, (1)Applied and Pure Research Institute17 Newcastle Drive, Nashua, NH03060USA
June 2007
It is found that the Reissner-Nordstrom metric enables to show that general relativity + electrodynamics are not yet a close system. In gravity, there is a repulsive effect due to the electromagnetic energy. Analysis of this effect shows that the geodesic equation as the equation of motion is inadequate. To include the force of charge-mass repulsion, modifications of electromagnetic and gravitational theories are necessary. Moreover, it is shown that the unification within the theoretical framework of a five-dimensional theory would resolve this problem because of the additional metric elements. In the five-dimensional theory of Einstein and Pauli, those elements were disregarded as having no physical meaning. Concurrently, a limitation of the formula E = mc2 is proven and experimental verifications of the new force are discussed. Thus, although the so-called “covariance principle” is proven as not generally valid, the full meaning of relativity is still emerging after 100 years of Einstein’s creation.
04.20.-q 04.20.Cv
Einstein initiated the unification of electromagnetism and gravity [1], but the need of unification has not yet been clarified [2].Einstein considered general relativity as logically complete [3]. Thus, in the five-dimensional theory of Einstein and Pauli [4], all the “extra” metric elements are regarded as having no physical meaning. Maxwell showed, however, that unification is a remedy to remove the shortcomings of the theories to be unified [1]. Accordingly, Einstein’s unification [2] would fail.
Nevertheless, the rise of Yang-Mills theory [5] results in a great advance of unifying the weak, the strong, and the electromagnetic [6]. Then, the unification with gravity is the next goal. Many claimed that the string theory would give this final unification. However, string theorists tried for more than a quarter of a century without any visible success [6]. Recently, critics started to openly question the validity of string theory, and even the relevance of unification [6, 7].
In this paper, it will be shown that general relativity is not a closed system as Einstein claimed [3]. Apart from the difficulty in mathematics, a hidden problem is that theorists do not understand general relativity and related theories yet. Thus, the real problem is that general relativity is not yet ready for the stage of unification. For instance, the editorial the Royal Society still rejects Einstein’s requirement on weak gravity [8, 9] since the “covariance principle” is proven invalid only recently [10]. Moreover, there are examples of unphysical metrics that can give the correct light bending,[11].
Most of those who work on the issue of unification are particle physicists or mathematicians, who are excellent in their own fields. Naturally, they rely on experts of relativity. Unfortunately, those perceived “experts” actually do not understand general relativity [12, 13, 14], and Feynman [15] was aware of their inadequacy.For instance, except in Einstein's original works, there are no textbooks or reference books (including the British Encyclopedia [2006]) that explained Einstein's equivalence principle correctly although this principle is stated squarely in page 57 of Einstein's book, “The Meaning of Relativity'” [2]. In addition, some of such theorists criticized Einstein without getting the facts straight first [12, 14].
About 25 year ago, we [16] conjectured that, as in the case of electromagnetism, the unification is due to internal inadequacy of such theories. However, it was very difficult to identify inadequacy in general relativity because things are not clearly defined. For instance, Einstein's equivalence principle [2, 17], was very difficult to apply. It took a long time to recognize that Einstein's theory was not even self-consistent because of two reasons. First, Einstein’s theory of measurement is actually inconsistent with his equivalence principle [18]. Moreover, he over-looked that his measuring instruments are in a free fall state and his method of measurement, in principle, may not be executable for the length of an extended object [20]. Second, the so-called “covariance principle” is not generally valid in physics [10]. In fact, Einstein’s argument for the justification of the “covariance principle” was actually not valid (see also Appendix).
However, problems seem to be rectifiable within the theoretical framework of general relativity [10, 12-14, 18-20].A major problem to be fixed is that Einstein's equation must be modified to have a dynamic solution [21, 22]. However, an equation of first order approximation, which was derived independent of the Einstein equation, would give dynamic solutions for massive sources [23]. It is interesting to note that Einstein and Rosen [22, 24] were the first who discovered the non-existence of wave solutions. Einstein also objected that Pauli’s version is a misinterpretation of his principle [25]. The famous formula E = mc2, 1)was “derived” in 1905 [17], but few other than Einstein saw the limitation of E = mc2 [26]. Moreover, Einstein did not see that his formula is inconsistentwith the notion that light consists of just electromagnetic waves [27].
Thus, when the Riessner-Nordstrom metric is used to show that m = E/c2 is not generally valid [28], some counter with misinterpretations [29-31]. However, a simple differentiation would uncover such interpretations invalid (seesection 2).
Moreover, the skeptics demand for additional experimental verification on the limitation of E = mc2.Then such an investigation leads to focusing attention to the Riessner-Nordstrom metric. This metric turns out to be a key to find shortcomings of the theoretical framework of relativity. In section 3, it will be shown that the geodesic equation as an equation of motion is inadequate and this cannot be fixed within the theoretical framework of general relativity + electromagnetism.
However, there are strong indications that such a problem can be resolved in the theoretical framework of a five-dimensional theory. Thus, the conjecture [1, 16] of more than 25 years ago that unification is due to internal inadequacy is proven. Now, it becomes obvious that any attempt to have a unification including gravity must study general relativity first. Thus, clearly the lack of progress in unification should not be blamed on string theorists alone.
2. The Reissner-Nordstrom Metric, and the Repulsive Effect
Ironically, the famous formula E = mc2 is also a formulathat many physicists do not understand properly [28]. Einstein himself has made clear that this formula must be understood in terms of energy conservation [26]. This formula means that there is energy related to a mass, but it does not mean that, for any type of energy, there is a related mass [26, 28, 30].
A root of misunderstanding E = mc2 is related to the fact that its derivation [17] has not been completed. A crucial step is Einstein’s implicit assumption of treating light as a bundle of massless particles. However, in Einstein’s derivation of 1905, gravity was not considered. Consequently, it was not aware that an electromagnetic energy-stress tensor is incompatible with the energy-stress tensor of massless particles [27, 29].
However, general relativity makes it explicit that the gravity generated by mass and that by the electromagnetic energy are different, as shown by the existence of repulsive effect in the Riessner-Nordstrom metric [32-34],
, (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 structure2) [19, 20]) from the particle center. In 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.
In fact, it is probably that the publication of this metric in 1916 and 1918 ended Einstein’s misconception starting from 1905 [35] that any energy related to a mass m = E/c2. However, such a misinterpretation [13, 14] is crucial to the unconditional universal couplingassumption3) for the singularity theorems of Hawking and Penrose [36]. Thus, some theorists would even ignore that the Hulse-Taylor experiment has proven the extended universal coupling is incorrect [13, 21]. Moreover, in his book Will [37] continue to use his misinterpretations m = E/c2 eight years after it has been proven incorrect [28].
Some argued that the effective mass in metric (1) is M – q2/2r (in the units, the light speed c = 1) since the total electric energy outside a sphere of radius r is q2/2r.4) However, they overlooked that the gravitational forces would be different. From metric (1), the gravitational force is different from the force created by the “effective mass” M – q2/2r because
.(2)
They achieved only exposing further an inadequate understanding in the theory of relativity [30-32].
Moreover, the nonequivalence between electromagnetic energy and mass can be obtained without detailed calculations. From Einstein equation Rmn – ?gmnR = 8pTmn [37], R is invariant with an additional electromagnetic energy-stress tensor T(E)mn in the source [31]. Thus, according to general relativity, the electromagnetic energy cannot be equivalent to mass.
To show the repulsive effect, one needs to consider only gtt in metric (1). According to Einstein [2],
where
(3)
and
are defined by the metric gmn. However, we consider only the static case. Thus,
,where
(4)
since gmn is also static. (One need not worry whether the gauge of the Reissner-Nordstrom metric is physically valid since the gauge affects only the second order approximation of gt t [38].) For a particle P with mass m at r, the force on P is
(5)
in the first order approximation since gr r@ -1. Thus, the second term is a repulsive force.
3. The Interaction with a Charge and Five-dimensional Theory
If the particles are at rest, then the force acts on the charged particle Q has the same magnitude
(
)
,where
is a unit vector(6)
since the action and reaction forces are equal and in the opposite directions. However, for the motion of the charged particle Q 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 forthe equation of motion. Moreover, since the second term is proportional to q2, it is not a Lorentz force either.5) In other words, it is necessary to have a repulsive force with the coupling q2 to the charged particle Q in a gravitational field generated by masses.
In conclusion, force (6) to particle Q is beyond current theoretical framework of gravitation+ electromagnetism. 6)However, in a five-dimension theory [1], the geodesic equation would include the coupling of q2. The geodesic is
(7a)
(7b)
where
,m, n = 0, 1, 2, 3, 5(
;k, l = 0, 1, 2, 3).
If instead of s, tis used in (8), the Lorentz force suggests
Thus,
,
and
(8)
where K is a constant. It thus follows that
(9a)
(9b)
Currently, theorists claimed that the higher dimensions are curl up. Our position is that the physical meaning the fifth dimension is not yet very clear [1], except some physical meaning is given in equation (8). The fifthdimension is assumed [1] as part of the physical reality, and the metric signature is (+,-,-,-,-). However, 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 “wenderbar”, in memorial of its Germany origin). Unlike mathematics, 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, it follows from (9) and (6), we have the forces on the charged particle Q in the
-direction 6)
,and
(10a)
and
where 
(10b)
in the (-r)-direction. Here particle P is at the origin of spatial coordinate system (r, q’, j’). It is interesting that the same force would come from different type of metric element depending on the test particle used. Thus,
,and
constant(11)
In other words, g55 is a repulsive potential plus a constant. Since g55depends on M, it is a function of local property, and thus is difficult to calculate. This is different from the metric element gt t that depends on a distant source of mass m.
On the other hand, since g55 is independent of q, (?g55/?r)/M depends only on the distant source with mass m. Thus, this force though acting on a charged particle, would penetrate electromagnetic screening. This would make such a force easier to be identified. From (11), 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 Gk, 55) on P.
4. Experimental Verification of Mass-Charge Repulsive Force.
The repulsive force in (6) can be detected with a neutral mass. To see the effect of repulsive gravity, one must have
0 (12)
Thus, repulsive gravity would be observed at r < q2/M. For the electron the repulsive gravity would exist only inside the classical electron radius r0 (= 2.817
10–13cm). It would be very difficult to test a single charged particle. 7)
However, the existence of repulsive gravity can actually be verified with a charged metal ball. The reason is that the added attractive effect in gravity is proportional to mass related to the number of electrons, but the repulsive effect in gravity is proportional to square of charge related to the square of the number of electrons. Thus, when the electrons are numerous enough accumulated in a metal ball, the effect of repulsive gravity will be shown in a macroscopic distance.8)
Now, consider the charge q and mass M is consist of N electrons, i.e.,
, where M0 is the mass of the metal ball, m and e are the massand charge of an electron. To have sufficient electrons, the necessary condition is
,where 
= 2.817
10–13 cm.(13)
For example, if r = 10 cm, then it requires N > 3.550
1013. Thus q = 5.683
10-7Coulomb.Then, one would see the attractive and repulsive additional forces change hands, although this experiment is difficult just like other small effects.
Similarly, the mass-to-charge repulsive force in (7) can be detected with a charge particle. However, since the repulsive force is very small, the interference of electricity would be comparatively large. Thus, it would be necessary to screen the electromagnetic effects out. The modern capacitor is such a piece of simple equipment that can do this screening.
When a capacitor is charged, it separates the electron from the atomic nucleus, but there is no change of mass. Thus, the capacitor would have less weight after being charged. 9) This is a nonlinear force towards charges. This simple experiment would confirm the existence of mass-charge repulsive forces, and thus the unification in term of a five-dimensional theory.
5. Conclusions and Discussions
It has been shown that the theoretical framework of general relativity is inadequate, and modification is necessary. However, it should be noted that no new theory is compelling. All you can show is that it is a feasible way to solve the problem.
One may ask whether the force Fcmof charge to mass repulsion and the force Fmc of mass to charge repulsion are the same kind of force. Since they have different origin; according to Einstein’s equation, the repulsive term in gt t is due to the electromagnetic energy, where the term in g55 is due to mass alone. Since the electromagnetic energy is subjected to electromagnetic screening, the force Fcm would also be subjected to screening although the force Fmc would not.
It should be pointed out that the screening effect to the force Fcm is only a result of the current four-dimensional theory. From the viewpoint of the five-dimensional theory, the charge would create an independent field to react with the mass. To test this, one should observe whether there is a repulsive force from a charged capacitor to a mass particle. For instance, one can have a large spherical capacitor to do the testing. From the viewpoint of five-dimensional theory, an additional repulsive force on the test mass would be observed after sufficiently charging up.
In other words, the charge-mass repulsive force mq2/r3 is a prediction of the five-dimensional theory and is independent of the four known forces. It should be noted also that in electrodynamics the term -Gk, 55 (dx5/dt)2 is also necessary because it has been shown in 1981 that the terms ?g5k/?x5 are related to the radiation reaction force [1]. Moreover, if the investigation of electric energy leads to a charge-mass repulsive force, it is expected that the magnetic energy would generate an added current-mass force. However, this is beyond the scope of this paper.
Gravitation was considered as producing attractive force only, and all the coupling constants were assumed to have the same sign. Recently, it is proven that for the radiation of binary pulsars the coupling constants must have different signs [8, 20].Now, it is shown that even the electromagnetic energy would produce repulsive forces. Thus, the physical picture provided by Newton is just too simply for a phenomenon as complicated as gravity that relates to everything.
It should be noted, however, that the five-dimensional theory is far from a theory of everything since the issues of particle creation and annihilation are not addressed. Moreover, in this paper only the static case is considered, and formula (7) is essentially derived from general relativity. This would make this calculation on a very firm ground. For the dynamic cases, a five-dimensional theory would help the necessary modification of the field equation of general relativity [1, 39].
Moreover, since many still do not understand E = mc2, this manifests that misunderstandings actually started from special relativity and electromagnetism. They also ignored issues such as the conflict between the “covariance principle” and Einstein’s requirement on weak gravity, 10) and they believed this invalid principle [10]. Newtonian invalid notions are still dominating, and the current theory of general relativity is not yet a self-consistent theory [10, 13, 14, 18]. Thus, it is unrealistic to expect the string theorists to perform a miracle in unification. Einstein is really a genius and the full meaning of general relativity is still emerging after 100 years of its creation. Now, it is clear that unification is a necessity.
In closing, we quote a remark by Einstein and Pauli [4], 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 turns out that their observation would be a prophecy for the future advancement of such unification.11) Moreover, since a theory of weak interaction must be unified with electromagnetism, the necessity of unifying gravitation and electromagnetism would imply also that the goal of the string theorists is, independent of their desire, a realistic problem.
Acknowledgments
The authors gratefully acknowledge stimulating discussions with L. Borissova, David P. Chan, S.-J. Chang, A. J. Coleman, S. Crothers, Z. G. Deng, Richard C. Y. Hui, G. R. Goldstein, A. Napier, J. J. Pi, D. Rabounski, and Eric J. Weinberg. This work is supported in part by Innotec Design, Inc., U. S. A. This presentation is supported by the Chan Foundation, Hong Kong.
Appendix: Invalidity of the “Covariance Principle”
The so-called “covariance principle” is a favorite among applied mathematicians, who often over-looked physical requirements. In fact, the creation of such a principle is due to Einstein’s failure to identify adequately the physical meaning of the coordinates. Einstein called it the “principle of covariance” [17], "The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant)." This leads to the notion of Lorentz manifolds [37] that cannot be one-one corresponding to a four-dimensional Minkowski space. Then, for such a manifold, Einstein’s requirement for weak gravity may not be applicable since a mathematical coordinate system may not relate to a physical frame of reference.
The crucial point of the covariance principle is the validity of any Gaussian coordinate system as a space-time coordinate system in physics. For this, Einstein’s supporting arguments [17] are as follows:
"That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion. All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurings are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock dial, and observed point-events happening at the same place at the same time. The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences."
Einstein’s arguments, though convinced many, are actually false. Note that the meaning of measurements is crucially omitted. First, his arguments are incompatible with his earlier argument for defining time relating to local clocks [17]. Moreover, in order to predict events, one must be able to relate events of different locations in a definite manner [10]. Moreover, Zhou correctly pointed out that coordinates must have physical meaning [40].
Zhou argued [40], “When we come to solve the field equation of moving matter, we must first define the geometrical con
Zhou argued [40], “When we come to solve the field equation of moving matter, we must first define the geometrical configuration of matter, the symmetry of the configuration, its density distribution, pressure, and velocity of motion in space-time. All of them have to be expressed in terms of coordinates.” Note that all physical predictions, including Einstein’s own three tests, must be understood in terms of the physical meaning of coordinates [10].
In view of that the “principle of covariance” is an interim assumption due to Einstein’s certain ignorance of the space-time coordinates [17], it is only natural that such an invalid “principle” is a source of theoretical inconsistence in Einstein’s theory [14]. For instance, although the bending of light can be derived from different metrics, these metrics give different formulas for the de Sitter precession [10]. Since the root of such covariance is due to Einstein’s failure in laying down coordinates in a definite manner [17], to resolve this problem, one must identify the physical meaning of space-time coordinates [20].
ENDNOTES
.
This result of Weinberg [42] is correct, since the distance in the frame of reference is decided by the Euclidean-like structure [8, 20]; whereas the metric determines only the space contractions [13, 18]. A similar calculation of total mass [36] gives a continuation from the internal to the external of the Schwarzschild solution. However, if a factor (gr r)1/2 is added to the integration [2, 36], this results in a larger mass, and would lead to another inconsistency [18, 27]. Nevertheless, some theorists objected the above calculation as incorrect. They even claimed Einstein’s requirement on weak gravity were incorrect because their knowledge in general relativity was out dated [43].
10–34 cm,me = 6.764
10–56 cm). Nevertheless some theorists [46] claimed that(q2/M) is always inside of an event of horizon.