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Misinterpretation of E = [mc.sup.2] and Einstein's theory of general relativity.


General relativity is established on two principles (Einstein et al 1923, Einstein 1954), 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 universal coupling (Lo 1997).

In Newtonian theory of gravity, the universal coupling is limited to massive matter. Einstein extended the universal coupling to include the electromagnetic energy-stress tensor. On the other hand, as pointed out by Pauli (1958), the theoretical framework of general relativity actually allows the existence of the (antigravity) coupling of different sign. However, some theorists rejected such a possibility because they interpreted E = [mc.sup.2] as and unconditional equivalence between mass and energy. Although for the universal coupling to include electromagnetism such an interpretation is not needed (Pauli 1958), there is no other justification for the unconditional universality.

Moreover, the unconditional universal coupling is a crucial assumption (1) for the singularity theorems of Hawking and Penrose (Hawking and Ellis 1973). Thus, theorists who believe in the notions of black holes and big bang, would accept the interpretation that E = [mc.sup.2] as and unconditional equivalence between mass and energy.

Naturally, such an interpretation was not questioned until the universal coupling was proven incorrect (Lo 1995, 2000) by the Hulse-Taylor experiment. By then, such a misinterpretation of E = [mc.sup.2] has become a prevailingly accepted assumption. For instance, this assumption is implicitly used in Will's interpretation of the Kreuzer Experiment(Will 1976) . An examination of Will's interpretation led to a paper (Lo 1997) that criticized his interpretation of the Kreuzer Experiment and E = [mc.sup.2] as invalid. After more than four years of deliberation, the paper (Lo 1997) that criticizes misinterpretation of special relativity was published in the Astrophysical Journal and Will was an open referee.

However, although Will was defeated because he cannot defend his interpretation of m = E/[c.sup.2], he did not admit his mistake. This is evident since Will did not make any change in his book (Will 1981), Theory and experiment in gravitational physics that used his misinterpretations m = E/[c.sup.2] to the Kreuzer Experiment and other experiments. In fact, after more than eight years, not only he did not add a note on his mistake to help his readers, but also keep the comments of Nature and Science in the cover that have the effect of misleading the readers. Such a behavior seems very strange for a scientist. (2) Recently, I discovered an explanation. I found a new misinterpretations of the Reissner-Nordstrom metric (Weinberg 1972, Wald 1984) that Will failed to reconcile with m = E/[c.sup.2]. These new interpretations seems to be able to reconcile the metric and m = E/[c.sup.2]. Unfortunately, these new interpretations are also inconsistent with general relativity.


An implicit assumption, i.e., any type of energy is equivalent to mass

m = E/[c.sup.2], (1)

is used in Will's book and papers. Unfortunately, according to general relativity, this is simply not true. This is due to that the source of an Einstein equation is an energy-stress tensor (Weinberg 1972), and thus the equivalence of energy-mass is restricted. For example, the electromagnetic energy and mass are not equivalent, since an electromagnetic stress tensor is traceless. This has been explicitly manifested by the Reissner-Nordstrom metric (Weinberg 1972, Wald 1984),


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) (Lo 2002, 2003) that Einstein (Einstein et al 1923) called as "in the sense of Euclidean geometry") from the particle center. In this metric (2), the gravitational components generated by mass and electricity have different signs and furthermore, very different radial coordinate dependence.

Some might argue that the total electric energy outside a sphere of radius r is [q.sup.2]/2r, and thus the effective mass is

M - [q.sup.2]/2r. (3)

Thus (3) could be interpreted as supporting m = E/[c.sup.2] at least for electromagnetic energy. Such a view aims at a narrow point but misses the whole picture. There are several difficulties raised from such a view:

1) 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 (3), decreases everywhere.

2) If the electric energy is assigned a 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 energy should assign a mass.

3) If any electromagnetic energy should assign a mass equivalence, then this means a rejection of the notion that a photon is massless and that special relativity is valid.

In the above, problems were created because one ignores that the electromagnetic energy-stress tensor is traceless, but the massive energy-stress tensor is not. However, to interpret that electromagnetic energy has a mass equivalence, the coincidence (3) is far from adequate (Lo 2006).

In fact, it is invalid in physics that a particle with additional energy would result in less mass since (3) was interpreted as the mass inside a sphere of radius r. Thus, such a frivolous conclusion of equivalence is a manifestation of inadequate understanding of general relativity.


If one interpreted M in (6) as a "total mass" that includes the electric energy, (4) problems 2) and 3) remain unsolved although this interpretation of M would alleviate problem 1), in addition to double counting of the electric energy. Thus, the general validity of m = E/[c.sup.2] is incorrect.

It is shown that their efforts achieved only exposing further their inadequate understanding in the theory of relativity. According to Einstein, the field equation for the metric is (Wald 1984),

[G.sub.[mu]v] [equivalent to] [R.sub.[mu]v] - [g.sub.[mu]v]R = -8[pi][T.sub.[mu]v], (4)


[R.sub.[mu]v] = -8[pi][[T.sub.[mu]v] - 1/2[g.sub.[mu]v]T], where T = [g.sup.[alpha][beta]][T.sub.[alpha][beta]].

In this equation, the energy stress tensor [T.sub.[mu]v] 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.

If one assumes that the metric has the following form,

[ds.sup.2] = f [dt.sup.2] - [hdr.sup.2] - [r.sup.2]([d[theta].sup.2] + sin[[theta].sup.2][d[phi].sup.2]), (5)

then, as shown by Wald (1984), at the region out side the particle (r > [r.sub.0]) we have




Moreover, outside the particle we have

[T(m).sub.[mu]v] = 0 for r > [r.sub.0]. (7a)


[T(m).sub.00] = [rho](r), [T(m).sub.11] = [T(m).sub.22] = [T(m).sub.33] = P(r), when r < [r.sub.0] (7b)

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

Because of the electric energy-stress tensor [T(m).sub.[mu]v] is traceless, we also have, for r > [r.sub.0],


is the electric field, according to Misner et al. [1973; p. 841]. If h = 1/f as in metric (2), then (6) is reduced to




Moreover, if = (1 - 2M/r + [q.sup.2]/[r.sup.2]) as in metric (2), then we have, in consistent with (8),

[q.sup.2]/[r.sup.2] = [r.sup.2][E.sup.2] (10)

Thus, it seems there is no restriction on M of metric (2). However, from (7), it is clear that M in metric (2) cannot include the electric energy (out side the particle) since it has been represented in (8). In other words, to interpret (M - [q.sup.2]/2r) as representing the mass inside a sphere of radius r is incorrect.


In physics, the most famous formula is probably E = [mc.sup.2]. Ironically, it is also this formula that many physicists do not understand properly. Einstein himself has made clear that this formula must be understood in terms of energy conservation (Einstein 1982). This formula means that there is an energy related to a mass, but it does not means that, for any type of energy, there is a related mass. Moreover, general relativity also makes it explicit that the gravity generated by mass and that by the electromagnetic energy are different as shown by the Riessner-Nordstrom metric.

Since not every type of energy is equivalent to mass, Will's assumption (1) is actually invalid. Nevertheless, this nonequivalence remains compatible with Einstein's famous equation, relating the total energy [E.sub.T] and mass [M.sub.T],

[E.sub.T] = [M.sub.T][c.sup.2] (11)

It is crucial to note that [E.sub.T] is the total energy of the particle. (However, [E.sub.T] does not include the energy of the field out side the particle.) The massive energy-stress tensor has a very specific form. This means that there should be cancellations among gravitational effects due to different types of energy (Lo 1995, 2000).

One should not interpret eq. (11) as having no observable consequences from the misinterpretation. Logically, this is the same that one cannot claim the pieces of a puzzle as rectangles since they form a rectangular and thus having no observable consequences from the irregulars of the pieces. Nevertheless, one might ask whether there is an inconsistency between general relativity and E = [mc.sup.2].

It will be pointed out that the problem is not general relativity, but the misinterpretation of E = [mc.sup.2]. For example, it has been observed that the particle [[pi].sup.0] meson decays into two photons (i.e., [[pi].sup.0] [right arrow] [gamma] + [gamma]). Then, there would be a conflict if a photon includes only electromagnetic energy. However, misinterpretation based on experiment is not new. A well-known example is that Madame Curie misinterpreted momentum was not conserved in decay because she did not realize the existence of a particle called neutrino. The correct interpretation is that a photon includes non-electromagnetic energy, since the current electromagnetism is valid.

In fact, one needs not rely on general relativity to show the equivalence between electromagnetic energy and mass as invalid. Independent of general relativity, they are not equivalent because the trace of an electromagnetic energy-stress tensor is zero, but the trace of an energy-stress tensor of massive matter is non-zero. Note that an electromagnetic energy-stress tensor is necessarily different from the energy-stress tensor of massless particles. The latter can become having a non-zero trace because the momentum of two massless particles can cancel each other. Thus, as Einstein showed (Einstein et al 1923), massless particles can be equivalent to mass, but the electromagnetic energy cannot.

The misinterpretation of E = [mc.sup.2] as any energy has a mass correspondence, i.e., m = E/[c.sup.2] is a rather common one. For instance, Tolman (Tolman 1987 ; p. 49), Fock (1964; p.111), and Hawking (1988, p. 107] have also made this mistake explicitly. In view of that Will's book (Lo 1997) gets only positive comments from two top journals (5) (6), this error seems to have been rather universally committed. Nevertheless, there are exceptions, such as Bohm (1996).

Although the problem of misinterpretation was identified in 1997 (Lo 1997, 1997), there is little improvement in understanding this issue. A hidden agenda of the implicit assumption of universal equivalence of energy and mass is to justify the universal coupling that is a vital assumption of the singularity theorems (Wald 1984). However, the universal coupling has already been proven to be incorrect (Lo 1995, 2000) by the Hulse-Taylor experiment. This illustrates that self-interest has a powerful influence on the judgment of scientists.

Another problem is what makes the radiating energy differs from the electromagnetic energy. This remaining issue will be addressed in a subsequent paper (Lo 2006). In that paper, it will explain also the historical limitations that prevented Einstein from addressing this issue. It is hope that the present paper would call the attention to this important issue of over simplifying the relationship between mass and energy.


The author gratefully acknowledges stimulating discussions with Professors S. J. Chang, A. J. Coleman, Eric J. Weinberg, and C. Wong. Special thanks are to D. Rabounski for valuable comments. This work is supported in part by Innotec Design, Inc., U. S. A.


[1.] Bohm, D., 1996. Special Theory of Relativity (Taylor & Francis, Inc., New York).

[2.] Cheng, T. P., 2005. Relativity, Gravitation, and Cosmology-a basic introduction (Oxford University Press).

[3.] Einstein, A., Lorentz, H. A., Minkowski, H. and Weyl, H. 1923. The Principle of Relativity (Dover, New York,).

[4.] Einstein, A. 1954. The Meaning of Relativity (1921) (Princeton Univ. Press).

[5.] Einstein, A. 1982. 'E = [mc.sup.2]' (1946) in Ideas and Opinions (Dover, NEW York).

[6.] Fock, V. A. 1964. The Theory of Space Time and Gravitation, translated by N. Kemmer (Pergamon Press).

[7.] Hawking, S. 1988. A Brief History of Time (Bantam Books, New York).

[8.] Hawking, S. W. & Ellis, G. F. R. 1973. The Large Scale Structure of Space-time (Cambridge Univ. Press).

[9.] Lo, C. Y. 1995. Astrophys. J. 455: 421-428.

[10.] Lo, C. Y. 1997. Phys. Essays, Vol. 10 (4): 540-545

[11.] Lo, C. Y. 2000. Phys. Essays, 13 (4) : 527-539.

[12.] Lo, C. Y. 1997. Astrophys. J. 477: 700-704

[13.] Lo, C. Y. 2002. Phys. Essays, 15 (3) : 303-321.

[14.] Lo, C. Y. 2003. Chinese J. of Phys. (Taipei), Vol. 41(No. 4): 233-343.

[15.] Lo, C. Y. 2005. Phys. Essays, 18 (4), (December).

[16.] Lo, C. Y. 2006."Completing Einstein's proof of E = [mc.sup.2]," (in preparation).

[17.] Lo, C. Y. 2006.Remarks on Interpretations of the Eotvos Experiment and Misunderstandings of E = [mc.sup.2], (in preparation).

[18.] Misner, C. W., Thorne, K. S., and Wheeler, J. A. 1973. Gravitation (Freeman, San Francisco).

[19.] Pauli, W. 1958. Theory of Relativity (Pergamon, London).

[20.] Tolman, R. C. 1987. Relativity, Thermodynamics, and Cosmology (Dover, New York).

[21.] Wald, R. M. 1984. General Relativity (The Univ. of Chicago Press, Chicago).

[22.] Will, C. M. 1976. Astrophysical J. 204: 224.

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

[24.] Weinberg, S. 1972. Gravitation and Cosmology (John Wiley Inc., New York).


(1.) Hawking in his recently (June 2006) visit to China, still misleadingly told his audience that his theory was based on general relativity only. The root of his problem would be that he still does not understand the formula E = [mc.sup.2].

(2.) Clifford M. Will, who is the President of the International Society on General Relativity and Gravitation (2004-2007), was an open referee of Lo (1997).

(3.) The existence of a Euclidean-like structure in the frame of reference is a necessary condition for a physical space (Lo 2003) . For example, the Schwarzschild solution in quasiMinkowskian coordinates [11Weinberg 1972; p.181) is the following:


where [[rho].sup.2] = [x.sup.2] + [y.sup.2] + [z.sup.2], x = [rho] sin[theta] cos[phi], y = [rho] sin[theta] sin[phi], and z = [rho] cos[theta].

Thus, is space coordinates satisfy the Pythagorean theorem, and has the Euclidean-like struc In this system a light speed ([ds.sup.2] = 0) is defined in terms of dx/dt, dy/dt, and dz/dt (Einstein et al 1923, Einstein 1954).

(4.) This approach ignores that in general relativity gravity is decided by a tensor metric, instead of a scalar gravitational potential. For example, Einstein's equivalence principle was commonly mistaken (Cheng 2005), as shown in the British Encyclopedia, to be the same as the 1911 preliminary application of equivalence between acceleration and Newtonian gravity. A clear difference is that the metric of a uniform gravity must be time-dependent because of Einstein's equivalence principle (Lo 2005), but in Newtonian theory such a potential can be static.

(5.) "Consolidates much of the literatures on experimental gravity and should be invaluable to researchers in gravitation."--Science on Will's book (Will 1981).

(6.) "A concise meaty book ... and a most useful reference work ... researchers and serious students of gravitation should be pleased with it."--Nature on Will's book (Will 1981).

C.Y. Lo

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Author:Lo, C.Y.
Publication:Bulletin of Pure & Applied Sciences-Physics
Date:Jan 1, 2006
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