The Mass-Charge Repulsive Force and Space-Probes Pioneer Anomaly

C. Y. Lo

Applied and Pure Research Institute

7 Taggart Drive, Unit E, Nashua, NH 03060 USA

April 2009

Abstract

It is reported that something has gradually dragged two of America's oldest space probes-Pioneer 10 and Pioneer 11- a quarter-million miles off course. This is called the pioneer anomaly. Astrophysicists have struggled for15 years in vain to identify the infinitesimal force at play. It is shown that the anomaly should be due to a newly discovered force from the sun. Such a force is due to a charge - mass interaction, derived from general relativity. For a charge q and another particle of mass m, with a distance r between them, the static repulsive force is q²m/r³. Furthermore, such a force is not subjected to electromagnetic screening. Because of the r ^-3- dependence, the repulsive force becoming negligible faster with distance r increases. This effect would be observed as the pioneer anomaly. It is conjectured that the anomaly of a planetary probe would also be due to charge-mass interaction that includes also the current-mass attractive force. Thus, all the anomalies are related to the mass-charge interaction, and thus natural consequences of general relativity. The numerical confirmation of the fifth force by the pioneer anomaly would help further understanding of Einstein's unification, and Einstein's formula E = mc² as conditionally valid.

04.20.-q, 04.20.Cv

Key Words: pioneer anomaly, repulsive force, charge-mass interaction, charged capacitors, E = mc².

1. Introduction 

         It was reported [1] that beyond the edge of the solar system, something has gradually dragged two of America's oldest space probes-Pioneer 10 and Pioneer 11- a quarter-million miles off course. Astrophysicists have struggled for 15 years in vain to identify the infinitesimal force at play. The pioneer anomaly, as it is called, throws a monkey wrench into celestial mechanics.

         The anomaly officially materialized in 1988, 16 years after NASA Pioneer 10 traveled toward the outer planets. The 568-pound spacecraft has been designed to stay in radio contact with Earth just 21 months, time enough for it to become the first spacecraft to pass through the asteroid belt, the first to fly past Jupiter and the first to visit the outer solar system. The plutonium-powered probe, however, transmitted data 31 years until 2003.

         As it sped through space, a specialist in radio-wave physics named John Anderson at NASA's Jet Propulsion Laboratory noticed an odd thing. The spacecraft was drifting off course. The discrepancy was less than a few hundred-millionths of an inch per second for every second of space-flight, accumulating year after year across billions of miles. Then Pioneer 11, an identical probe escaping the solar system in the opposite direction, also started to veer off course at the same rate.

         Dr. Anderson monitored the trajectories six years before calling attention to the matter. "I'm a little like an accountant," Dr. Anderson said. "We have Newton's theory and Einstein's theory, and when you apply them to something like this- and it doesn't add up - it bothers me."

         Not everything in solar system adds up, of course. The moon's actual orbit is off its calculated course by about six millimeters a year. No one knows why. The standard yardstick for length on an interplanetary scale, the Astronomical Unit, grows by about seven centimeters a year. Scientists have yet to agree on an explanation. At least four recent planetary probes experienced such unaccountable changes in velocity as they passed Earth, Dr. Anderson and his colleagues reported.

         None prompted the scrutiny given the Pioneer anomaly. In hundreds of technical papers, Dr. Slava Turyshev and scores of other space scientists considered and eliminated most mundane explanations, including fuel leaks, software bugs, mechanical flaws, navigation errors, fading plutonium power, planetary influences, the solar wind, even the effect of the ocean tides and local plate tectonics on the placement of ground antennas. Others proposed more far-fetched scenarios; the tug of the dark matter, the accelerating expansion of the universe or a break down of gravity's most fundamental laws.

         Indeed, Dr. Turyshev at the Jet Propulsion Laboratory and his colleagues around the world regard the pioneer probes as the largest test of Newton's law of gravity ever conducted. By that axiom, refined by Einstein, any two objects in the universe exert gravitational attraction on each other proportional to their mass and affected predictably by the distance between them.

         "We would expect the two spacecraft to follow Newton's law of gravity," Dr. Turyshev said, "but they in fact fail to confirm Newton's law, If Newton is wrong, Einstein is wrong too."

         After six years of work, the researchers expect to finish restoring the last data filed next month. Based on a partial analysis of the data that took six years to restore, Dr. Turyshev reported in April at a meeting of the American Physical Society in St. Louis that at least 30% of the force can be attributed to heat radiating from the probe. "The rest is unknown," he said.

         In the year ahead, Dr. Turyshev and his colleagues plan to use the vintage data to create a computer flight simulation of the two Pioneer missions with a precision never before possible. That may finally lay it to rest. There is some hope that his would show a new physics," Dr. Turyshev said, "With the Pioneers, we are exploring uncharted territory."

         So far, all existing theories failed to explain the additional weak force, which appears at long distance. In this paper, it will be shown that the pioneer anomaly would be due to the interaction between charge and mass that has a very different dependency on distance [2, 3] because the Sun has charged particles. Thus, Dr. Anderson and Dr. Turyshev are essentially right.

2. Pioneer Anomaly, Flyby Anomaly, and a New Force beyond General Relativity.

        It is noticed that Space-Probes Pioneer 10 & 11 were heading for opposite directions. This would eliminate the possibility that the cause, as reported, were beyond the solar system. If the sources of this anomaly were beyond the solar system, the effect would appear to be attractive for one pioneer, but repulsive for the other. It has been observed that when far away from the sun, the slowing down of leaving speed from the solar system appears to be larger than that from Newtonian theory [4]. ¹⁾

Anderson and his colleagues discovered [5, 6] that four spacecraft each raced either a tiny bit faster or slower than expected when they flew past the Earth en route to other parts of the solar system. Anderson said, "There is something very strange going on with spacecraft motions. We have no convincing explanation for either the Pioneer anomaly or the flyby anomaly."

The only exception is the case of Messenger that the spacecraft approached the Earth at about latitude 31 degrees north and receded from the Earth at about latitude 32 degrees south. "This near-perfect symmetry about the equator seemed to result in a very small velocity change, in contrast to the five other flybys," Anderson explained - so small no anomaly could be confirmed. Another case of uncertainty is the second flight of Galileo spacecraft in December 1992, because any possible velocity increase was masked by atmospheric drag of the lower altitude of 303 km. The four other flybys involved flights whose incoming and outgoing trajectories were asymmetrical with each other in terms of their orientation with Earth's equator. For the other cases of Galileo, NEAR>, Cassini-Huygens>, and Rosetta>, all experienced an anomalous velocity increase after its Earth encounter

         Thus, Wikipedia, the free encyclopedia, defines the flyby anomaly as "an unexpected energy increase during Earth flybys> of spacecraft". Moreover, there are significant unaccounted velocity increments at infinity after at the perigees. These facts imply that an unknown force that causes the flyby anomaly from is a long range neutral repulsive force. Thus, it seems that these anomalies are beyond the reach of both Newton and Einstein. Thus, some prominent relativists claimed these anomalies must be due to experimental errors. (Even Turyshev and Antreasian, had said that most likely the anomaly is simply an error in the computer code that is used to shift between Earth-bound and space-based coordinate systems.) However, such a claim has been proven to be very unlikely since NASA seems to have exhausted all possibilities from existing theories [4].

         However, these two kinds of anomalies may not be related to each other. An obvious problem is that the flyby anomaly is an acceleration rather than a deceleration. Turyshev believes, is that both involve spacecraft, and both are called anomalies. "Nobody has established that the two are connected," he says. However, Anderson's not so sure. "Another thing in common between the Pioneer and these flybys is what you would call an unbound orbit around a central body," Anderson said. "For instance, the Pioneers are flying out of the solar system> - they're not bound to their central body, the sun. For the other flybys, the Earth is the central body. These kinds of orbits just don't occur very often in nature - it could be when you get into an unbound orbit around a central body, something goes on that's not in our standard models."

In summary, the new forces must satisfy the following requirements: 1) it is a neutral force; 2) it is a long range force; 3) it is a force much weaker than the Newtonian gravitational force; 4) it is a repulsive force for the flyby anomaly; 5) it appears in the unbound orbits though not noticeable for a closed orbit, 6) it leads to the additional constant deceleration on space-probe pioneers at very long distance. The requirement 5), observed by Anderson, implies that this new force has a distance dependency very different from r^-2. Thus, in addition to being a repulsive force, this requirement implies that the anomalies are definitely beyond current theory of general relativity. However, this does not necessarily mean that these anomalies are not related to general relativity. In this paper, it will be shown that they are closely related to general relativity.

Currently, there are four known forces, namely, as follows: the electromagnetic, the gravitational, the weak and the strong forces. The last two are short-range forces that operate in the nuclei scale, and the first two are long-range forces that can be seen in the macroscopic scale. The electromagnetic force is a force between charges, and the gravitational force is a force between masses. However, a natural question is, is there a force between a mass and a charge?

Recent theoretical research shows that there is indeed a repulsive force between the mass and the charge. It is found [2, 3] that this fifth force between two point-like particles is mq²/r³, where m is the mass of one particle, q is the charge of the other particle, and r is the distance between them. This is a very weak, long-range, repulsive, neutral force that satisfies requirement 5). The remaining problem is could requirement 6) also be satisfied. The existence of such a repulsive force has been preliminary verified by weighting charged capacitors because a charged capacitor would have less weight, although the distance dependency cannot be verified with such an experiment [2, 3]. In contrast, according to existing theory, the capacitor after charged would be considered as becoming heavier because many failed to learn that E = mc² is only conditionally valid [23] and there is a new force. In fact, general relativity has never been well understood since 1916 (see Section 5).

A new explanation would be that there is a neutral weak force due to the mass-charge interaction, which is repulsive and reducing faster than a Newtonian force as the distance from the sun increases. Then, at a very long distance, the net effect may appear as a constant additional weak force that observation suggests. Recently, based on general relativity, a very weak repulsive neutral force of charge-mass interaction has been derived [2, 3]. Since the sun has charged particles, this neutral force would be a suitable candidate. This infinitesimal weak force would produce the anomaly since it is reducing faster than the Newtonian force (see Section 6). If only charge and mass are able of producing long rang fields, this is the only possibility.

On the other hand, if general relativity is essentially correct, it should lead to understanding the observed anomaly. In this paper, it will be shown that the charge-mass interaction derived from general relativity would provide the natural explanation, which has been failed so far, for the pioneer anomaly as well as other planetary anomalies [1, 4]. Thus, what is being called as anomaly would actually be normal, and such a term just reminds us about our oversight in the past.

3. The Charge - Mass Interaction and Conditional Validity of E = mc²

        Of course, this involves new physics, but it is a new physics discovered from a study of general relativity [8, 9]. Then both theories of Newton and Einstein are only inadequate. Moreover, this very weak neutral charge-mass interaction is not subjected to electromagnetic screening, and thus seems to be uniquely suitable for the explanation of the pioneer anomaly. However, the discovery of such an interaction takes a long way involving the resolves of some fundamental issues in general relativity.

         First, in 1993 it is discovered [10, 11] that, for the dynamic case, linearization of the Einstein equation is not valid although it is valid for the static cases. Subsequently, it is found that the Einstein equation of 1915 does not have a dynamic solution just as Gullstrand [12] suspected in his report to the Nobel Committee. Thus, modification of the 1915 equation is necessary by adding a term for the gravitational energy-stress tensor, which has a coupling of different sign [11, 13]. This new coupling sign is necessary [11, 13] to explain the binary pulsars experiment of Hulse and Taylor [14]. 

         Since coupling signs can be different, the formula E = mc² cannot be generally valid. Moreover, it should be noted that the electromagnetic energy-stress tensor has its trace being zero, but the massive energy-stress tensor has a non-zero trace. Thus the electromagnetic energy is not equivalent to mass. Subsequently, it is found that Einstein's proof is incomplete because he has assumed only but did not prove [15] that an electromagnetic wave is equivalent to massless particles, the photons. On the other hand, experimentally it is observed that the meson p₀ can decay to two photons. Thus, the energies of photons and the electromagnetic wave are not equivalent. It turns out that the photons actually include also gravitational energy [16]. 

         The non-equivalence between electromagnetic energy and mass is also confirmed by general relativity. This is shown by the Reissner-Nordstrom metric [17-19] (with c = 1) as follows:

         ,                                              (1)

where q and M are the charge and mass of a particle and r is the radial distance (in terms of the Euclidean-like structure ²⁾ [20, 21]) from the particle center. In this metric (1), the gravitational components generated by electricity have not only a very different radial coordinate dependence but also a different sign that makes it a new repulsive gravity in general relativity. After the publication this metric, Einstein [22, 23] changed to that validity of E = mc² is only conditional. Nevertheless, some theorists still hold on unconditional validity [8], because of inadequate understanding of Einstein's equivalence principle (see Appendix).

    Moreover, some argued that the effective mass could be considered as               

         M - ,                                                                             (2)

since the total electric energy outside a sphere of radius r is q²/2r [24, 25], and thus (2) could be interpreted as supporting m = E/c² for electromagnetic energy. However, there are several difficulties raised from such a view: For example, if any energy has a mass equivalence, an increase of energy should lead to an increment of gravitational strength. However, although energy increases by the presence of a charge, the strength of a gravitational force, as shown by metric (1), decreases everywhere.

         Nevertheless, theorists such as Herrera, Santos & Skea [26] argued that M in (1) includes the external electric energy. They overlooked that this would create a double counting of the electric energy in two different ways [9, 17, 25]. ³⁾ Moreover, the gravitational forces would be different from the force created by the "effective mass" M - q²/2r because

         .                             (3)

In addition, if M included the external electric energy, then the inertial mass of the particle would be smaller than M since an acceleration of the charged particle would not immediately affect the electric energy at long distances. (Note that the radius r_e of an electron is much smaller than a half of its classical radius e²/m₀c² [27], where e is the charge of the electron and m₀ is its inertial mass. Accordingly, the total external electric energy e²/2r_e would be much larger than m₀.)

     Moreover, according to Einstein, the field equation for the metric with a source energy stress tensor T_mn is [28],

         G_mn º R_mn - g_mnR = - 8p T_mn ,  where                R = .                      (4)

For the Reissner-Nordstrom metric, T_mn it includes at least the sum of massive energy-stress tensor and the electromagnetic energy-stress tensor. Since the electromagnetic energy-stress tensor is traceless R (= ) is independent of the electromagnetic energy-stress tensor, the electric energy cannot be equivalent to a mass. ⁴⁾

         If the external electric energy of a particle were included to the mass M in (1), the gravitational mass would be larger than the inertial mass. Thus, the unconditional validity of E = mc² is a misinterpretation.

4. The Charge-Mass Repulsive Force and Extension of Einstein's Theory.

         To show the repulsive effect, one needs to consider only g_tt in metric (1). According to Einstein [7, 15],

              where                  (5)

and . Let us consider only the static case, dx/ds = dy/ds = dz/ds = 0. Thus,

         ,      where                 (6)

since g_mn would also be static. One need not worry whether the gauge is physically valid since the gauge affects only the second order approximation of g_{t t} [29]. For a particle of mass M, in the harmonic gauge

                and                                                 (7)

in the Schwarzschild solution, but the second order term is negligible.

   For a particle P with mass m at r, since g^{r r} @ -1, the force on P in the first order approximation is

            .                                                                             (8a)

Thus, the second term is a repulsive force. If the particles are at rest, then since the action and reaction forces are equal and in the opposite directions, the force acts on the charged particle Q has the same magnitude

        ( ) ,      where     is a unit vector.                                                   (8b)

However, for the motion of the charged particle with mass M, if one calculates the metric according to the particle P of mass m, only the first term is obtained. Thus, the geodesic equation is inadequate for the equation of motion. Moreover, since the second term is proportional to q², it is not a Lorentz force.⁵⁾ nor the radiation reaction force since the charged particle remains static.

         Thus, it is necessary to have a repulsive force with the coupling q² to the charged particle Q in a gravitational field generated by masses. It thus follows that, force (8b) to particle Q is beyond current theoretical framework of gravitation + electromagnetism. ⁶⁾ In other words, as predicted by Lo, Goldstein, and Napier [28], Einstein's general relativity leads to a realization of the inadequacy of general relativity just as electricity and magnetism lead to the exposition of their shortcomings.

         For two point-like particles of respectively charge q and mass m, the charge-mass repulsive force is mq²/r³ , where r is the distance between these two particles.⁷⁾ Clearly, this force is independent of the charge sign. Such characteristics would make the repulsive effects easier to be verified [9] since a concentration of electrons would increase such repulsion.

         The term of the repulsive force in (1) comes from the electric energy [2, 9]. An immediate question would be whether such a charge-mass repulsive force mq²/r³ is subjected to electromagnetic screening. It is conjectured that this force, being independent of a charge sign, would not be subjected to such a screening [2] although it should be according to general relativity. From the viewpoint of physics, this force can be considered as a result of a field created by the mass m and the field interacts with the q². Thus such a field is independent of the electromagnetic field and is beyond general relativity [2]. In fact, this has been tentatively confirmed since a charged capacitor does change its weight [31].

However, the r^-3-dependence (unlike a r^-2-dependence) is difficult to test because it would be sensitive to the near surroundings. Thus, being a long distance effect, the pioneer anomaly provides an excellent opportunity to test such dependence. Obviously, to accommodate the mass-charge interaction, unification between gravity and electromagnetism is necessary.

To this end, Kaluza [32] proposed a five-dimensional general relativity to reproduce gravitation and electromagnetism from metric element g_k5 (k = x, y, z, t). This started the Kaluza-Klein theories [33, 34] that assumed g₅₅ as a constant. In stead of Kaluza's cylindrical condition that reduces the five variables to four, Klein speculated the metric elements are periodic function of the fifth variable. However, nothing that can be verified comes out from such unification either. Another deficiency of their theory is the inability to deal with radiation reaction force just as Maxwell's theory [27]. Understandably, the theories of Kaluza and Klein are essentially abandoned and were criticized as unification in name only [30].

However, the theory of Lo et al. [30] has no cylindrical condition, no negligence of the metric elements, and the radiation reaction force can be included as essentially a function of the fifth variable. Thus, such a theory would provide a theoretical framework for unification. Moreover, based on their theory, the static charge-mass interaction would be generated from the metric element g₅₅ [2, 3] that others disregarded [33]. Moreover, in the static case, one does not have to worry about the fifth variable yet [30]. Since the static charge-mass interaction can be generated from g₅₅, one can claim theoretically that this new force is not subjected to the electromagnetic screening [2, 3], and this is subsequently verified experimentally [31].

5. Problems in General Relativity and Einstein's self-inconsistency

The Reissner-Nordstrom metric was first published in 1916, the same year that Einstein published his first paper in general relativity. Normally, the necessary unification of gravitation and electromagnetism should have been recognized in a year or so since Einstein advocates such unification [7]. However, this was not recognized until 2006 [9], a good 90 years after ward. A reason is that in Einstein's theory there are invalid and even inconsistent claims that are obstacles to theoretical progresses. Of course, a main problem was that E = mc² was mistakenly regarded as unconditionally valid [8]. A related problem was that it would be difficult to clarify confusions and to rectify errors because the coordinates were ambiguous [2, 3].

Moreover, Einstein's accurate predictions created a faith, and this makes a critical analysis over due. Also, the observational confirmations were exaggerated since his equation has no dynamic solutions [10, 11] as conjectured by Gullstrand [12]. Einstein's covariance principle has been found invalid [35], and the failure to recognize this invalidity is due to logical immaturity [36]. To defend the covariance principle, the editorial of the Royal Society [37] regarded that the shortest distance r₀ from the sun center as an arbitrary label although Einstein [7] showed that the light deflection angle is 4kM/r₀, where k = 7.425x10^-29cmg^-1 and M is th mass of the sun. Nevertheless, Einstein's equivalence principle, which is inconsistent with the covariance principle, would play a crucial role in rectifying the shortcoming of his theory [20, 21]. However, few understand this principle since there is no textbook or reference, other than Einstein's own works, that explains this principle correctly [38].

Moreover, there is no general consensus on general relativity due to logical immaturity. Bondi, Pirani & Robin [39] incorrectly rejected Einstein's requirement on weak gravity [40] with the backup of the editorial of the Royal Society [41]. On the other hand, theorists such as Wald [28], and Will [42], backup by the editorial of the Physical Review D, believe in both Einstein's requirement on weak gravity and his covariance principle. Nevertheless, they failed responding to this inconsistence [39] since 1959. Moreover, Penrose [43] accepted a metric solution with unphysical parameters [13]. Hawking [44] claimed, "In relativity, there is no real distinction between the space and time coordinates". However, he also claimed, "an arrow of time, something that distinguishes the past from the future, giving a direction to time", and thus there are differences.

In fact, such confusion and inconsistencies actually have a long history. Although the International Society on General Relativity and Gravitation was formed, founders of the society such as Bergmann [45], Bondi [39], Fock [46], Synge [47], Wheeler [17], and etc. have never reached a general consensus on general relativity. Under the auspices of this society, "General Relativity and Gravitation" is published. Thus, an incorrect paper [26] that (against Einstein's view [23]) claims unconditional validity of E = mc² was published. Also, members of the Editorial Board do not sufficiently understand physical principles [38, 48, 49]. Some theorists even incorrectly criticized Einstein without getting the facts straight first [38, 46, 50].

The unconditional validity of E = mc² is the justification for the physical assumption, the unique sign for coupling constants, in the singularity theorems of Penrose and Hawking [28]. Thus, more errors were created at its defense. For instance, that the binary pulsar experiment implies the existence of different coupling signs [11] is ignored. And ‘t Hooft attempted to challenge the Nobel Committee on the non-existence of dynamic solutions with an invalid counter example [49]. Apparently, they did not know that Einstein's proof of E = mc² actually involves an implicit assumption that is not always valid [16]. 

Einstein's difficulties are due to incorrectly adapt the mathematical notion of local distance in Riemannian geometry as if valid in physics [21, 37]. Whitehead [51, p.83], strongly objected,

"By identifying the potential mass impetus of a kinematic element with a spatio-temporal measurement Einstein, in my opinion, leaves the whole antecedent theory of measurement in confusion, when it is confronted with the actual conditions of our perceptual knowledge. The potential impetus shares in the contingency of appearances. It therefore follows that measurement on his theory lacks systematic uniformity and requires a knowledge of the actual contingent physical field before it is possible."

Since Einstein's notion of distance depends on the metric, which in turn depends on the motion of matter, using such a theory to describe motion can only lead to confusion. Moreover, Einstein's theory of measurement is inconsistent with the observed light bending [52, 53], and leads to ambiguity of coordinates and the need of his covariance principle as an interim measure [15].

However, Einstein's theory of measurement may not be an integral part of general relativity [37] since such a theory was not used for his predictions [7, 15]. Moreover, such a theory is actually based on invalid applications of special relativity [7]. ⁸⁾ Fortunately, based on Einstein's equivalence principle, it is found that the local distance is determined by the Euclidean-like structure that is compatible with physical measurements [21]. Thus, Einstein's errors in general relativity are rectifiable.

For a physical space, Einstein's equivalence principle provides a measurement of the space contractions and the time dilation. Einstein's error is that he overlooked that his equivalence principle provides only a dynamic method of measurement for space contractions since the measurements are done in a movable local space resting but under the influence of gravitational acceleration, but are not done in the frame of reference with a static method of attachment.

Thus, so far as measurements are possible in principle, the Euclidean-like structure is operationally defined in terms of spatial measurements essentially the same as Einstein defined the frame of reference for special relativity [21]. If the measuring rods are attached to the frame of reference, since the measuring rods and the coordinates being measured are under the same influence of gravity, a Euclidean-like structure emerges as if gravity did not exist [21, 22]. Hence, independent of the space-time metric, a physical space must have a frame of reference with a realizable Euclidean-like structure. An example is metric of Einstein's rotating disk [54]. Thus, Einstein's equivalence principle rectifies Einstein's invalid theory of measurement.

6. The Charge-Mass Repulsive Force on a Space Probe Pioneer    

Thus, the calculation of (8a) is essentially based on general relativity after related invalidities are removed. The five-dimensional theory is invoked only to justify that the new force is not subjected to electromagnetic screening. However, this is theoretically crucial to establish a charge-mass repulsive force, which is independent of electromagnetism.

Then, the charge-mass repulsive force between a point charge q and a point mass m is,

         F =                                                                                                 (9)

in the r-direction. This formula essentially comes from general relativity. The five-dimensional theory supports that it is not subjected to electromagnetic screening, and this is supported by the experiment of weighting charged capacitors. This new force would behave very differently from an attractive force, which is inverse proportional to the square of the distance r_.  However, due to the q² term, this formula should be modified for the case of a composite object consisting of many charged particles.

The space probes would give a good opportunity to check the mass-charge interaction. If the repulsive force comes from the sun, then m in (9) would be m_p the mass of the pioneer, and distance r would be R the distance between the sun and the space probe. However, the charge term is not clear since for the sun we do not know how the non-linear term q² should be.

Nevertheless, since such forces act essentially in the same direction, we could use a parameter P_s to represent the collective effect of the charges. ⁹⁾ Then, the effective repulsive force F_p would be (see also Section 6)

         F_p = .                                                                                                      (15)

If the data fit well with an appropriate parameter P_s, then this would be another confirmation of the charge-mass interaction.

         Since this force is much smaller than the gravitational force from the sun, in practice the existence of such a repulsive would result in a very slightly smaller mass M_ss for the sun, i.e.

F = ,          and                           for   R₀              (16)

Then, we have

F = .                                                                                     (17)

Thus, there is an additional attractive force for R > R₀, the distance the earth from the sun. Of course, if the space probe is charged, then there is another repulsive force with M_s being the mass of the sun and P_q due to such charges.

Moreover, such a force would not be noticeable from a closed orbit since the variation of the distance from the sun is small. However, for open orbits of the pioneers, there are great variations. When the distance is very large, the repulsive force becomes negligible, and thus an additional attractive force would appear as the anomaly. Such a force would appear as a constant over a not too long distance. Thus, the repulsive fifth force satisfies the over all requirements according to the data [4].

         When the four recent planetary probes experienced unaccountable changes in velocity as they passed Earth, they experienced an additional repulsive force from the Earth because the core of the globe has charged currents. Moreover, depending on the way of approaching the globe, a planetary probe would also experience an additional attractive force due to current-mass interaction (see next section). Thus, a planetary probe would experience an additional acceleration or de-acceleration.

7. Conclusions and Discussions

         In conclusion, all anomalies would have a unified cause just as Anderson [1, 4-6] expected. The charge-mass interaction provides a theoretical explanation of the space-probe anomaly qualitatively that so far no other theory could provide [1, 4]. This interaction would be further confirmed with the data of NASA in her future theoretical simulation. This simulation would not be simple since there are other minor uncertain effects [4]. However, since this new force is based on general relativity, its chance of success would be great. Moreover, unlike others, such a force was not proposed just to solve the pioneer anomaly. It is amazing that such a force explain the requirements from the data so well, and thus deserves further investigation.

In current theory, the charge-mass repulsive force would be subjected to electromagnetic screening. From the viewpoint of physics, it is unnatural that a neutral force could be screened in such a way. From the viewpoint of the five-dimensional theory, the charge-mass repulsive force would be understood as that the charge interacts with a new field created by a mass [2, 3]. Therefore, the repulsive force would have the characteristic of not subjecting to such screening.

Current theory would predict that the weight increases slightly because the increment of energy.¹⁰⁾ However, in a five-dimensional theory, the charge-mass repulsive force is not subjected to screening [2, 3], and thus would make the charged capacitor lighter. In a charged capacitor, both the positive and the negative charges are concentrated, and thus an effect of the repulsive force would be observed as a lighter weight for the charged capacitor.^{11), 12)} Moreover, since the existence of such a repulsive force has been verified [31], validity of the characteristic of this new force is independent of the five dimensional theory.

Thus, the five-dimensional theory would predict the existence of the charge-mass repulsive force, and it is a fifth force that is independent of the four known forces. Moreover, if the investigation of electric energy leads to a charge-mass repulsive force, the magnetic energy would similarly generate a current-mass force. According to the effect of a magnetic field in general relativity [49, p. 263], it is expected that the current-mass force would be an attractive force that is perpendicular to the current. ¹³⁾ Naturally such a current related force would cancel part of the repulsive force.

         Although general relativity is not yet a self-consistent complete theory [12, 54], its completion naturally leads to the necessary existence of the fifth force. Thus the existence of orbital anomaly would follow after its characteristic is verified. In other words, pioneer anomaly actually is not an exception, but a natural extension of general relativity.           

Einstein is really a genius and the full meaning of general relativity is still emerging after 100 years of its creation although he also made mistakes just like other great physicists. Einstein is essentially right although a small force is neglected. Moreover, Einstein and Pauli [34] have predicted that a five dimensional theory would be a strong candidate for a unified theory, and now a new weak repulsive force has been derived within this theoretical framework of relativity. These strongly support that the newly discovered interaction should be the cause of this pioneer anomaly.

On the other hand, if general relativity is essentially correct, it must lead to verification of these anomalies. It is very lucky that such experiments were inadvertently done many years ago. Nevertheless, an existence of the fifth force would be greatly resisted since it would loudly declare Einstein's view that the formula E = mc² is only conditionally valid [8, 9, 16, 23, 24]. Thus, the logical immaturity and systematical errors of current dominating theorists would be unequivocally exposed.

         The weight reduction of charged capacitors supports the existence of a neutral mass-charge interaction. The spinning superconducting top experiments [55] support the mass-current interaction. ¹³⁾ The pioneer anomaly would further confirm the existence of a mass-charge repulsive force with r^-3-dependence. Moreover, since the r ^-3-dependency is derived from general relativity, this would also be a test of the Einstein's general relativity.  

Moreover, the fifth force has been established, although the five-dimensional theory is still at a preliminary stage [2, 3]. Kaluza & Klein had convinced many [33] that assuming g₅₅ as constant would be valid. Now, it is further reaffirmed that such an assumption is incorrect [2, 3]. Klein's assumption that metric elements are periodic functions of the fifth dimension is a speculation that has no experimental supports, and is essentially another form of the cylindrical condition. A confirmation of the fifth force by the data of pioneer anomaly would help resurrecting the five-dimensional theory after Klein had put it to almost death. It is also hoped that this paper would be useful for NASA to understand the pioneer anomaly better. In any case, it is clear that the pioneer anomaly does not necessarily imply that general relativity is invalid.

Acknowledgments

         The author gratefully acknowledge stimulating discussions with S. -J. Chang, A. J. Coleman, Z. G. Deng, G. R. Goldstein, C. S. Hui, A. Napier, D. Rabounski, Eric J. Weinberg, and C. Wong. Special thanks are to Sharon Holcombe for valuable suggestions. This work is supported in part by Innotec Design, Inc., USA and the Chan Foundation, Hong Kong.

Appendix: Einstein's Principle of Equivalence, the Einstein-Minkowski Condition

         Einstein's equivalence principle is stated clearly in "The Meaning of Relativity" [7] as follows:

‘Let now K be an inertial system. Masses which are sufficiently far from each other and from other bodies are then, with respect to K, free from acceleration. We shall also refer these masses to a system of co-ordinates K', uniformly accelerated with respect to K. Relatively to K' all the masses have equal and parallel accelerations; with respect to K' they behave just as if a gravitational field were present and K' were unaccelerated. Overlooking for the present the question as to the "cause' of such a gravitational field, which will occupy us latter, there is nothing to prevent our conceiving this gravitational field as real, that is, the conception that K'; is "at rest" and a gravitational field is present we may consider as equivalent to the conception that only K is an "allowable" system of co-ordinates and no gravitational field is present. The assumption of the complete physical equivalence of the systems of coordinates, K and K', we call the "principle of equivalence;" this principle is evidently intimately connected with the law of the equality between the inert and the gravitational mass, and signifies an extension of the principle of relativity to coordinate systems which are non-uniform motion relatively to each other.'

In the above, no Newtonian gravity was mentioned, and gravity is generated from a space-time metric. Moreover, Einstein's equivalence principle should not be confused with Einstein's 1911 assumption of equivalence with Newtonian gravity [15].

         What is new in Einstein's equivalence principle is the Einstein-Minkowski condition as a consequence [15, p. 161]. The Einstein-Minkowski condition has its foundation from theorems [46] in Riemannian geometry as follows:

     Theorem 1. Given any point P in any Lorentz manifold (whose metric signature is the same as a Minkowski space) there always exist coordinate systems (x^m) in which ¶g_mn/¶x^l = 0 at P.

     Theorem 2. Given any time-like geodesic curve G there always exists a coordinate system (so-called Fermi coordinates) (x^m) in which ¶g_mn/¶x^l = 0 along G.

In these theorems, the local space of a particle is locally constant, although not necessarily Minkowski.

However, after some algebra, a local Minkowski metric exists at any given point and along any time-like geodesic curve G. Thus, Pauli's version [56] is essentially a corrupted version of these theorems. (Note that few states and explain Einstein's equivalence principle correctly [41].) In fact, Einstein [15, p. 144] has given an example that illustrates Pauli's errors. Nevertheless, theorists commonly but mistakenly regarded [57] Pauli's version the same as Einstein's equivalence principle.

What Einstein added to these theorems is that physically such a locally constant metric must be Minkowski. Such a condition is needed for special relativity as a special case [7]. In a uniformly accelerated frame, the local space in a free fall is a Minkowski space according to special relativity. A consequence of inadequate understanding of Einstein's equivalence principle is that the meaning of the coordinates remains ambiguous and theorists would make mistakes inadvertently [8-14, 37, 42].

Einstein's equivalence principle rectifies this confusion and it is proven that a physical space must have a frame of reference with a Euclidean-like structure [20, 21]. Then, physical principles such as the principle of causality can be clearly applied to general relativity. Nevertheless, Einstein's equivalence principle was not well understood because his derivation [15] of the space contractions and the time dilation for the case of a rotating disk were actually based on invalid applications of special relativity. Recently, however, it has been shown that these effects can be derived from Einstein's equivalence principle [54].

ENDNOTES

• 1) This additional force could be attractive and reduced slower than the Newtonian force as the distance increases, or a repulsive force reducing faster than a Newtonian force.
• 2) The existence of a Euclidean-like structure in the frame of reference is necessary [20, 21].
• 3) Some theorists often take a conditionally valid mathematical expression as physically absolute and thus out of contact with the physical reality. This is a form of confusion on mathematics and physics.
• 4) For a metric generated by massive matter, the Ricci scalar R could be zero in vacuum, but not in the whole space. For instance, in the Schwarzschild solution, R is not zero in the interior solution [28].
• 5) Currently, for a charged particle under the influence of gravity, the Lorentz force and the radiative reaction force are added to the geodesic equation to form an equation of motion. However, since there is no external electromagnetic field for this case, the Lorentz force is absent. Also, for the static case, the radiative reaction force is also absent [27].
• 6) We calculate the field generated by charge particle Q, then the force acting at particle P; and the field generated by P, then the force acting at Q. Although one should consider the field generated by both, this approach (used in electrodynamics) is valid because the field generated by a particle, does not make itself move. For the metric generated by P, the metric would be ds² = (1- 2m/r)dt² - (1- 2m/r)^-1dr² - r²d W‘², where (r, q', j') is a new coordinate system with P at the center. Thus, the force on Q in the r-direction would be only -M(m/r²) since the Lorentz force and the radiation reaction force are absent. Since the distance between P and Q is r = r, there should be another term in the r-direction as q²(m/r³).
• 7) According to the Reissner-Nordstrom metric, the event of horizon would be M ± (M² - q²)^1/2. However, M² > q² may not be valid, and the electron does not have an event of horizon because e > m_e (e = 1.381 10^-34 cm, m_e = 6.764 10^-56 cm).
• 8) When such invalidities were discovered, I wonder why nobody had noticed this earlier?
• 9) The formula (15) is based on the assumption that the total force is the sum of each individual forces calculated separately. Of course, one cannot consider such an approach as completely accurate. However, we believe that this is an accurate approximation since similar approach to the Newtonian gravity has been successful.
• 10) By combining the electromagnetic energy with other energy such as in the case of photons [16], the combined energy can be equivalent to mass. In other word, for total energy E_T, Einstein's formula, E_T = m_T c², is still valid [24].
• 11) Experimentalist W. Q. Liu (http://www.cqfyl.com/>) performed the weighting in a Chinese Laboratory of the Academy of Science, and got certified results of lighter capacitors after charged [31] although previously he got mixed results of both lighter and heavier weights. Also, his weighting of magnets is consistent with the claim of J. A. Wheeler [49, p. 263].
• 12) According to m = E/c², the mass increment of a charged capacitor is negligible. For a capacitor of 200mF charged to 1000 volt, the related mass increment would be about 10^-12 gram.
• 13) Recently, Martin Tajmar and Clovis de Matos [53], from the European Space Agency, found that a spinning ring of superconducting material increases its weight much more than expected. Thus, they believed that general relativity had been proven wrong. However, according to quantum theory, spinning superconductors should produce a weak magnetic field. Thus, they are measuring also the interaction between an electric current and the earth, i.e. an effect of the fifth force!

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