On the equivalence principle and gravitational and inertial mass relation of classical charged particles

ar
X
iv
:1
00
7.
48
46
v1
[
gr
-q
c]
27
Ju
l 2
01
0
On the Equivalence Principle and gravitational
and inertial mass relation of classical charged
particles
Mario Goto
Departamento de F´ısica, Universidade Estadual de Londrina
86051-990, Londrina, PR, Brazil
(mgoto@uel.br)
P. L. Natti
Departamento de Matema´tica, Universidade Estadual de Londrina
86051-990 Londrina, PR, Brazil
(plnatti@uel.br)
E. R. Takano Natti
Pontif´ıcia Universidade Cato´lica do Parana´
Rua Jo´quei Clube, 458, 86067-000, Londrina, PR, Brazil
(erica.natti@pucpr.br)
July 29, 2010
Abstract
We show that the locally constant force necessary to get a stable hy-
perbolic motion regime for classical charged point particles, actually, is a
combination of an applied external force and of the electromagnetic ra-
diation reaction force. It implies, as the strong Equivalence Principle is
valid, that the passive gravitational mass of a charged point particle should
be slight greater than its inertial mass. An interesting new feature that
emerges from the unexpected behavior of the gravitational and inertial
mass relation, for classical charged particles, at very strong gravitational
field, is the existence of a critical, particle dependent, gravitational field
value that signs the validity domain of the strong Equivalence Principle.
For electron and proton, these critical field values are gc ' 4.8×1031m/s2
and gc ' 8.8× 1034m/s2, respectively.
PACS: 04.20.Cv, 03.50.De
1

1 Introduction
The problem of the electromagnetic radiation reaction force on the charged
particle dynamics, as given by the Lorentz-Abraham-Dirac (LAD) equation [1]-
[11], has been a subject of active investigation. There are a lot of works about
this subject accumulated since the first attempt was made by Dirac [1].
There is now a renewed interest on this subject, with works pointing to some-
thing new, which should affect the validity of the Weak Equivalence Principle
at some circumstances [12]-[21]. Perhaps because the main experimental justi-
fication that led Einstein to formulate the Equivalence Principle (EP), which
is one of the foundations of his General Theory of Relativity, is the numerical
equality between inertial and gravitational mass, nowadays they are taken quite
as synonymous, so we have to be aware to avoid misleading conclusions. Ac-
cording to Weinberg [5], we distinguish the Weak Equivalence Principle (WEP)
of the Strong Equivalence Principle (SEP). The Strong Equivalence Principle
postulates that at every space-time point in a arbitrary gravitational field it
is possible to choose a locally inertial coordinate system such that, within a
sufficiently small region of the point in question, the laws of the nature take the
same form as in unaccelerated Cartesian coordinate systems in the absence of
gravitation. On the other hand, the Weak Equivalence Principle is nothing but
a restatement of the observed equality of gravitational and inertial mass.
About the verification of the WEP, there is a surprising richness in the va-
riety of experimental techniques and choice of the test bodies which have been
used so far. The equality of gravitational and inertial mass is in fact what the
experiments, since the famous Eo¨tvos balance until recents experiments, actu-
ally measure. We show a brief review. The most obvious way to proof the WEP
is to compare the motion of two bodies during free fall. These experiments,
limited by rather short free falling periods, reached an accuracy of about 1 part
in 10−10 [22]. The Bremen drop tower experiments, using SQUID displacement
sensors, will provide a much longer time for free fall, allowing to reach an accu-
racy of about 10−12− 10−13 [23, 24]. Torsion balance experiments have reached
an accuracy of few parts in 10−13 [25]. A planned experiment using a cryo-
genic balance claims an accuracy of 10−14 [26]. The most sensitive long-range
measurements have used the Sun as the source and Earth and Moon type test
bodies. The lunar laser ranging (LLR) techniques reach an accuracy of 5×10−13
[27, 28]. On the other hand, future space experiments promise much better pre-
cision in this measurement. The MICROSCOPE mission [29] aims to test, on a
microsatellite of the MYRIADE series developed by CNES/FRANCE, the WEP
with a 10−15 accuracy. The Galileo Galilei-GG is a proposed experiment in low
orbit around the Earth aiming to test the WEP to the level of 1 part in 10−17
[30]. STEP, using pairs of concentric free-failing proof-masses, will be able to
test the WEP to a sensitivity at 1 part in 10−18 [31].
On the other hand, notice that these mentioned experiments don’t investi-
gate the WEP in the case of charged particles. The reason is that electromag-
2

netic fields influence gravitation experiments with charged particles and must
be shielded carefully. The experiments for freely falling electrons carried out by
Witteborn and Fairbank [32], with an accuracy of 10−1, is the only one cited
in literature. Nowadays, Dittus and La¨mmerzahl [33] showed that an experi-
ment in space with the Witteborn-Fairbank set-up may be well suited to test
the WEP and to improve the results for free fall test with charged particles by
orders of magnitude.
Some important comments should be made on these two formulations of the
Equivalence Principle (EP). The SEP is valid only in static and homogeneous
gravitational field, but it is always possible to choose a sufficiently small space-
time region where the gravitational field can be locally approximated by a static
homogeneous field, so that the SEP is valid locally. For a scalar particle, the
Pauli formulation of EP proposes that a homogeneous gravitational field can
always be transformed away globally so that in a suitable reference frame there
is only Minkowski space - no gravitational field. On the other hand, Audretsch
[34] observes that if one takes a particle with spin, the equation of motion
for such a particle will inevitably involve the curvature tensor, which can not
be eliminated by any transformation of coordinates. Some authors ignore the
influence of curvature (second derivates) or tidal effects, but this means that
they get rid of gravitational field. Finally, some results has been obtained for an
infinite homogeneous gravitational field (in the entire space) or for an uniformly
accelerated boundless reference frame. These gravitational fields are not a true
gravitational fields [14].
About the WEP, the equations of motion of a point mass in a curved back-
ground spacetime were investigated by Mino, Sasaki and Tanaka [15]. The
same equations of motion were later obtained by Quinn and Wald [16, 17] from
an axiomatic approach. Following Mino, Quinn and Wald, Haas and Poisson
[18] calculate the self-force acting on a point scalar charge in a wide class of
cosmological spacetimes. The self-force produce two effects: a time-changing
inertial mass and a deviation relative to geodesic motion. The work of Dewitt
and Brehme [19], corrected by Hobbs [20], showed that a point charge in true
gravitational field not follows a geodesic, so that WEP is violated for a charged
particle. Using the techniques of finite-temperature field theory, Donoghue et al.
[21] showed that the equality of inertial mass and gravitational mass, for charged
spin- 12 or spin-zero particles, is no valid in the context of quantum field theory
at finite temperature. Higuchi [35] calculated the position shift of the final-state
wave packet of the charged particle due the radiation and showed that it dis-
agrees with the result obtained using the Lorentz-Abraham-Dirac equation for
the radiation-reaction force. In an alternative approach, Spohn [36] and other
authors [37]-[39] changed the Lorentz-Abraham-Dirac equation for the force on
an accelerating charge, which avoids the pathologies of preacceleration and run-
away solutions. Yaghjian [40] suggest that these problems will be absent once
the finite-size effects are properly taken into account. Finally, the validity of the
WEP is very well tested for macroscopic bodies to a sensitivity of few parts in
3

10−13, but this does not necessarily imply that such principle continues to hold
at a microscopic scale and in the quantum regime.
The goal of this work is not to discuss about these papers, but, instead,
to add the possibility to analyze the problem of local motion of the classical
charged point particles in a different perspective, with emphasis in the EP,
which validity is used as a good starting point. As the subject of this work
is about the conditions of validity or not of the EP, it is just to notice that
results from General Relativity are not used at any moment, in order to avoid
any possibility to fall in a vicious causal recurrence.
This text is a review of an old work of Goto [41]. What we have to do
is to figure out the condition that we have to provide such that a classical
charged point particle can reach a locally stable hyperbolic motion. We show
that is necessary to furnish a balance between an applied external force and the
electromagnetic radiation reaction force to get an hyperbolic motion regime. An
important consequence is that, taking account the SEP, it implies in a passive
gravitational mass that is slight greater than the inertial mass. From this result,
one show that what seems to be uncomfortable, as the presence of radiation
for the charged particle performing hyperbolic motion and its absence for one
supported at rest in an uniform gravitational field [42, 43], both equivalent
situations as the SEP is valid, lead to a new physical feature performed by
charged particles. As consequence of the unexpected behavior of the passive
gravitational and inertial mass relation, at a very strong gravitational field, we
find the presence of a divergence that indicates a critical field value that signs
the validity domain of the SEP.
This paper is organized as follows. In section II, we show that the locally
external force necessary to produce an hyperbolic motion in neutral particles
is smaller than the locally external force necessary to give the same hyperbolic
motion in classical charged particles. In section III, we figure out that, to
the SEP to be valid, the WEP is violated for classical charged point particles in
stable local hyperbolic motion regime. More, we show that there exists a critical,
particle dependent, gravitational field value that signs the validity domain of
the SEP. The section IV is devoted to a final discussion and conclusions.
2 Local hyperbolic motion of charged particles
Hyperbolic motion is the natural generalization of the concept of the Newto-
nian uniformly accelerated motion due to a constant force applied to a particle,
which might be due to an uniform gravitational field. At relativistic level, as
the velocity is upper limited by the light velocity, constant force don’t imply
in constant acceleration; instead, it results in the above mentioned hyperbolic
motion, which denomination comes from the hyperbola that it is drawn in the
zt-plane by this kind of motion.
An one dimensional hyperbolic motion of a particle of mass m occurs as a
4

solution of the relativistic equation of motion [6, 7]
md
2xµ
dτ2 = f
µ(τ), (1)
when external force F is parallel to velocity v and it is locally constant in the
proper referential frame. In (1) fµ is the relativistic force defined as
fµ = γ
(v · F
c , F
)
with γ = 1√
1− β2
and β = v/c , (2)
where c is the velocity of light.
Supposing the motion along the z-axes, the trajectory of hyperbolic motion
is given by
(z0, z) = c
2
a (sinhλτ, coshλτ) , (3)
where a = F/m is a constant proper acceleration and λ = a/c. From (3) the
velocity and acceleration are given by
( .z0, .z) = c(coshλτ, sinhλτ) = γc (1, β) (4)
and
(..z0, ..z) = a(sinhλτ, coshλτ) , (5)
respectively, so that the relativistic force responsible by the hyperbolic motion
is
fµ(τ) = m(..z0, ..z) = ma(sinhλτ, coshλτ) . (6)
The choice of the metric tensor gµν is such that vµvµ = −c2 for four velocity
vµ = .xµand, at non relativistic limit, aµaµ = a2 for four acceleration aµ =
..xµ.
The equation of motion of a classical charged point particle, including elec-
tromagnetic radiation reaction force, is given by the well known Lorentz-Abraham-
Dirac equation [1]-[11], [36]-[39],
maµ(τ) = fµext(τ) + fµrad(τ), (7)
where fµext(τ) is the external four-force and
fµrad(τ) = mτ0
(
.aµ − 1c2 a
νaνvµ
)
, (8)
with
5