Non-inertial effects in fusion reactions of astrophysical interest
aa r X i v : . [ nu c l - t h ] F e b Non-inertial effects in reactions of astrophysical interest
C. A. Bertulani, J.T. Huang, and P. G. Krastev ∗ Department of Physics, Texas A&M University, Commerce, TX 75429, USA † (Dated: January 14, 2019)We discuss the effects of non-inertial motion in reactions occurring in laboratory, stars, and else-where. It is demonstrated that non-inertial effects due to large accelerations during nuclear collisionsmight have appreciable effects nuclear and atomic transitions. We also explore the magnitude of thecorrections induced by strong gravitational fields on nuclear reactions in massive, compact stars,and the neighborhood of black holes. PACS numbers: 26.20.+f, 26.50.+x, 24.10.-i
I. INTRODUCTION
Extremely large accelerations occur when atomic nu-clei collide. For instance, two lead nuclei in a head on col-lision with a center of mass kinetic energy of 500 MeV,reach a closest distance of 19.4 fm before they bounceback and move outward. At this distance each nucleusaccelerates with an intriguing ∼ m/s . Very fewother physical situations in the Universe involve nucleiundergoing such large accelerations, usually related toastrophysical objects, as in the vicinity of neutron starsand black holes, where huge gravitational fields exist. Inthis article we explore the effects of large accelerationsand large gravitational fields, and their possible influenceon nuclear reactions in the laboratory and in astrophys-ical environments. Nuclear reactions are crucial for theformation of stellar structures and their rates could beaffected by various factors. To our knowledge, the effectof large gravitational fields on nuclear reaction rates instars has not been considered so far.As mentioned in the previous paragraph, atomic andnuclear systems undergo large accelerations during re-actions. The effect of acceleration is observed in termsof excitations followed by decay of these systems. If weconsider two-body reactions, there are two systems ofreference which are often used to describe the effects ofthe collision: (a) the center-of-mass (cm) system of thetwo nuclei and (b) the system of reference of the excitednucleus. System (b) is appropriate to use when the in-trinsic properties of the excited nucleus is described insome nuclear model. A typical example is the case ofCoulomb excitation. One assumes that the nuclei scatterand their cm wave functions are described by Coulombwaves due to the Coulomb repulsion between the nuclei.Then one considers the residual effect of the Coulombpotential on the motion of the nucleons inside the nuclei.This is done by expanding the Coulomb potential in mul-tipoles and using the high order terms (higher than firstorder) as a source of the excitation process. In this ap-proach one illustrates the privileged role of the cm of thenuclear system: the net effect of the external forces is to ∗ Current address: San Diego State University, Department ofPhysics, 5500 Campanile Dr., San Diego, CA 92182-1233 † e-mail: carlos˙[email protected], [email protected], [email protected] (i) accelerate all the particles together, along with thecm of the system, and (ii) to change the intrinsic quan-tum state of the system through the spatial variation ofthe interaction within the system. Thus the theoreticaltreatment of accelerated many-body systems is well un-der control in non-relativistic dynamics.In the non-relativistic case, the separation of variablesinto intrinsic motion and relative motion between the cmof each nucleus is a simple algebraic procedure. A prob-lem arises when one wants to extend the method to de-scribe intrinsic excitations of relativistic many-body sys-tems. Very few works exist in the literature addressingthis problem. The reason is that for nuclear reactions inthe laboratory, the effect is expected to be very small,a common belief which must be tested. Another otherreason is that in stellar environments where the gravi-tational fields are large, huge pressures develop, ”crush-ing” atoms, stripping them from their electrons, and ul-timately making nuclei dissolve into their constituents.Effects of nuclear excitation are not relevant in the pro-cess. But, on the other hand, nuclear reactions are cru-cial for the formation of stellar structures and their ratescould be affected by minor effects such as those exploredin this article.Nuclei participating in nuclear reactions in a gaseousphase of a star follow inertial trajectories between colli-sions with other nuclei. Such trajectories are “free fall”trajectories in which all particles within the nucleus havethe same acceleration. That is surely true in the non-relativistic case, but not in the relativistic one becauseretardation effects lead to corrections due to the nuclearsizes. The central problem here is the question regard-ing the definition of the center of mass of a relativisticmany body system. We have explored the literature ofthis subject and found few cases in which this problem isdiscussed. Based on their analysis we show that relativis-tic effects introduce small corrections in the Lagrangianof a many-body system involving the magnitude of theiracceleration. We follow Refs. [1, 2, 3], with few modifi-cations, to show that a correction term proportional tothe square of the acceleration appears in the frame of ref-erence of the accelerated system. To test the relevanceof these corrections, we make a series of applications tonuclear and atomic systems under large accelerations. II. HAMILTONIAN OF AN ACCELERATEDMANY-BODY SYSTEM
Starting with a Lagrangian of a free particle in an in-ertial frame and introducing a coordinate transformationinto an accelerated frame with acceleration A , a “ficti-tious force” term appears in the Lagrangian when writtenin coordinates fixed to the accelerated frame. Thus, inan accelerated system the Lagrangian L for a free parti-cle can be augmented by a (non-relativistic) interactionterm of the form − m A z , that is L = − mc + 12 mv − m A z, (1)where z is the particle’s coordinate along the direction ofacceleration of the reference frame [1].In the relativistic case, the first step to obtain theLagrangian of a many body system in an acceleratedframe is to setup an appropriate measure of space-timein the accelerated frame, i.e. one needs to find outthe proper space-time metric. The free-particle action S = − mc R ds requires that ds = ( c − v / c + A z ) dt ,which can be used to obtain ds . To lowest order in 1 /c one gets ds = c (cid:18) A zc (cid:19) dt − dx − dy − dz = g µν dξ ν dξ µ , (2)where v dt = d r was used, with dξ µ = ( cdt, dx, dy, dz )and g µν = ( g , − , − , − g = (cid:0) A z/c (cid:1) . Theindices µ run from 0 to 3. Eq. (2) gives a general formfor the metric in an accelerated system. This approachcan be found in standard textbooks (see, e.g. ref. [1], § H = p · v − L ,with p = ∂L/∂ v = m v / p g − v /c , and using theaction with the metric of Eq. (2), after a straightforwardalgebra one finds H = g mc q g − v c = c p g ( p + m c ) . (3)Expanding H in powers of 1 /c , one obtains H = p m (cid:18) − p m c (cid:19) + m A z (cid:18) p m c (cid:19) + O (cid:18) c (cid:19) . (4)This Hamiltonian can be applied to describe a systemof particles with respect to a system of reference movingwith acceleration A , up to order 1 /c . For an acceler-ated nucleus the obvious choice is the cm system of thenucleus. But then the term carrying the acceleration cor-rection averages out to zero in the center of mass, as onehas ( P i m i A z i = 0). There is an additional small contri-bution of the acceleration due to the term proportional to p . Instead of exploring the physics of this term, one hasto account for one more correction as explained below.The above derivation of the Hamiltonian for particlesin accelerated frames does not take into account that thedefinition of the cm of a collection of particles is also modified by relativity. This is not a simple task as mightseem at first look. There is no consensus in the literatureabout the definition of the cm of a system of relativisticparticles. The obvious reason is the role of simultaneityand retardation. Ref. [2] examines several possibilities.For a system of particles it is found convenient to definethe coordinates q µ of the center of mass as the mean ofcoordinates of all particles weighted with their dynami-cal masses (energies). The relativistic (covariant) gener-alization of center of mass is such that the coordinates q µ must satisfy the relation [2] P q µ = X i p i z µi , (5)where the coordinates of the i th particle with respect tothe center of mass are denoted by z µi and the total mo-mentum vector by P µ = X i p µi . Ref. [2] chooses eq. (5)as the one that is most qualified to represent the defini-tion of cm of a relativistic system, which also reduces tothe non-relativistic definition of the center of mass. Wedid not find a better discussion of this in the literatureand we could also not find a better way to improve onthis definition.The above definition, Eq. (5), leads to the compactform, to order 1 /c , X i m i r i q g − v i c = X i m i r i (cid:18) v i c − z i A c + O ( 1 c ) (cid:19) = 0 , (6)where r i = ( x i , y i , z i ) is the coordinate and v i is the ve-locity of the i th particle with respect to the cm.For a system of non-interacting particles the conditionin Eq. (6) implies that, along the direction of motion, X i A m i z i = − X i A m i z i (cid:18) v i c − z i A c (cid:19) . (7)Hence, the Hamiltonian of Eq. (4) for a collection of par-ticles becomes H = X i p i m i (cid:18) − p i m i c (cid:19) + A c X i m i z i + U ( r i )+ O ( 1 c ) , (8)where we have added a scalar potential U ( r i ), whichwould represent a (central) potential within an atom, anucleus, or any other many-body system.Notice that the term proportional to − m A z com-pletely disappears from the Hamiltonian after the rela-tivistic treatment of the cm. This was also shown in Ref.[3]. It is important to realize that non-inertial effectswill also carry modifications on the interaction betweenthe particles. For example, if the particles are charged,there will be relativistic corrections (magnetic interac-tions) which need to be added to the scalar potential U ( r i ) = P j = i Q i Q j / | r i − r j | . As shown in Ref. [3], thefull treatment of non-inertial effects together with rela-tivistic corrections will introduce additional terms pro-portional to A and A in the Hamiltonian of Eq. (8), toorder 1 /c . Thus, a more detailed account of non-inertialcorrections of a many-body system requires the inclusionof A -corrections in the interaction terms, too. We referthe reader to Ref. [3] where this is discussed in moredetails. Here we will only consider the consequences ofthe acceleration correction term in Eq. (8), H nin = A c X i m i z i . (9) III. REACTIONS IN STARS
Nuclei interacting in a plasma or undergoing pycnonu-clear reactions in a lattice can experience different ac-celerations, allowing for an immediate application ofEq. (9). But in order to use this equation to measurechanges induced by the gravitational fields in stars, weassume that one can replace A by a local gravitationalfield, g . This assumption requires a few comments at thispoint. If we consider two nuclei participating in a nuclearreaction in a star, they are, most likely, in a gaseous phasefollowing inertial trajectories in between collisions. Theeffect of gravity is to modify slightly the inertial trajec-tories of the two nuclei due to the difference in the grav-itational field strength in their initial and final positions.Thus the best way to study the reaction problem is tocalculate reaction rates in terms of a local metric at apoint within the star. This metric can be deduced fromGeneral Relativity at the reaction observation point. Tofirst-order one can also use Eq. (2), which is shown inRef. [1] to describe particles in a gravitational field.Here instead, we will adopt the Hamiltonian of Eq. (8)as representative of the same problem. Here we will notattempt to prove the equality between the two proce-dures, and several other issues (e.g., time-dependence ofaccelerations, modification of interactions in presence ofa gravitational field, etc.), leaving this for future studies.Our goal here is to estimate the magnitude of the grav-itational field which could produce sizable “non-inertialcorrections” and study physical cases where such correc-tions might be important and could change appreciablythe reaction rates and/or the internal structure of manybody systems. A. Nuclear fusion reactions
Nuclear fusion reactions in stars proceed at low ener-gies, e.g., of the order of 10 KeV in our Sun [4, 5]. Dueto the Coulomb barrier, it is extremely difficult to mea-sure the cross sections for charged-particle-induced fusionreactions at laboratory conditions. The importance ofsmall effects such as the correction of Eq. (9) in treatingfusion reactions is thus clear because the Coulomb bar-rier penetrability depends exponentially on any correc-tion. To calculate the effect of the term given by Eq. (9)we use, for simplicity, the WKB penetrability factor P ( E ) = exp " − ~ Z R C R N dr | p ( r ) | , (10) where p ( r ) is the (imaginary) particle momentum insidethe repulsive barrier. The corrected fusion reaction isgiven by σ = σ C · R , (11)where σ C is the Coulomb repulsion cross section and R = P corr ( E ) /P ( E ) is the correction due to Eq. (9).The non-inertial effect is calculated using | p ( r ) | = p m [ V C ( r ) − E ] and | p corr ( r ) | = s m (cid:20) V C ( r ) + A mr h cos θ i c − E (cid:21) (12)where (cid:10) cos θ (cid:11) = 1 / V C = Z Z e /r . In orderto assess the magnitude of the acceleration A for which itseffect is noticeable, we consider a proton fusion reactionwith a Z = 17 nucleus (chlorine) at E = 0 . R C = Z Z e /E = 245 fm and take R N = 3 . g = A =10 − c /R C ≈ × m/s , which is about 26 ordersof magnitude larger than the acceleration due to grav-ity on Earth’s surface and 15 orders of magnitude largerthan the one at the surface of a neutron star (assuming M ns = M ⊙ and R ns = 10 km). It appears that the effectis extremely small in stellar environments of astrophys-ical interest where nuclear fusion reactions play a role.Such large gravitational fields would only be present inthe neighborhood of a black-hole. Under such extremeconditions nuclei are likely to disassemble, as any otherstructure will. B. Atomic transitions
As an example in atomic physics, we consider the en-ergy of the 2p / level in hydrogen which plays an impor-tant role in the Lamb shift and probes the depths of ourunderstanding of electromagnetic theory. We calculatethe energy shift of the 2p / level within the first-orderperturbation theory and we get∆ E p / nin = (cid:10) p / | H nin | p / (cid:11) = 24 a H A m e c , (13)where a H = ~ /m e e = 0 .
529 ˚A. One should comparethis value with the Lamb splitting which makes the 2p / state slightly lower than the 2s / state by ∆ E Lamb =4 . × − eV. One gets ∆ E p / nin ≃ ∆ E Lamb for
A ≃ m/s , which is 9 orders of magnitude larger thangravity at the surface of a neutron star. Thus, even fortiny effects in atomic systems, the effect would only benoticeable for situations in which electrons are bound inatoms. A R C / c R FIG. 1: Suppression factor due to the non-inertial effects, R ,for fusion reactions of protons on chlorine at E = 0 . IV. REACTIONS IN THE LABORATORY
The logical conclusion from the last section is that itis very unlikely that non-inertial effects due to gravita-tional fields are of relevance in stars. Nowhere, exceptin the vicinity of a black-hole, accelerations are of orderof ∼ m/s , which would make the effect noticeable.However, there is another way to achieve such large ac-celerations and that is nothing else but the huge accelera-tions which occur during nuclear reactions. For example,for a nuclear fusion reaction, at the Coulomb radius (dis-tance of closest approach, R C ) the acceleration is givenby A C = Z Z e R C m , (14)where m = m N A A / ( A + A ) is the reduced massof the system and m N is the nucleon mass. For typicalvalues, E = 1 MeV, Z = Z = 10, and A = A =20, one obtains R C = Z Z e /E = 144 fm and A C =6 . × m/s . This is the acceleration that the cm ofeach nucleus would have with respect to the laboratorysystem.As we discussed in the introduction, the cm of theexcited nucleus is the natural choice for the referenceframe. This is because it is easier to adopt a descrip-tion of atomic and nuclear properties in the cm frame ofreference. Instead, one could also chose the cm of thecolliding particles. This later (inertial) system makes itharder to access the acceleration effects, as one wouldhave to boost the wave functions to an accelerated sys-tem, after calculating it in the inertial frame. This is amore difficult task. Therefore we adopt the cm referenceframe of the excited nucleus, using the Hamiltonian ofsection II. This Hamiltonian was deduced for a constantacceleration. If the acceleration is time-dependent, themetric of Eq. (2) also changes. Thus, in the best casescenario, the Hamiltonian of Eq. (8) can be justified in an adiabatic situation in which the relative velocity be-tween the many-body systems is much smaller than thevelocity of their constituent particles with respect to theirindividual center of masses. If we accept this procedure,we can study the effects of accelerated frames on the en-ergy shift of states close to threshold, as well as on theenergy location of low-lying resonances. A. Reactions involving halo nuclei
The nuclear wave-function of a (s-wave) loosely-bound,or “halo”, state can be conveniently parameterized byΨ ≃ r α π exp ( − αr ) r , (15)where the variable α is related to the nucleon separationenergy through S = ~ α / m N . In first order perturba-tion theory the energy shift of a halo state will be givenby ∆ E Nnin = h Ψ | H nin | Ψ i = 18 S (cid:18) Z Z e ~ R C m c (cid:19) . (16)Assuming a small separation energy S = 100 keV, andusing the same numbers in the paragraph after Eq. (14),we get ∆ E Nnin = 0 .
024 eV, which is very small, exceptfor states very close to the nuclear threshold, i.e. for S →
0. But the effect increases with Z for symmetricsystems (i.e. Z = Z = A / E Nnin = 1 −
10 eV for larger nuclear systems.There might exist situations where this effect could bepresent. For instance, the triple-alpha reaction whichbridges the mass = 8 gap and forms carbon nuclei instars relies on the lifetime of only 10 − s of Be nuclei.It is during this time that another alpha-particle meets Be nuclei in stars leading to the formation of carbon nu-clei. This lifetime corresponds to an energy width of only5 . ± .
25 eV [6]. As the third alpha particle approaches Be, the effects of linear acceleration will be felt in thereference frame of Be. This will likely broaden the widthof the Be resonance (which peaks at E R = 91 . ± . Be, and that the effects of acceler-ation internal to the Be nucleus arise from the differentdistances (and thus accelerations) between the third al-pha and each of the first two. To our knowledge, thiseffect has not been discussed elsewhere and perhaps de-serves further investigation, if not for this particular re-action maybe for other reactions of astrophysical interestinvolving very shallow nuclear states.
B. Nuclear transitions
Many reactions of astrophysical interest are deducedfrom experimental data on nucleus-nucleus scattering.Important information on the position and widths of res-onances, spectroscopic factors, and numerous other quan-tities needed as an input for reaction network calculationsin stellar modeling are obtained by the means of nuclearspectroscopy using nuclear collisions in the laboratory.During the collision the nuclei undergo huge accelera-tion, of the order of
A ≃ m/s . Hence, non-inertialeffects will be definitely important.A simple proof of the statements above can be obtainedby studying the Coulomb excitation. The simplest treat-ment that one can use in the problem is a semi-classicalcalculation. The probability of exciting the nucleus to astate f from an initial state i is given by a if = − i ~ Z V if e iωt dt, (17)where ω = ( E f − E i ) / ~ , is the probability amplitudethat there will be a transition i → f. The matrix element V if = R Ψ ∗ f V Ψ i dτ contains a potential V of interactionbetween the nuclei. The square of a if measures the tran-sition probability from i to f and this probability shouldbe integrated along the trajectory.A simple estimate could be obtained in the case ofthe excitation of a initial, J = 0, state of a deformednucleus to an excited state with J = 2 as a result of ahead on collision with scattering angle of θ = 180 ◦ . Theperturbation V is due to the interaction of the charge Z e of the projectile (one of the two nuclei) with thequadrupole moment of the target (of the other) nucleus.This quadrupole moment should work as an operator thatacts between the initial and final states. One finds that V = Z e Q if / r , with Q if = e i (cid:10) Ψ ∗ f (cid:12)(cid:12) z − r (cid:12)(cid:12) Ψ i (cid:11) ≃ e i (cid:10) Ψ ∗ f (cid:12)(cid:12) z (cid:12)(cid:12) Ψ i (cid:11) , (18)where e i is the effective charge of the transition.The amplitude is then written as a if = Z e Q if i ~ Z e iωt r dt. (19)At θ = 180 ◦ the separation r , the velocity v , the initialvelocity v and the distance of closest approach s , arerelated by v = dr/dt = ± v (cid:0) − s/r (cid:1) , which is obtainedfrom energy conservation. Furthermore, if the excitationenergy is small, we can assume that the factor e iωt inEq. (19) does not vary much during the time that theprojectile is close to the nucleus. Then the remainingintegral is easily solved by substitution and one gets a if = 4 Z e Q if i ~ v s . (20)Following the same procedure as above, we can calcu-late the contribution of the Hamiltonian of Eq. (9). Inthis case, A = Z Z e /m r and the equivalent potential V is given by V nin = (cid:18) Z Z e m (cid:19) Xm N c r , (21) where we assume that X nucleons participates in thetransition. One then finds a ninif = (cid:18) Z Z e m (cid:19) Xm N Q if is ~ v c . (22)The ratio between the two transition probabilities is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ninif a if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) Xm N Z Z e sm c (cid:19) . (23)Applying eq. 23 to the lead-lead collision at 500 MeV,as mentioned in the introduction, we find (cid:12)(cid:12)(cid:12) a ninif /a if (cid:12)(cid:12)(cid:12) =(0 . X ) . This yields very small results for the relativeimportance of non-inertial effects in single particle tran-sitions ( X ≃ X ≫
1. This result is intriguing to say theleast. We think that it deserves more studies, assumingthat the physics of non-inertial effects described in sec-tion II is right. We have made a preliminary study oftheses effects in the excitation of giant resonances in rel-ativistic heavy ion collisions using Eq. (9) which seem toconfirm this statement.
C. Electron screening of fusion reactions
In laboratory measurements of nuclear fusion reactionsone has found enhancements of the cross sections due tothe presence of atomic electrons. This screening effectleads to an enhancement in the astrophysical S-factor, orcross section: S lab ( E ) = f ( E ) S ( E ) = exp (cid:20) πη ∆ EE (cid:21) S ( E ) , (24)where η ( E ) = Z Z e / ~ v , and v is the relative veloc-ity between the nuclei. The energy ∆ E is equal to thedifference between the electron binding energies in the( Z + Z )-system and in the target atom ( Z ). For lightnuclei it is of the order of 100 eV, enhancing fusion crosssections even for fusion energies of the order of 100 KeV.For more details we refer the interested reader to Ref. [7].An intriguing fact is that this simple estimate, whichis an upper value for ∆ E , fails to reproduce the experi-mental data for a series of cases. In Ref. [8] several smalleffects, ranging from vacuum polarization to the emis-sion of radiation, have been considered but they cannotexplain the experimental data puzzle. Besides vacuumpolarization, atomic polarization is one of the largest ef-fects to be considered (among all other small effects [8]).Non-inertial corrections contribute to polarization po-tential V pol = − X n =0 |h | H nin | n i| E n − E . (25)An estimate based on hydrogenic wave functions for theatom yields V pol ( r ) ≃ − E n (cid:18) Z Z e ~ m c (cid:19) exp ( − αr ) r . (26)Assuming α ∼ = 1 /a H , E n = E n − E ∼ = 10 eV and usingEqs. (10) and (11) to calculate the modification of thefusion cross sections due to this effect, we find the crosssection for D(d,p)T and Li( d, α ) He can increase by upto 10%. This is surprising compared with the smallervalues reported on Table 1 of Ref. [8]. It is not a veryaccurate calculation as it relies on many approximations.But it hints for a possible explanation of the differencebetween the experimental and theoretical values of ∆ E ,as discussed in Ref. [7].In stars, reactions occur within a medium rich in freeelectrons. The influence of dynamic effects of theseelectrons was first mentioned in Ref. [9] and studied inRef. [10]. The underlying assumption is that the Debye-Hueckel approximation, based on a static charged cloud,does not apply for fast moving nuclei. In fact, mostof the nuclear fusion reactions occur in the tail of theMaxwell-Boltzmann distribution. For these nuclear ve-locities Ref. [10] finds that an appreciable modification ofthe Debye-Hueckel theory is necessary. One has to add tothis finding the fact that the nuclei get very strongly ac-celerated as they approach each other, and this increasesfurther the deformation of the Debye-Hueckel cloud. V. CONCLUSIONS
In summary, assuming that the Hamiltonian for a sys-tem of particles moving in an accelerated frame containsa correction term of the form given by Eq. (9), we haveexplored the non-inertial effects for a limited set of nu-clear reactions in stars and in the laboratory. These re-sults are somewhat surprising and present a challenge toour understanding of accelerated many-body systems. In the case of stellar environments, we have shownthat only in the neighborhood of black-holes would non-inertial effects become relevant. But then the wholemethod adopted here is probably not rigorous enough,as one may have to use the full machinery of generalrelativity. Nonetheless, it is very unlikely (and perhapsunimportant, except maybe for science-fiction-like time-traveling) that internal structures of any object is of anyrelevance when it is extremely close to a black-hole.The apparent reason for the appearance of non-inertialeffects in many-body systems is that the non-inertialterm of Eq. (9) only appears when relativistic correctionsare included, what has precluded its consideration in pre-vious studies, especially for reactions that are thought tobe fully non-relativistic such as fusion reactions in stars.The main question is whether the relativistic definitionof the center of mass, through Eq. (5) as proposed byPryce in Ref. [2] contains the right virtue of describingcorrectly the center of mass frame of relativistic many-body systems.Even in the case of high energies nuclear collisions theintrinsic structure of the nuclei are sometimes an impor-tant part of the process under study. Fictitious forceswill appear in this system which might not average outand appreciably influence the structure or transition un-der consideration. It is surprising that, for a reason notquite understood, this effect has been overseen in the lit-erature so far.
Acnowledgments
This work was partially supported by the U.S. DOEgrants DE-FG02-08ER41533 and DE-FC02-07ER41457(UNEDF, SciDAC-2), the NSF grant PHY0652548, andthe Research Corporation under Award No. 7123. [1] L.D. Landau and E.M. Lifshitz, Classical theory of fields(Second Edition), Pergamon Press, p.286[2] M. H. L. Pryce, Proc. Roy. Soc. A195, 62 (1948).[3] M.P. Fewell, Nucl. Phys.