5.5.
Electromagnetism.
The relativistic version of Maxwell’s equa
tions uses the 4vectors
A
μ
and
j
μ
, where
A
0
=
V
, its space components
are those of the usual vector potential;
j
0
= 4
πρ
, and
j
i
=
4
π
c
J
i
for
i
= 1
,
2
,
3. Then Maxwell’s equations with the Lorenz gauge
∂
μ
A
μ
= 0
become:
(5.9)
∂
μ
∂
μ
A
α
=
j
α
.
The Minkowski force on a moving charge is given by
φ
μ
=
q
c
F
μν
u
ν
,
where
(5.10)
F
μν
=
∂
μ
A
ν

∂
ν
A
μ
is the
fieldstrength tensor
, and
u
μ
=
dx
μ
/dτ
.
In terms of
F
μν
, Maxwell’s equations are:
(5.11)
∂
ν
F
μν
=
j
μ
∂
θ
F
μν
+
∂
μ
F
νθ
+
∂
ν
F
θμ
= 0
.
Equivalently, they can be expressed as
*
F
[
λμ,ν
]
=
j
μ
and
F
[
λμ,ν
]
= 0
respectively, where
*
F
μν
=
1
2
F
αβ
αβμν
is the dual tensor of
F
μν
and the
brackets denote antisymmetrization.
These equations can be derived from a variational principle with the
following action:
(5.12)
S
=

1
4
Z
F
μν
F
μν
d
3
x
dt.
Here the components of the tensor
F
μν
play the role of generalized
coordinates.
On the other hand, the action of a charged particle in an electro
magnetic field is
(5.13)
S
=
q
c
Z
A
μ
dx
μ
.
12
MIGUEL A. LERMA
6.
General Relativity
6.1.
Newtonian Gravitation.
Newton’s gravitational law states that
the force between two masses
m
1
and
m
2
at a distance
r
is
(6.1)
F
=

k
m
1
m
2
r
2
,
where
k
= 6
.
670
×
10

11
N m
2
kg

2
is Newton’s gravitational constant.
The corresponding potential produced by a mass
m
is
φ
=
km/r
, and
for a continuous fluid of density
ρ
it is given by the
Poisson’s equation
:
(6.2)
∇
2
φ
= 4
πk ρ.
Since Poisson’s equation is not Lorentz invariant, it is not appropriate
for a relativistic theory of gravity.
There are various ways of developing a covariant theory of gravita
tion in the frame of Special Relativity, but they are inconsistent or
lead to predictions that do not match experimental observations. So a
completely new approach is needed.
6.2.
The Principle of Equivalence.
In General Relativity it is pos
tulated that a gravitational field is locally indistinguishable from an
accelerated frame. In Relativity non inertial frames are represented by
non Cartesian coordinates in 4space. So gravitation is not a “force”,
but a change in the geometry of spacetime. Particles that are not un
der the action of some other field behave like free particles, and their
paths are still geodesics in spacetime.
6.3.
Geometry in a curved spacetime.
Spacetime is represented
by a pseudoRiemannian 4dimensional manifold. Its geometry is given
by the (symmetric)
metric tensor
g
μν
.
4
The line element is
(6.3)
ds
2
=
g
μν
dx
μ
dx
ν
.
The usual derivatives of the components of a tensor respect to the co
ordinates are not covariant in a general frame, particularly in a curved
spacetime. This is due to the fact that the basic vectors
e
μ
=
∂
∂x
μ
are
not constant. So, in particular
(6.4)
∂
ν
(
A
μ
e
μ
) = (
∂
ν
A
μ
)
e
μ
+
A
μ
∂
ν
e
μ
= (
∂
ν
A
μ
+ Γ
μ
αν
A
α
)
e
μ
,
where Γ
μ
αν
are the components of
∂
ν
e
α
.
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 Physics, MIGUEL A. LERMA