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P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 12:34 Style file version May 27, 2002 C 2004) 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 ° P1: IKH General Relativity and Gravitation (GERG) 572 PP1066-gerg-477709 December 22, 2003 12:34 Style file version May 27, 2002 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. P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 12:34 Testing the Universality of Free Fall for Charged Particles in Space Style file version May 27, 2002 573 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, P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 574 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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) à ! 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 P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 12:34 Testing the Universality of Free Fall for Charged Particles in Space Style file version May 27, 2002 575 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) P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 576 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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 ! à 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. P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 Testing the Universality of Free Fall for Charged Particles in Space 12:34 Style file version May 27, 2002 577 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. P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 578 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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 − P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 Testing the Universality of Free Fall for Charged Particles in Space 12:34 Style file version May 27, 2002 579 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) P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 580 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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 P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 Testing the Universality of Free Fall for Charged Particles in Space 12:34 Style file version May 27, 2002 581 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. P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 582 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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 P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 Testing the Universality of Free Fall for Charged Particles in Space 12:34 Style file version May 27, 2002 583 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. P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 584 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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) P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 Testing the Universality of Free Fall for Charged Particles in Space 12:34 Style file version May 27, 2002 585 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. P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 586 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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 à 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 à 0 eE SB = − m eg − m ei m atom g m atom i ! g (36) P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 12:34 Testing the Universality of Free Fall for Charged Particles in Space Style file version May 27, 2002 587 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 à ! 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 à ! 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 P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 588 12:34 Style file version May 27, 2002 Dittus, Lämmerzahl, and Selig 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 P1: IKH General Relativity and Gravitation (GERG) PP1066-gerg-477709 December 22, 2003 Testing the Universality of Free Fall for Charged Particles in Space 12:34 Style file version May 27, 2002 589 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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