The Reissner-Nordstrom Metric, Invalidity of E = mc2, and Einstein’s Unification

C. Y. Lo

Applied and Pure Research Institute17 Newcastle Drive, Nashua, NH 03060

August 2011

Abstract

The formula E = mc2 is often misinterpreted as unconditionally valid. This unverified speculation is often used for support -

ing invalidity in physics. This error is, in part, responsible for the misinterpretations of the Reissner-Nordstrom metric of a

charged particle. Thus, the charge-mass interaction that could have been discovered from this metric is over looked for about

90 years. Recently, experiments have verified the existence of such a new charge-mass repulsive force, and thus reject the mis-

interpretations. Concurrently, the nonequivalence of mass and electric energy is also clearly verified, and thus the photons

must included non-electromagnetic energy. Therefore, current theories based on unconditionally validity of this formula

should be revised with new supports. This new force demands a new coupling from the unification of gravitation and electro -

magnetism, and the existence of a five-dimensional space that Einstein conjectured. Then, the weight reduction of a charged

capacitor, which is beyond general relativity, can be understood. Thus, a new chapter for both experimental and theoretical

physics is now opened.

04.20.-q, 04.20.Cv

Key Words: Riessner-Nordstrom metric, Pekeris metric, E = mc2, the charge-mass interaction, weight reduction, charged ca-

pacitor, space probe pioneer anomaly, volcano, five-dimensional physical space.

“Science sets itself apart from other paths to truth by recognizing that even its greatest practitioners sometimes err. … We rec -

ognize that our most important scientific forerunners were not prophets whose writings must be studied as infallible guides—

they were simply great men and women who prepared the ground for the better understandings we have now achieved.”

-- S. Weinberg, Physics Today, November 2005.

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1. Introduction

General relativity is established on two principles [1, 2], namely: 1) Einstein’s equivalence principle, which requires

the Einstein-Minkowski condition that a free falling point-like particle in a gravitational field is along a geodesic and results in

a co-moving local Minkowski space; and 2) the principle of general relativity, that is "The law of physics must be of such a

nature that they apply to systems of reference in any kind of motion." However, in current theory, there are other additional

implicit assumptions such as the formula E = mc2 is unconditionally valid [3-5]. In fact, Einstein had tried very hard for years

(1905-1909) to prove this formula to be generally valid, but failed [6].

The formula can be traced back to special relativity, which suggested a rest mass m 0 has the rest energy of m0c2. This

is supported by the nuclear fissions with ΔE = Δmc 2, where Δm is the mass difference after the fission and ΔE the total energy

created and is usually a combination of different types of energy. However, for an arbitrary energy E, the reverse relation m =

E/c2 is only an unverified speculation. Many believed that the equivalence of electromagnetic energy and mass was verified

[7]. However, this is in conflict with electromagnetism because the trace of an electromagnetic tensor is zero, but the trace of

an energy-stress tensor of massive matter is non-zero. 1)

On the other hand, it is observed that a π0 meson would decay into two γ rays. However, this only means that the pho -

tons must include non-electromagnetic energy, which has been identified later as the gravitational energy [8]. If the photons

have only electromagnetic energy, the sum is also electromagnetic energy, but the photons, being massless particles, can cre -

ate a massive energy-stress tensor. 2) Thus, Einstein’s proposal of a photon being a quantum of electromagnetic energy [1] is

actually inadequate. However, Einstein was limited by the fact that his general relativity has not been created then.

The unverified speculation E = mc2, being unconditionally valid, would imply that gravity would always generate an

attractive force since masses attract each other. This will be proven to be unequivocally invalid by the Riessner-Nordstrom

metric, with the help of experiments. However, misinterpretations were created in attempts to save such a situation. In this pa -

per, it will be shown first that such effects are actually in conflict with the derivation of the Riessner-Nordstrom metric.

2. The Reissner-Nordstrom Metric.

Now, let us examine the Reissner-Nordstrom metric [3, 9, 10] (with c = 1) as follows:

, (1)

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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 3) [11,

12]) from the particle center. In this metric (1), the gravitational components generated by electricity have not only a very dif -

ferent radial coordinate dependence but also a different sign that makes it a new repulsive gravity in general relativity.

Nevertheless, 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 [13, 14], and thus (2) could be interpreted as supporting m =

E/c2 at least for electromagnetic energy. If the electric energy has a mass equivalence, an increase of such energy should lead

to an increment of gravitational strength. However, the strength of a gravitational force, from metric (1), decreases every-

where.

Moreover, the gravitational forces would be different from the force created by the “effective mass” M – q2/2r because

. (3)

Nevertheless, it is difficult to overcome a prevailingly accepted speculation. Consider also the static Einstein equation [5],

G R – 1

2gR = – 8 T , where R = , (4)

and T is the sum of energy-stress tensors. Since the electromagnetic energy-stress tensor is traceless R (= ) is in-

dependent of the electromagnetic energy-stress tensor. Thus, according to general relativity, the electric energy cannot be

equivalent to a mass. 4) Thus Will was defeated because he cannot defend his interpretation of m = E/c2 [15].

The general validity of E = mc2 was questioned since the binary pulsars experiment that the coupling constants neces-

sarily have different signs [16]. Nevertheless, with supports from editorials of Nature, the Physical Review D, and Science,

Will continued to misinterpret [17] the formula. Also, theorists such as Herrera et al. incorrectly argued that M in (1) includes

the external electric energy [18]. To see this clearly, one has to go through the derivation of the Reissner-Nordstrom metric.

3. Derivation the Reissner- Nordstrom Metric and its Misinterpretations

It seems that mass M in (1) as a “total mass” that includes the electric energy, would be allowed if you are careless.

However, a close examination shows that this invalid.

According to Einstein, the static field equation for the metric is [3],

G R – 1

2gR = – 8 T , (5)

or

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R = – 8[T – g T], where T = .

In this equation, the energy stress tensor T 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 elec -

tromagnetic energy-stress tensor is traceless whereas the massive energy-stress tensor is not.

If one assumes that the metric has the following form,

,

(6)

then, as shown by Wald (1984), at the region out side the particle (r > r0) we have

– R00 = , (7a)

– R11 = , (7b)

– R22 = (7c)

Moreover, outside the particle we have

T(m) = 0 for r > r0 . (8a)But

T(m)00 = ( r), T(m)11 = T(m)22 = T(m)33 = P( r), when r < r0 (8b)

where P(r) is the pressure of the perfect fluid model.

Because of the electric energy-stress tensor T(E) is traceless, we also have, for r > r0,

R00 = –R11 = R22 = – E2, where (9)

is the electric field, according to Misner et al. [1973; p. 841]. If h = 1/f in metric (6), then (7) is reduced to

– R00 = R11 = = (10a)

And– R22 = = (10b)

Moreover, if as in metric (1), then we have, in consistent with (9),

(11)

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Thus, from the above derivation, it seems there is no restriction on the mass M of metric (6). However, from (9), it is clear that

M in metric (6) cannot include the electric energy (out side the particle) since it has been represented in (9).

Nevertheless, Herrera et al. argued that M in (1) involves the electric energy [18]. They follow the error of Whittaker

[19] and Tolman [20] who believed the equivalence of mass and electric energy. They defined the active gravitational mass

density μ with the electromagnetic energy tensor Eαβ as and the active mass in a volume Va is given by

, (12)

where g is the four dimensional determinant of the metric. It thus follows that, for a particle with charge Q, one has

, and , where (13)

Thus ma(r) would be in agreement with that the total force is proportion to

(14a)

since = , (14b)

where r0 is the radius of the particle. However, (13) does not agree with (2) since

(14c)

Eq. (14a) implies that the weight of a charged metal ball would increase when the charge Q is increased, According to eq.

(12), ma(r0) would only increases as the charge Q increases. Thus, no repulsive effects can be detected. However, as shown in

(14b) a problem is that, in conflict with (9), M includes energy outside the particle. If the mass M is just the inertial mass of

the particle, the weight of a charged metal ball can be reduced [21]. Thus, as expected, experiments of two metal balls [22] re -

ject eq. (13).

The inertial mass of the particle should be smaller than M defined in (14b) since an acceleration of the charged parti -

cle would not immediately affect the electric energy at long distances. Note that the radius re of an electron is much smaller

than a half of its classical radius e2/m0c2 [23], where e is the charge of the electron and m0 is its inertial mass. Accordingly, the

total external electric energy e2/2re would be much larger than m0.

Note that the problem started from the assumption of the incorrect assumption of equivalence between mass and elec -

tric energy. If electric energy is assumed as equivalent to mass, should it be considered as part of the gravitational mass of the

particle or not. If it is, then gravitational mass and inertial mass are different. If it is not that means any electromagnetic en-

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ergy should assign a mass. If any electromagnetic energy should assign a mass equivalence, then this also reject the notion that

a photon is massless and also special relativity. Thus, the electric energy should not be equivalent to mass.

The approach of Herrera et al. is essentially the same as that of Pekeris [24], who gets a similar metric earlier as fol-

lows:

where R3 = r3 + r03 (15a)

, where Mem = Q2/r0, and M = Mmat + Mem (15b)

The difference is due to that Pekeris (1982) requires that . Thus, the approach of Herrera et al. [18] is essen-

tially what Pekeris had done. Apparently, theorists have run out of ways that can be used to against the repulsive force.

Nevertheless, some claimed that the Pekeris metric still has the repulsive effort. This is clearly a lack of physical un-

derstanding of the whole situation. 5) In summary, although the Riessner-Nordstrom metric and the other two metrics look the

same, they are different because the mass M means different in different metric. However, the Riessner-Nordstrom metric can

explain this force only for a special case. For the case of charge capacitor, the observed repulsive force is beyond general rela-

tivity.

4. The Charge-Mass Repulsive force and Unification.

To show the static repulsive effect, one needs to consider only gtt in metric (1). According to Einstein [1, 2],

where (16)

and . Let us consider only the static case. (One need not worry whether the gauge is physically valid since

the gauge affects only the second order approximation of gt t [25].) For a particle P with mass m at r, the force on P is

(17)

in the first order approximation since gr r -1. Thus, the second term is a repulsive force.

If the particles are at rest, then the force acts on the charged particle Q has the same magnitude

( ) , where is a unit vector (18)

since the action and reaction forces are equal and in the opposite directions. 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 geo-

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desic equation is inadequate for the equation of motion. Moreover, since the second term is proportional to q2, it is not a

Lorentz force.6) It is also not a 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 gener-

ated by masses. It thus follows that, force (18) to particle Q is beyond current theoretical framework of gravitation + electro-

magnetism. 7) In other words, as predicted by Lo, Goldstein, & Napier [26], general relativity leads to a realization of the inad-

equacy of general relativity, just as electricity and magnetism lead to the exposition of their shortcomings.

The charge-mass repulsive force mq2/r3 for two point-like particles is inversely proportional to the cube power (instead of

the square) of the distances between the two particles.8) This would mean that such a repulsive force is much weaker faster

than gravity at long distance, although it would be much stronger at very small distance. Moreover, this force is proportional

to the square of the charge q, and thus is independent of the charge sign. Such characteristics would make the repulsive effects

verifiable [21] since a concentration of electrons would increase such repulsion.

The term of repulsive force in (1) comes from the electric energy [5]. 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 indepen -

dent of a charge sign, would not be subjected to such a screening 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.

5. Extension of Einstein’s Equivalence Principle and the Five-Dimensional Relativity

If we consider the coupling with q2, this natural leads to a five dimensional space. Kaluza [27] proposed a five-dimen -

sional general relativity, and this maintains the equation of motion as being a geodesic equation. Based on the cylindrical con -

dition 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. Subsequently, Einstein and Pauli [28] wrote a paper to con -

tinue the work of Kaluza. 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

[26].

One may ask what the physical meaning of the fifth dimension is. Many theorists claimed that those high dimensions are

curl up. Our position is that the physical meaning of the fifth dimension is not yet very clear [1], except some physical mean-

ing is given in the equation, dx5/dτ = q/Mc2K where M and q are respectively the mass and charge of a test particle, and K is a

constant. This equation relates the fifth variable x5 to .

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The fifth dimension is assumed [1] as part of the physical reality, and the metric signature is (+, –, –, –, –). Our approach

is to find out the full physical 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, we have the forces on the charged particle Q in the -direction 7)

, and (19a)

and

where (19b)

in the (-r)-direction. The meaning of (19b) 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。 (20)

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 cal-

culate. This is different from the metric element g t t that depends on a distant source of mass m.

On the other hand, since g55 is independent of q, (g55/)/M 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 k, 55) on P.

At the end of this section, we quote a remark by Einstein and Pauli [28], who wrote in 1943

“When one tries to find a unified theory of gravitational and electromagnetic fields, he cannot help feeling

that there is some truth in Kaluza’s five-dimension theory.”

It may turn out that their observation would be a prophecy for the future advancement of such unification.

5. The Static Repulsive Charge-Mass Interaction

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

F= , (21)

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and it would behave very differently from an attractive force, which is inverse proportional to the square of the distance r . In

fact, it has been shown that the repulsive force on the charge q is sensitive to the surrounding of the charge [29]. Note that the

dependence of r –3 in (21) is derived from general relativity. Thus, this force can be used to test general relativity.

Although such a force (21) is small between a charged particle and a massive particle, such a force between a metal

ball charged with Q and a massive particle with mass m can be tested experimentally [21] if the charge is large enough. This is

so because such a force would be Q2m/R3, where R is the distance from the center of the metal ball [21]. According to the

Riessner-Nordstrom metric, the repulsive force would increase with the square of the number of free electrons in the ball, but

the attractive force would increase with only the number of electrons [21].

The existence of such a repulsive force would be an unequivocal evidence for the non-equivalence between electric energy

and mass. Also, a repulsive force on a charged capacitor would be an evidence for the fifth dimension since such force acting

on a charged capacitor is beyond general relativity. Although it would be difficult to calculate such a force it is expected that

the weight reduction of a charged capacitor would be increased with the increment of the charge voltage being not too high.

In current theory, the charge-mass repulsive force would exist on a charged capacitor because the electromagnetic field is

subjected to screening. Thus, current theory of Einstein would predict the weight of a charged capacitor would increase

slightly because the increment of energy.10) From the viewpoint of physics, however, 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 un -

derstood 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 test for the existence of a five-dimensional space. Moreover, this force

can be verified by simply weighting a capacitor before and after charged. In a charged capacitor, both the positive and the neg -

ative charges are concentrated, and thus an effect of the repulsive force would be observed as a lighter weight for the charged

capacitor.11), 12)

6. Conclusions and Discussions

Moreover, if the investigation of electric energy leads to a charge-mass repulsive force, the magnetic energy would simi -

larly generate a current-mass force. According to the effect of a magnetic field in general relativity [30, p. 263], it is expected

that the current-mass force would be an attractive force, and details of this force need further investigation. Nevertheless, such

a new attractive force has been verified by the experiments of Tajmar and Matos [31] from the European Space

Agency.13) Thus, the charge-mass interaction and the current-mass interaction would cancel each

other in the normal state.

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Now, it is clear that the charged capacitor becomes lighter because the balance has been dis-

rupted. Thus, the repulsive force is due to the state of motion for the electrons have been changed al -

though the number of charges remains the same. Thus, it is expected that it would take a while be -

fore a discharged capacitor to recover back to its original weight. Accordingly, one can expect that

the metal would reduce its weight after being heated up. This prediction would be directly in dis -

agreement with Einstein who predicted a piece of heated up metal would be heavier.

In conclusion, general relativity is not yet ready for the stage of an overall unification. Thus, it is unrealistic to expect the

string theorists to perform a miracle in unification. Moreover, it should be noted that many string theorists based their works

on the positive mass theorem of Yau [32] and the positive energy theorems of Witten [33]. However, these theorems are based

on an implicit but invalid assumption that the signs of coupling constants are unique [16]. This invalid assumption was used

earlier by Penrose and Hawking [5]. However, the origin of such an error is also the unconditional validity of the formula E =

mc2.

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 mistake [34-37]. Since journals specialized on gravitation failed to identify such prob-

lems, understandably there is little progress on some areas of gravitation. Now, since the existence of the charge-repulsive

force is established, the unification of gravity and electromagnetism is clearly necessary. It should be noted that the weight re -

duction experiments have been started around 1960, but nobody was able to explain them before the unification is considered

[38-40].

Since the charge-mass interaction is a long range interaction, it must have some interesting consequences in astrophysics.

A well known problem is NASA’s discovery of the Space Probe Pioneer Anomaly [41-43]. Currently, the repulsive charge-

mass force is the only candidate that has not been rejected by the data [44, 45]. It seems a simulation to test this force would

be useful. Moreover, since there are two forces, one attractive another repulsive with different strengths and distance depen -

dences, it is possible that these forces would form a coupling that would have an effect on the spins of the planets. Another

possible speculation is that such a coupling would supply the energy that heats up the planets. Current explanation for this as

due to radiation is not satisfactory since there is no radioactive material discovered from the volcano. However, as Einstein

[46] pointed out, every theory is speculative and needs experimental verifications.

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Acknowledgments

The author gratefully acknowledges stimulating discussions with S. Y. Auyang, David P. Chan, S. -J. Chang, A. J. Coleman,

S. Crothers, Z. G. Deng, Richard C. Y. Hui, G. R. Goldstein, C. C. Lo, A. Napier, J. J. Pi, D. Rabounski, Eric J. Weinberg,

and D. X. Yue. Special thanks are to S. Holcombe for valuable suggestions. This work is supported in part by Innotec Design,

Inc., USA. and the Chan Foundation, Hong Kong.

ENDNOTES

1. The electric energy cannot be equivalent to a mass since they are related to tensors that are not equivalent [8, 13, 17].

2. Many theorists have overlooked this crucial point, and thus incorrectly regarded that the equivalence of electromagnetic

energy and mass had been proved [7] as Einstein mistakened [1].

3. The existence of a Euclidean-like structure in the frame of reference is necessary [11. 12].

4. For example, in the Schwarzschild solution, R is zero at vacuum, but not zero in the interior solution [5].

5. In the Pekeris metric, repulsive effects due to the term Q2/R2 is canceled by the misinterpretation of the mass term M.

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 [23].

7. We calculate the field of generated by charge particle Q, then the force acting on 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, which is often 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/)dt2 – (1- 2m/)-1d2 – 2d ‘2, where (, ’, ’) is a new coordinate system with

P at the center. Thus, the force on Q in the -direction would be only –M(m/2) since the Lorentz force and the radiation

reaction force are absent. Since the distance between P and Q is r = , there should be another term in the -direction as

q2(m/3).

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 > me (e = 1.381 10–34 cm, me = 6.764 10–56 cm).

9. A rigorous cylindrical condition may not be compatible with Kaluza’s theory [47].

10. Although the electromagnetic energy is not equivalent to mass, it can be equivalent to mass by combining with other en -

ergy such as in the case of photons [8]. In other word, for total energy ET, Einstein’s formula, ET = mT c2, is still valid [13,

14]. In a charged capacitor, the increased energy is not pure electromagnetic energy since an electron has mass, and etc.

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11. Experimentalist W. Q. Liu (http://www.cqfyl.com) performed the weighting of capacitors in a Chinese Laboratory of the

Academy of Science, and got certified results of lighter capacitors after charged [48] although previously he got mixed

results, both lighter and heavier weights after a capacitor being charged because the situations were not stable. Also, his

results on weighting of magnets are consistent with the claim of J. A. Wheeler [27, p. 263].

12. According to m = E/c2, the mass increment of a charged capacitor is negligible. For a capacitor of 200F charged to 1000

volt, the related mass increment would be about 10 -12 gram. Thus, many rejected experimental results of Liu as just errors

although they also found unstable results of weighting that should not have been there according to current theory.

13. Martin Tajmar and Clovis de Matos [31], from the European Space Agency, found that a spinning ring of superconduct -

ing material increases its weight much more than expected. However, according to quantum theory, spinning supercon -

ductors should produce a weak magnetic field. Thus, they are measuring also the interaction between a current and the

earth!

References:

1. A. Einstein, H. A. Lorentz, H. Minkowski, & H. Weyl, The Principle of Relativity (Dover, New York, 1923).

2. A. Einstein A, The Meaning of Relativity (1921) (Princeton Univ. Press, 1954).

3. C. W. Misner, K. S. Thorne, & J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

4. W. Pauli, Theory of Relativity (Pergamon, London, 1958).

5. R. M. Wald, General Relativity (The Univ. of Chicago Press, Chicago, 1984).

6. "Einstein's Miraculous Year" edited by John Stachel, Princeton Press (1998), p. 118.

7. H. C. Ohanian, “Einstein's Mistakes: The Human Failings of Genius (Norton, New York, 2008).

8. C. Y. Lo, Completing Einstein’s Proof of E = mc2, Progress in Phys., Vol. 4, 14-18 (2006).

9. H. Reissner, Ann. Phy. (Germany) 50, 106-120 (1916).

10. G. Nordstrom, “On the energy of gravitational field in Einstein’s theory,” Proc. Kon. Ned. Akad. Wet. 20, p. 1238 (1918).

11. C. Y. Lo, Phys. Essays, 15 (3), 303-321 (2002).

12. C. Y. Lo, Chinese J. of Phys., Vol. 41, No. 4, 332-342 (2003)

13. C.Y. Lo, Chin. Phys., 16 (3) 635-639 (March 2007).

14. C. Y. Lo, Bulletin of Pure and Applied Sciences, Vol. 25D, No.1, 41-47, 2006.

15. C. M. Will, Theory and experiment in gravitational physics (Cambridge Univ. press, Cambridge, 1981).

16. C. Y. Lo, Astrophys. J. 455, 421 (1995); Editor S. Chandrasekhar, a Nobel Laureate, suggests and approves the Appen-

dix: The Gravitational Energy-Stress Tensor for the necessity of modifying Einstein equation.

17. C. Y. Lo, Astrophys. J. 477, 700 (1997).

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18. L. Herrera, N. O. Santos and J. E. F. Skea, Gen. Rel. Grav. Vol. 35, No. 11, 2057 (2003).

19. E. T. Whittaker, Proc. R. Soc. (London) A 149, 384, (1935).

20. R. Tolman, Phys. Rev. 35, 875 (1930).

21. C. Y. Lo & C. Wong, Bull. of Pure and Applied Sciences, Vol. 25D (No.2) P.109-117 (2006).

22. D. Yu. Tsipenyuk, V. A. Andreev, Physical Interpretations of the Theory of Relativity  Conference, Bauman Moscow

State Technical University, 4 – 7 July 2005. e-mail：andrvlad@yandex.ru.

23. J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, New York, 1962).

24. C. L. Pekeris, Proc. Nat. L Acad. Sci. USA Vol. 79, pp. 6404-6408 (1982).

25. S. Weinberg, Gravitation and Cosmology (John Wiley Inc., New York, 1972).

26. C. Y. Lo, G. R. Goldstein, & A. Napier, Hadronic Journal 12 (2) 75-89 (1989).

27. Th. Kaluza Sitzungsber, Preuss. Akad. Wiss. Phys. Math. Klasse 966 (1921).

28. A. Einstein & W. Pauli, Ann. Math. 44, 133 (1943).

29. C. Y. Lo, Phys. Essays 21 (1), 44-51 (March 2008).

30. K. S. Thorne, Black Holes and Time Warps (Norton, New York, 1994).

31. http://news.softpedia.com/news/The-First-Test-That-Proves-General-Theory-of-Relativity-Wrong-20259.shtml M. Tajmar

and C. de Matos. 2006.

32. R. Schoen and S.-T. Yau, “Proof of the Positive Mass Theorem. II,” Commun. Math. Phys. 79, 231-260 (1981).

33. E. Witten, “A New Proof of the Positive Energy Theorem,“ Commun. Math. Phys., 80, 381-402 (1981).

34. C. Y. Lo, Phys. Essays 18 (1), 112-124 (March 2005).

35. Zhou, Pei-Yuan, in Proc. of the Third Marcel Grossmann Meetings on Gen. Relativ. ed. Hu Ning, Sci. Press/North Hol-

land. (1983), 1-20.

36. P. Y. Zhou, Proc. of the International Symposium on Experimental Gravitational Physics, Guang Zhou, China (1987).

37. C. Y. Lo, Phys. Essays, 23 (3), 491-499 (Sept. 2010).

38. T. Musha and T. Kanamoto, Proc. of the 38th Space Science and Technology Conference, JSASS, 1994, pp. 31–32.

39. T. Musha, Proceedings of the 37th Conference on Aerospace Propulsion, JSASS, 1997, pp. 342–349.

40. D. R. Buehler, J. of Space Mixing 2, 1 (2004); see also electrogravitics in Wikipedia.

41. Robert Lee Hotz, “Newton, Einstein Lost in Space?—Scientist May be Getting Warmer in Finding Why Pioneer Probes

Veered off Course”, Wall Street Journal (May 16, 2008), PA7.

42. Charles Q. Choi, NASA Baffled by Unexplained Force Acting on Space Probes, SPACE.com, 3 March 2008.

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43. NASA’s Space-Probes Pioneer Anomaly and the Mass-Charge Repulsive Force, 17th Annual Natural Philosophy Al-

liance Conference, California State University, Long Beach, California June 23-26, 2010.

44. S. G. Turgshev, V. Toth, L. R. Kellogy, E. L. Lau, and K. J. Lee, “The Study of the Pioneer Anomaly: New Data Objec-

tives for New Investigation”, IJMPD, 15(1), 1-55 (2006).

45. C. Y. Lo, The Mass-Charge Repulsive Force and Space-Probes Pioneer Anomaly, Physical Interpretation of Relativity

Theory: Proceedings of International Meeting.  Moscow, 6 – 9 July 2009 / Edited by M. C. Duffy,  V. O. Gladyshev, 

A. N. Morozov, P. Rowlands – Moscow: BMSTU.

46. A. Einstein, ‘On the Generalized Theory of Gravitation’ in Ideas and Opinions (Crown, New York, 1954), p. 349.

47. C. Y. Lo, Hadronic Journal 12 (6) 275-276 (1989).

48. W. Q. Liu, private communication (August 2007).

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