A possible Reinterpretation of Einstein's Equations

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A possible Reinterpretation of
Einstein’s Equations
A. Bouda∗ and A. Belabbas †
Laboratoire de Physique The´orique, Universite´ de Be´ja¨ıa,
Route Targa Ouazemour, 06000 Be´ja¨ıa, Algeria
December 13, 2010
Abstract
In this paper, we first review Huei’s formulation in which it is shown that
the linearized Einstein equations can be written in the same form as the
Maxwell equations. We eliminate some imperfections like the scalar po-
tential which is ill linked to the electric-type field, the Lorentz-type force
which is obtained with a time independence restriction and the undesired
factor 4 which appears in the magnetic-type part. Second, from these
results and in the light of a recent work by C.C. Barros, we propose an
extension of the equivalence principle and we suggest a new interpreta-
tion for Einstein’s equations by showing that the electromagnetic Maxwell
equations can be derived from a new version of Einstein’s ones.
PACS: 04.20.-q; 04.20.Cv; 04.40.Nr
Key words: Linearized Einstein’s equations, Principle of Equivalence, Maxwell’s
equations, Lorentz force.
∗Electronic address: bouda a@yahoo.fr
†Electronic address: belabbas.moumene@gmail.com
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1 Introduction
There are several attempts to describe gravitation and electromagnetism in a
unified field theory. The electromagnetic field was sometimes presented as the
nonsymmetric part of the metric [1, 2], sometimes as some additional compo-
nents of the metric in five dimensional space-time [3]. In this paper, we suggest
that the electromagnetic field is contained in the four dimensional Einstein equa-
tions themselves. Our approach is based on two things.
The first is Huei’s formulation [4] in which it is shown that the linearized
Einstein equations can be written in the same form as the Maxwell equations.
In this formulation, developed also by Wald [5], there are however some imper-
fections to point out:
1. The usual relation between the potential and the electric-type field as
known in electromagnetism is reproduced only in the harmonic gauge.
2. The geodesic equation predicts a Lorentz-type force with an undesired
factor 4 in the magnetic-type part compared to the usual electromagnetic
Lorentz force.
3. The Lorentz-type force is obtained only in the case where the fields are
independent on time.
We would like to add that Carroll [6] has redefined the fields in such a way as
to eliminate the undesired factor 4. However, Maxwell-type equations are not
satisfied in his formulation. In what follows, with the use of a subtle gauge,
we will review the linearized gravity so as to get to a strong similarity with
electromagnetism.
The second thing is the recent work by C.C.Barros in which he suggested
to study the effect of the metric in the subatomic systems instead of trying to
quantize gravity [7, 8, 9]. He assumed that interactions, even non gravitational
ones, affect the space-time structure. In the context of Schwarzschild’s solution,
by introducing Coulomb’s potential instead of the gravitational one into the
metric, he derived the spectrum of hydrogen atom. Although the author con-
sidered a subatomic system, however the use of Schwarzschild’s solution implies
that he implicitly assumed the existence of versions of Einstein’s equations and
geodesic equations for the Coulomb interaction. What is even more surprising
is that his results for hydrogen atom spectrum, confirmed in more detail in [10],
are extremely close to those predicted by Dirac’s equation. These intriguing re-
sults encourage us to wonder about the extension of the principle of equivalence
(PE).
The Barros results and the extraordinary similarity between gravitation and
electromagnetism resulting from the revised version of linearized gravity pre-
sented here suggest that the electromagnetic Maxwell equations can be followed
from a new version of Einstein’s equations in the linear approximation within
the framework of the electromagnetic interaction. We will see that the higher
order terms of this new version of Einstein’s equations are negligible in the usual
application domains of the electromagnetism.
The paper is organized as follows. In section 2, we will review the linearized
gravity. In section 3, we will present several arguments to show that the current
version of the PE is restrictive and suggest how it can be extended to include
2

other interactions. We consolidate our arguments by showing in the context of
the electromagnetic interaction that Maxwell’s equations are contained as a first
order approximation in Einstein’s equations. Section 4 is devoted to conclusion.
2 Linearized gravity revisited
Our goal here is to review the linearized gravity so as to avoid the imperfec-
tion cited above and then to get to a strong similarity with electromagnetism.
For this purpose, let us decompose the metric into the Minkowski one plus a
perturbation
gµν = ηµν + hµν , (1)
where ηµν = (+1,−1,−1,−1) and hµν ≪ 1. To first order, setting
gµν = ηµν − hµν (2)
and taking into account Eq. (1), it is easy to check that indices of hµν can be
raised and lowered by using ηµν and ηµν . In this approximation, the Christoffel
symbols,
Γλµν =
1
2g
λρ [∂µgρν + ∂νgµρ − ∂ρgµν ] , (3)
and the curvature tensor,
Rλ µνρ = ∂ρΓλµν − ∂νΓλµρ + ΓσµνΓλρσ − ΓσµρΓλνσ, (4)
take the forms
Γλµν =
1
2η
λρ [∂µhρν + ∂νhµρ − ∂ρhµν ] (5)
and
Rλµνρ =
1
2 [∂ρ∂µhλν + ∂ν∂λhµρ − ∂λ∂ρhµν − ∂µ∂νhλρ] . (6)
The infinitesimal coordinate transformation xµ → x′µ = xµ + ξµ induces on the
perturbation of the metric a transformation δhµν = −∂νξληµλ− ∂µξληνλ which
leaves invariant expression (6) of the curvature tensor. This invariance allows
us to fix a gauge
gµνΓρµν = 0, (7)
known as the harmonic gauge [5, 6, 11]. In the linear approximation, this
condition takes the following form
∂µhµν −
1
2∂νh = 0, (8)
where h ≡ hµµ is the trace of the perturbation. Huei defined the electric-type
field and the potential vector as
Eig =
c2
4
[
∂ihˆ00 + ∂j hˆij
]
(9)
and
Aig =
c
4 hˆ
0i, (10)
3

where hˆµν = hµν − hηµν/2, c is the velocity of light and i, j, ... = 1, 2, 3. It is
assumed that the Einstein convention for repeated indexes works also for i and
j. In this formulation, although expression (9) satisfies Maxwell-type equations,
however it does not allow to reproduce the expected relation
−→E g = −−→∇φ−
∂−→A g
∂t , (11)
where φ is the newtonian potential. Nevertheless, if we use the gauge condition
(8), expression (9) turns out to be
Eig =
c2
4 ∂
ihˆ00 − c
2
4 ∂0hˆ
0i, (12)
which is compatible with (10) and (11) if we set φ = c2hˆ00/4. This feature is
unsatisfactory since relation (11) should be valid in any gauge. We also notice
that in the Lorentz-type force,
d2−→r
dt2 =
−→E g + 4−→v ×−→B g, (13)
obtained from the geodesic equation, there is in the magnetic-type part an
undesired factor 4 compared to the electromagnetic case. In addition, this
equation is obtained only if the fields do not depend on time.
In what follows, we will remedy these weakness by defining the scalar and
vector potentials as
A0g =
c
2h
00, (14)
Aig = ch0i (15)
and introducing the tensor
Fµνg = ∂µAνg − ∂νAµg , (16)
as in electromagnetism. We notice that definition (14) of the scalar potential is
compatible with the well-known expression obtained when the geodesic equation
is used in the newtonian approximation. Contrary to Huei’s formulation, by
defining Eig = −cF 0ig as in electromagnetism, relation (11) is automatically
satisfied without using any gauge. It is also the case for
−→B g = −→∇ ×−→A g, (17)
where Big = −ǫijk(Fg)jk/2 is the magnetic-type field and ǫijk is the usual Levi-
Civita antisymmetric tensor
(
ǫ123 = +1
)
. Concerning the first group of the
Maxwell-type equations,
∂λFµνg + ∂νFλµg + ∂µF νλg = 0, (18)
with the use of definition (16), it is automatically satisfied. With regard to the
second group, let us consider the Einstein tensor
Gµν ≡ Rµν −
1
2gµνR (19)
4

and use the following definition for the Ricci tensor
Rµν = gλρRλµρν . (20)
In the linear approximation, by using (1), (2), (6) and (20), expression (19)
turns out to be
Gµν =
1
2 [∂µ∂νh +2hµν − ηµν2h
−∂λ∂µhλν − ∂λ∂νhλµ + ηµν∂ρ∂λhρλ
]
, (21)
where 2 = ∂λ∂λ is the d’Alembertian. Carroll [6] showed that the trace hii is
not a propagating degree of freedom. Then, instead of the time component of
(8), let us impose an alternative condition
hii = 0 (22)
and keep in the gauge condition (8) only the three space components
∂µhµi −
1
2∂ih = 0. (23)
From relation (22), we observe that h = h00 and expressions (14) and (15) can
be written in a unified form
Aµg = c
(
h0µ − 12η
0µh
)
. (24)
With the use of (22), we can deduce from (21) that
G00 =
1
2∂i∂jh
ij (25)
and
G0i =
1
2
[
∂j∂jh0i − ∂0∂jhij − ∂i∂jh0j
]
. (26)
Relations (22) and (23) allow to write (25) and (26) as follows
G00 = 12∂i
[1
2∂
ih00 − ∂0h0i
]
(27)
and
G0i = 12
[
∂j
(
∂jh0i − ∂ih0j
)
+ ∂0
(
∂0h0i − 12∂
ih00
)]
. (28)
Taking into account (16) and (24), relations (27) and (28) become
G00 = 12c∂iF
i0
g =
1
2c∂µF
µ0
g (29)
and
G0i = 12c∂µF
µi
g . (30)
Finally, we have
G0ν = 12c∂µF
µν
g . (31)
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