


Einstein’s Formula andVariations of Redundancies E = mc^{2} E^{2} = m^{2} c^{4} E^{2} = m^{2} c^{4} + p^{2} c^{2} Charles William Johnson ©December, 2010 Copyrighted It has been stated that Einstein’s popularly cited formula, E = m c^{2} , is for lay people and the sophisticated version, E^{2} = m^{2} c^{4} + p^{2} c^{2}, is for physicists. In my view, tautologies and redundancies abound irrespective of the version chosen for analysis. In fact, the sophisticated version appears to even be theoretically deficient. The concept of momentum is already suggested in the m times c term, yet added to that is p, which is said to also represent momentum. Momentum of mass in motion is added to momentum of rest mass as though it were a case of simple addition of terms. E^{2} = m^{2} c^{4} + p^{2} c^{2} E^{2} = m^{2} c^{4} + (mv)^{2} c^{2} A certain inherent equivalency/tautology is understood regarding the concept of momentum as mc and/or mv, based on the directional distinction between speed [c] and velocity [v]. In previous essays, I have treated the more popular versions of Einstein’s formula, E = m c^{2} and E^{2} = m^{2} c^{4} . Time and again I ask myself the question, why is the speed of light squared in Einstein’s formula? Here is a commonplace answer: “It has to do with the nature of energy. When something is moving four times as fast as something else, it doesn’t have four times the energy but rather 16 times the energy in other words, that figure is squared. So the speed of light squared is the conversion factor that decides just how much energy lies captured within a walnut or any other chunk of matter. And because the speed of light squared is a huge number 448,900,000,000,000,000 in units of mph the amount of energy bound up into even the smallest mass is truly mindboggling...”. [Source: Peter Tyson, The Legacy of E = mc^{2} .] I missed something here; the implication in the explanation is that the speed of light is traveling four times faster than what? And, that’s why it is squared? If the speed of light is taken as 299792458 meters/second, then the square of the speed of light in a vacuum is 8.98751787 x 10^{16} . That means that the speed of light, c, is 299,792,458 times greater than itself. The idea of “4 or 16 times greater” in the previous statement makes little sense; as in relation to what? Often I wonder whether scientists actually understand that they are affirming that the speed of light can be or may be or could represent mathematically 299,792,458 times itself. The fact that it is squared, the power of n2, may cause them to think possibly that the amount is not much, for the power 2 gives the impression of being the smallest power possible as it were [aside from one]. But what is actually happening behind that power of two is the derivation of a multiplier of 299,792,458! I’ll state it again: they are multiplying the speed of light, 299792458 meters/second by 299,792,458 times itself and that is just the power of two. Imagine the speed of light taken to the seventh and ninth power as is implied/demanded by the respective fractal numerical values of 2.17644 and 1.9561! {Please excuse me, but I can’t help myself with the exclamation points here.} The seventh power of c, 2.176431087 x 10^{59} or, 299792458^{7} is 7.259792663 x 10^{50} times greater than the speed of light in a vacuum The ninth power of c, 1.956078711 x 10^{76} or, 299792458^{9} is 6.524776253 x 10^{67} times greater than the speed oflight in a vacuum Now, it is necessary to examine the more specialized and supposedly headier version of Einstein’s formula: E^{2} = m^{2} c^{4} + p^{2} c^{2}. The observations that I made previously with regard to the first two versions also are applicable to this latter, lengthier version of Einstein’s formula. Additional comments may be made about this latter version that were not made regarding the previous two versions, since this latter version contains additional terms in the equation, and combinations of powers of c are employed. E^{2} = m^{2} c^{4} 1.9561^{2} = 2.17644^{2} times c^{4} 3.82632721 fractal = 4.736891074 times c^{4} Remember, the numerical value, 1.9561 is the fractal product of c^{9} [19560.78711]; and, the numerical value of 2.17644 is the product of c^{7} [2176.431087]. So, when these terms are thus additionally squared in this particular presentation of Einstein’s formula, well, one is even further removed from any relationship of possible equivalency between energy and mass. In this example, c = 299792458.0 meters/second: E^{2} = m^{2} c^{4} [c^{9} ] ^{2} = [ c^{7} ]^{2} c^{4} [1.956078711 x 10^{76}]^{2} = [2.176431087 x 10^{59}]^{2} times [8.077608713 x 10^{33}] ^{4} Hence, in this expanded version of Einstein’s formula, now even larger powers of c, the speed of light in a vacuum are derived. The best I can do here is use 299792.458 as the base fractal value. I used kilometers for the expression of c in this example, since my pocket calculator will not allow me to derive the numbers involved in squaring 1.956078711 x 10^{76} and 2.176431087 x 10^{59} . [ c^{9 }]^{2} is actually c^{18 }= 3.826243921 x 10^{98} and, [ c^{7} ]^{2} is actually c^{14} = 4.736852277 x 10^{76} Forget about the diminished csquare term [8.987551787]. The c^{14} mterm is 1.580043844 x 10^{71} times greater than 299792.458 as c. The c^{18} Eterm is 1.276297592 x 10^{93} times greater than 299792.458 as c. There is no reason to reexamine the myriad of critical observations about using powers of c, the speed of light in a vacuum, which is supposedly an upper limit on the velocity of energymass in spacetime. But, a concern now exists as to the aggregate in the third version of Einstein’s formula: the compound term noted as: + p^{2} c^{2}. Supposedly this additional set of terms reflects the momentum of mass in the equation. And, this accounts for the total energy by adding on this concept for kinetic energy. The simple versions, E = m c^{2} , and, E^{2} = m^{2} c^{4} supposedly represent rest mass. Supposedly, p = momentum; “when the particle is at rest its momentum (p) equals zero”. Hence, E^{2} = m^{2} c^{4} derives finally E = m c^{2} . “And, the complete equation for relativistic energy is”: E^{2} = m^{2} c^{4} + p^{2} c^{2} Source: Summarized from numerous blogs and comments appearing on physics fora on the Internet. “Kinematics is the study of motion without reference to mass or force, and it figures in a more elaborate form of Einstein’s equation that unlike plain old E = mc^{2} , which concerns mass at rest also takes into account mass in motion. (If you must know, it’s E^{2} = m^{2} c^{4} + p^{2} c^{2}, where p equals momentum.) “his bigger equation plays an enormous part in our understanding of how light works, and how energy and light can be transferred and transformed from one place to another, and that sort of thing’, Gates says. ‘So if you consider the larger context, the part of the equation that’s not in the public eye, it has an even larger legacy in science’.” [Source: Peter Tyson, The Legacy of E = mc^{2} .] In fact, throughout the science literature there is not much agreement about the meaning of these repetitive equations and their terms. Randomly, one can find references to “mass”, “relative mass”, “relativistic mass”, “rest mass”, “relativistic energy”, “energy”, “kinetic energy”, etc., the list is too long to go through; and unnecessary, for it is too well known. One can find an infinite array of interpretations as to what each one of these formulae mean, and often opposing and contradictory views. One can find superlative remarks about the value of E = mc^{2} , as well as remarks denying it any value whatsoever. One can even find accusations of plagiarism on the part of Einstein in deriving his formula. Yet, no one appears to present a page view of where Einstein actually listed this simple formula; it is derived and suggested in his writing. Various theoretical issues arise with the expanded version of Einstein’s formula. And, just how much of these formulae are attributable actually to Einstein is another matter for debate. [Consult numerous essays on this subject by, Roger Anderton, The General Science Journal; http://wbabin.net/anderton.htm] The original formula, whereby m times c, would actually be recording the momentum of the mass cited, inasmuch as momentum is m times velocity. But, then, the aggregate part of the expanded equation now proposes adding an additional computation [+ p^{2} c^{2} ] for momentum to the original one, since now p also contains an expression of m times v [c, speed] for the same mass in the equation. This is probably the first visible redundancy, not taking into account the redundancies reviewed regarding the short versions of the cited formula. Another significant observation concerns that of the aggregate of supposed concept for momentum of [+ p^{2} c^{2} ], since it in fact proposes the idea of “remainder math” to the original formula. The addition of the remainder math eliminates the possibility of finding roots to the powers of c in the original equation. The roots will now be expressed as a certain number of multiples of c, plus a fractional percentage thereof [a certain number of psquare times csquare]. As noted in other essays, one may find in the original equation such relationships among powers of c as c^{9} = c^{7 }c^{2}. Such a relationship of powers among c is no longer available with the aggregate of the remainder math summed to the original equation, which has the addon of: + p^{2} c^{2}. The results, due to the remainder math, will always be fractional root expressions of c and its multiples with an aggregate value. With regard to the equation, E^{2} = m^{2} c^{4} + p^{2} c^{2} , if the object has zero momentum, then the object/mass is at rest: E^{2} = m^{2} c^{4} + p^{2} c^{2} E^{2} = m^{2} c^{4} + 0 i.e., E^{2} = m^{2} c^{4} i.e., E = m c^{2} A physics blogger comment: “The proper, general equation to use is E^{2} = m^{2} c^{4} + p^{2} c^{2}. So, in the case of a photon, m = 0 so E = pc or p = E/c. On the other hand, for a particle with mass m at rest (i.e., p = 0), you get back to the famous E = mc^{2} .[Source: www.starmind. com/question/15314.] One could probably go on for decades attempting to distinguish the subtleties among the different equations purported to be Einstein’s formula which is actually what has happened since 1905! E = mc^{2} E^{2} = m^{2} c^{4} E^{2} = m^{2} c^{4} + p^{2} c^{2} Nonetheless, in the final instance, each equation is a distinct manipulation of powers of c, the speed of light in a vacuum. This either occurs because numerical values for powers of c are employed for energy [E] and mass [m], together with c, and/or, the numerical values of mass [m] and momentum [p] represent the number of times one multiplies or increases the term c. For example, read as the m number of times of c, and the p number of times of c. Plus, one is faced with the squares and the power of c as defined by the cterms. And, in as much as numerical values for c [1.9561 and 2.17644] are employed for energy (E) and mass (m) respectively, together with c, it is in that very same manner that all terms and values of the equations reflect massless events. There is no trace of a mass event as of those assigned numerical values. When mass is employed as numerical value 2.17644, as in Planck mass, the seventh power of c, then the energy term (E), whether squared or not in the equations produces a fractal multiple of Planck energy, 1.9561, the ninth power of the speed of light in a vacuum. E = mc^{2} 19560.78711 = 2176.431087 x 8.987551787 19560.78711 = 19560.78711 E^{2} = m^{2} c^{4} E^{2} = 4736852.276 x 80.77608713 E^{2} = 382624392.2 19560.78711^{2} = 382624392.2 E^{2} = m^{2} c^{4} + p^{2} c^{2} E^{2} = 4736852.276 x 80.77608713 + 0 E^{2} = 382624392.2 19560.78711^{2} = 382624392.2 Simply squaring the original formula, E = mc^{2} , derives that equation squared: E^{2} = m^{2} c^{4} . The first and second equations listed immediately above represent simply fractal multiples of one another; no new knowledge is added to either one. Now, the remainder math, with the aggregate +p^{2} c^{2} as nonzero, complicates the manipulation of powers of c by deriving fractional roots that are not roots of c as such. Rather they are roots of c plus an additional aggregate amount (remainder math through addition). And, when mass (m) is any numerical value other than that of Planck mass, 2.17644, or is not a numerical value for the powers of c, then there derives a fractional root in its energy (E) expression. Now, if you substitute mass (m) for a nonzero amount that is a nonc term as of one of the powers of c, a result is obtained that evidently is not a root expression of c, the speed of light. For example, use 2.388 = mass_{defect} which is not a power/root of c, the speed of light in a vacuum. E = m c^{2} E = 2.388 times csquare E = 21.46227367 The fractal numerical value, 21.46227367, is not a root expression of the speed of light. In fact, it is 0.388 times greater than twice csquare. 8.987551787 + 8.987551787 + 3.487170093 = 21.46227367 c^{2} + c^{2} + 0.388 of c^{2} = 21.46227367 For this reason, it becomes obvious how the terms mass (m) and momentum (p) [squared or not] simply represent by how many times csquare is increased within the equation. An interesting aspect obtains from these computations. It now becomes evident where the 3.4870+ numerical value derives from with regard to the constant Planck mass, 2.17644. Previously, I have shown how certain pairs of fundamental physical constants derive other specific constants, especially the Planck constants. For example, the reciprocal of the speed of light in a vacuum times the proton’s mass derives the Planck implied mass constant: 3.4870407. This latter constant in turn divided by the elementary charge derives Planck mass [2.17644]. The reciprocal of the speed of light in a vacuum times Proton mass = Planck implied mass 0.333564095 times 1.672621637 = 0.557926523 halves to 3.4870407 Precisely, Planck implied mass divided by Elementary charge = Planck mass 3.4870407 divided by 1.602176487 = 2.1674398 From the analytical observations in this essay, it is now evident that the 3.4870+ numerical value relates to the mantissa of mass_{defect}, 2.388. When the mantissa of the mass_{defect} is taken as a percentage , 0.388, to the square of the speed of light in a vacuum, the fractal 3.4870+ numerical value is derived. The posits of spacetime/motion have it such that all forms of matterenergy are related. Hence, it is not surprising to find numerical values of different fundamental physical constants that are relational in this manner. The question is to know whether these relationships reflect actual spacetime/motion events, and whether there exist redundancies and tautologies in the numbers. If the numerical values are duplicated in that sense, then the supposed distinct fundamental physical constants that they represent may not be so distinct. For example, if the numerical value of one supposed constant is a fractal multiple of another fundamental physical constant, then possibly there is no need for two distinct constants, but rather only one distinction is required, as one may be a manifestation of the other. In previous analyses, I have illustrated how various fundamental physical constants represent in fact fractal multiples of other constants. The tendency to identify specific numerical values and then assign a particular name to that numerical value as a fundamental physical constant has been predominant within the history of physics. Six CODATA Fundamental Constants That Are Near Fractal Multiples of One Another:
Magnetic Flux Quantum is the fractal reciprocal of the Conventional Value of Josephson Constant. The Conventional Value of the Josephson Constant is fractally double the numerical value of the Elementary Charge. The Elementary Charge doubles fractally to the Compton Wavelength. The Compton Wavelength doubles fractally to the Bohr Magneton. And, finally, the Bohr Magneton doubles fractally to the Elementary Charge Squared. In a sense, then, these six supposedly independent and unique fundamental physical constants represent fractal multiples of the same reciprocal fractal numbers: 2.067833667  4.835978908549. Immediately, the question arises as to why scientists should think that the varied manifestations of these reciprocal values would require a particular identifier, three of which are named after scientists themselves: Josephson, Compton and Bohr. Consider an analytical analogy as follows: 1 divided by 7 equals 0.142857142 0.142857142 divided by 2 equals 0.071428571 0.071428571 doubles to 0.285714284 0.285714284 doubles to 0.571428568 0.571428568 doubles to 1.142857136 And with that, we could then proceed to assign a particular name to the six numerical values that are underlined immediately above. However, the six numbers illustrated there represent members of a series of a particular set of reciprocals; nothing more. As far as numerical values go within a number series, there is no special significance to any one of the particular member numbers per se of the given series. They are simply fractal multiples of the same reciprocals. Of course, many more exist infinitely so; only six have been chosen to maintain the analogy. It is possible to isolate particular values within a series for a given meaning, but difficult to identify any given value as distinct and unique in meaning as carried out by the assignment of fundamental physical constants. For example, the numerical value of 1.142857136 has been assigned in my work the name of diametian. I chose this identifier due to the relationship of this quasidimensionless number with regard to a circle that is consciously divided into 360 sectors/units/degrees and related to the concept of pi: 3.141592654.... 360 / pi = 114.591559 In ancient times, 3.15 was employed as pi in some cultures: 360 / 3.15 = 114.2857143 [the reciprocal of seven number series] But, even this case of assigning the wordconcept diametian to the fractal value 1.142857142 of the circle’s diameter does not represent anything special. One could call it two radians; that’s the same thing in a sense. It simply represents a selected numerical value within a particular series and assigned in relation to a specific derivation made by human beings: that of dividing a circle into 360 degrees in relation to pi, which in a sense is an historically arbitrary choice. Divide the circle into 260 degrees/sectors [as the ancients did], and a distinct numerical value/series appears: 260 / 3.15 = 82.53968254 Or, relationally and symbolically, invert the terms: 260
/ 1.14591559 = 226.8928028 [2268 is an ancient Now, other number series appear within the Einstein formulae as presented in their different formats. The numerical series employed and emphasized concerns the powers and roots of c, the speed of light in a vacuum. Aside from generating fractal multiples as of the cterms within the different variations of the E = mc equation, the mterm and the pterm are also being employed as multipliers against the cterms. There is nothing really significant being carried out with regard to the variations of the Einstein equations as such. The variations in the Einstein equations are basically manipulating then powers and roots of the cterms in relation to the multipliers mterm and pterm. In the examples of Einstein’s formula offered in my analyses, relations of equivalency have gone as high to the sixteenth power of c, the speed of light in a vacuum. Obviously, one can go as high as infinitely allowed, depending upon the assigned values of the mterm and the pterm which can be infinitely large. One might expect an infinitely large result of the cterm in relation to the amount of massmomentum in the entire Universe. A numerical translation of the energy in the Universe, however, represents a futile exercise in mathematical reasoning for many reasons, one of which is that we still do not know for certain and exactly what are the boundaries [if any] of the Universe. In a sense, it is unnecessary to square the speed of light in a vacuum in Einstein’s formula. The simple multiplication of a given mass times 299,792,458 is sufficient to create the idea of an extremely large number. Just imagine taking all of the mass in the Universe [whatever that may be] and multiplying it by the numerical value 299,792,458; that would produce an unimaginable number in itself. It is understandable that one may theoretically propose that a given mass when converted to energy has its rest mass liberated, and shoot out, as in the rays of the Sun, or in the explosion of energy of an atomic bomb, is so liberated at the phenomenal speed of 299,792,458 meters/second; i.e., at the speed of light. But, to consider that the energy liberated from a given mass propagates away from itself at a speed 299,792,458 times that 299,792,458 meters/second seems like a theoretical overkill, reflecting some kind of desire in producing an imaginary number that cannot even be imagined. Anyone who knaps a piece of obsidian and watches the sparks fly as the stone is struck realizes that the light liberated from that chipping action is traveling at 299,792,458 meters/second reaching our eyes in a flash. But to imagine that the light liberated from that stone is reaching our eyes over and above that speed 299,792,458 times faster than the 299,792,458 meters/second the flash is traveling, is simply the product of an imaginative mind. Light traveling at the speed it travels from the far corners of the Universe has not been known to supercede its own velocity of 299,792,458 meters/second, neither here on Earth, nor anywhere in the Universe. Scientists need to calm down and stop inventing these imaginary numbers that only confuse the picture of spacetime/motion and its forms of matterenergy. It’s time they get back to basics. The layperson is impressed with the numerical values of matterenergy events as they exist [299792458 meters/second]. These are difficult enough as it is for the mind to comprehend. Multiplying them zillions of times over is undoubtedly even more impressive; but this only confuses things. It is time to set the historical record straight in physics. Possibly a layperson’s view may be required at this juncture, with less emphasis on those sophisticated, redundantly progressive formulae that increase the numerical values of their terms beyond spacetime. ©December 20102013 Copyrighted by Charles William Johnson. All rights reserved. Reproduction prohibited. Earth/matriX: Science Today, www.earthmatrix.com johnson@earthmatrix.com ISSN15263312 Einstein's Formula: Mass Confusion
