w:Henri Poincaré (July 1905, published January 1906) defined the Lorentz force and its Lorentz transformation:[R 1]
X
=
ρ
f
+
ρ
(
η
γ
−
ζ
β
)
…
X
′
=
ρ
′
f
′
+
ρ
′
(
η
′
γ
′
−
ζ
′
β
′
)
…
{
X
′
=
k
(
X
+
ϵ
T
)
,
T
′
=
k
(
T
+
ϵ
X
)
,
Y
′
=
Y
,
Z
′
=
Z
;
{\displaystyle {\begin{matrix}X=\rho f+\rho (\eta \gamma -\zeta \beta )\dots \\X'=\rho 'f'+\rho '(\eta '\gamma '-\zeta '\beta ')\dots \\{\begin{cases}X^{\prime }=k(X+\epsilon T),\quad &T^{\prime }=k(T+\epsilon X),\\Y^{\prime }=Y,&Z^{\prime }=Z;\end{cases}}\end{matrix}}}
with
T
=
Σ
X
ξ
{\displaystyle T=\Sigma X\xi }
and
k
=
1
1
−
ϵ
2
{\displaystyle k={\frac {1}{\sqrt {1-\epsilon ^{2}}}}}
.
in which (X,Y,Z,T) represent the components of four-force density (b1, d1) because
ϵ
=
v
c
,
(
X
,
Y
,
Z
)
=
ρ
{
E
+
v
×
B
}
=
d
,
T
=
Σ
X
ξ
=
d
⋅
v
=
v
⋅
E
{\displaystyle \epsilon ={\frac {v}{c}},\ \left(X,\ Y,\ Z\right)=\rho \left\{\mathbf {E} +\mathbf {v} \times \mathbf {B} \right\}=\mathbf {d} ,\ T=\Sigma X\xi =\mathbf {d} \cdot \mathbf {v} =\mathbf {v\cdot E} }
.
Additionally, he explicitly obtained the four-force per unit charge by setting
X
1
X
=
Y
1
Y
=
Z
1
Z
=
T
1
T
=
1
ρ
{\displaystyle \scriptstyle {\frac {X_{1}}{X}}={\frac {Y_{1}}{Y}}={\frac {Z_{1}}{Z}}={\frac {T_{1}}{T}}={\frac {1}{\rho }}}
:[R 2]
(
k
0
X
1
,
k
0
Y
1
,
k
0
Z
1
,
k
0
T
1
)
{\displaystyle \left(k_{0}X_{1},\quad k_{0}Y_{1},\quad k_{0}Z_{1},\quad k_{0}T_{1}\right)}
with
T
1
=
Σ
X
1
ξ
{\displaystyle T_{1}=\Sigma X_{1}\xi }
and
k
0
=
1
1
−
ϵ
2
{\displaystyle k_{0}={\tfrac {1}{\sqrt {1-\epsilon ^{2}}}}}
equivalent to (b, d) and obtained its Lorentz transformation by multiplying the transformation of (X,Y,Z,T) by
ρ
ρ
′
=
1
k
(
1
+
ξ
ϵ
)
=
δ
t
δ
t
′
{\displaystyle {\frac {\rho }{\rho ^{\prime }}}={\frac {1}{k(1+\xi \epsilon )}}={\frac {\delta t}{\delta t^{\prime }}}}
.
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 four-force density X,Y,Z,iA as the product of the electromagnetic tensor and the four-current with
ϱ
{\displaystyle \varrho }
as charge density and
v
{\displaystyle {\mathfrak {v}}}
as velocity:[R 3]
X
,
Y
,
Z
,
i
A
X
j
=
ϱ
1
ψ
j
1
+
ϱ
2
ψ
j
2
+
ϱ
3
ψ
j
3
+
ϱ
4
ψ
j
4
A
=
X
v
x
+
Y
v
y
+
Z
v
z
{
ψ
23
,
ψ
31
,
ψ
12
⇒
H
x
,
H
y
,
H
z
ψ
14
,
ψ
24
,
ψ
34
⇒
−
i
E
x
,
−
i
E
y
,
−
i
E
z
ψ
j
j
=
0
}
{\displaystyle {\begin{matrix}X,Y,Z,iA\\\hline X_{j}=\varrho _{1}\psi _{j1}+\varrho _{2}\psi _{j2}+\varrho _{3}\psi _{j3}+\varrho _{4}\psi _{j4}\\A=X{\mathfrak {v}}_{x}+Y{\mathfrak {v}}_{y}+Z{\mathfrak {v}}_{z}\\\left\{{\begin{aligned}\psi _{23},\ \psi _{31},\ \psi _{12}&\Rightarrow {\mathfrak {H}}_{x},{\mathfrak {H}}_{y},{\mathfrak {H}}_{z}\\\psi _{14},\ \psi _{24},\ \psi _{34}&\Rightarrow -i{\mathfrak {E}}_{x},-i{\mathfrak {E}}_{y},-i{\mathfrak {E}}_{z}\\&\psi _{jj}=0\end{aligned}}\right\}\end{matrix}}}
equivalent to (b1, d1, e1) because
ϵ
=
v
c
,
(
X
,
Y
,
Z
)
=
d
,
A
=
d
v
=
v
E
{\displaystyle \epsilon ={\frac {v}{c}},\ \left(X,Y,Z\right)=\mathbf {d} ,\ A=\mathbf {d} \mathbf {v} =\mathbf {vE} }
.
In another lecture from December 1907, he used the symbols
e
,
m
{\displaystyle {\mathfrak {e}},{\mathfrak {m}}}
(i.e.
E
,
B
{\displaystyle \mathbf {E} ,\mathbf {B} }
) and
E
,
M
{\displaystyle {\mathfrak {E}},{\mathfrak {M}}}
(i.e.
D
,
H
{\displaystyle \mathbf {D} ,\mathbf {H} }
) which he represented in the form of “vectors of second kind”, i.e. the electromagnetic tensor f and its dual
f
∗
{\displaystyle f^{*}}
, from which he derived the electric rest force
Φ
{\displaystyle \Phi }
and magnetric rest force
Ψ
{\displaystyle \Psi }
as the product with four-velocity w , which in turn can be used to express F and f and four-conductivity:[R 4]
Φ
=
−
w
F
(
Φ
1
,
Φ
2
,
Φ
3
)
=
E
+
[
w
M
]
1
−
w
2
,
Φ
4
=
i
(
w
E
)
1
−
w
2
Ψ
=
i
w
f
∗
,
Φ
1
=
w
2
F
12
+
w
3
F
13
+
w
4
F
14
,
Φ
2
=
w
1
F
21
+
w
3
F
23
+
w
4
F
24
,
Φ
3
=
w
1
F
31
+
w
2
F
32
+
w
4
F
34
,
Φ
4
=
w
1
F
41
+
w
2
F
42
+
w
3
F
43
.
|
Ψ
1
=
−
i
(
w
2
f
34
+
w
3
f
42
+
w
4
f
23
)
,
Ψ
2
=
−
i
(
w
1
f
43
+
w
3
f
14
+
w
4
f
31
)
,
Ψ
3
=
−
i
(
w
1
f
24
+
w
2
f
41
+
w
4
f
12
)
,
Ψ
4
=
−
i
(
w
1
f
32
+
w
2
f
13
+
w
3
f
21
)
.
w
F
=
−
Φ
,
w
F
∗
=
−
i
μ
Ψ
,
w
f
=
−
ε
Φ
,
w
f
∗
=
−
i
Ψ
F
=
[
w
,
Φ
]
+
i
μ
[
w
,
Ψ
]
∗
,
f
=
ε
[
w
,
Φ
]
+
i
[
w
,
Ψ
]
∗
,
s
+
(
w
s
¯
)
w
=
−
σ
w
F
.
{\displaystyle {\begin{matrix}{\begin{matrix}\Phi =-wF\\\left(\Phi _{1},\Phi _{2},\Phi _{3}\right)={\frac {{\mathfrak {E}}+[{\mathfrak {wM}}]}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ \Phi _{4}={\frac {i({\mathfrak {wE}})}{\sqrt {1-{\mathfrak {w}}^{2}}}}&\Psi =iwf^{*},\\\hline \left.{\scriptstyle {\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|&{\scriptstyle {\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 ]^{*},\\s+(w{\overline {s}})w=-\sigma wF.\end{matrix}}}
which becomes the covariant Lorentz force (d,e) when multiplied with the charge. He gave the general expression of four-force density as
K
+
(
w
K
¯
)
w
K
=
lor
S
=
−
s
F
+
N
K
1
=
∂
X
x
∂
x
+
∂
X
y
∂
y
+
∂
X
z
∂
z
−
∂
X
t
∂
t
=
ϱ
E
x
+
s
y
M
z
−
s
z
M
y
−
1
2
Φ
Φ
¯
∂
ε
∂
x
−
1
2
Ψ
Ψ
¯
∂
μ
∂
x
+
ε
μ
−
1
1
−
w
2
(
W
∂
w
∂
x
)
K
2
=
∂
Y
x
∂
x
+
∂
Y
y
∂
y
+
∂
Y
z
∂
z
−
∂
Y
t
∂
t
=
ϱ
E
y
+
s
z
M
x
−
s
x
M
z
−
1
2
Φ
Φ
¯
∂
ε
∂
y
−
1
2
Ψ
Ψ
¯
∂
μ
∂
y
+
ε
μ
−
1
1
−
w
2
(
W
∂
w
∂
y
)
K
3
=
∂
Z
x
∂
x
+
∂
Z
y
∂
y
+
∂
Z
z
∂
z
−
∂
Z
t
∂
t
=
ϱ
E
z
+
s
x
M
y
−
s
y
M
x
−
1
2
Φ
Φ
¯
∂
ε
∂
z
−
1
2
Ψ
Ψ
¯
∂
μ
∂
z
+
ε
μ
−
1
1
−
w
2
(
W
∂
w
∂
z
)
1
i
K
4
=
−
∂
T
x
∂
x
−
∂
T
y
∂
y
−
∂
T
z
∂
z
−
∂
T
t
∂
t
=
s
x
E
x
+
s
y
E
y
+
s
z
E
z
+
1
2
Φ
Φ
¯
∂
ε
∂
t
+
1
2
Ψ
Ψ
¯
∂
μ
∂
t
−
ε
μ
−
1
1
−
w
2
(
W
∂
w
∂
t
)
[
lor
=
|
∂
∂
x
1
,
∂
∂
x
2
,
∂
∂
x
3
,
∂
∂
x
4
|
]
{\displaystyle {\begin{matrix}K+(w{\overline {K}})w\\\hline K={\text{lor }}S=-sF+N\\\hline {\begin{aligned}K_{1}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial X_{t}}{\partial t}}=\varrho {\mathfrak {E}}_{x}+{\mathfrak {s}}_{y}{\mathfrak {M}}_{z}-{\mathfrak {s}}_{z}{\mathfrak {M}}_{y}-{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \varepsilon }{\partial x}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial x}}+{\frac {\varepsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial x}}\right)\\K_{2}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial Y_{t}}{\partial t}}=\varrho {\mathfrak {E}}_{y}+{\mathfrak {s}}_{z}{\mathfrak {M}}_{x}-{\mathfrak {s}}_{x}{\mathfrak {M}}_{z}-{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \varepsilon }{\partial y}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial y}}+{\frac {\varepsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial y}}\right)\\K_{3}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial Z_{t}}{\partial t}}=\varrho {\mathfrak {E}}_{z}+{\mathfrak {s}}_{x}{\mathfrak {M}}_{y}-{\mathfrak {s}}_{y}{\mathfrak {M}}_{x}-{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \varepsilon }{\partial z}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial z}}+{\frac {\varepsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial z}}\right)\\{\frac {1}{i}}K_{4}&=-{\frac {\partial T_{x}}{\partial x}}-{\frac {\partial T_{y}}{\partial y}}-{\frac {\partial T_{z}}{\partial z}}-{\frac {\partial T_{t}}{\partial t}}={\mathfrak {s}}_{x}{\mathfrak {E}}_{x}+{\mathfrak {s}}_{y}{\mathfrak {E}}_{y}+{\mathfrak {s}}_{z}{\mathfrak {E}}_{z}+{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \varepsilon }{\partial t}}+{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial t}}-{\frac {\varepsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial t}}\right)\end{aligned}}\\\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}}}
equivalent to (f).
In his lecture “space and time” (1908, published 1909), Minkowski defined the “moving force vector” as[R 5]
t
˙
X
,
t
˙
Y
,
t
˙
Z
,
t
˙
T
T
=
1
c
2
(
x
˙
t
˙
X
,
y
˙
t
˙
Y
,
z
˙
t
˙
Z
)
{\displaystyle {\begin{matrix}{\dot {t}}X,\ {\dot {t}}Y,\ {\dot {t}}Z,\ {\dot {t}}T\\T={\frac {1}{c^{2}}}\left({\frac {\dot {x}}{\dot {t}}}X,\ {\frac {\dot {y}}{\dot {t}}}Y,\ {\frac {\dot {z}}{\dot {t}}}Z\right)\end{matrix}}}
equivalent to (b) because
x
˙
,
y
˙
,
z
˙
,
t
˙
{\displaystyle {\dot {x}},\ {\dot {y}},\ {\dot {z}},\ {\dot {t}}}
is the four-velocity
γ
[
u
,
c
]
{\displaystyle \gamma [\mathbf {u} ,c]}
and
f
=
[
X
,
Y
,
Z
]
{\displaystyle \mathbf {f} =[X,Y,Z]}
.
w:Max Born (1909) summarized Minkowski's work using index notation, defining the equations of motion in terms of rest mass density
μ
{\displaystyle \mu }
and rest charge density
ϱ
0
∗
{\displaystyle \varrho _{0}^{\ast }}
[R 6]
c
i
μ
∂
2
x
α
∂
ξ
4
2
=
ϱ
0
∗
c
∑
β
=
1
4
f
α
β
∂
x
β
∂
ξ
4
μ
∂
2
x
∂
τ
2
=
ϱ
0
∗
{
E
x
∂
t
∂
τ
+
1
c
(
∂
y
∂
τ
M
z
−
∂
z
∂
τ
M
y
)
}
μ
∂
2
y
∂
τ
2
=
ϱ
0
∗
{
E
y
∂
t
∂
τ
+
1
c
(
∂
z
∂
τ
M
x
−
∂
x
∂
τ
M
z
)
}
μ
∂
2
z
∂
τ
2
=
ϱ
0
∗
{
E
z
∂
t
∂
τ
+
1
c
(
∂
x
∂
τ
M
y
−
∂
y
∂
τ
M
z
)
}
μ
∂
2
t
∂
τ
2
=
1
c
2
ϱ
0
∗
{
E
x
∂
x
∂
τ
+
E
y
∂
y
∂
τ
+
E
z
∂
z
∂
τ
}
{\displaystyle {\begin{matrix}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}}}\\\hline {\begin{aligned}\mu {\frac {\partial ^{2}x}{\partial \tau ^{2}}}&=\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{x}{\frac {\partial t}{\partial \tau }}+{\frac {1}{c}}\left({\frac {\partial y}{\partial \tau }}{\mathfrak {M}}_{z}-{\frac {\partial z}{\partial \tau }}{\mathfrak {M}}_{y}\right)\right\}\\\mu {\frac {\partial ^{2}y}{\partial \tau ^{2}}}&=\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{y}{\frac {\partial t}{\partial \tau }}+{\frac {1}{c}}\left({\frac {\partial z}{\partial \tau }}{\mathfrak {M}}_{x}-{\frac {\partial x}{\partial \tau }}{\mathfrak {M}}_{z}\right)\right\}\\\mu {\frac {\partial ^{2}z}{\partial \tau ^{2}}}&=\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{z}{\frac {\partial t}{\partial \tau }}+{\frac {1}{c}}\left({\frac {\partial x}{\partial \tau }}{\mathfrak {M}}_{y}-{\frac {\partial y}{\partial \tau }}{\mathfrak {M}}_{z}\right)\right\}\\\mu {\frac {\partial ^{2}t}{\partial \tau ^{2}}}&={\frac {1}{c^{2}}}\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{x}{\frac {\partial x}{\partial \tau }}+{\mathfrak {E}}_{y}{\frac {\partial y}{\partial \tau }}+{\mathfrak {E}}_{z}{\frac {\partial z}{\partial \tau }}\right\}\end{aligned}}\end{matrix}}}
equivalent to (c1, d1, e1).
w:Philipp Frank (1909) discussed “electromagnetic mechanics” by defining four-force (X,Y,Z,T) as follows:[R 7]
m
d
d
σ
d
t
d
σ
=
T
,
m
d
2
x
d
σ
2
=
X
,
d
2
y
d
σ
2
=
Y
,
d
2
z
d
σ
2
=
Z
T
=
m
d
d
σ
d
t
d
σ
=
m
1
(
1
−
w
2
)
2
(
w
x
d
w
x
d
t
+
w
y
d
w
y
d
t
+
w
z
d
w
z
d
t
)
d
2
x
d
σ
2
=
1
1
−
w
2
d
w
x
d
t
+
w
x
(
1
−
w
2
)
2
(
w
x
d
w
x
d
t
+
w
y
d
w
y
d
t
+
w
z
d
w
z
d
t
)
e
t
c
.
{\displaystyle {\begin{matrix}m{\frac {d}{d\sigma }}{\frac {dt}{d\sigma }}=T,\ m{\frac {d^{2}x}{d\sigma ^{2}}}=X,\ {\frac {d^{2}y}{d\sigma ^{2}}}=Y,\ {\frac {d^{2}z}{d\sigma ^{2}}}=Z\\\hline {\begin{aligned}T&=m{\frac {d}{d\sigma }}{\frac {dt}{d\sigma }}=m{\frac {1}{\left(1-w^{2}\right)^{2}}}\left(w_{x}{\frac {dw_{x}}{dt}}+w_{y}{\frac {dw_{y}}{dt}}+w_{z}{\frac {dw_{z}}{dt}}\right)\\{\frac {d^{2}x}{d\sigma ^{2}}}&={\frac {1}{1-w^{2}}}{\frac {dw_{x}}{dt}}+{\frac {w_{x}}{\left(1-w^{2}\right)^{2}}}\left(w_{x}{\frac {dw_{x}}{dt}}+w_{y}{\frac {dw_{y}}{dt}}+w_{z}{\frac {dw_{z}}{dt}}\right)\\&{\rm {etc}}.\end{aligned}}\end{matrix}}}
corresponding to (b, c).
While Minkowski used the four-force density as
K
+
(
K
w
¯
)
w
{\displaystyle K+(K{\overline {w}})w}
with
K
=
lor
S
{\displaystyle K={\text{lor }}S}
, w:Max Abraham (1909) directly used
K
{\displaystyle K}
as four-force density, and expressed it in terms of “momentum equations” and an “energy equation” using momentum density
g
{\displaystyle {\mathfrak {g}}}
, energy density
ψ
{\displaystyle \psi }
, Poynting vector
S
{\displaystyle {\mathfrak {S}}}
, Joule heat Q [R 8]
K
x
=
∂
X
x
∂
x
+
∂
X
y
∂
y
+
∂
X
z
∂
z
−
∂
g
x
∂
t
K
x
=
∂
X
x
∂
x
+
∂
X
y
∂
y
+
∂
X
z
∂
z
−
∂
X
t
∂
l
K
y
=
∂
Y
x
∂
x
+
∂
Y
y
∂
y
+
∂
Y
z
∂
z
−
∂
g
y
∂
t
⇒
K
y
=
∂
Y
x
∂
x
+
∂
Y
y
∂
y
+
∂
Y
z
∂
z
−
∂
Y
t
∂
l
K
z
=
∂
Z
x
∂
x
+
∂
Z
y
∂
y
+
∂
Z
z
∂
z
−
∂
g
z
∂
t
K
z
=
∂
Z
x
∂
x
+
∂
Z
y
∂
y
+
∂
Z
z
∂
z
−
∂
Z
t
∂
l
w
K
+
Q
=
−
d
i
v
S
−
∂
ψ
∂
t
K
t
=
−
∂
T
x
∂
x
−
∂
T
y
∂
y
−
∂
T
z
∂
z
−
∂
T
t
∂
l
(
d
i
v
S
=
∂
S
x
∂
x
+
∂
S
y
∂
y
+
∂
S
z
∂
z
,
g
=
S
c
2
)
{\displaystyle {\begin{matrix}{\begin{aligned}{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}}&{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial X_{t}}{\partial l}}\\{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}}\quad \Rightarrow &{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial Y_{t}}{\partial l}}\\{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}&{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial Z_{t}}{\partial l}}\\{\mathfrak {wK}}+Q&=-\mathrm {div} \,{\mathfrak {S}}-{\frac {\partial \psi }{\partial t}}&{\mathfrak {K}}_{t}&=-{\frac {\partial T_{x}}{\partial x}}-{\frac {\partial T_{y}}{\partial y}}-{\frac {\partial T_{z}}{\partial z}}-{\frac {\partial T_{t}}{\partial l}}\end{aligned}}\\\hline \left(\mathrm {div} \,{\mathfrak {S}}={\frac {\partial {\mathfrak {S}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {S}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {S}}_{z}}{\partial z}},\ {\mathfrak {g}}={\frac {\mathfrak {S}}{c^{2}}}\right)\end{matrix}}}
or alternatively by introducing “relative stresses” and the “relative energy flux”:
K
x
=
∂
X
x
′
∂
x
+
∂
X
y
′
∂
y
+
∂
X
z
′
∂
z
−
∂
g
x
∂
t
K
y
=
∂
Y
x
′
∂
x
+
∂
Y
y
′
∂
y
+
∂
Y
z
′
∂
z
−
∂
g
y
∂
t
K
z
=
∂
Z
x
′
∂
x
+
∂
Z
y
′
∂
y
+
∂
Z
z
′
∂
z
−
∂
g
z
∂
t
w
K
+
Q
=
−
d
i
v
{
S
−
w
ψ
}
−
∂
ψ
∂
t
(
X
x
′
=
X
x
+
w
x
g
x
X
y
′
=
X
y
+
w
y
g
x
X
z
′
=
X
z
+
w
z
g
x
Y
x
′
=
Y
x
+
w
x
g
x
Y
y
′
=
Y
y
+
w
y
g
x
Y
z
′
=
Y
z
+
w
z
g
x
Z
x
′
=
Z
x
+
w
x
g
x
Z
y
′
=
Z
y
+
w
y
g
x
Z
z
′
=
Z
z
+
w
z
g
x
)
{\displaystyle {\begin{matrix}{\begin{aligned}{\mathfrak {K}}_{x}&={\frac {\partial X_{x}^{\prime }}{\partial x}}+{\frac {\partial X_{y}^{\prime }}{\partial y}}+{\frac {\partial X_{z}^{\prime }}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}}\\{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}^{\prime }}{\partial x}}+{\frac {\partial Y_{y}^{\prime }}{\partial y}}+{\frac {\partial Y_{z}^{\prime }}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}}\\{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}^{\prime }}{\partial x}}+{\frac {\partial Z_{y}^{\prime }}{\partial y}}+{\frac {\partial Z_{z}^{\prime }}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}\\{\mathfrak {wK}}+Q&=-\mathrm {div} \,\left\{{\mathfrak {S}}-{\mathfrak {w}}\psi \right\}-{\frac {\partial \psi }{\partial t}}\end{aligned}}\\\hline \left({\begin{aligned}X_{x}^{\prime }&=X_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{x}&X_{y}^{\prime }&=X_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{x}&X_{z}^{\prime }&=X_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{x}\\Y_{x}^{\prime }&=Y_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{x}&Y_{y}^{\prime }&=Y_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{x}&Y_{z}^{\prime }&=Y_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{x}\\Z_{x}^{\prime }&=Z_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{x}&Z_{y}^{\prime }&=Z_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{x}&Z_{z}^{\prime }&=Z_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{x}\end{aligned}}\right)\end{matrix}}}
all equivalent to (f1).
In a subsequent paper (1909) he formulated these relations as follows[R 9]
K
x
=
∂
X
x
∂
x
+
∂
X
y
∂
y
+
∂
X
z
∂
z
−
∂
g
x
∂
t
K
x
=
∂
X
x
∂
x
+
∂
X
y
∂
y
+
∂
X
z
∂
z
+
∂
X
u
∂
u
K
y
=
∂
Y
x
∂
x
+
∂
Y
y
∂
y
+
∂
Y
z
∂
z
−
∂
g
y
∂
t
⇒
K
y
=
∂
Y
x
∂
x
+
∂
Y
y
∂
y
+
∂
Y
z
∂
z
+
∂
Y
u
∂
u
K
z
=
∂
Z
x
∂
x
+
∂
Z
y
∂
y
+
∂
Z
z
∂
z
−
∂
g
z
∂
t
K
z
=
∂
Z
x
∂
x
+
∂
Z
y
∂
y
+
∂
Z
z
∂
z
+
∂
Z
u
∂
u
Q
+
(
w
K
)
=
−
∂
S
x
∂
x
−
∂
S
y
∂
y
−
∂
S
z
∂
z
−
∂
ψ
∂
t
K
u
=
∂
U
x
∂
x
+
∂
U
y
∂
y
+
∂
U
z
∂
z
+
∂
U
u
∂
u
(
u
=
i
c
t
)
{\displaystyle {\begin{matrix}{\begin{aligned}{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}}&{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}+{\frac {\partial X_{u}}{\partial u}}\\{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}}\quad \Rightarrow &{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}+{\frac {\partial Y_{u}}{\partial u}}\\{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}&{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}+{\frac {\partial Z_{u}}{\partial u}}\\Q+({\mathfrak {wK}})&=-{\frac {\partial {\mathfrak {S}}_{x}}{\partial x}}-{\frac {\partial {\mathfrak {S}}_{y}}{\partial y}}-{\frac {\partial {\mathfrak {S}}_{z}}{\partial z}}-{\frac {\partial \psi }{\partial t}}&{\mathfrak {K}}_{u}&={\frac {\partial U_{x}}{\partial x}}+{\frac {\partial U_{y}}{\partial y}}+{\frac {\partial U_{z}}{\partial z}}+{\frac {\partial U_{u}}{\partial u}}\end{aligned}}\\\hline \left(u=ict\right)\end{matrix}}}
In addition, Abraham gave arguments in favor of his choice to use
K
=
lor
S
{\displaystyle K={\text{lor }}S}
directly as four-force density: He argued that Minkowski's definition of force as the product of constant rest mass with four-acceleration together with a complementary force is disadvantageous at non-adiabatic motion, because mass-energy equivalence according to which mass depends on its energy content would suggest a variable rest mass
m
0
{\displaystyle m_{0}}
, and showed that
m
0
{\displaystyle m_{0}}
is compatible with the general definition of force density
K
{\displaystyle {\mathfrak {K}}}
over volume dv as the rate of momentum change without the need of complementary components[R 10]
d
d
t
{
m
0
c
q
1
−
q
2
}
=
K
d
v
{\displaystyle {\frac {d}{dt}}\left\{{\frac {m_{0}c{\mathfrak {q}}}{\sqrt {1-{\mathfrak {q}}^{2}}}}\right\}={\mathfrak {K}}\,dv}
.
As he showed in 1910, this implies that the equations of motion in terms of four-force density
K
{\displaystyle {\mathfrak {K}}}
and rest mass density
ν
{\displaystyle \nu }
and four-velocity assume the form:[R 11]
d
d
τ
(
ν
d
x
d
τ
)
=
K
x
d
d
τ
(
ν
d
y
d
τ
)
=
K
y
d
d
τ
(
ν
d
z
d
τ
)
=
K
z
d
d
τ
(
ν
d
u
d
τ
)
=
K
u
(
u
=
i
c
t
)
{\displaystyle {\begin{aligned}{\frac {d}{d\tau }}\left(\nu {\frac {dx}{d\tau }}\right)&={\mathfrak {K}}_{x}\\{\frac {d}{d\tau }}\left(\nu {\frac {dy}{d\tau }}\right)&={\mathfrak {K}}_{y}\\{\frac {d}{d\tau }}\left(\nu {\frac {dz}{d\tau }}\right)&={\mathfrak {K}}_{z}\\{\frac {d}{d\tau }}\left(\nu {\frac {du}{d\tau }}\right)&={\mathfrak {K}}_{u}\ (u=ict)\end{aligned}}}
This is equivalent to (a1) defining four-force density as the rate of change of four-momentum density.
The first discussion in an English language paper of four-force in terms of integral forms (even though in the broader context of w:spherical wave transformations ), was given by w:Harry Bateman in a paper read 1909 and published 1910, who defined the following invariants (with
λ
2
=
1
{\displaystyle \lambda ^{2}=1}
in relativity):[R 12]
ρ
λ
2
[
(
E
x
−
w
z
H
y
+
w
y
H
z
)
d
y
d
z
d
t
+
(
E
y
−
w
x
H
z
+
w
z
H
x
)
d
z
d
x
d
t
+
(
E
z
−
w
y
H
z
+
w
x
H
y
)
d
x
d
y
d
t
−
(
w
x
E
x
+
w
y
E
y
+
w
z
E
z
)
d
x
d
y
d
z
]
,
ρ
λ
4
[
(
E
x
+
w
y
H
z
−
w
z
H
y
)
δ
x
+
(
E
y
+
w
z
H
x
−
w
x
H
z
)
δ
y
+
(
E
z
+
w
z
H
y
−
w
y
H
x
)
δ
z
−
(
w
x
E
x
+
w
y
E
y
+
w
z
E
z
)
δ
t
]
,
ρ
d
x
d
y
d
z
d
t
[
(
E
x
+
w
y
H
z
−
w
z
H
y
)
δ
x
+
(
E
y
+
w
z
H
x
−
w
x
H
z
)
δ
y
+
(
E
z
+
w
x
H
y
−
w
y
H
x
)
δ
z
−
(
w
x
E
x
+
w
y
E
y
+
w
z
E
z
)
δ
t
]
.
{\displaystyle {\begin{matrix}{\frac {\rho }{\lambda ^{2}}}\left[\left(E_{x}-w_{z}H_{y}+w_{y}H_{z}\right)dy\ dz\ dt+\left(E_{y}-w_{x}H_{z}+w_{z}H_{x}\right)dz\ dx\ dt\right.\\\left.+\left(E_{z}-w_{y}H_{z}+w_{x}H_{y}\right)dx\ dy\ dt-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)dx\ dy\ dz\right],\\\\{\frac {\rho }{\lambda ^{4}}}\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.\\\left.+\left(E_{z}+w_{z}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right],\\\\\rho \ dx\ dy\ dz\ dt\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.\\\left.+\left(E_{z}+w_{x}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right].\end{matrix}}}
equivalent to (e) because it can be seen as the product of charge, four-velocity and the electromagnetic tensor, which is the definition of the covariant Lorentz force.
w:Wladimir Ignatowski (1910) formulated the four-force in terms of the Lorentz force as well as the equation of motion, and following Abraham (1909) he assumed variable rest mass in the case of non-adiabatic motion[R 13]
(
K
1
−
n
v
2
,
n
v
K
1
−
n
v
2
)
d
m
0
v
1
−
n
v
2
d
t
=
K
⇒
(
K
1
−
n
v
2
,
n
v
K
1
−
n
v
2
+
d
m
0
d
t
)
K
=
e
E
+
[
e
v
H
]
K
′
=
e
′
E
′
+
[
e
′
v
′
H
′
]
m
0
d
v
1
−
n
v
2
d
t
=
K
[
n
=
1
c
2
]
{\displaystyle {\begin{matrix}\left({\frac {\mathfrak {K}}{\sqrt {1-n{\mathfrak {v}}^{2}}}},\ {\frac {n{\mathfrak {vK}}}{\sqrt {1-n{\mathfrak {v}}^{2}}}}\right)\\{\frac {d{\frac {m_{0}{\mathfrak {v}}}{\sqrt {1-n{\mathfrak {v}}^{2}}}}}{dt}}={\mathfrak {K}}\quad \Rightarrow \quad \left({\frac {\mathfrak {K}}{\sqrt {1-n{\mathfrak {v}}^{2}}}},\ {\frac {n{\mathfrak {vK}}}{\sqrt {1-n{\mathfrak {v}}^{2}}}}+{\frac {dm_{0}}{dt}}\right)\\\hline {\begin{matrix}{\mathfrak {K}}=e{\mathfrak {E}}+[e{\mathfrak {vH}}]\\{\mathfrak {K}}'=e'{\mathfrak {E}}'+[e'{\mathfrak {v'H}}']\end{matrix}}\quad m_{0}{\frac {d{\frac {\mathfrak {v}}{\sqrt {1-n{\mathfrak {v}}^{2}}}}}{dt}}={\mathfrak {K}}\end{matrix}}{\scriptstyle \left[n={\frac {1}{c^{2}}}\right]}}
equivalent to (b, d). He also derived the four-force density in terms of the density of three-force or Lorentz force
K
1
{\displaystyle {\mathfrak {K}}_{1}}
by integrating three-force
K
{\displaystyle {\mathfrak {K}}}
with respect to volume:[R 14]
(
K
1
,
n
K
1
v
)
K
1
=
E
ϱ
+
[
ϱ
v
H
]
K
1
′
=
E
′
ϱ
+
[
ϱ
′
v
′
H
′
]
[
K
d
v
=
K
1
,
n
=
1
c
2
]
{\displaystyle {\begin{matrix}\left({\mathfrak {K}}_{1},\ n{\mathfrak {K}}_{1}{\mathfrak {v}}\right)\\\hline {\begin{matrix}{\mathfrak {K}}_{1}={\mathfrak {E}}\varrho +[\varrho {\mathfrak {vH}}]\\{\mathfrak {K}}'_{1}={\mathfrak {E}}'\varrho +[\varrho '{\mathfrak {v'H}}']\end{matrix}}\left[{\mathfrak {K}}\ dv={\mathfrak {K}}_{1},\ n={\frac {1}{c^{2}}}\right]\end{matrix}}}
equivalent to (b1, d1).
In an influential paper, w:Arnold Sommerfeld translated Minkowski's matrix formalism into a four-dimensional vector formalism involving four-vectors and six-vectors. He defined the four-force density as the covariant product of a six-vector denoted as “field vector” f (now known as electromagnetic tensor) and the four-current P , which he related to Lorentz force density
F
{\displaystyle {\mathfrak {F}}}
:[R 15]
[
P
f
]
F
x
=
(
P
f
x
)
=
ϱ
(
v
x
c
f
x
x
+
v
y
c
f
x
y
+
v
z
c
f
x
z
+
i
f
x
l
)
=
ϱ
(
v
y
c
H
z
−
v
z
c
H
y
+
E
x
)
…
F
l
=
(
P
f
l
)
=
ϱ
(
v
x
c
f
l
x
+
v
y
c
f
l
y
+
v
z
c
f
l
z
+
i
f
l
l
)
=
i
ϱ
c
(
v
x
E
x
+
v
y
E
y
+
v
z
E
z
)
F
j
=
ϱ
(
E
+
1
c
[
v
H
]
)
,
F
l
=
i
ϱ
c
(
E
v
)
(
j
=
x
,
y
,
z
;
l
=
i
c
t
)
{\displaystyle {\begin{matrix}[Pf]\\\hline {\begin{array}{ll}{\mathfrak {F}}_{x}=\left(Pf_{x}\right)&=\varrho \left({\frac {{\mathfrak {v}}_{x}}{c}}f_{xx}+{\frac {{\mathfrak {v}}_{y}}{c}}f_{xy}+{\frac {{\mathfrak {v}}_{z}}{c}}f_{xz}+if_{xl}\right)\\&=\varrho \left({\frac {{\mathfrak {v}}_{y}}{c}}{\mathfrak {H}}_{z}-{\frac {{\mathfrak {v}}_{z}}{c}}{\mathfrak {H}}_{y}+{\mathfrak {E}}_{x}\right)\\&\dots \\{\mathfrak {F}}_{l}=\left(Pf_{l}\right)&=\varrho \left({\frac {{\mathfrak {v}}_{x}}{c}}f_{lx}+{\frac {{\mathfrak {v}}_{y}}{c}}f_{ly}+{\frac {{\mathfrak {v}}_{z}}{c}}f_{lz}+if_{ll}\right)\\&={\frac {i\varrho }{c}}\left({\mathfrak {v}}_{x}{\mathfrak {E}}_{x}+{\mathfrak {v}}_{y}{\mathfrak {E}}_{y}+{\mathfrak {v}}_{z}{\mathfrak {E}}_{z}\right)\end{array}}\\\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 (b1, d1, e1).
In the first textbook on relativity, w:Max von Laue (1911) followed Abraham in defining the four-force density (Viererkraft) F as the product of a six-vector denoted as “field vector”
M
{\displaystyle {\mathfrak {M}}}
(now known as electromagnetic tensor) and the four-current related to Lorentz force density
F
{\displaystyle {\mathfrak {F}}}
, and alternatively as the divergence of the electromagnetic stress-energy tensor T ,[R 16] which included some printing errors corrected in the 1913 edition[R 17]
F
=
[
P
M
]
=
−
Δ
i
v
T
[
P
M
]
x
=
F
x
,
[
P
M
]
y
=
F
y
,
[
P
M
]
z
=
F
z
,
[
P
M
]
l
=
i
ϱ
c
(
q
E
)
=
i
c
(
q
F
)
[
F
=
ϱ
(
E
+
1
c
[
q
H
]
)
,
Δ
i
v
=
divergence six-vector
,
l
=
i
c
t
]
{\displaystyle {\begin{matrix}{\begin{aligned}F&=[P{\mathfrak {M}}]\\&=-\varDelta iv\,T\end{aligned}}\\\hline {}[P{\mathfrak {M}}]_{x}={\mathfrak {F}}_{x},\ [P{\mathfrak {M}}]_{y}={\mathfrak {F}}_{y},\ [P{\mathfrak {M}}]_{z}={\mathfrak {F}}_{z},\ [P{\mathfrak {M}}]_{l}={\frac {i\varrho }{c}}({\mathfrak {qE}})={\frac {i}{c}}({\mathfrak {qF}})\\\left[{\mathfrak {F}}=\varrho ({\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {qH}}]),\ \varDelta iv={\text{divergence six-vector}},\ l=ict\right]\end{matrix}}}
equivalent to (b1, d1, e1, f1).
In 1913, Laue also showed that four-force density F and three force density
F
{\displaystyle {\mathfrak {F}}}
can be used to derive the “Minkowskian force vector” (i.e. four-force) K and three-force
K
{\displaystyle {\mathfrak {K}}}
per unit charge by defining the volume
δ
V
:
c
2
−
q
2
{\displaystyle \delta V:{\sqrt {c^{2}-q^{2}}}}
:[R 18]
K
⇒
K
x
=
K
x
c
2
−
q
2
,
K
y
=
K
y
c
2
−
q
2
,
K
z
=
K
z
c
2
−
q
2
,
K
l
=
i
c
(
q
K
)
c
2
−
q
2
,
⇒
d
(
m
Y
)
d
τ
⇒
m
d
Y
d
τ
{\displaystyle K\Rightarrow {\begin{aligned}K_{x}&={\frac {{\mathfrak {K}}_{x}}{\sqrt {c^{2}-q^{2}}}},&K_{y}&={\frac {{\mathfrak {K}}_{y}}{\sqrt {c^{2}-q^{2}}}},\\K_{z}&={\frac {{\mathfrak {K}}_{z}}{\sqrt {c^{2}-q^{2}}}},&K_{l}&={\frac {i}{c}}{\frac {({\mathfrak {qK}})}{\sqrt {c^{2}-q^{2}}}},\end{aligned}}\Rightarrow {\frac {d(mY)}{d\tau }}\Rightarrow m{\frac {dY}{d\tau }}}
Silberstein (1911-14)
edit
w:Ludwik Silberstein in 1911 and published 1912, devised an alternative 4D calculus based on w:Biquaternions which, however, never gained widespread support. He defined the "force-quaternion" (i.e. four-force density) as the electrical part
P
e
{\displaystyle \mathrm {P} _{e}}
of P , which he related to the “current-quaternion” (i.e. four-current) C and the “electromagnetic bivector” (i.e. w:Weber vector )
F
{\displaystyle \mathbf {F} }
}[R 19]
P
=
C
F
=
D
[
F
⋅
F
]
P
=
ρ
{
ι
M
+
E
+
1
c
p
M
−
ι
c
p
E
}
=
P
e
+
ι
P
m
P
e
=
ρ
{
ι
c
(
p
E
)
+
E
+
1
c
V
p
M
}
P
m
=
ρ
{
ι
c
(
p
M
)
+
M
+
1
c
V
p
E
}
[
D
=
∂
∂
l
−
∇
]
{\displaystyle {\begin{matrix}\mathrm {P} =\mathrm {C} \mathbf {F} =\mathrm {D} [\mathbf {F} \cdot \mathbf {F} ]\\\hline {\begin{aligned}\mathrm {P} &=\rho \left\{\iota \mathbf {M} +\mathbf {E} +{\frac {1}{c}}p\mathbf {M} -{\frac {\iota }{c}}p\mathbf {E} \right\}\\&=\mathrm {P} _{e}+\iota \mathrm {P} _{m}\\\mathrm {P} _{e}&=\rho \left\{{\frac {\iota }{c}}(p\mathbf {E} )+\mathbf {E} +{\frac {1}{c}}\mathrm {V} p\mathbf {M} \right\}\\\mathrm {P} _{m}&=\rho \left\{{\frac {\iota }{c}}(p\mathbf {M} )+\mathbf {M} +{\frac {1}{c}}\mathrm {V} p\mathbf {E} \right\}\end{aligned}}\\\left[\mathrm {D} ={\frac {\partial }{\partial l}}-\nabla \right]\end{matrix}}}
equivalent to (d1,e1,f1).
In his textbook on quaternionic special relativity written 1914, he defined four-force density using stress-energy tensor
S
{\displaystyle {\mathfrak {S}}}
like Abraham and Laue, as well as in terms of Minkowski's alternative force definition
F
M
n
k
{\displaystyle F_{\mathrm {Mnk} }}
using four-velocity Y :[R 20]
F
=
ρ
{
ι
c
(
p
E
)
+
E
+
1
c
V
p
M
}
=
ι
c
(
P
p
)
+
P
=
−
1
c
R
[
D
]
L
=
−
l
o
r
S
F
M
n
k
=
1
2
[
F
+
1
c
2
Y
F
c
Y
]
[
P
=
ρ
{
E
+
1
c
V
p
M
}
,
D
=
∂
∂
l
−
∇
,
l
o
r
=
|
∂
∂
x
,
∂
∂
y
,
∂
∂
z
,
∂
i
∂
t
|
]
{\displaystyle {\begin{matrix}{\begin{aligned}F&=\rho \left\{{\frac {\iota }{c}}(p\mathbf {E} )+\mathbf {E} +{\frac {1}{c}}\mathrm {V} p\mathbf {M} \right\}\\&={\frac {\iota }{c}}(\mathbf {Pp} )+\mathbf {P} \\&=-{\frac {1}{c}}\mathbf {R} [D]\mathbf {L} \\&=-\mathrm {lor} {\mathfrak {S}}\\F_{\mathrm {Mnk} }&={\frac {1}{2}}\left[F+{\frac {1}{c^{2}}}YF_{c}Y\right]\end{aligned}}\\\left[\mathbf {P} =\rho \left\{\mathbf {E} +{\frac {1}{c}}\mathrm {V} \mathbf {p} \mathbf {M} \right\},\ D={\frac {\partial }{\partial l}}-\nabla ,\ {\rm {lor}}=\left|{\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}},\ {\frac {\partial }{i\partial t}}\right|\right]\end{matrix}}}
equivalent to (d1,e1,f1).
Lewis and Wilson (1912)
edit
w:Gilbert Newton Lewis and w:Edwin Bidwell Wilson explicitly defined a four-vector called “extended momentum” (i.e. four-momentum)
m
0
w
{\displaystyle m_{0}\mathbf {w} }
, deriving the “extended force” (i.e. four force) using
c
{\displaystyle \mathbf {c} }
as four-acceleration:[R 21]
m
0
c
=
d
m
0
w
d
s
=
d
m
v
d
s
k
1
+
d
m
d
s
k
4
=
1
1
−
v
2
(
d
m
v
d
t
k
1
+
d
m
d
t
k
4
)
{\displaystyle 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)}
equivalent to (a, c).
While formulating electrodynamics in a generally covariant way, w:Friedrich Kottler expressed the “Minkowski force”
F
α
{\displaystyle F_{\alpha }}
in terms of the electromagnetic field-tensor
F
α
β
{\displaystyle F_{\alpha \beta }}
, four-current
P
(
β
)
{\displaystyle \mathbf {P} ^{(\beta )}}
, stress-energy tensor
S
α
β
{\displaystyle S_{\alpha \beta }}
:[R 22]
F
α
(
y
)
=
∑
β
F
α
β
(
y
)
P
(
β
)
(
y
)
1
−
w
2
/
c
2
[
∑
β
F
α
β
(
y
)
P
(
β
)
(
y
)
=
∑
β
F
α
β
(
y
)
∑
γ
∂
∂
y
(
γ
)
F
β
γ
(
y
)
=
∑
β
∂
∂
y
(
β
)
S
α
β
]
{\displaystyle {\begin{matrix}F_{\alpha }(y)=\sum _{\beta }{\frac {F_{\alpha \beta }(y)\mathbf {P} ^{(\beta )}(y)}{\sqrt {1-{\mathfrak {w}}^{2}/c^{2}}}}\\\left[{\underset {\beta }{\sum }}F_{\alpha \beta }(y)\mathbf {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 }\right]\end{matrix}}}
equivalent to (e, f).
In 1914 Kottler derived the equations of motion from the “Nordström tensor” (i.e. dust solution of the stress energy tensor) in terms of rest mass density
ν
0
{\displaystyle \nu _{0}}
, which he equated to the action of a constant external electromagnetic field in terms of electromagnetic tensor
F
(
h
k
)
{\displaystyle F^{(hk)}}
and charge density
ϱ
{\displaystyle \varrho }
:[R 23]
∑
k
=
1
4
∂
∂
x
(
k
)
(
ν
d
x
(
h
)
d
t
d
x
(
k
)
d
t
)
=
ν
d
2
x
(
h
)
d
t
2
=
K
(
h
)
=
ϱ
c
∑
k
=
1
4
F
(
h
k
)
x
(
k
)
{\displaystyle \sum _{k=1}^{4}{\frac {\partial }{\partial x^{(k)}}}\left(\nu {\frac {dx^{(h)}}{dt}}{\frac {dx^{(k)}}{dt}}\right)=\nu {\frac {d^{2}x^{(h)}}{dt^{2}}}=K^{(h)}={\frac {\varrho }{c}}\sum _{k=1}^{4}F^{(hk)}x^{(k)}}
equivalent to (f).
In an unpublished manuscript on special relativity, written between 1912-1914, w:Albert Einstein wrote the four-force density
(
K
μ
)
{\displaystyle \left(K_{\mu }\right)}
in terms of electromagnetic tensor
(
F
μ
ν
)
{\displaystyle \left({\mathfrak {F}}_{\mu \nu }\right)}
, four-current
(
J
ν
)
{\displaystyle \left({\mathfrak {J}}_{\nu }\right)}
, stress-energy tensor
(
T
μ
ν
)
{\displaystyle \left(T_{\mu \nu }\right)}
:[R 24]
(
F
μ
ν
)
(
J
ν
)
=
(
K
μ
)
K
1
=
ρ
(
e
x
+
q
y
c
h
z
−
q
z
c
h
y
)
K
2
=
ρ
(
e
y
+
q
z
c
h
x
−
q
x
c
h
z
)
K
3
=
ρ
(
e
z
+
q
x
c
h
y
−
q
y
c
h
x
)
K
4
=
i
c
ρ
(
q
x
e
x
+
q
y
e
y
+
q
z
e
z
)
(
K
μ
)
=
−
(
∂
∂
x
ν
)
(
T
μ
ν
)
K
1
=
−
∂
p
x
x
∂
x
1
−
∂
p
x
y
∂
x
2
−
∂
p
x
z
∂
x
3
−
∂
i
c
s
x
∂
x
4
…
…
K
4
=
−
∂
i
c
s
x
∂
x
1
−
∂
i
c
s
y
∂
x
2
−
∂
i
c
s
z
∂
x
3
−
∂
(
−
w
)
∂
x
4
{\displaystyle {\begin{matrix}\left({\mathfrak {F}}_{\mu \nu }\right)\left({\mathfrak {J}}_{\nu }\right)=\left(K_{\mu }\right)\\\hline {\begin{aligned}K_{1}&=\rho \left({\mathfrak {e}}_{x}+{\frac {{\mathfrak {q}}_{y}}{c}}{\mathfrak {h}}_{z}-{\frac {{\mathfrak {q}}_{z}}{c}}{\mathfrak {h}}_{y}\right)\\K_{2}&=\rho \left({\mathfrak {e}}_{y}+{\frac {{\mathfrak {q}}_{z}}{c}}{\mathfrak {h}}_{x}-{\frac {{\mathfrak {q}}_{x}}{c}}{\mathfrak {h}}_{z}\right)\\K_{3}&=\rho \left({\mathfrak {e}}_{z}+{\frac {{\mathfrak {q}}_{x}}{c}}{\mathfrak {h}}_{y}-{\frac {{\mathfrak {q}}_{y}}{c}}{\mathfrak {h}}_{x}\right)\\K_{4}&={\frac {i}{c}}\rho \left({\mathfrak {q}}_{x}{\mathfrak {e}}_{x}+{\mathfrak {q}}_{y}{\mathfrak {e}}_{y}+{\mathfrak {q}}_{z}{\mathfrak {e}}_{z}\right)\end{aligned}}\\\hline \left(K_{\mu }\right)=-\left({\frac {\partial }{\partial x_{\nu }}}\right)\left(T_{\mu \nu }\right)\\\hline {\begin{aligned}K_{1}&=-{\frac {\partial p_{xx}}{\partial x_{1}}}-{\frac {\partial p_{xy}}{\partial x_{2}}}-{\frac {\partial p_{xz}}{\partial x_{3}}}-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{x}}{\partial x_{4}}}\\&\dots \\&\dots \\K_{4}&=-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{x}}{\partial x_{1}}}-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{y}}{\partial x_{2}}}-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{z}}{\partial x_{3}}}-{\frac {\partial (-w)}{\partial x_{4}}}\end{aligned}}\end{matrix}}}
equivalent to (d1,e1,f1).