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

C. Y. Lo

Applied and Pure Research Institute

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

November 2008

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 15 years in vain to identify the infinitesimal force at play. Our analysis shows a new possibility that the anomaly should be due to a newly discovered force from the sun. We show that such a force would be due to a charge - mass interaction, which is based on general relativity. For a charge q and another particle of mass m, the static repulsive force is derived to be q²m/r³, where r is the distance between them. Furthermore, such a neutral force is very weak and is not subjected to electromagnetic screening. Naturally, such a force from the sun always exists in celestial mechanics since the sun has charged particles. Then, because of the r ^-3- dependence, the pioneer anomaly should be due to the repulsive force becoming negligible faster at a long distance. It is conjectured also that the anomaly of a planetary probe would be due to charge-mass interaction that includes also the current-mass attractive force. In conclusion, all the anomalies are related to the mass-charge interaction, and thus are natural consequences of general relativity.

04.20.-q, 04.20.Cv

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

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 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 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 years after year across billions of miles. Then Pioneer 11, an identical probe escaping the solar system in the opposite direct, 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 this past March in Physical Review Letters.

         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."

         In this paper, it will be shown that the pioneer anomaly would be due to the interaction between charge and mass [2, 3]. The Pioneer anomaly appears because the Sun has charged particles. Thus, Dr. Turyshev is essentially right.

2. Pioneer Anomaly and 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. If the source is in the solar system, the cause should be an "additional" force originated from the Sun. 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⁾

Currently, NASA seems to have exhausted all possibilities except that the additional force is repulsive and reducing faster than a Newtonian force when far away from the sun [4]. When the reduction is going very slowly as the distance increases, the net effect may appear as a constant additional force that observation suggests. Recently, a very weak repulsive neutral force of charge-mass interaction has been derived from general relativity [2, 5]. Since the sun has charged particles, this neutral force would be a suitable candidate. The force would be the infinitesimal force at play to produce the anomaly since it is reducing faster than the Newtonian force at a long distance as the distance increases (see Section 6).

On the other hand, if general relativity is essentially correct, then it must be able to produce the observed anomaly. In this paper, it will be shown that so far the charge-mass interaction derived from general relativity would provide the most natural explanation for the pioneer anomaly as well as other planetary anomalies [1, 4]. Thus, what is being called as anomaly is actually normal, and such a term just minds 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 [6, 7]. 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 [8, 9] 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 [10] 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 [9, 11]. This new coupling sign is necessary [9, 11] to explain the binary pulsars experiment of Hulse and Taylor [12]. 

         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 [13] 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 [14]. 

         The non-equivalence between electromagnetic energy and mass is also confirmed by general relativity. This is shown by the Reissner-Nordstrom metric [15-17] (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⁾ [18, 19]) 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. It is probably due to the publication this metric that convinced Einstein [20, 21] that validity of E = mc² is only conditional. Nevertheless, because of inadequate understanding of Einstein's equivalence principle (see Appendix), theorists such as Will ³⁾ still hold on the incorrect interpretation of unconditional validity, especially in his highly praised book.

    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 [22, 23], 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 [24] 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 [7, 15, 23]. ⁴⁾ 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 [25], 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 is [26],

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

In this equation, the energy stress tensor T_mn is the sum of any type of energy-stress tensor. For the Reissner-Nordstrom metric, it includes at least the massive energy-stress tensor and the electromagnetic energy-stress tensor. They differ by that the electromagnetic energy-stress tensor is traceless whereas the massive energy-stress tensor is not. Since R (= ) is independent of the electromagnetic energy-stress tensor, the electric energy cannot be equivalent to a mass. ⁵⁾

         In short, the electric energy cannot be equivalent to mass. 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 existence of a charge-mass repulsive effect is firmly established and the unconditional validity of E = mc² is a misinterpretation.

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

         To show the repulsive effect, one needs to consider only g_tt in metric (1). According to Einstein [5, 13],

              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} [27]. 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.⁶⁾ 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. ⁷⁾ 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.⁸⁾ Clearly, this force is independent of the charge sign. Such characteristics would make the repulsive effects easier to be verified [7] since a concentration of electrons would increase such repulsion.

         The term of the repulsive force in (1) comes from the electric energy [2, 7]. 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 [29]. 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.

5. Extension of Einstein's Theory and the Five-Dimensional Relativity 

         Obviously, to accommodate the mass-charge interaction, unification between gravity and electromagnetism is necessary. In fact, Kaluza [30] proposed a five-dimensional general relativity, and this maintains the equation of motion as being a geodesic equation. However, based on the cylindrical condition ⁹⁾ that reduces the five variables to four, this theory reproduces the Einstein equation and the Maxwell equation if the "extra" metric elements are considered as constant or negligible [31]. However, the theory of Lo et al. [28] has no cylindrical condition, and thus would include the charge-mass interaction [2, 3].

The five-dimensional relativity does have the coupling with the square of a charge if the "extra" metric elements are retained. If cylindrical condition is not imposed, the radiation reaction force would also be accounted for [28]. Now let us give a brief introduction of the five-dimensional relativity. The geodesic is

         ，  (9a)

                                            (9b)

where ,    m, n = 0, 1, 2, 3, 5        ( ;  k, l = 0, 1, 2, 3)     .

If instead of ds, dt is used in (9), the Lorentz force suggests

         .

Thus,

         ,       and                            (10)  

where K is a constant. It thus follows that

         ，                   (11a)

                                                  (11b)

One may ask what the physical meaning of the fifth dimension is. Note that although the string theorists talk about space of much higher dimensional, they have no physical reason except for mathematical validity of their speculation. They claimed that those dimensions are curl up. Our position is that the physical meaning the fifth dimension is not yet very clear [28], except some physical meaning is given in equation (10) that relates the fifth variable x⁵ to t.

         The fifth dimension is assumed [28] as part of the physical reality, and the metric signature is (+, -, -, -, -). Our approach is to find out the full meaning of the fifth dimension as our understanding gets deeper. Nevertheless, we shall denote the fifth axis as the w-axis (w stands for "wunderbar", in memorial of Kaluza). In Physics things are not defined right at the beginning. For example, it takes us a long time to understand the physical meaning of energy-momentum conservation.

         For a static case, from (11) and (8), we have the forces on the charged particle Q in the -direction ⁸⁾

         ,       and                           (12a)

and

            where                                  (12b)

in the (-r)-direction. The meaning of (12b) is the energy momentum conservation. It is interesting that the same force would come from different type of metric element depending on the test particle used. Thus,

         ,                 and            constant。                                 (13)

In other words, g₅₅ is a repulsive potential. Since g₅₅ depends on M, it is a function of local property, and thus is difficult to calculate. From (12a), however, we need only g₅₅/Mc² = K²m/r² + constant.

         On the other hand, since g₅₅ is independent of q, (¶g₅₅/¶r)/Mc² depends only on the distant source m. Thus, this force, though acting on a charged particle, would penetrate electromagnetic screening. From (13), it is possible that a charge-mass repulsive potential would exist for a metric based on the mass M of the charged particle Q. However, since P is neutral, there is no charge-mass repulsion force (from G_{k, 55}) on P.

       Since the sun has charged particles and this force has a long range, it should participate in celestial mechanics. We conjecture that it would be the cause of the pioneer anomaly. This will be explained in the next section.

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

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

         F =                                                                                                (14)

in the r-direction. It would behave very differently from an attractive force, which is inverse proportional to the square of the distance r_. Thus, its effect would lead to the off course of the Pioneers. Please note also that this force is not subjected to electromagnetic screening.

         The data of space probes-Pioneer 10 and Pioneer 11 would give a good opportunity to check validity of the mass-charge interaction. If the repulsive force comes from the sun, then m in (14) 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 we do not know for this case of the sun 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. Then, the effective repulsive force F_p would be

         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, ¹⁰⁾ i.e.

F = ,        and                    for   R₀              (16)

Then, we have

F = .                                                                       (17)

Thus, there is an additional attractive force for R > R₀. 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 the orbit of a planet if such an orbit is essentially a closed path and thus the variation of the distance from the sun is small. However, for open orbits such as the pioneers, there is a great variation of the distance from the sun. 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. Thus, our conclusion that the charge-mass interaction is the unified cause of all anomalies is just as Anderson [1, 4] predicted.

7. Conclusions and Discussions

         It has been shown that the theoretical framework of general relativity is inadequate, and modification is necessary. Then, the necessity of unification is firmly established. Thus, the limitation of Einstein's general relativity for a four-dimensional space leads to extending Einstein's theory to a five-dimensional space.

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. Therefore, the repulsive force would not be subjected to such screening. It thus follows that such a force is a good test for the existence of a five-dimensional space. Moreover, this can be verified by simply weighting a capacitor before and after charged.

         Current theory of Einstein would predict that the weight would increase slightly because the increment of energy.¹¹⁾ However, in a five-dimensional theory, the charge-mass repulsive force is not subjected to screening, 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.^{12), 13)}

Thus, the five-dimensional theory would predict the existence of the charge-mass repulsive force (see also Sections 5 & 6), 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 [32, p. 263], it is expected that the current-mass force would be an attractive force. ¹⁴⁾ However, in the space probe, the current-mass interaction for this case would be negligible. Details of this issue will be discussed in a separate paper. Nevertheless, such a repulsive force has been established although its r^-3-depencence remains to be verified experimentally. Moreover, the parameter P_s in (15) due to the charges of the Sun can be obtained from measurements.

         Gravitation was considered as producing attractive force only, and all the coupling constants were assumed to have the same sign. Such an assumption could be considered as natural because it was misunderstood that all types of energy are equivalent to mass. Recently, it is proven that for the radiation of binary pulsars the coupling constants must have different signs [9, 11]. Now, it is clear that the electromagnetic energy would not be equivalent to massive energy. Thus, general relativity implies that the Newtonian picture is just too simple for a phenomenon as complicated as gravity that relates to everything.

         Moreover, this paper further confirms that general relativity of Einstein was not yet a self-consistent complete theory [10, 33, 34]. It follows that general relativity is not yet ready for the stage of an overall unification. Its completion naturally leads to the necessary existence of the fifth force, and thus the existence of orbital anomaly follows. In other words, pioneer anomaly actually is not an exception, but a natural consequent 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 Einstein's "covariance principle" was a major mistake [35, 36], due to inadequate deliberation distinguishing the difference between mathematics and physics.¹⁵⁾ Newton and Einstein are essentially right although a small force is neglected. Moreover, a new weak repulsive force would still be derived within the theoretical framework of relativity. This new force implies also that the unification of gravity and electromagnetism is clearly necessary. These strongly manifest that the newly discovered force 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.

         The weight reduction of charged capacitors supports the existence of a neutral mass-charge interaction. The spinning superconducting top experiments [37] support the mass-current interaction.¹⁴⁾ The pioneer anomaly, which is due to the repulsiveness and r^-3-dependence, further confirms the existence of a mass-charge repulsive force. Note that when r is very large, the r ^-2 and r^-3-dependence over a not too long distance would make the effect as if a constant "additional" force as data suggested. Moreover, since the r ^-3-dependency is derived from general relativity, this would also be a test of the Einstein's general relativity. In other words, if such a dependency were not supported, general relativity would have to be reexamined thoroughly.

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

         Einstein's equivalence principle is stated clearly in "The Meaning of Relativity" [4] 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 preliminary application on the notion of equivalence with uniform Newtonian gravity [13]. Nevertheless, Fock [38] attempted to show such a metric as follows:

ds² = g _{t t} (x) c²dt² - dx² - dy² - dz².     Fock [38] also assumed that the metric is static and this would guarantee his failure [39].

         What is new in Einstein's equivalence principle is the Einstein-Minkowski condition as a consequence [13, p. 161]. Einstein used the Einstein-Minkowski condition as an assumption for his calculations [4; p. 91, 13; p. 161].

The Einstein-Minkowski condition has its foundation from theorems [40] 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, but 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 [41] is essentially a corrupted version of these theorems. In fact, Einstein [13, p. 144] has given an example that illustrates Pauli's errors. Nevertheless, theorists commonly failed to recognize the errors of Pauli, and mistaken [42] Pauli's version the same as Einstein's equivalence principle. In fact, there was no reference, other than Einstein's own work, that states and explains Einstein's equivalence principle correctly [39].

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 [4]. In a uniformly accelerated frame, the local space in a free fall is a Minkowski space according to special relativity. ⁶⁾ However, Einstein over-looked that the Einstein-Minkowski condition is actually in conflict with his interim "covariance principle", which has recently been proven as invalid [35, 36].

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 [18, 19].
• 3) Clifford M. Will was the President of the International Society on General Relativity and Gravitation. He incorrectly insisted on the unconditionally validity of E = mc² [6, 33, 34]. Moreover, just as Pauli [41], C. M. Will also misunderstand Einstein's equivalence principle. Following Will, some theorists incorrectly claimed [11] that the difference between Einstein's equivalence principle and Pauli's version is just a matter of philosophy.
• 4) 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.
• 5) 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 [26].
• 6) 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 [25].
• 7) 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).
• 8) 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).
• 9) A rigorous cylindrical condition may not be compatible with Kaluza's theory [42].
• 10) These problems are routinely neglected in calculating the obits of a planet.
• 11) By combining the electromagnetic energy with other energy such as in the case of photons [14], 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 [22].
• 12) 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 [28] although previously he got mixed results of both lighter and heavier weights. Also, his weighting of magnets are consistent with the claim of J. A. Wheeler [31, p. 263].
• 13) 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.
• 14) Recently, Martin Tajmar and Clovis de Matos [37], 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!
• 15) Some claimed [44] that the so-called physically measurable values (or gauge invariant measurable quantities as Logunov [45] called it) are used for experimental verification of the theory. However, when the measurement is considered, such values actually manifest gauge-dependence [35], and thus are just mathematical illusions.

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先锋号如果受到来自太阳的斥力, 它的速度就会变快, 事实是它们的速度部变慢了,

这与斥力作用的结果相反, 鲁先生把方向搞错了, 你的英语水平很高, 但是英国人也

有出错的, 不能因为你写了很多英文我就说你正确! 不过, 一开始时我还是大惊失色

过, 以为是英国科学家来了。