arXiv:physics/9906059v4 [physics.gen-ph] 9 Jul 1999 eeainadices fulk hre’acceleration charges’ unlike of ac- charges’ increase like and to resistance princi- celeration of equivalence effects the opposite By the ple acceleration. their enhance- of further in ment ac- results of in- charges attraction opposite as unbalanced celerating known the acceleration Alternatively, its ertia. to that resistance particle charged the accelerating causes an of elements ume r hti steunbalanced the- this the from is follows It it 12]. that [11, ory Fermi Rohrlich [8], and Abraham 10] 7], Searle Poincar´e[9, [6, [2], 5], Heaviside [4, by de- Lorentz mostly was [3], theory concept full electromag- This a of in particles. veloped concept charged the of observa- of mass His netic origin the particle. marked than neutral tion accelerated identical charged being otherwise a to that an resistant realized more first was [1] Thomson particle J. J. 1881 In Introduction inertia altering of possibility a gravitation. constitutes and effect it this of that significance is The uni- been an field. have in gravitational they supported form or that accelerated provided uniformly charges initially accelerated the the of enhances at- motion charges electromagnetic opposite accelerated of of its effect traction to the of Conversely, resistance elements the volume motion. the produces of charge repulsion the means mutual This the respectively. that mass, iner- gravitational charge’s for and the accounts tial to field sup- elements contribution gravitational charge electromagnetic volume uniform a the an the or charge in of accelerating ported repulsion uniformly the an of known is As Abstract ltl aacdadteeoeteei ontfreatn o acting force net no is there charges. therefore the and balanced pletely 1 h uulrplino w nrillk hre scom- is charges like inertial two of repulsion mutual The RPLINTRUHEETOANTCSELF-SUSTAINED ELECTROMAGNETIC THROUGH PROPULSION hsc eatet ocri University Concordia Department, Physics 1 euso ftevol- the of repulsion 45D asnev olvr West Boulevard Maisonneuve De 1455 otel ubc aaaHG1M8 H3G Canada Quebec, Montreal, -al [email protected] E-mail: ACCELERATION eslnPetkov Vesselin n 1 frsot-tevlct flgt.Snei snow two of discuss is curvature light shall it the of we from Since here velocity results spacetime, the bodies light). of massive anisotropy of around the velocity that the signals believed electromagnetic - the of in short itself velocity (for manifests the is It of question effects. bod- anisotropy those this massive what causes around to anisotropy that into spacetime answer ies insight a The is any it field provide that there. gravitational not them a does causes in but present well be as effects requires these a principle in that equivalence are The charges re- the field. charges, when gravitational unlike occur also of should attraction spectively unbalanced charges the like of and repulsion unbalanced the from resulting hw htteei one o pctm curvature spacetime for just need anisotropy result no spacetime first is since the there While that anisotropy, shows spacetime curvature. of spacetime terms not of in interpretation is correct relativity the general that indicate which sults n i)ieta(n nrilms)a ecie nan in space- the described from as originates mass) frame reference inertial accelerating (and inertia objects (ii) massive caused around and spacetime is of anisotropy attraction the the gravitational by to (i) leads relativity that general conclusion the- with mass conjunction electromagnetic in classical ory the of analysis gravitation. An and inertia called traditionally phenomena be principle) in least (at manipulated. that can electromagnetically is far-reaching gravitation spacetime and One of inertia anisotropy gravitation. the the from and into consequence insight inertia deeper of effects gaining nature the for also of for but utilization only discussed not possible crucial and is curvature that understanding spacetime implication no The is there experiments. grav- redshift the contradicts itational relativity in- general curved-spacetime directly of terpretation standard one the second that the demonstrates effects, gravitational and tial ti h nstoyo pctm htcue the causes that spacetime of anisotropy the is It alone consfraliner- all for accounts AIAA-99-2144 independent re- time anisotropy in that frame [13, 14]. The essence gravitational attraction and inertia of all matter can of this analysis is as follows. Consider a classical2 be accounted for as well if it is assumed that there are electron in the Earth’s gravitational field. Due to the no elementary neutral particles in nature. A direct anisotropy of the velocity of light in a non-inertial ref- consequence from here is that only charged particles erence frame (supported in the Earth’s gravitational or particles that consists of charged constituents pos- field) the electric field of an electron on the Earth’s sess inertial and passive gravitational mass7. Stated surface is distorted which gives rise to a self-force origi- another way, it is only elementary charges that com- nating from the interaction of the electron charge with prise a body; there is no such fundamental quantity its distorted electric field3. This self-force tries to force as mass. This means that a body’s (inertial or passive the electron to move downwards4 and coincides with gravitational) mass corresponds to the energy stored what is traditionally called the gravitational force. in the electric fields of all elementary charged parti- The electric self-force is proportional to the gravita- cles comprising the body. However, the inertial and tional acceleration g and the coefficient of proportion- passive gravitational mass of a body manifest them- ality is the mass ”attached” to the electron’s electric selves as such - as a measure of the body’s resistance field which proves to be equal to the electron mass. to being accelerated - only if the body is subjected The anisotropy of the speed of light in the Earth’s to an acceleration (kinematic or gravitational). This vicinity is compensated if the electron is falling to- resistance originates from the unbalanced mutual re- ward the Earth’s surface with an acceleration g. In pulsion of the volume elements of every elementary other words, the electron is falling in order to keep its charged constituent of the body. The active gravita- electric field not distorted. A Coulomb (not distorted) tional mass of a body proves to be also electromag- field does not give rise to any self-force acting on the netic originating from its charged constituents8 since electron; that is why the motion of a falling electron is there is no mass but only charges. As there is no need non-resistant (inertial, or geodesic)5. If the electron is for spacetime curvature since all gravitational effects prevented from falling it can no longer compensate the are fully accounted for by the electromagnetic nature anisotropy of the speed of light, its field distorts and of the passive gravitational mass and the anisotropic as a result a self-force pulling the electron downwards velocity of light in the vicinity of a massive object arises. The resistance which an electron offers to being [13, 14], it follows that it is the object’s charges (and accelerated is similarly described in an accelerating ref- their fields) that cause that anisotropy of spacetime erence frame as caused by the anisotropy of the speed around the object. of light there. When the electron accelerates its elec- One thing concerning the electromagnetic mass the- tric field distorts and the electron resists that defor- mation. In such a way, given the fact that the speed of ported by the fact that the electromagnetic mass of the classical light is anisotropic in non-inertial reference frames, all electron is equal to its observable mass. On the other hand, the inertial and gravitational effects (including the equiva- electromagnetic mass raises the question of stability of the elec- lence of inertial and gravitational mass) of the electron tron (what keeps its charge together). This question, however, cannot be adequately addressed until a quantum-mechanical and the other elementary charged particles are fully model of the electron structure is obtained. An important fea- and consistently accounted for if both the inertial and ture of the electromagnetic mass theory is that the stability passive gravitational mass of the elementary charged problem does not interfere with the derivation of the self-force particles are entirely6 electromagnetic in origin. The (acting on a non-inertial electron) containing the electromag- netic mass [13, 14]. This hints that perhaps there is no real problem with the stability of the electron (as a future quan- 2At present quantum mechanical treatment of the electro- tum mechanical model of the electron itself may find); if there magnetic mass is not possible since quantum mechanics does were one it would inevitably emerge in the calculation of the not offer a model for the quantum object itself. self-force. 3That explanation of the origin of the self-force is another 7It is evident that in this case the electromagnetic mass the- way of saying that the force arises from the mutual unbalanced ory predicts zero neutrino mass and appears to be in conflict repulsion of the volume elements of the electron charge. with the apparent mass of the Z0 boson which is involved in 4The self-force which starts to act on the electron whenever the neutral weak interactions. The resolution of this apparent its electric field distorts effectively resists this distortion. That is conflict could lead to either restricting the electromagnetic mass why the self-force strives to make the electron move downwards theory (in a sense that not the entire mass is electromagnetic) with an acceleration g in order to compensate the spacetime or reexamining the facts believed to prove (i) that the Z0 boson anisotropy which in turn will eliminate the distortion of the is a fundamentally neutral particle (unlike the neutron), and electron’s electric field. (ii) that it does possess inertial and gravitational mass if truly 5It is clear from here that a falling electron does not radiate neutral. since its electric field is the Coulomb field and therefore does 8Given that the inertial and passive gravitational masses are not contain the radiation r−1 terms [13]. electromagnetic the electromagnetic nature of the active gravita- 6On the one hand, the entirely electromagnetic mass of an tional mass follows immediately since the three kinds of masses elementary charged particle - an electron for example - is sup- are equal.

2 ory which is often overlooked should be especially curved-spacetime interpretation of general relativity stressed: even if the mass is viewed as only partly the gravitational effects observed in a non-inertial ref- electromagnetic, as presently believed, it still follows erence frame N g on the Earth’s surface are caused that inertia and gravitation are electromagnetic in ori- by the curvature of spacetime originating from the gin but in part. It should be also noted that once the Earth’s mass. The principle of equivalence requires fact of the partly electromagnetic origin of inertia and that what is happening in N g be happening in a non- gravitation is fully realized a thorough analysis of this inertial (accelerating) frame N a as well. The gravi- open issue can be carried out which will most proba- tational effects in general relativity include time and bly lead to the result that electromagnetic interaction length effects in addition to the pre-relativistic ones is the only cause behind inertia and gravitation which (falling of bodies and their weight). These, accord- are now regarded as separate phenomena9. ing to the standard interpretation of general relativ- The concept of spacetime curvature is in direct con- ity, are also caused by the spacetime curvature around tradiction with experiments. A recently obtained re- the Earth. By the principle of equivalence the time a sult [15] shows that the gravitational redshift contra- and length effects must be present in N as well. An dicts the curved-spacetime interpretation of general inertial observer can explain those effects happening a relativity. It has not been noticed up to now that in N by employing only special relativity [17]. An a both frequency and velocity of a photon change in the observer in N , however, can explain them neither by gravitational redshift experiment. In such a way the directly making use of the frame’s acceleration nor by measurement of a change in a photon frequency is in the anisotropic velocity of light there since the con- fact an indirect measurement of a change in its local cepts of time and space are more fundamental than velocity in this experiment. This shows that the lo- the concepts of velocity and acceleration. The non- a cal velocity of a photon depends upon its pre-history inertial observer in N has to prove that spacetime in a a (whether it has been emitted at the observation point N is anisotropic due to N ’s acceleration by obtain- a or at a point of different gravitational potential) - a ing the spacetime interval in N . Only then the ob- a result that contradicts the standard curved-spacetime server can derive the time and length effects in N by interpretation of general relativity which requires that using the anisotropic spacetime interval there. There- the local velocity of light be always c [16]. Therefore fore the standard interpretation of general relativity the gravitational redshift demonstrates that general leads to a picture involving the principle of equivalence relativity cannot be interpreted in terms of spacetime that is not quite satisfying: if one cannot distinguish a g curvature. This situation calls for another interpreta- between the effects in N and in N then why is the a g tion of the mathematical formalism of general relativ- spacetime in N anisotropic while in N it is curved. ity. Such a possibility of interpreting the Riemann ten- Taking into account the two results discussed above sor not in terms of spacetime curvature but in terms of the picture becomes perfectly consistent: spacetime in a g a spacetime anisotropy has always existed since the cre- both N and N is anisotropic (in N the spacetime ation of general relativity but received no attention. In anisotropy is caused by the frame’s acceleration while g such an interpretation the Riemannian geometry de- in N it originates from the elementary charges that scribes not a curved but an anisotropic spacetime thus comprise the Earth). linking gravitation to the anisotropy of spacetime [13]. The result that spacetime is anisotropic (in which An additional indication that spacetime around the velocity of light is different in different directions) massive bodies is anisotropic (not curved) comes from has enormous implications for both understanding the the following argument. According to the standard nature of inertia and gravitation and the possibility of controlling them since both inertia and gravitation 9Such an analysis will be presented in another paper. The turn out to be electromagnetic in origin (at least in 10 basic idea of this analysis is to demonstrate that it is highly part ). There exist two theoretical possibilities for unlikely that Nature has invented two drastically different and electromagnetic manipulation of inertia and gravita- independent causes of gravitation - an anisotropic spacetime for tion: elementary charged particles and a curved spacetime for elemen- tary neutral ones (such as the Z0 boson if it turns out to be a (i) Changing the anisotropy of spacetime. Since one truly neutral particle). If the mass of the elementary charged of the corollaries of the electromagnetic mass theory particles is regarded as only partly electromagnetic, the phe- is that the anisotropy of spacetime around a body nomenon of gravitation becomes even more complicated. Space- time must be anisotropic (to account for the gravitational inter- is caused by the body’s charged constituents (and action of the electromagnetic part of the mass of the particles) as well as curved (to account for the gravitational interaction of 10It is now an established (but unexplainably ignored [18]) the non-electromagnetic part of the mass of the charged parti- fact from the electromagnetic mass theory that the inertial mass cles). and inertia are at least partly electromagnetic in origin [12, 19].

3 their electromagnetic fields), the employment of strong +q is, is distorted11 [13] electromagnetic fields can create a local spacetime −q n− g · n− g anisotropy which may lead to a body being propelled g + + E−+ = 2 − 2 n−+ + 2 (1) without being subjected to a direct force. 4πǫo  d c d c d

where n−+ is a unit vector pointing from the nega- (ii) Using the spacetime anisotropy. Due to the tive charge toward the positive charge and n−+ = −ˆx anisotropic velocity of light in an accelerating refer- where ˆx is a unit vector along the x axis. Since ence frame the electromagnetic attraction of the oppo- g · n− = 0 (g is orthogonal to n−+) the electric field site charges of an accelerating electric dipole enhances + (1) reduces to its accelerated motion - it leads to a self-sustaining accelerated motion perpendicular to the dipole’s axis g q q E−+ = 2 ˆx − 2 g. [20]. According to the principle of equivalence an elec- 4πǫod 4πǫoc d tric dipole supported in an uniform gravitational field should levitate [21]. In such a way a strong electro- The force with which the negative charge attracts the g magnetic attraction between oppositely charged parts positive one in the anisotropic spacetime in N is [13, of a non-inertial device may lead to its propulsion or 14]

at least to a reduction of its mass; thus allowing for g g · n+− g F− = q 1 − E− the external force that accelerates the device (or the + 2c2 + weight of the device) to be reduced.   The possibility of manipulating inertia and gravita- where n+− is a unit vector pointing from the posi- tion by changing the anisotropy of spacetime was re- tive charge toward the negative charge. Noting that ported in [14]. This paper deals with the possibility of g · n+− = 0 we can write altering inertia and gravitation by using the spacetime q2 q2 anisotropy. g F−+ = 2 ˆx− 2 g. (2) 4πǫod 4πǫoc d The first term in (2) is the ordinary force with which the negative charge attracts the positive one. The second term represents the vertical component of the force (2) that is opposite to g and has a levitating effect on the positive charge. Self-Sustained Acceleration The calculation of the force with which the negative charge of the dipole is attracted by the positive one gives The equations of classical electrodynamics applied to 2 2 an accelerating electric dipole show that it can undergo g q q F+− = − 2 ˆx− 2 g. (3) self-sustaining accelerated motion perpendicular to its 4πǫod 4πǫoc d axis, meaning that not only does the electromagnetic The net (self) force acting on the dipole as a whole is attraction of the opposite charges of a dipole not resist directly obtained from (2) and (3) its accelerated motion but further increases it. The ap- 2 plication of the principle of equivalence shows that an g g g q Fself = F−+ + F+− = − 2 g. (4) electric dipole supported in an uniform gravitational 2πǫoc d field will be also subjected to a self-sustained accelera- Therefore, unlike the attraction of the charges of an tion which may lead to the dipole’s levitation. Here we inertial dipole which does not produce a net force act- shall derive this effect in a gravitational field directly ing on the dipole, the mutual attraction of the charges without applying the equivalence principle. of a dipole in a gravitational field becomes unbalanced Consider a non-inertial reference frame N g sup- 11 ported in a gravitational field of strength g. The This is the electric field of a charge at rest in a gravitational − y field. If the charge is uniformly accelerated with a = g its gravitational field is directed opposite to the axis. electric field at a distance d from the negative charge is [22, 23] A dipole with a separation distance d between the − · a q n−+ a n−+ − a two charges is laying along the x axis. Due to the E−+ = + n−+ . 4πǫo d2 c2d c2d spacetime anisotropy in N g (manifesting itself in the   anisotropic velocity of light in N g) the electric field This is the electric field as described in an inertial reference frame. The calculation of the electric field in the accelerated of the negative charge −q with coordinates (d, 0) at a frame in which the dipole is at rest gives the same expression point with coordinates (0, 0), where the positive charge due to the anisotropy of spacetime in that frame [13].

4 and results in a self-force which opposes the dipole’s of mass matt accelerates on its own (due to the un- weight. The effect of the self-force (4) on the dipole balanced attraction of the two charges) which results can be explained in a sense that a fraction of the dipole in a reduction of the dipole’s resistance to being ac- of mass celerated by the external force. Therefore in order to 2 q maintain the same acceleration a the external force matt = 2 , (5) 2πǫoc d accelerating the dipole should be reduced. Stated an- resulting from the unbalanced attraction of the two other way, the dipole mass is effectively reduced and charges, is subjected to an acceleration −g as long as the resistance which the dipole offers to being acceler- the dipole stays in a gravitational field of strength g. ated will reduce as well. When matt becomes equal to While the mass (5) remains smaller than the dipole the dipole mass the resistance of the dipole to being mass the effect of the self-force (4) will be a reduction accelerated will cease and consequently there will be of the dipole mass by matt since the self-force is oppo- no external force needed to accelerate it. The dipole site to the dipole’s weight. When matt becomes equal will continue to maintain its acceleration entirely on its g own - it will be in a state of self-sustained accelerated to the dipole mass (i.e. when Fself becomes equal to the weight of the dipole), the dipole starts to levitate. motion. This type of motion resembles the inertial mo- Further increase of matt will result in lifting of the tion of an object: as a free object continues to move dipole. with constant velocity until being prevented from do- If the charges of the dipole are an electron and a ing so, a dipole (initially accelerated by an external positron its weight is F =2meg, where me is the mass force) whose charges and separation distance ensure of the electron (and the positron). Using the electron that matt is equal to the dipole mass will continue to electromagnetic mass [5] move with constant acceleration on its own until being prevented from doing so. e2 If the charges of the dipole are an electron and a me = 2 , (6) 4πǫoc r0 positron the external force accelerating the dipole will be where r0 is the classical electron radius, we can calcu- Fext =2mea. late the resultant force acting on the dipole supported in a gravitational field Taking into account (8) the dipole will mantain a con- stant acceleration if 2 g e 1 1 Fres = F + Fself = 2 − g. (7) a 2πǫoc r0 d Fext + Fself =2mea.

As seen from (7) the dipole will start to levitate when Noting that for an electron and a positron the mass in the separation distance d between its charges becomes (8) will be 2 equal to r0. However, this could hardly be achieved in e −15 matt = 2 a laboratory since r0 ∼ 10 m. 2πǫoc d Consider now a reference frame N a which is uni- we can write formly accelerating with an acceleration a = −g. Let the dipole be at rest in N a placed in such a way that Fext = (2me − matt) a. the acceleration a is perpendicular to its axis. In a similar fashion to what we have done in the case of a If we assume that matt > 2me (which is unlikely to g dipole in N here too can be shown that due to the be achieved since the separation distance d between a anisotropic speed of light in N there is a self-force the charges should be smaller than the dimension r0 acting on the dipole as a whole which is given by of the classical electron considered) an external force would be needed to slow down the dipole in order that q2 a it maintains its uniform acceleration a. Fself = 2 a (8) 2πǫoc d The effect of mass reduction caused by the mutual (the self-force (8) can be directly obtained from (4) attraction of the accelerating dipole’s charges can be by applying the equivalence principle and substitut- described in the following way as well. Instead of re- ing g = −a). A fraction of the dipole of mass matt garding the self-force (8) as subjecting only a part of resulting from the unbalanced attraction of the two the dipole of mass matt to the acceleration a it is also a charges will be subjected to an acceleration a as long possible to say that Fself subjects the whole dipole as the whole dipole is experiencing the same acceler- of mass 2me to an acceleration aatt originating from ation a. This means that the fraction of the dipole the unbalanced attraction between the electron and

5 the positron. Then using the electron electromagnetic [4] H. A. Lorentz, Proceedings of the Academy of Sci- mass (6) we obtain the relation between aatt and a ences of Amsterdam 6, 809 (1904).

e2 e2 [5] H. A. Lorentz, Theory of Electrons, 2nd ed. 2 aatt= 2 a (Dover, New York, 1952). 2πǫoc r0 2πǫoc d or [6] H. Poincar´e, Compt. Rend. 140, 1504 (1905).

r0 [7] H. Poincar´e, Rendiconti del Circolo Matematico a = a. (9) att d di Palermo 21, 129 (1906). As seen from (9) the dipole will experience a self- [8] M. Abraham, The Classical Theory of Electricity sustained acceleration aatt ≥ a if d ≤ r0. and Magnetism, 2nd ed. (Blackie, London, 1950). [9] E. Fermi, Phys. Zeits. 23, 340 (1922). Conclusion [10] E. Fermi, Atti Acad. Nazl. Lincei 31, 184; 306 At present it does not appear realistic to expect that a (1922). self-sustained acceleration of a body (equal or greater [11] F. Rohrlich, Am. J. Phys. 28, 639 (1960). than its initial acceleration) can be achieved. How- ever, the possibility for eventual practical applications [12] F. Rohrlich, Classical Charged Particles of the effect of mass reduction can be assessed if macro- (Addison-Wesley, New York, 1990). scopic charge distributions are considered. It appears that most promising will be the use of specially de- [13] V. Petkov, Ph.D. Thesis, Concordia University, signed capacitors like the commercial ones which con- Montreal, 1997. sist of alternatively charged layers of metal foil rolled [14] V. Petkov, 34th AIAA/ASME/SAE/ASEE into the shape of a cylinder with the cylinder axis par- Joint Propulsion Conference, Cleveland, allel to a or g. Such capacitors can be charged to Ohio, July 12-15, 1998; Paper AIAA-98-3142 large amounts of charge. There are capacitors already (http://xxx.lanl.gov/abs/physics/9805028). available on the market that can carry a charge well above 1C. With such (and greater) amounts of charge [15] V. Petkov, Found. Phys. Lett., submitted the experimental testing of the mass reduction effect (http://xxx.lanl.gov/abs/gr-qc/9810030). appears now possible although it is not a real issue since this effect is a direct consequence of the classical [16] C. W. Misner, K. S. Thorne and J. A. Wheeler, electrodynamics when applied to a non-inertial dipole Gravitation (Freeman, San Francisco, 1973), p. and for this reason there should be no doubt that it 385. will be experimentally confirmed. The purpose of this [17] L. Schiff, Am. J. Phys. 28, 340 (1960). paper is to demonstrate that the practical applicabil- ity of the mass reduction effect may be within reach if [18] P. Pearle, in D. Teplitz, ed. Electromagnetism: proper technological effort is invested. Paths to Research (Plenum Press, New York, 1982), p. 213. Acknowledgement [19] J. W. Butler, Am. J. Phys. 37, 1258 (1969). [20] F. H. J. Cornish, Am. J. Phys. 54, 166 (1986). I would like to thank Dr. G. Hathaway for bringing to my attention the papers of Cornish [20] and Griffiths [21] D. J. Griffiths, Am. J. Phys. 54, 744 (1986). [21] in a discussion of the effects considered here. [22] J. D. Jackson, Classical Electrodynamics, 3rd ed., (Wiley, New York, 1999), p. 664. References [23] D. J. Griffiths, Introduction to Electrodynamics, [1] J. J. Thomson, Phil. Mag. 11, 229 (1881). 2nd ed., (Prentice Hall, New Jersey, 1989), p. 424.

[2] O. Heaviside, The Electrician 14, 220 (1885).

[3] G. F. C. Searle, Phil. Mag. 44, 329 (1897).

6