In a general relativistic spacetime
E
{\boldsymbol {E}}
, we consider the gravitational field and the electromagnetic field, which are represented, respectively, by the Levi–Civita connection and the scaled 2–form
K
♮
:
T
E
→
T
∗
E
⊗
T
T
E
and
F
:
E
→
(
L
1
/
2
⊗
M
1
/
2
)
⊗
Λ
2
T
∗
E
.
\begin{equation*} K{}^{\natural }{}: T{\boldsymbol {E}} \to T^*{\boldsymbol {E}} \otimes TT{\boldsymbol {E}} \qquad \text {and}\qquad F: {\boldsymbol {E}} \to (\mathbb {L}^{1/2} \otimes \mathbb {M}^{1/2}) \otimes \Lambda ^2T^*{\boldsymbol {E}}. \end{equation*}
Then, with reference to a charged particle, with mass and charge
(
m
,
q
)
(m, q)
, we obtain a minimal coupling of the gravitational connection
K
♮
K{}^{\natural }{}
with the electromagnetic field
F
F
, so yielding the joined (non linear) connection
K
≔
K
♮
+
K
e
:
T
∗
⊗
T
E
→
T
∗
E
⊗
T
(
T
∗
⊗
T
E
)
,
\begin{equation*} K ≔K{}^{\natural }{} + K^{\mathfrak {e}}: \mathbb {T}^* \otimes T{\boldsymbol {E}} \to T^*{\boldsymbol {E}} \otimes T(\mathbb {T}^* \otimes T{\boldsymbol {E}}), \end{equation*}
where we have set
K
e
≔
−
q
m
t
F
^
a
m
p
;
:
T
∗
⊗
T
E
→
T
∗
E
⊗
(
T
∗
⊗
T
E
)
,
F
^
≔
g
♯
2
(
F
)
a
m
p
;
:
E
→
(
L
−
3
/
2
⊗
M
1
/
2
)
⊗
T
∗
E
⊗
T
E
,
t
a
m
p
;
:
T
∗
⊗
T
E
→
R
:
v
→
−
1
c
2
g
(
v
,
v
)
.
\begin{align*} K^{\mathfrak {e}} ≔- {\frac {q}{m}} {t}\widehat {F} &: \mathbb {T}^* \otimes T{\boldsymbol {E}} \to T^*{\boldsymbol {E}} \otimes (\mathbb {T}^* \otimes T{\boldsymbol {E}}),\\ \widehat {F} ≔g^{\sharp }{}^2 (F) &: {\boldsymbol {E}} \to (\mathbb {L}^{-3/2} \otimes \mathbb {M}^{1/2}) \otimes T^*{\boldsymbol {E}} \otimes T{\boldsymbol {E}},\\ {t} &: \mathbb {T}^* \otimes T{\boldsymbol {E}} \to \mathbb {R}: v \to - \frac {1}{c^2}g(v,v). \end{align*}
Actually, the standard Lorentz law of motion
∇
♮
d
s
d
s
=
−
q
m
g
♯
(
d
s
⌟
F
)
\begin{equation*} \nabla {}^{\natural }{}_{ds}ds = - {\frac {q}{m}} g^{\sharp }(ds \lrcorner F) \end{equation*}
turns out to be equivalent, in terms of the joined connection
K
K
, to the law
∇
d
s
d
s
=
0.
\begin{equation*} \nabla _{ds} ds = 0. \end{equation*}
Then, chosen a general observer
o
o
, we rephrase the above joined objects in terms of the observed electric and magnetic fields
E
→
[
o
]
\vec {E}[o]
and
B
→
[
o
]
\vec {B}[o]
.
The above results extend to an einsteinian general relativistic framework a minimal coupling of gravitational and electromagnetic fields, which has been found for classical and quantum mechanics in the galileian framework.