^{1}

^{2}

^{*}

The paper suggests that E = mc
^{2} may be open to misinterpretation and that in this form it is not what Einstein advanced first. It is further suggested to return to the slightly less compact formula E = Kmc
^{2} where a < K < 1 which has the merit of accounting for the measured ordinary energy density of the cosmos (K = 1/22) and the conjectured missing dark energy density of the universe (K = 21/22) from the view point of economical notation.

The present paper started by the following innocent but for at least historical reasons, a justified question [

DOES THE FORMULA E = mC^2 BELONGS TO EINSTEIN?

Now those of us who are passionate readers of the classical work of Einstein [

It’s hard to nail down those responsible for transforming the Einstein original but they probably proceeded in the following way. As l is a very small mass (defect of mass in reality), it was suggested to rename l = dm, and the small energy, associated with dm was taken as dE. As a result the following formula appeared:

There is nothing wrong with this formula as it corresponds to Einstein’s result in a slightly different notation. The misleading error occurred upon integration of dm from 0 to entire mass m of the body of arbitrary Chemistry, and dE is integrated to give the energy E, corresponding to the mass of the entire body. The fact is however this formula cannot be integrated, unless dm is treated as small portion of the defect of mass, then the integration limits are from 0 to l, resulting in the initial Einstein formula.

Unfortunately, this practically incorrect result E = mC^2 [

The present article has grown out of an extensive open discussion invited by a question posed by the first author (A.J.B.) on Research Gate which is nowadays a leading and popular forum for research in general science, engineering, medicine and technology as well as humanities and social sciences [

L.B. Okun, a world renowned expert on Einstein relativity and particularly on E = mc^{2}, put his misgivings plainly in a formal scientific paper by entitling it “The Einstein formula; E_{o} = mc^{2}. Isn’t the Lord laughing?” published in 2008 in a leading Russian physics journal [^{2} has a long history and an even longer list of published papers [^{2} is first not due to Einstein and second Albert Einstein is not responsible for all these misunderstandings in the first place. Subsequently we show that writing E = Kmc^{2} is more general and accounts for the deeper meaning of Einstein’s intention as well as for ordinary energy and dark energy by setting K = 1/22 for the first and K = 21/22 for the second so that at the end we find E = (mc^{2}/22) + mc^{2}(21/22) = mc^{2} as should be [

The most common way used in deriving E = mc^{2} is via the following example of electron-positron production and annihilation:

If a photon enters the coulomb field of a nucleus or another charged particle, this photon can disappear by so called pair production if the energy of the photon is above an energetic threshold, the pair production minimal energy. For the production of an electron-positron pair this minimal energy is 1022 keV plus a small rest of recoil energy for the coulomb field spending nucleus or another particle like atomic shell electron or free unbound electrons. The value of m_{e}/M_{n} is typically about 10^{−6}, the threshold formula (

The symbolic formula for the process is

This is a well known elementary particle process. Its probability depends on the square of the atomic number of the field producing nucleus and increases with about the logarithm of the photon energy above the threshold.

The energy of the photon is changed into particle masses, and if then a difference of energy remains as kinetic energy of the two particles. From conservation laws this particle pair must consist of a matter-antimatter pair to keep the particle numbers etc. at zero.

In the case of zero remainingkineticenergy,wefinddirectly the famous Einstein mass- energy relation

which means the direct equivalence of energy and mass.

The revised process of pair production is pair annihilation. It happens if a particle and the corresponding antiparticle collide. The example here is the electron-positron annihilation.

If this process happens with zero rest kinetic energy of the two colliding leptons, we immediately get the equation

The two annihilation photons are emitted in a typical 180 degree angle, which is applied in nuclear medicine imaging as PET technology. Or in a more symbolic way you can write again

the direct equivalence again of mass and energy (

If the annihilation takes place with remaining energy of the electron or the positron, the two annihilation photons are emitted under a sharp labor angle smaller than 180 degree.

Let us start from the generally accepted De Broglie-Bohm pilot wave picture, which is probably the only quantum theory which Einstein was ready to at least partially “swallow” to put it mildly. We take this picture because it agrees completely with von Neumann-Connes’ pointless noncommutative geometry as our spacetime and consequently also with the fundamental equation [

which shows that the “pilot” quantum wave is the cobordism, i.e. “surface” of the quantum particle. In particular the “pilot” pre-quantum wave would be fixed by the empty set bi-dimension

while the pre-quantum particle would be fixed by the zero set

Let us calculate the quasi volume of the pre-quantum particle and the quasi area of its surface, i.e. the surface constituting the “pilot” wave. Noting that the basic dimensionality of spacetime is that of Kaluza-Klein five dimensions [

where

and

Adding ordinary energy E(O) to dark E(D) it is delightful to see that we retrieve Einstein's correct formulas [

where

It is not always easy to set an exact limit for where we become relativistic or render quantum starting from classical physics. Consequently we should not be astonished when we see that at the edge of the universe at hyperbolic infinity of the boundary of the holographic boundary we have an accumulation of dark energy which accounts for 95.5% of the total energy density of the universe given by^{2} which we propose to replace with E = mc^{2}. However for K = 1 we have a very special situation indeed for which by chance or providence we found E(max) = (1)(mc^{2}) and that by a totally ingenious leap of faith rather than a systematic derivation as noted by the grand connoisseur of relativity, Professor Wolfgang Rindler [^{2}, ordinary energy and dark energy density of the universe but in a very subtle language and quite different from the language used in the present work. Never the less, E = Kmc^{2} is the physical incarnation of the old proverb which says that all roads lead to Rome or E = Kmc^{2}.

The celebrated formula E = mc^{2} has a long history dating back to before Einstein, for instance Poincare. Not only that but it is also somewhat misleading and better replaced by E = Kmc^{2}. In this second form E becomes the sum of two types of energy, the measurable ordinary energy E(O) = mc^{2}/22 and the supposedly missing dark energy E(D) = mc^{2}(21/22) [^{2}. In fact classical relativity already implies the total unity of Einstein’s relativity, classical mechanics and quantum mechanics. This is so because it is the relative size of what we are measuring vis-à-vis our own size and the size of our experimental device is what makes things relatively very large where relativity is applicable or relatively very small where quantum mechanical description takes over. No wonder then that the quantum relativity formula E(O) = mc^{2}/22 when added to the quantum cosmology formula E(D) = mc^{2}(21/22) gives us the famous formula E = mc^{2} which is only a scaling of Newton’s E = 1/2mv^{2} sending 1/2v^{2} to c^{2}. Nothing could possibly be more astoundingly simple than that.

The authors are deeply indebted to all who participated in the relevant discussion on Research Gate questions and answers of Ref. [

A. J.Babchin,M. S. ElNaschie, (2016) On the Real Einstein Beauty E = Kmc^{2}. World Journal of Condensed Matter Physics,06,1-6. doi: 10.4236/wjcmp.2016.61001