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General Relativity and Gravitation, Vol. 36, No. 3, March 2004 (°
Testing the Universality of Free Fall
for Charged Particles in Space
Hansjörg Dittus,1 Claus Lämmerzahl,1,2 and Hanns Selig1
Received September 19, 2003
At first a short analysis of the notion of the Universality of Free Fall (UFF) for charged
matter is given. Even if neutral bound systems of charged particles are in full accordance
with the UFF, there is still a possibility that an isolated charge couples anomalously to
gravitational fields. The experiment of Witteborn and Fairbank aimed at testing the UFF
for electrons is shortly reviewed emphasizing the various additional disturbing gravity
induced electromagnetic fields. Since these additional gravity induced fields are not
very well under control, a space borne version of this experiment will reduce these
disturbances considerably. The corresponding estimates for these kinds of tests in space
are presented. As a result, gravity–induced stray field can be reduced considerably. Furthermore, also patch–effects can be reduced efficiently due to novel coating techniques.
Therefore, due to microgravity conditions and new techniques the UFF for charged
particles may be tested with much higher accuracy than in previous experiments.
KEY WORDS: Equivalence principle; Universality of Free Fall; charged particles;
Schiff–Barnhill effect.
1. INTRODUCTION
The Einstein Equivalence Principle is the basis for establishing the gravitational
interaction as a metric theory [1]. It consists of the Universality of Free Fall (UFF)
also called the Weak Equivalence Principle, of Local Lorentz Invariance (LLI),
and of the Universality of the Gravitational Redshift (UGR), also called Local
Position Invariance. The UFF states that in a gravitational field all structureless
pointlike particles follow the same path. The UFF has been confirmed for neutral
bulk matter with an accuracy of 10−12 [2] and for quantum matter with an accuracy
1 ZARM,
2 E-mail:
University of Bremen, 28359 Bremen, Germany.
[email protected]
571
C 2004 Plenum Publishing Corporation
0001-7701/04/0300-0571/0 °
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Dittus, Lämmerzahl, and Selig
of 10−9 [3]. At least for neutral bulk matter it is planned to increase this accuracy
by 6 orders of magnitude with the Satellite Test of Equivalence Principle (STEP)
[4, 5]. The main consequence of the UFF is that the gravitational interaction can
be geometrised.
LLI states that locally Special Relativity should be valid which has as consequence that at each space–time point one can define a Lorentzian metric. Within
a widely used kinematical framework [6, 7], see [8] for a review, the validity
of Special Relativity is connected with the isotropy of light propagation, the independence of the velocity of light from the state of motion of the laboratory,
and with a specific outcome for Doppler shift experiments. Today, the best tests
confirm Special Relativity in terms of modified Mansouri–Sexl–parameters [8]
with an accuracy of |β + δ − 12 | ≤ 4 · 10−9 [9], |α − β − 1| ≤ 2 · 10−6 [10], and
|α| ≤ 2 · 10−7 [11], respectively.
UGR describes that each clock, irrespective of the physical interaction and
matter they are built of, is influenced by gravity in the same way. This implies that
all kinds of matter and non–gravitational interactions couple to gravity in the same
way. Today, this is confirmed to an accuracy of 10−2 to 10−4 [12–14] depending
on the type of clocks used in the experiment. UGR is also connected with the
constancy of physical ‘constants’ [15, 16]. Very recently, a variation of the fine
structure constant has been inferred from the analysis of astrophysical data [17].
Here we are considering the UFF for charged particles. If all electromagnetic fields are shielded, then charged particles which contain no internal structure
should fall along the same path as neutral particles3 . Until now, there is only one
single experiment which was dedicated to a test of the UFF for charged matter,
the Witteborn–Fairbank experiment [23]. The reported accuracy was 0.1 which,
compared with neutral matter, is poor. The reason for that was the appearance
of stray electric fields and of gravity–induced electric fields of which the Schiff–
Barnhill effect [24] is the most popular one. For a thorough survey of experimental
constraints on free fall experiments with charged matter, see [25].
The reason for testing the UFF for charged particles is to extend the tests of
UFF with respect to baryon number to the charge property of matter. The original
motivation to develop these experiments as a test version for tests of the UFF for
3 For
completeness we mention that there is an additional theoretical problem concerning the UFF for
charged matter: Accelerated charged particles are subject to a back reaction of the field of the accelerated charge to its motion. This additional acceleration is ∼ 23 (e2 /c3 )˙˙˙
x(t). The question is whether
a charged particle attached to a fixed position on the surface of the Earth (described by a homogenous gravitational field), or a freely falling particle radiates. This problem has been controversially
discussed, see e.g. [18–20]. Beside the above tail–terms, the equation of motion of a charged particle
in a curved space–time will be furthermore modified by an additional charge–dependent coupling
to curvature [21, 22]. In any case, since these effects are of relativistic order or are connected to
curvature, they are too small to be of experimental importance in low energy laboratory experiments
and can be safely neglected. The terms we are considering are independent of these couplings.
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anti–matter in the form of anti–protons became obsolete in the meantime, because
the production of neutral antimatter in form of anti–Hydrogen opens up much
better possibilities to test the UFF for antimatter (see the paper of J. Walz and T.
Hänsch in this issue [26]).
A further issue of tests of the UFF with charged matter is that within certain
frameworks a violation of the UFF is connected with charge non–conservation [27].
Furthermore, there is also a connection between UFF and UGR [28, 15, 16, 29]. The
most favourable model for a violation of UGR is a time–dependent fine structure
constant and the most believed reason for that is a time–varying electron charge,
what has to be interpreted as charge non–conservation. Therefore, tests of the
UFF of charged matter may also be interpreted as test of UGR and of charge
conservation.
In this paper (a preliminary version of this work has appeared in [30]) we
review the only experiment, which has been devoted to test the UFF for charged
matter, describe its functionality and its limitations and discuss whether it makes
sense to do this kind of experiment in space. According to the exhaustive analysis
of the Witteborn and Fairbank as well as of the Los Alamos/CERN set up “most
spurious interactions can be rendered negligible. However, uncertainty remains
over electric fields produced by the patch effect and gravitationally induced strain
gradients in the drift tube (the DMRT effect)” [25]. The result of the present
paper is, that it is possible to considerably reduce the gravity–induced errors by
performing this experiment in a microgravity environment. Furthermore, due to
novel techniques in coating surfaces, also patch effects should play no role in this
kind of experiment. As a result, we claim that with present techniques it might
be possible to test the UFF for charged matter (except electrons, as we shall see)
in a microgravity environment to an accuracy of 10−5 (in terms of the Eötvös
coefficient).
2. AN EÖTVÖS COEFFICIENT FOR CHARGED PARTICLES
At the beginning we want to introduce a simple frame and define notions,
which quantify the degree of validity of UFF for charged particles. As a result the
Eötvös coefficient will be split into a charge–independent part characterizing the
UFF for neutral particles or due to the mass only, and a part, which scales with the
charge–to–mass ratio and thus characterizes the influence of charge on a hypothetical violation of UFF. This is motivated by the idea that any charge–induced
violation of the UFF should be larger the bigger the charge–to–mass ratio is.
Since a charged particle possesses an electromagnetic field, the total rest
energy of a charged particle consists of its bare rest mass and the energy of the
electromagnetic field. In general, this electromagnetic energy may contribute to a
(hypothetical) violation of the UFF in a different way than the bare mass. Since a
violation of the UFF is described with the help of inertial and gravitational masses,
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we describe a charge–dependent violation of the UFF by splitting the inertial mass
of a particle into a bare part and a part depending on the electric charge:
µ
¶
q
0
m i = m i 1 + κi 0 .
(1)
mi
A similar modification of the inertial mass of a charged particle in a gravitational
field has been found by Rohrlich [31] by taking into account the self–force of the
charge. We are considering anomalous pendants of such terms in the same way
as one is searching for e.g. anomalous spin–couplings – beside the ordinary spin–
curvature couplings, which are always present for particles with spin moving in
gravitational fields.
The same splitting will be introduced in the gravitational mass since in general it might be possible that the electromagnetic energy reacts to the gravitational
attraction in a different way than the ‘bare’ gravitational mass (what may be considered as an electromagnetic analogue of a violation of the strong equivalence
principle)
Ã
!
q
0
(2)
m g = m g 1 + κg 0 .
mg
In a special case, this point of view is supported by the T H ²µ–formalism. In
this formalism it has been shown [1, 32] that the acceleration of a bound system
of charged particles in general violates the UFF. The reason for that is that the
electromagnetic binding energy depends on the position and on the state of motion.
This violation can be related to an Eötvos–coefficient which is proportional to the
charge of the particles.
If we take these modified inertial and gravitational masses, then the equation
of motion for a charged particle in a gravitational field (all electromagnetic fields
are shielded) is
m i ẍ = −m g ∇U,
(3)
where U is the Newtonian gravitational potential and where we neglected the
very small non–local tail and curvature terms described in footnote 1. The Eötvös
coefficient describing the difference in the acceleration of two charged particles
then turns out to be
q2
q1
ẍ 2 − ẍ 1
≈ η0 + (κg2 − κi2 ) 0 − (κg1 − κi1 ) 0 ,
(4)
η=2
ẍ 1 + ẍ 2
m2
m1
where we inserted (3) and the charge–dependent ansatz for m i and m g , used the
usual Eötvös–coefficient
¡ 0 ± 0¢ ¡ 0 ± 0¢
m g2 m i2 − m g1 m i1
0
(5)
η = 2¡ 0 ± 0 ¢ ¡ 0 ± 0 ¢
m g2 m i2 + m g1 m i1
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for the ‘bare’ masses, made an expansion to first order in the κ’s, and neglected
combined violations due to ‘bare’ masses and to charges. Most important is, of
course, the case of only one charged particle. Then we end up with the modified
Eötvös coefficient
η = η0 + κ
q2
,
m 02
(6)
where κ = κg2 − κi2 . A charge–induced violation of the UFF is encoded in the
parameter κ. Only if the electromagnetic energy of charged particles contributes
in the same way to their inertial and the gravitational mass (which means κg =
κi ), then there will be no charge–induced violation of the UFF. To first order of
approximation, charge–induced violations of UFF are independent of any ‘bare’
violation of the UFF described by η0 .
Consequently, a comparison of a charged particle with a neutral or a second
charged particle gives an estimate on the total η. Only if we can change the charge
of the body and thus the charge–to–mass ratio then we can also make statements
about the two contributions η0 and κg − κi . For a single charged particle like the
electron it is possible that η0 and δκ mq compensate. If we use atoms and ionise
them to different degrees, then we can get information about both parts of η.
It is also obvious that for smaller charge–to–mass ratios the estimate for
κ will become worse. In the case of electrons, which has been used in the
Witteborn–Fairbank experiment, we have e/m = 1.7 · 1011 C/kg. Then |η0 + κ ·
1.7 · 1011 C/kg| ≤ 0.1. Assuming no cancellation of the two terms, we can conclude |η0 | ≤ 0.1 and |κ| ≤ 6 · 10−13 kg/C. If, instead, we take protons with a mass
approximately 2,000 times larger than the mass of electrons, then the estimate on
κ is 2,000 times worse.
One should also discuss the effect of these charge–dependent contributions
to the inertial and gravitational mass on bound neutral systems like the Hydrogen atom. We start with the Lagrange function of two electromagnetically bound
particles of total charge zero
L=
1
e2
1
m i1 v12 + m i2 v22 − m g1 U (x 2 ) − m g2 U (x 1 ) −
.
2
2
|x 2 − x 1 |
(7)
We define the relative coordinate x = x 2 − x 1 and the center–of–mass coordinate
X so that x 1 = X + αx and x 2 = X + (1 + α)x and require the vanishing of mixed
terms in the kinetic energy. This yields α = −m i2 /Mi where Mi = m i1 + m i2 is the
total inertial mass. If we expand the Newtonian potential U (X + αx) = U (X) +
αx · ∇U (X) then the Lagrangian is
L=
1
e2
1
2
Mi Ẋ + µred,i ẋ 2 + Mg U (X) + µred,i η21 x · ∇U (X) −
,
2
2
|x 2 − x 1 |
(8)
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where we also introduced the reduced inertial mass µred,i = m i1 m i2 /Mi and the
total gravitational mass Mg = m g1 + m g2 . η21 = m g2 /m i2 − m g1 /m i1 is the Eötvös
coefficient. The equations of motion for X and x read
Mi Ẍ = −Mg ∇U (X) + µred,i η21 x · ∇ ⊗ ∇U (X)
µred,i ẍ =
e2
x − µred,i η21 x · ∇U (X).
|x|3
(9)
(10)
Since the electromagnetic bounding forces are much stronger than the gravitational
force, the coordinate X in (10) is adiabatic and can be treated as constant. Therefore, the two equations decouple and (10) can be solved for x. As a by–product we
observe that for a non–vanishing Eötvös coefficient the internal dynamics depends
on the gravitational potential. Since the internal dynamics defines the time scale
of e.g. an atomic clock, this shows that a violation of UFF is related to a violation of UGR. However, due to the weakness of the gravitational influence, clock
experiments are no substitute for UFF tests [29].
We insert the solution x into (9) and average over the timescale T of the
internal motion which is much shorter than the timescale of the center–of–mass
motion X. However, for a bound system the time average hxiT vanishes. Only the
second order approximation of the Newtonian potential in x, that is the gravity
gradient, will contribute, what, however, can safely be neglected here. Therefore,
the center–of–mass motion is given by the usual equation
Mi Ẍ = −Mg ∇U (X).
(11)
Using the expansion of the inertial and gravitational masses according to (1) and
(2) this finally gives
!
Ã
Mg0
e
+ ζ 0 ∇U (x com ),
(12)
Ẍ = −
Mi0
Mi
where ζ = κ1,g − κ2,g + κ2,i − κ1,i . For a second neutral bound system, e.g. a second kind of atom, we have another total mass what changes the second term.
Therefore, in order to be compatible with current free fall experiments with neutral
matter which give the estimate η ≤ 10−12 [2], we have to require ζ e/m p ≤ 10−12 ,
where m p is the mass of the proton. That means ζ ≤ 10−20 kg/C. This is fulfilled
if, e.g., the coefficients κg and κi , respectively, are the same for protons, electrons
and all other charged particles, κe,g = κ p,g , etc. (For a more elaborate model of the
atom including the dynamics of the nucleus, more complicated conditions might
be discussed). This means that the coefficients κ are really attributed to a charge
and not to a particle. However, for a comparison of a single charged particle like a
proton with a neutral atom, the coefficients κg and κi of the proton will still have
influence on the comparison of the free fall of the proton and the atom. This proves
that even in the case that neutral electromagnetically bound systems respect the
UFF, there is still room for a violation of UFF for charged particles.
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3. THE WITTEBORN–FAIRBANK SETUP
The so far only experiment devoted to test the UFF for charged matter has
been carried through in 1967 by Witteborn and Fairbank [23] who measured the
net force on electrons freely falling in a copper tube. The experimental set–up
consisted of a vacuum tank cooled down to liquid helium temperature of 4.2 K as
well as a vertical copper tube (drift tube with length of about 1 m and diameter of
5 ± 0.0003 cm) inside the Dewar vessel to shield stray electrical fields. Electrons
moved along the drift tube’s symmetry axis and had been forced to do so by a
magnetic field of a coaxial solenoid. A cathode at the bottom emitted the electrons
and accelerated them upwards. Electrons having passed the drift tube have been
detected by an electron multiplier detector (Fig. 1). The electrons are emitted as
short bursts, so that a mean time–of–flight between the emission from the cathode
and the detection can be measured.
First we analyse this experiment in the case that only gravity and no
electric field is acting. Our test particle is an arbitrary particle with inertial and gravitational masses m i and m g and charge q. Energy conservation
E = 12 m i v 2 − m g gx = const. gives the equation of motion m g ẍ = m g g which
m
has the general solution x = 12 mgi gt 2 + v0 t + x0 where v0 and x0 are the initial
velocity and position, respectively. There is a maximum time of flight tmax which
is given by the condition that the electrons arrive the detector with zero velocity.
This is characterized by E = 12 m i v02 and x − x0 = h for t = tmax where h is the
height of the drift tube. Solving these equations for tmax yields
s
tmax =
m i 2h
.
mg g
(13)
Figure 1. Schematic view of the Witteborn–Fairbank
set–up. Electrons move against gravity inside a metallic
shielding. An additional tuneable electric field in a drift
tube allows determining the maximum time of flight.
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m
For known h and measured tmax we can determine mgi g. Here g is the gravitational
acceleration of the test particle as seen in the frame of the metallic shield (what
in the present case is the same as the rest frame of the Earth). A comparison of
the same quantity with another kind of particle or with neutral matter gives the
corresponding Eötvös–coefficients.
However, the purely gravitational case is not realized in the Witteborn–
Fairbank set–up. There are three additional electric fields, which are connected
with the existing gravitational environment and with the measurement process:
r The Schiff–Barnhill field which is unavoidable on Earth. Schiff and
Barnhill [24] calculated (see below) that electrons bound in the metallic shield of the drift tube create an electric field E SB = m eg g/e ∼
5 · 10−11 V/m which, for electrons, balances the gravitational field exactly. Here g is the gravitational acceleration acting on the electrons in the
metallic shield.
r The so–called DMRT field, introduced by and named after Dessler, Michel,
Rorschach and Trammel [33], which comes from the differential compression of the metallic shielding. If the surfaces of this shielding are aligned
to the gravitational acceleration, alsoatomthis field is proportional to the gravm
itational acceleration: E DMRT = γ ge g, where m atom
is the gravitational
g
atomic mass of the material the drift tube is made from. There is still some
debate about the value of γ for various substances. Witteborn and Fairbank
who carried through their experiment at 4.2 K, experienced an only later
recognized vanishing of this extra field. Later experiments showed that
the DMRT field at least in that particular set–up vanished for temperatures
below 4.5 K [34].
r Another external uniform electric field E a < 2.5 · 10−10 V/m directed parallel to the drift tube’s symmetry axis is applied in order to decelerate the
electrons. With this additional field, the electrons experience an additional
acceleration (or deceleration) which leads to a variation of the time of
flight of the electrons. This is necessary in order to be able to determine
the maximum time of flight tmax .
With these additional fields Eq.(13) must be modified: The observed flight
times of charged particles of mass m and charge q now is
s
tmax =
mg
g
mi
−
2h
.
+ E DMRT − E a )
q
(E SB
mi
By varying E a and measuring tmax one can determine the difference
q
(E SB + E DMRT ) for electrons.
mi
(14)
mg
g
mi
−
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If we insert the general valueatom
of the Schiff–Barnhill field E SB = m eg g/e and
m
of the DMRT–field, E DMRT = γ ge g, then we get for the maximum time of flight
v
u
2h
u
tmax = t m − q (m + γ m atom )
.
(15)
g e
eg
q
g
g
+
E
a
mi
mi
By varying the external electric field E a , the quantity
¡
¢
m g − qe m eg + γ m atom
g
g
mi
(16)
can be measured which contains information about a modified ratio of the gravitational to inertial mass, only.
For elementary particles with the charge of the electron, q = ±e we get
v
u
2h
u
.
(17)
tmax = t m ∓ (m + γ m atom )
g
eg
q
g
g
+
E
a
mi
mi
If the charge of the particle has the same (opposite) sign than the electron charge,
like protons, positron, or antiprotons, only the difference (sum) of the gravitational
mass of the particle and the electron can be tested. For positrons the term in question
m eg + m ēg + γ m atom
g
is 2
g.
m ēi
For electrons we have q = e and m = m e so that the first part, and thus the
gravitational mass of the electron, completely disappears. Therefore, for electrons
it is not possible to make statements about a relation between the inertial and
gravitational mass. This is only possible for particles different from electrons.
However, the statement is valid only if the Schiff–Barnhill field indeed is of
the form E SB = m eg g/e, that is, if the gravitational acceleration of the electrons
inside the metallic tube gbulk is the same as for free electrons gfree . If we distinguish
between these two accelerations by setting gbulk 6= gfree , then we get from (14)
s
2h
tmax = m eg
.
(18)
(gfree − gbulk ) + meei (E DMRT + E a )
m ei
In the Witteborn–Fairbank experiment there happened to be a fortuitous and temperature dependent absence of the DMRT–effect, resulting in γ ≈ 0. By varying
E a and measuring tmax the Witteborn–Fairbank experiment aimed to measure the
m
possible difference megei (gfree − gbulk ) for electrons. This value, the subtraction of
two large numbers, is extremely small and has been observed to be:
m eg
(gfree − gbulk ) ≤ (0.13 ± 0.47) · 10−11 eV/m,
m ei
(19)
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setting an upper limit for the Eötvös coefficient for charged matter in the sense of
η = 2(gfree − gbulk )/(gfree + gbulk ) to
η < 0.1 ,
(20)
where one has to insert the calculated value for the Schiff–Barnhill field.
Our analysis shows that Witteborn and Fairbank did not really measure the
UFF for charged particles but instead the equality of accelerations of bulk and free
electrons so that the significance of the result is rather limited. Tests of the UFF
with this device can only be performed for particles different from electrons.
4. SIDE EFFECTS IN UFF EXPERIMENTS
FOR CHARGED PARTICLES
In the following we distinguish between (1) gravity–induced side effects
which scale with the acceleration of the apparatus and (2) other errors.
4.1. Gravity Induced Side–Effects
4.1.1. The Schiff–Barnhill Effect
The Schiff–Barnhill effect [24] was the first unwanted side–effect calculated
in order to analyse the Witteborn–Fairbank set–up. It can be understood very easily
in terms of solving the Poisson equation with metallic boundary condition in an
external homogeneous gravitational field.
A metal is characterized by the mobility of electrons. If there is an external
gravitational field, then the electrons move downwards. However, the excess of
electrons at the down–side of the metal creates a repulsive force with the result
that both forces in equilibrium should cancel,
0 = −eE + m eg g .
(21)
This relation holds inside the whole metal. Consequently, since E = −∇φ, the
m
electrostatic potential φ inside the metal is given by φ = − eeg g · x. This is also
the potential at the (inner and outer) boundary of the metal.
The electric potential inside a hole in the metal, see Fig. 2, is given by the
m
Dirichlet problem 1φ = 0 with φ|boundary = − eeg g · x. The unique solution to this
m
problem φ = − eeg g · x yields the electric field
E SB = −∇φ =
m eg
g
e
(22)
whith E SB ∼ 6 · 10−11 V/m on Earth. The induced electric field outside the shield
can be calculated for a spherical shape of radius R, as the dipole moment d =
m eg
g R3.
e
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Figure 2. The Schiff–Barnhill effect. An external gravitational field acting on the electrons in the metal changes
the boundary condition thus inducing an electric field inside the metal.
Consequently, a charged particle with charge q and mass m g experiences the
total force
µ
¶
¶
µ
m eg
mg
mg
q
F = F grav + F SB = q
+
+
g = m eg
g.
(23)
q
e
m eg
e
For an electron, q = −e, the resulting force vanishes – the electron stays within
the metallic hole without falling down, even if the UFF is not valid, that is, even
for m ei 6= m eg .
4.1.2. The DMRT–Field
The DMRT–field [33] is a dominating error source. It results from the differential compression of the crystal layers of the metal of the drift tube in the
gravitational field. In the lower part of the metal the crystal layers have smaller
distances than in the upper parts which leads to more positive charges per unit
volume in the lower than in the upper part.
The value of an electric field induced by lattice deformation can be calculated
from E = 1e ∇W where W is the work function of the metal. One gets as result
[25]
E≈
2n 0 m atom
g
²F
g,
9K
e
(24)
is the (gravitational) atomic mass of the metallic solid, n 0 the density
where m atom
g
of the atoms in the shield without gravitational deformation, K the bulk modulus
of compressibility, and ² F the Fermi energy of the free electron gas in the metal.
Eq. (24) is the so–called DMRT–field [33].
As expected, the DMRT–field scales with the gravitational acceleration g.
Therefore, in principle, one can take this as additional electric field in the equation
0
² is still in debate. It is
of motion. However, the value for the quantity γ = 2n
9K F
expected that under normal conditions this field for copper is up to 2 · 104 larger
than the Schiff–Barnhill field. Some other estimates give values which are one
order smaller, and other even give an opposite sign.
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One way to solve this problem is to measure the DMRT–field for varying
accelerations, e.g. on a turntable with varying angular velocities or in free fall (in a
drop tower), and use this in order to determine the coefficient γ . In the experiment
carried through by Lockhart, Witteborn and Fairbank [34] they observed a vanishing DMRT field below 4.5 K. There seems to exist up to now no explanation for
this behaviour of the DMRT–field at low temperatures.
4.2. Other Side Effects and Errors
Beside the disturbances by gravity induced electric fields there are many other
influences which have to be treated in detail. Some of the possible error sources
have been already discussed by the experimenters themselves [23], but the most
comprehensive analyses can be found in [25]. Here we mention most of the error
sources and show how they can be kept under control.
4.2.1. Patch Effects
Serious disturbances are caused by inhomogeneities of the surface of the
shielding drift tube. These inhomogeneities consist in small crystal facets of
slightly different electric properties, which cause a small variation in the electric potential between neighbouring facets. Furthermore, surface contaminations
like oxidation layers also will modify the electrostatic potential. These small variations in the electric potential on the metallic surface extends to small variations
along the tube’s axis which are not known explicitly and thus induce errors in the
theoretical interpretation of the time–of–flight of the charged particle along the
axis.
For a theoretical description we start with the condition of total energy conservation E = 12 mv 2 + φ, where φ is the sum of all potentials, gravitational and
electrostatic. The potential φ is now split into a mean part φm and a random part φr
Rh
(compare [25]) where the random part is defined by the property h1 0 φr dz → 0
for h being large enough. Then we get in the case of large velocities, that is φ ¿ E,
the variation of the time–of–flight
r
3 m h 2­ 2®
1t ≈
e φr
(25)
8 2E E 2
where
p
hφ 2 i :=
1
h
Z
0
h
φr2 dz
(26)
is the mean square root of the random potential. (In [25] the quantity E has
been replaced by mgh which is, strictly speaking, not admissible in view of
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the approximation made.) In our case we have charged particles which move
through the cylindrical metallic shield with a constant velocity so that 12 mv 2 À eφr .
Therefore, E ≈ E kin = 12 mv 2 and, with the undisturbed time–of–flight given by
t0 = h/v,
­ ®
3 e2 φr2
1t
≈
.
(27)
2
t0
8 E kin
The relative variation of the time–of–flight is thus characterized by the strength
of the msr (mean square root) of the random potential compared with the kinetic energy. This is a reasonable result: if the velocity is very small, then
the small variations of the random potential may be strong enough to stop the
particle.
The random potential has been calculated in [25]
µ
¶2
­ 2®
lpatch
(28)
V0 ,
φr = C
R
where lpatch is the characteristic length of the patch patterns, R is the radius of the
tube, V0 the voltage between two facets, and C a constant of the order 0.5. It has been
assumed lpatch ¿ R. Typically, V0 has been found to be of the order 0.1 V which
is lowered by one order of magnitude due to surface contaminations and oxide
layers. For the tube used by Witteborn and Fairbank with tube radius of
p 2.5 cm,
lpatch has been observed to be around 1 µm, resulting in hφr2 i ≈ 0.3 µV hφr2 i. ≈
These patch effects seem to be much too big for the result claimed by Witteborn
and Fairbank4 .
Due to novel techniques for machining very clean surfaces and for vapour
deposition, today patch effects may be much smaller than those reported for the
Witteborn–Fairbank apparatus. For example, CVD (Chemical Vapour Deposition)
is a method for coating surfaces with metals so that no facet structure can be
identified even with electron microscopy [35]. That means, the facets are smaller
than 10 nm. Consequently, for these coatings the msr of the potential is smaller than
30 pV. For protons with an energy of 10 K = 10−3 eV (v ≈ 400 m/s) this means
that 1t/t0 ≤ 3 · 10−16 which in this experiment clearly is beyond any observational
importance.
To sum up, the patch effects which have been an important error source in
previous experiments are under control with modern techniques.
4 However,
Darling et al. [25] have given two explanations why these disturbances might not have
played a dominant role in the experiment: First, exposure to air would have produced an amorphous
surface layer masking underlying crystal structures in the copper surface, and second, the shielding
appearing at cryogenic temperatures observed by Lockart et al. [34] could also have caused a reduction
of the patch effects.
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4.2.2. Electric Field Gradients
Electric field gradients are only affecting polarisable systems like ions and are
therefore negligible for electrons. The error calculated for ions is only Fgrad /Fgrav ≤
3 · 10−9 [25].
4.2.3. Thermoelectric Fields
Additional disturbing electric fields can also be produced by temperature
gradients in the shielding drift tube. The thermoelectric field is given by [25]
µ
¶
1 ∂W
+ S ∇T,
(29)
ET =
e ∂T
where W is the work function of the metal. S is the thermoelectric coefficient
which for copper is at liquid helium temperature not greater than 0.05 µV/K. The
experimental requirements for ∇T ¿ 10−4 K/m for electrons and 10−1 K/m for
protons should be satisfied within the Dewar vessel despite the thermal isolation of
the vacuum inside the drift tube. It is interesting that residual helium gas producing a
monolayer on the copper drift tube could produce a strong temperature dependence
near helium boiling temperature and could explain the unexpected temperature
dependent shielding observed by Lockart et al. [34] due to helium desorption
above helium boiling temperature [25].
4.2.4. Magnetic Fields and Field Gradients
In the Witteborn–Fairbank experiment guide magnetic fields were needed in
order to keep the electrons on the way to the detector. It has been shown that the
guide solenoid can be aligned precisely enough with the symmetry axis of the
drift tube. A measure of this estimate might be the ratio between the maximum
horizontal displacement of the freely falling particle from the axis and the length of
the drift tube. This ratio has to be at least as small as 1 part in 10,000 for electrons
and requires a homogeneity ratio of the horizontal disturbing magnetic field to the
aligned vertical field of the solenoid at least one order of magnitude smaller. The
analysis showed that this requirement has been fulfilled for the Witteborn-Fairbank
experiment.
The spin of electrons or protons couples to magnetic fields and causes deviations of the vertical path by gyromagnetic forces. The deviation from the vertical axis Fz /Fgrav , is to first order proportional to the magnetic field gradient
0z = ∂ Bz /∂z (Fgrav = m g g is the gravitational force.). The ratio can be written:
µ
Fz
=
0z ,
Fgrav
Bz
(30)
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where µ is the effective magnetic moment of the test particle depending on the
particle mass, its charge, and its quantum state. In the Witteborn–Fairbank experiment µ is a very small value; because of experimental reasons [23] all freely falling
electrons are in the ground state, and the influence by magnetic field gradients can
be neglected [25]. By carrying out the same experiment with protons or antiprotons
µ has to be averaged over all states for a given temperature. Therefore, Eq. (30)
must be modified [25]:
kT 1
Fz
=
0z ,
Fgrav
m g g Bz
(31)
which gives reason to cool down the protons to at least 10 K.
4.2.5. Radiation Pressure
The influence of radiation pressure on the free fall behaviour of the test particle
can be estimated by a photon scattering analysis. It can be shown that even for
room temperature and for electrons (with a much larger cross section than protons),
radiation pressure is a negligible effect.
4.2.6. Gas Scattering in the Tube
Residual gas in the drift tube could cause a serious systematic error due to
scattering on helium atoms. Darling et al. [25] calculated the influence for varying
residual gas pressure in the tube. The sufficient vacuum conditions inside the tube
is determined by the time τ between two scatterings which should be longer than
the maximum flight time tmax of Eq. (17):
τ=
kT
pσ (vtest particle )
vhelium < tmax ,
(32)
where p is the effective pressure at temperature T , k is the Boltzmann constant, and
σ is the cross section of the test particle. This value is small enough at a pressure
of about 10−10 Pa (attained in the Witteborn–Fairbank experiment) for electrons
with drift velocities vtest particle of the order of m/s compared to the drift velocities
of helium atoms vhelium = 140 m/s. To carry out the experiment with protons or
antiprotons an ultrahigh vacuum of at least 10−12 Pa has to be attained.
5. EXPERIMENTS IN SPACE
5.1. The Idealized Case
As we have seen above, the main error in the Witteborn–Fairbank experiment is the DMRT–effect that cannot be predicted accurately enough by theory.
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However, since this effect, and also the Schiff–Barnhill effect, scales with the external gravitational field, the idea is to perform such experiments in space under
microgravity conditions. This is what we analyse in the following.
The various accelerations appearing in tmax , Eq. (14), and in the Schiff–
Barnhill– and DMRT–fields have to be carefully distinguished. The point is that
three kinds of particles are involved, the charged test particle, the ions of the
metallic shield and the electrons inside this shield, and for each pair of these
particles we have to take a hypothetical violation of the UFF into account.
m atom
m
m
To be more precise: The accelerations mgi g, megei g, and m gatom g are the acceli
erations of the charged test particle, of the electrons, and of the atomic ions of
the shield, respectively, as described in the rest frame of the gravitating body (g
is the gradient of the Newtonian potential U = G M/r ). In order to calculate the
Schiff–Barnhill effect in space, which may result from a violation of the UFF
between electrons and atoms of the metallic shield, we consider the forces and
the equations of motion of the electrons and the atoms of the metallic shield. The
forces on the atoms and electrons in the reference system of the Earth are given by
ẍatom = m atom
g
Fatom = m atom
i
g
(33)
Fe = m ei ẍe = m eg g .
(34)
The question now is: what is the force m ei ẍ 0 of the electron in the rest system of the
atoms? We get this force if we perform a coordinate transformation to the frame
which accelerates with ẍ atom with respect to the Earth’s system. That is, we look
0
for new coordinates x 0 = f (x) so that ẍatom
= 0.
Such a transformation is given by x 0 = x − 12 ẍatom t 2 (the acceleration, of
course, is constant). Then we get for the force on the electron
m ei ẍe0
d2
= m ei 2
dt
Ã
µ
1
xe − ẍ atom t 2
2
!
m atom
g
= m eg − m ei atom g .
mi
¶
(35)
This force on the electron with respect to the electric shield is induced by a violation
0
in the rest frame
of UFF and has to be balanced by an electric field Fe0 = −eE SB
of the atoms. This electric field is given by
Ã
0
eE SB
= − m eg − m ei
m atom
g
m atom
i
!
g
(36)
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This Schiff–Barnhill field has to be used in Eq. (14) (in the rest frame of the
atoms):
s
2h
tmax =
,
(37)
0
ẍ 0 − mq i (E SB
− Ea )
where m i and m g are inertial and gravitational mass of the test particle, m g ẍ 0 is the
force on the test body and ẍ 0 its acceleration in the rest frame of the atom, that is,
with respect to the atoms of the electric shield. We insert the Schiff–Barnhill field
and get
v
u
2h
u
³
´
tmax = t
.
(38)
m atom
m
ẍ 0 + qe mmeii megei − m gatom g + mq i E a
i
Here ẍ 0 is the relative acceleration between the test particle and the shield. The
corresponding Eötvös coefficient then is ηtest−shield = ẍ 0 /g. Since
Ã
!
m atom
q m ei m eg
g
0
− atom g
(39)
aeff = ẍ +
e m i m ei
mi
is the acceleration which is determined by the time–of–flight measurement, the
Eötvös coefficient is then given by
Ã
!
m atom
q m ei m eg
q m ei
aeff
aeff
g
−
−
− atom =
ηe−shield .
(40)
ηtest−shield =
g
e m i m ei
mi
g
e mi
At first, this means that if there is an aeff 6= 0, then there must be a violation of the
UFF either for the electron–shield or for the test mass–shield comparison.
We may isolate the charge–induced UFF violating term using (4) and the
result, that the κg and κi , respectively, are all identical for the participating charged
particles, that is κe,g = κtest,g = κshield,g , and similar for the κi . This gives
0
κg − κ i
(µshield µe − 2µtest µe + µtest µshield )
µe
aeff
µtest 0
=
−
η
,
g
µe e−shield
ηtest−shield +
(41)
where the µ are the corresponding charge–to–mass ratios. The difference
1κ = κg − κi indicates a charge–induced violation of the UFF. If the merely
mass–induced violations of the UFF, 0 ηtest−shield and 0 ηe−shield , are of comparable order, then qe mmeii ηe−shield ¿ 0 ηtest−shield provided q is of the order of e.
If the test particles are simply and twice ionised atoms of the same kind,
then we have two equations (41) with µtest2 = 2µtest1 , so that we can eliminate the
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unknown 0 ηtest−shield and get
1κ
1aeff
µtest1
(2µe − µshield ) =
.
µe
g
(42)
Since µshield ¿ µe this simplifies
1κ =
1 1aeff
.
2µtest1 g
(43)
Therefore, a charge–induced violation of the UFF is directly related to the comparison of the effective accelerations of simply and twice ionised atoms.
5.2. Errors
We need the acceleration aeff in order to determine the Eötvös coefficient.
Any estimate on the Eötvös coefficient is thus induced by errors of aeff . From (38)
and (39) we get aeff = 2h/t 2 − (q/m i )E and thus the total error
s
¶ µ ³ ´ ¶2 µ
¶
µ
µ
¶ µ ¶2 µ ¶2
δaeff
δx0
δv0
h δt 2
q E
qδ E 2
δh 2
=
+
+
+4
+ δ
+
,
g
gt 2
gt 2
gt
gt 2 t
m g
mg
(44)
where δh, δx, δv0 , δtmax , δ(q/m), and δ E are the uncertainties in the determination
of the path length, of the initial position, of the initial velocity, the time of flight,
the specific charge and the electric field, respectively. Errors due to non–ideal
microgravity conditions will be discussed below.
From (44) it is clear that a long time of flight, that is a small velocity, will
improve the accuracy. If the time of flight should be about 0.1 s, then the velocity
has to be of the order 10 m/s, that is, the energy of the charged particle has to
be, according to its mass, in the mK or even µK range (for one atomic mass unit
this is 50 µK). Therefore cooled particles are needed. The best for this is to use
laser cooled ions, or, in the case that these ions cannot be cooled directly due to
the lack of an appropriate level structure, sympathetically cooled ions. For our
purposes a large charge–to–mass ratio is preferable, implying that the ions should
have small masses. It has been shown recently by numerical simulations [36], that
by means of directly laser coolable 9 Be it is possible to sympathetically cool down
ions with masses as small as 2 amu to a temperature in the µK range. Accordingly,
we take ions with a mean velocity of 10 m/s and with a velocity spread of 1 cm/s.
Furthermore, the initial position of the ions can be confined to a region of 1 mm
width. Since the uncertainties in the initial position and the initial velocity as they
appear in (44) are inverse with the square root of the number of runs it can be made
small. We assume a 106 runs, so that δx0 = 10−6 m and δv0 = 10−5 m/s.
Furthermore, we take for our error analysis the length of the drift tube to be
1 m. The relative uncertainty of the specific charge is less than 10−7 and the electric
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field, which is of the order E ∼ 10−4 V/m can be determined5 with an accuracy
up to 10−10 . The uncertainty in the determination of the length of the drift tube
is 10−6 m, and the time of flight can measured very precisely, say δt = 10−10 s.
Therefore, the first, second, third and the fifth term are limit the accuracy with 10−5 .
Consequently, with the above specifications the error in the determination of
the effective acceleration aeff turns out to be δaeff /g ∼ 10−5 . In the case that no
effect will be observed, this implies with (43) that |1κ| ≤ 5 · 10−14 kg/C which
gives for the corresponding contribution to the Eötvos̈ coefficient η = 1κµtest ≤
10−5 .
5.3. Errors from Microgravity Conditions
Since the effective acceleration of a charged particle moving in vacuum is
the main experimental quantity to be measured, any residual acceleration of the
platform the apparatus is attached to will influence this measured quantity directly.
It is clear that the residual acceleration has to be smaller than 10−4 m/s2 in order
not to spoil the otherwise obtainable accuracy discussed above.
However, as can be seen from Fig. 2 in [38] about the conditions on the ISS,
the residual acceleration at the frequency 10 Hz, which is the timescale of our
experiment, is too large by at least one order of magnitude. That means, we need
either an isolation rack or a free flyer platform with drag–free conditions for this
experiment to be performed onboard of the ISS.
6. SUMMARY AND OUTLOOK
In this article we have indicated by rough estimates that a test of the equivalence principle for charged matter might be feasible on the ISS with an accuracy
of 10−5 in terms of the Eötvös coefficient. However, an additional experimental
facility like a free flyer accompanying the ISS is needed. We also have shown that
there might be an anomalous interaction of charge with the gravitational field even
if neutral bound systems like the Hydrogen atom, exactly fulfil the Universality of
Free Fall. Therefore it might be a valuable goal to test the UFF for isolated charged
particles.
Since in all conceivable set–ups for tests of the UFF for charged matter,
e.g. [39], one needs an electromagnetic shield, a roughly similar analysis as above
5 A very precise way to measure electric fields is based on the measurement of a length and the voltage,
where the latter can be based on the Josephson effect, see e.g. [37], which enables the measurement
of voltages with a relative accuracy of 10−10 . This is a quantum interference effect of electrons in
superconductors. Furthermore, the voltage has to be applied to the drift tube what again is realized
by electrons in a conducting material. If the electric field defined in this way is applied to electrons,
then again a circularity occurs so that probably no effect will be seen. However, in the case that the
test particle is a proton or an atomic ion, then no such circularity is present.
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applies to all these tests. Such experiments are ion interferometry or charged
particles in a trap. In each case, the induced errors which are mainly due to the
physics of the electromagnetic shielding and to the residual acceleration add up to
the same error for each of these experiments in the measured Eötvös coefficient.
ACKNOWLEDGMENTS
We acknowledge support of the German Space Agency DLR.
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