Minimal coupling of gravitational and electromagnetic fields in General Relativity
Authors: Janyška Josef , Modugno Marco 
Published: October 2023

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript normal ♮ Baseline colon upper T bold-italic upper E right-arrow upper T Superscript asterisk Baseline bold-italic upper E circled-times upper T upper T bold-italic upper E and upper F colon bold-italic upper E right-arrow left-parenthesis double-struck upper L Superscript 1 slash 2 Baseline circled-times double-struck upper M Superscript 1 slash 2 Baseline right-parenthesis circled-times normal upper Lamda squared upper T Superscript asterisk Baseline bold-italic upper E period">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K colon-equal upper K Superscript normal ♮ Baseline plus upper K Superscript German e Baseline colon double-struck upper T Superscript asterisk Baseline circled-times upper T bold-italic upper E right-arrow upper T Superscript asterisk Baseline bold-italic upper E circled-times upper T left-parenthesis double-struck upper T Superscript asterisk Baseline circled-times upper T bold-italic upper E right-parenthesis comma">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column upper K Superscript German e Baseline colon-equal minus StartFraction q Over m EndFraction t ModifyingAbove upper F With caret 2nd Column a m p semicolon colon double-struck upper T Superscript asterisk Baseline circled-times upper T bold-italic upper E right-arrow upper T Superscript asterisk Baseline bold-italic upper E circled-times left-parenthesis double-struck upper T Superscript asterisk Baseline circled-times upper T bold-italic upper E right-parenthesis comma 2nd Row 1st Column ModifyingAbove upper F With caret colon-equal g Superscript normal ♯ squared left-parenthesis upper F right-parenthesis 2nd Column a m p semicolon colon bold-italic upper E right-arrow left-parenthesis double-struck upper L Superscript negative 3 slash 2 Baseline circled-times double-struck upper M Superscript 1 slash 2 Baseline right-parenthesis circled-times upper T Superscript asterisk Baseline bold-italic upper E circled-times upper T bold-italic upper E comma 3rd Row 1st Column t 2nd Column a m p semicolon colon double-struck upper T Superscript asterisk Baseline circled-times upper T bold-italic upper E right-arrow double-struck upper R colon v right-arrow minus StartFraction 1 Over c squared EndFraction g left-parenthesis v comma v right-parenthesis period EndLayout">

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.