta \frac{{𝔖}_y}{c}\\ \varkappa X_z^{\prime }& =X_z+\beta \frac{{𝔖}_z}{c}\\ \varkappa Y_x^{\prime }& =Y_x+\beta c{𝔤}_y\\ Y_y^{\prime }& =Y_y\\ Y_z^{\prime }& =Y_z\\ \varkappa Z_x^{\prime }& =Z_x+\beta c{𝔤}_z\\ Z_y^{\prime }& =Z_y\\ Z_z^{\prime }& =Z_z\\ {\varkappa }^{2}c{𝔤}_x^{\prime }& =c{𝔤}_x-\beta u+\beta X_x+{\beta }^2\frac{{𝔖}_x}{c}\\ \varkappa c{𝔤}_y^{\prime }& =c{𝔤}_y+\beta Y_x\\ \varkappa c{𝔤}_z^{\prime }& =c{𝔤}_z+\beta Z_x\end{array}\\ & & & [\varkappa =\sqrt{1-\frac{{𝔳}^2}{c^2}}]\end{array}$

However, in order to derive the ponderomotive force from this tensor, he chose Abraham's approach of variable rest mass over Minkowski's assumption of constant rest mass.

In 1911 he wrote this tensor and its Lorentz transformation as follows (the indices $𝔳$ and $\overline{𝔳}$ indicate the components parallel and perpendicular to the velocity, respectively):

$\begin{array}{c}\begin{array}{c}\begin{array}{cccccccc}{𝐓}_{11}& ={𝔗}_{xx}& {𝐓}_{12}& ={𝔗}_{xy}& {𝐓}_{13}& ={𝔗}_{xz}& {𝐓}_{14}& =-ic{𝔤}_x\\ {𝐓}_{21}& ={𝔗}_{yx}& {𝐓}_{22}& ={𝔗}_{yy}& {𝐓}_{23}& ={𝔗}_{yz}& {𝐓}_{24}& =-ic{𝔤}_y\\ {𝐓}_{31}& ={𝔗}_{zx}& {𝐓}_{32}& ={𝔗}_{zy}& {𝐓}_{33}& ={𝔗}_{zz}& {𝐓}_{34}& =-ic{𝔤}_z\\ {𝐓}_{41}& =-\frac{i}{c}{𝔖}_x& {𝐓}_{42}& =-\frac{i}{c}{𝔖}_y& {𝐓}_{43}& =-\frac{i}{c}{𝔖}_z& {𝐓}_{44}& =u\end{array}\\ \begin{array}{cc}{𝔗}_{jk}& =\frac{1}{4\pi }({𝔈}_j{𝔇}_k+{ℌ}_j{𝔅}_k)-\frac{(𝔧𝔨)}{8\pi }\{(𝔈𝔇)+(ℌ𝔅)\}\\ 𝔤& =\frac{1}{4\pi c}[𝔇𝔅]\\ 𝔖& =\frac{c}{4\pi }[𝔈ℌ]\\ u& =\frac{1}{8\pi }\{(𝔈𝔇)+(ℌ𝔅)\}\\ & {𝔗}_{xx}+{𝔗}_{yy}+{𝔗}_{zz}+u=0\end{array}\end{array}\\ \\ \begin{array}{cc}{T}_{𝔳𝔳}^{\mathfrak{\prime }}& =\frac{1}{1-\frac{{𝔳}^2}{c^2}}({𝔗}_{𝔳𝔳}+𝔳\frac{{𝔖}_{𝔳}}{c^2}+𝔳{𝔤}_{𝔳}-\frac{{𝔳}^2}{c^2}u)\\ T_{\overline{𝔳}\overline{𝔳}}^{\mathfrak{\prime }}& ={𝔗}_{\overline{𝔳}\overline{𝔳}}\\ T_{𝔳\overline{𝔳}}^{\mathfrak{\prime }}& =\frac{1}{\sqrt{1-\frac{{𝔳}^2}{c^2}}}({𝔗}_{𝔳\overline{𝔳}}+𝔳\frac{{𝔖}_{\overline{𝔳}}}{c^2})\\ T_{\overline{𝔳}𝔳}^{\mathfrak{\prime }}& =\frac{1}{\sqrt{1-\frac{{𝔳}^2}{c^2}}}({𝔗}_{\overline{𝔳}𝔳}+𝔳{𝔤}_{\overline{𝔳}})\\ \frac{S_{𝔳}^{\mathfrak{\prime }}}{c}& =\frac{1}{c(1-\frac{{𝔳}^2}{c^2})}({𝔖}_{𝔳}+𝔳{𝔗}_{𝔳𝔳}-𝔳u-{𝔳}^2{𝔤}_{𝔳})\\ \frac{S_{\overline{𝔳}}^{\mathfrak{\prime }}}{c}& =\frac{1}{c\sqrt{1-\frac{{𝔳}^2}{c^2}}}({𝔖}_{\overline{𝔳}}+𝔳{𝔗}_{𝔳\overline{𝔳}})\\ c{g}_{𝔳}^{\mathfrak{\prime }}& =\frac{1}{c(1-\frac{{𝔳}^2}{c^2})}(c^2{𝔤}_{𝔳}-𝔳u+𝔳{𝔗}_{𝔳𝔳}+{𝔳}^2\frac{{𝔖}_{𝔳}}{c^2})\\ c{g}_{\overline{𝔳}}^{\mathfrak{\prime }}& =\frac{1}{c\sqrt{1-\frac{{𝔳}^2}{c^2}}}(c^2{𝔤}_{\overline{𝔳}}+𝔳{𝔗}_{\overline{𝔳}𝔳})\\ u^{\prime }& =\frac{1}{1-\frac{{𝔳}^2}{c^2}}(u-𝔳\frac{{𝔖}_{𝔳}}{c^2}-𝔳{𝔤}_{𝔳}-{𝔳}^2\frac{{𝔗}_{𝔳𝔳}}{c^2})\end{array}\end{array}$

Einstein (1912-14)

In an unpublished manuscript on special relativity, written between 1912-1914, w:Albert Einstein followed Abraham and Laue in defining the symmetric stress energy tensors in vacuum (upper index 0) and matter (upper index $e$ electric polarization, $m$ magnetic polarization), in which the upper asterisk represent the dual, $𝔤$ momentum density, $𝐬$ energy flux:

$\begin{array}{c}{T}_{\mu \nu }=\begin{array}{cccc}{p}_{xx}& p_{xy}& p_{xz}& ic{𝔤}_x\\ p_{yx}& p_{yy}& p_{yz}& ic{𝔤}_y\\ p_{zx}& p_{zy}& p_{zz}& ic{𝔤}_z\\ \frac{i}{c}{𝐬}_x& \frac{i}{c}{𝐬}_y& \frac{i}{c}{𝐬}_z& -\eta \end{array}\\ [p_{xy}=p_{yx}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.},𝔤\mathrm{=}\frac{1}{c^2}𝐬]\\ \left(T_{\mu \nu }\right)=\left(T_{\mu \nu }^0\right)+\left(T_{\mu \nu }^e\right)+\left(T_{\mu \nu }^m\right)\\ \begin{array}{cc}\left(T_{\mu \nu }^0\right)& =\frac{1}{2}\{\left({𝔉}_{\mu \sigma }\right)\left({𝔉}_{\nu \sigma }\right)-\left({𝔉}_{\mu \sigma }^{\ast }\right)\left({𝔉}_{\nu \sigma }^{\ast }\right)\}\\ \left(T_{\mu \nu }^e\right)& =\frac{1}{\epsilon -1}\{\left({𝔓}_{\mu \sigma }\right)\left({𝔓}_{\nu \sigma }\right)-\frac{1}{4}\left({\delta }_{\mu \nu }\right)\left({𝔓}_{\sigma \tau }\right)\left({𝔓}_{\sigma \tau }\right)\}\\ \left(T_{\mu \nu }^m\right)& =\frac{1}{\mu -1}\{\left({𝔐}_{\mu \sigma }\right)\left({𝔐}_{\nu \sigma }\right)-\frac{1}{4}\left({\delta }_{\mu \nu }\right)\left({𝔐}_{\sigma \tau }\right)\left({𝔐}_{\sigma \tau }\right)\}\end{array}\\ \begin{array}{cccc}{p}_{xx}^0& =-{𝔢}_x^2-{𝔥}_x^2+\frac{1}{2}({𝔢}^2+{𝔥}^2)& p_{xx}^e& =\frac{1}{\epsilon -1}\{-{𝔭}_x^{\ast 2}-{𝔭}_x^{\ast \ast 2}+\frac{1}{2}({𝔭}^{\ast 2}+{𝔭}^{\ast \ast 2})\}\\ p_{xy}^0& =-{𝔢}_x{𝔢}_y-{𝔥}_x{𝔥}_y& p_{xy}^e& =\frac{1}{\epsilon -1}\{-{𝔭}_x^{\ast }{𝔭}_y^{\ast }-{𝔭}_x^{\ast \ast }{𝔭}_y^{\ast \ast }\}\\ & \mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}& & \mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}\\ {𝐬}^0& =c^2{𝔤}^0=c[𝔢,𝔥]& {𝐬}^e& =c^2{𝔤}^e=\frac{c}{\epsilon -1}[{{𝔭}^{\mathfrak{\ast }\mathfrak{\ast }},𝔭}^{\ast }]\\ {\eta }^0& =\frac{{𝔢}^2+{𝔥}^2}{2}& {\eta }^e& =\frac{1}{\epsilon -1}\frac{{𝔭}^{\ast 2}+{𝔭}^{\ast \ast 2}}{2}\end{array}\\ T_{\mu \nu }^{\prime }=\sum _{\sigma \tau }{\alpha }_{\mu \sigma }{\alpha }_{\nu \tau }{T}_{\sigma \tau }\end{array}$

Silberstein (1912-14)

w:Ludwik Silberstein used Quaternions to give an alternative representation of the components of the electromagnetic stress-energy tensor. He derived the Poynting vector $𝔓$, energy density $u$ and the Maxwell stresses $f_n$ from the electromagnetic bivector (= w:Weber vector) $𝐆$ and its dual $𝐅$, and the Lorentz transformation of all these quantities:

$\begin{array}{c}\begin{array}{cc}𝔓& =-\frac{\iota c}{2}V𝐆𝐅=cV𝐄𝐌\\ u& =\frac{1}{2}(𝐆𝐅)=\frac{1}{2}(E^2+M^2)\\ f_n& =f𝐧=\frac{1}{2}(𝐆𝐅)𝐧-\frac{1}{2}𝐅(𝐆𝐧)-\frac{1}{2}𝐆(𝐅𝐧)\\ & =\frac{1}{2}V𝐆n𝐅\end{array}\\ [\frac{1}{2}𝐆𝐅=-u+\frac{\iota }{c}𝔓]\\ \begin{array}{cc}\frac{1}{{\gamma }^2}u& =u^{\prime }+\frac{2}{c^2}({𝔓}^{\prime }𝐯)+\frac{1}{c^2}(𝐯f^{\prime }𝐯)\\ \frac{1}{{\gamma }^2}ϵ𝔓& ={𝔓}^{\prime }+[\frac{1}{c^2}(𝐯{𝔓}^{\prime })+u^{\prime }+f^{\prime }]𝐯\\ \frac{1}{{\gamma }^2}ϵf& =f^{\prime }\frac{1}{ϵ}+\frac{1}{c}[{𝔓}^{\prime }+u^{\prime }𝐯](𝐯+\frac{1}{c^2}𝐯(\frac{1}{ϵ}{𝔓}^{\prime }\end{array}\end{array}$

Silberstein used equivalent expressions also in his textbook on quaternionic special relativity in 1914, in which he also discussed Minkowski's electrodynamics of media by giving the electromagnetic bivectors of an isotropic medium:

$\begin{array}{c}\begin{array}{cc}u& =\frac{1}{2}(𝐄𝔈+𝐌𝔐)=\\ 𝔓& =cV𝐄𝐌\\ 𝐠& =\frac{1}{c}V𝔈𝔐\\ {𝐟}_n& =u𝐧-𝐄(𝔈𝐧)-𝐌(𝔐𝐧)\end{array}\end{array}$

Lewis/Wilson (1912)

w:Gilbert Newton Lewis and w:Edwin Bidwell Wilson devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. They gave the dyadic $\mathrm{\Psi }$ (i.e. the stress-energy tensor) using unit dyadic $𝐈$ in terms of the electromagnetic 2-vector $𝐌$ (i.e. electromagnetic tensor) and its dual ${𝐌}^{\ast }$:

$\begin{array}{c}\mathrm{\Psi }=\frac{1}{2}(\mathrm{\Phi }+{\mathrm{\Phi }}^{\ast })\\ X_x{𝐤}_1{𝐤}_1+X_y{𝐤}_1{𝐤}_2+X_z{𝐤}_1{𝐤}_3-X_t{𝐤}_1{𝐤}_4\\ +Y_x{𝐤}_2{𝐤}_1+Y_y{𝐤}_2{𝐤}_2+Y_z{𝐤}_2{𝐤}_3-Y_t{𝐤}_2{𝐤}_4\\ +Z_x{𝐤}_3{𝐤}_1+Z_y{𝐤}_3{𝐤}_2+Z_z{𝐤}_3{𝐤}_3-Z_t{𝐤}_3{𝐤}_4\\ -T_x{𝐤}_4{𝐤}_1-T_y{𝐤}_4{𝐤}_2-T_z{𝐤}_4{𝐤}_3-T_t{𝐤}_4{𝐤}_4\\ \mathrm{\Phi }=(𝐈\cdot 𝐌)\cdot (𝐈\cdot 𝐌),{\mathrm{\Phi }}^{\ast }=(𝐈\cdot {𝐌}^{\ast })\cdot (𝐈\cdot {𝐌}^{\ast })\\ \begin{array}{cc}\frac{1}{2}({𝐤}_1\cdot 𝐌)\cdot 𝐌+\frac{1}{2}({𝐤}_1\cdot {𝐌}^{\ast })\cdot {𝐌}^{\ast }& =X_x{𝐤}_1+X_y{𝐤}_2+X_z{𝐤}_3-X_t{𝐤}_4\\ \frac{1}{2}({𝐤}_2\cdot 𝐌)\cdot 𝐌+\frac{1}{2}({𝐤}_2\cdot {𝐌}^{\ast })\cdot {𝐌}^{\ast }& =Y_x{𝐤}_1+Y_y{𝐤}_2+Y_z{𝐤}_3-Y_t{𝐤}_4\\ \frac{1}{2}({𝐤}_3\cdot 𝐌)\cdot 𝐌+\frac{1}{2}({𝐤}_3\cdot {𝐌}^{\ast })\cdot {𝐌}^{\ast }& =Z_x{𝐤}_1+Z_y{𝐤}_2+Z_z{𝐤}_3-Z_t{𝐤}_4\\ \frac{1}{2}({𝐤}_4\cdot 𝐌)\cdot 𝐌+\frac{1}{2}({𝐤}_4\cdot {𝐌}^{\ast })\cdot {𝐌}^{\ast }& =T_x{𝐤}_1+T_y{𝐤}_2+T_z{𝐤}_3+T_t{𝐤}_4\end{array}\\ \begin{array}{cc}{X}_x& =\frac{1}{2}(e_1^2-e_2^2-e_3^2+h_1^2-h_2^2-h_3^2)\\ Y_y& =\frac{1}{2}(e_2^2-e_3^2-e_1^2+h_2^2-h_3^2-h_1^2)\\ Z_z& =\frac{1}{2}(e_3^2-e_1^2-e_2^2+h_3^2-h_1^2-h_2^2)\\ T_t& =\frac{1}{2}(e_1^2+e_2^2+e_3^2+h_1^2+h_2^2+h_3^2)\\ X_y& =Y_x=e_1{e}_2+h_1{h}_2\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}\\ T_x& =X_t=e_2{h}_3-e_3{h}_2\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}\end{array}\end{array}$

Henschke (1912/13)

Erich Henschke (in a thesis from 1912, published 1913) derived the stress-energy tensor from a variational principle, obtaining different formulations of four-force density on the basis of a symmetric tensor equivalent to the one of Abraham and Laue (also using their notation). He started with the case of vacuum:

$\begin{array}{c}\{\begin{array}{cc}{X}_x& =\frac{1}{2}({ℌ}_x^2-{ℌ}_y^2-{ℌ}_z^2+{𝔈}_x^2-{𝔈}_y^2-{𝔈}_z^2)\\ Y_z& =\frac{1}{2}(-{ℌ}_x^2+{ℌ}_y^2-{ℌ}_z^2-{𝔈}_x^2+{𝔈}_y^2-{𝔈}_z^2)\\ Z_z& =\frac{1}{2}(-{ℌ}_x^2-{ℌ}_y^2+{ℌ}_z^2+{𝔈}_x^2+{𝔈}_y^2+{𝔈}_z^2)\\ U_u& =\frac{1}{2}({ℌ}_x^2+{ℌ}_y^2+{ℌ}_z^2+{𝔈}_x^2+{𝔈}_y^2+{𝔈}_z^2)\end{array}\\ \{\begin{array}{cc}{X}_y=Y_x& ={ℌ}_x{ℌ}_y+{𝔈}_x{𝔈}_y\\ X_z=Z_x& ={ℌ}_x{ℌ}_z+{𝔈}_x{𝔈}_z\\ Y_z=Z_y& ={ℌ}_y{ℌ}_z+{𝔈}_y{𝔈}_z\end{array}\\ \{\begin{array}{cc}{U}_x=X_u& =-i({𝔈}_y{ℌ}_x-{𝔈}_z{ℌ}_y)\\ U_y=Y_u& =-i({𝔈}_z{ℌ}_x-{𝔈}_z{ℌ}_z)\\ U_x=Z_u& =-i({𝔈}_x{ℌ}_y-{𝔈}_y{ℌ}_x)\end{array}\\ T=\left[\left[𝔐𝔐\right]\right],𝔉=-𝞓ivT\end{array}$

then in moving media

$\begin{array}{c}\{\begin{array}{cc}{X}_x& =\frac{1}{2}({𝔈}_x{𝔇}_x-{𝔈}_y{𝔇}_y-{𝔈}_z{𝔇}_z+{ℌ}_x{𝔅}_x-{ℌ}_y{𝔅}_y-{ℌ}_z{𝔅}_z+\frac{{𝔳}_x}{c}{𝔚}_x)\\ Y_z& =\frac{1}{2}(-{𝔈}_x{𝔇}_x+{𝔈}_y{𝔇}_y-{𝔈}_z{𝔇}_z-{ℌ}_x{𝔅}_x+{ℌ}_y{𝔅}_y-{ℌ}_z{𝔅}_z+\frac{{𝔳}_y}{c}{𝔚}_y)\\ Z_z& =\frac{1}{2}(-{𝔈}_x{𝔇}_x-{𝔈}_y{𝔇}_y+{𝔈}_z{𝔇}_z-{ℌ}_x{𝔅}_x-{ℌ}_y{𝔅}_y+{ℌ}_z{𝔅}_z+\frac{{𝔳}_z}{c}{𝔚}_z)\\ U_u& =\frac{1}{2}({𝔈}_x{𝔇}_x+{𝔈}_y{𝔇}_y+{𝔈}_z{𝔇}_z+{ℌ}_x{𝔅}_x+{ℌ}_y{𝔅}_y+{ℌ}_z{𝔅}_z+\frac{𝔳}{c}𝔚)\end{array}\\ \{\begin{array}{cc}{X}_y=Y_x& =\frac{1}{2}({𝔈}_x{𝔇}_y+{𝔈}_y{𝔇}_x+{ℌ}_x{𝔅}_y+{ℌ}_y{𝔅}_x+\frac{{𝔳}_x}{c}{𝔚}_y+\frac{{𝔳}_y}{c}{𝔚}_x)\\ X_z=Z_x& =\frac{1}{2}({𝔈}_x{𝔇}_z+{𝔈}_z{𝔇}_x+{ℌ}_x{𝔅}_z+{ℌ}_z{𝔅}_x+\frac{{𝔳}_x}{c}{𝔚}_z+\frac{{𝔳}_z}{c}{𝔚}_x)\\ Y_z=Z_y& =\frac{1}{2}({𝔈}_y{𝔇}_z+{𝔈}_z{𝔇}_y+{ℌ}_y{𝔅}_z+{ℌ}_z{𝔅}_y+\frac{{𝔳}_y}{c}{𝔚}_z+\frac{{𝔳}_z}{c}{𝔚}_y)\end{array}\\ \{\begin{array}{cc}{U}_x=X_u& =-i\frac{1}{2}([𝔈ℌ{]}_x+[𝔇𝔅{]}_x-{𝔚}_x-\frac{{𝔳}_x}{c}\left(\frac{𝔳}{c}𝔚\right))\\ U_y=Y_u& =-i\frac{1}{2}([𝔈ℌ{]}_y+[𝔇𝔅{]}_y-{𝔚}_y-\frac{{𝔳}_y}{c}\left(\frac{𝔳}{c}𝔚\right))\\ U_x=Z_u& =-i\frac{1}{2}([𝔈ℌ{]}_z+[𝔇𝔅{]}_z-{𝔚}_z-\frac{{𝔳}_z}{c}\left(\frac{𝔳}{c}𝔚\right))\end{array}\\ {𝔖}_x=c[𝔈ℌ{]}_x(𝔳=0)\end{array}$

Grammel (1913)

Richard Grammel developed three stress-energy tensors $T^{\prime },T^{″},T^{‴}$ in a dielectric satisfying the general four-force equation (using Laue's notation) $F^{\prime }=-𝞓iv{T}^{\prime },\dots$. The first one is identical to Abraham's, the second one a modification of Minkowski's so that it becomes symmetrical in the case of rest, and the third one is related to the other ones by $F^{‴}=2F^{\prime }-F^{″}$. He started with the case of rest

$\begin{array}{cc}E& =\frac{1}{2}\{(𝔈𝔇)+(ℌ𝔅)\}\\ 𝔖& =𝔤=[𝔈ℌ]\\ \overline{X_x^{\prime }}& ={𝔈}_x{𝔇}_x-\frac{1}{2}(𝔈𝔇)+{ℌ}_x{𝔅}_x-\frac{1}{2}(ℌ𝔅)\\ \overline{Y_y^{\prime }}& =\overline{Y_x^{\prime }}=\frac{1}{2}\{{𝔈}_x{𝔇}_y+{𝔈}_y{𝔇}_x+{ℌ}_x{𝔅}_y+{ℌ}_y{𝔅}_x\}\\ & \dots \\ \overline{X_x^{\prime \prime \prime }}& =\overline{X_x^{\prime \prime }}=\overline{X_x^{\prime }}\\ \overline{X_y^{\prime \prime \prime }}& =\overline{Y_x^{\prime \prime }}={𝔈}_y{𝔇}_x+{ℌ}_y{𝔅}_x\\ \overline{Y_x^{\prime \prime \prime }}& =\overline{X_y^{\prime \prime }}={𝔈}_x{𝔇}_y+{ℌ}_x{𝔅}_y\\ & \dots \end{array}$

leading to the expressions for moving bodies:

$\begin{array}{c}\{\begin{array}{cc}{X}_x^{\prime }& =\overline{X_x^{\prime }}+{𝔳}_x{𝔚}_x,Y_x^{\prime }=\overline{Y_x^{\prime }}+\frac{1}{2}\{{𝔳}_x{𝔚}_y+{𝔳}_y{𝔚}_x\}\\ X_y^{\prime }& =Y_x^{\prime }\dots ,\\ {𝔖}^{\prime }& ={𝔤}^{\prime }=\frac{1}{2}\{𝔘+𝔖-𝔚-𝔳(𝔳𝔚)\}\\ E^{\prime }& =E-(𝔳𝔚)\end{array}\\ \{\begin{array}{cc}{X}_x^{\prime \prime }& =\overline{X_x^{\prime \prime }}+{𝔳}_x{𝔚}_x,Y_x^{\prime \prime }=\overline{Y_x^{\prime \prime }}+{𝔳}_y{𝔚}_x,\\ X_y^{\prime \prime }& =X_y^{\prime \prime }+{𝔳}_y{𝔚}_y\dots \\ {𝔖}^{″}& =𝔖-𝔳(𝔳𝔚),{𝔤}^{″}=𝔘-𝔚,\\ E^{″}& =E-(𝔳𝔚)\end{array}\\ \{\begin{array}{cc}{X}_x^{\prime \prime \prime }& =\overline{X_x^{\prime \prime }}+{𝔳}_x{𝔚}_x,Y_x^{\prime \prime }=\overline{Y_x^{\prime \prime }}+{𝔳}_y{𝔚}_x,\\ X_y^{\prime \prime \prime }& =Y_x^{\prime \prime }\dots \\ {𝔖}^{‴}& ={𝔤}^{″},{𝔤}^{‴}={𝔖}^{″},\\ E^{‴}& =E^{″}\end{array}\\ \left[\begin{array}{c}{𝔚}^{\prime }={\varkappa }^2\{[{𝔈}^{\prime }{ℌ}^{\prime }]+{\varkappa }^2𝔳(𝔳[{𝔈}^{\prime }{ℌ}^{\prime }])\}\\ 𝔚=(ϵ\mu -1){𝔚}^{\prime },𝔘=[𝔇𝔅],\\ {𝔈}^{\prime }=𝔈+[𝔳𝔅],{ℌ}^{\prime }=ℌ-[𝔳𝔇]\\ \varkappa =\frac{1}{\sqrt{1-{𝔳}^2}}\end{array}\right]\end{array}$

References

• {{#section:History of Topics in Special Relativity/relsource|abra09elek}}
• {{#section:History of Topics in Special Relativity/relsource|abra09mech}}
• {{#section:History of Topics in Special Relativity/relsource|abra10elek}}
• {{#section:History of Topics in Special Relativity/relsource|einst12manu}}
• {{#section:History of Topics in Special Relativity/relsource|gram13}}
• {{#section:History of Topics in Special Relativity/relsource|hen13tens}}
• {{#section:History of Topics in Special Relativity/relsource|ishi10med1}}
• {{#section:History of Topics in Special Relativity/relsource|ishi11imp}}
• {{#section:History of Topics in Special Relativity/relsource|laue11prin}}
• {{#section:History of Topics in Special Relativity/relsource|lewis12non}}
• {{#section:History of Topics in Special Relativity/relsource|mink07b}}
• {{#section:History of Topics in Special Relativity/relsource|silber12quat}}
• {{#section:History of Topics in Special Relativity/relsource|silber14quat}}
• {{#section:History of Topics in Special Relativity/relsource|som10alg}}