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The Origin of the Equation E = mc2
The Origin of the Equation E = mc2
Because the issue of the source of this equation keeps coming up, here is a quick discussion of the origin of the famous formula, incorrectly attributed to Einstein. Thanks to Robert Sungenis for providing this.
Contrary to popular opinion, E = mc2 did not originate with Einstein. As van der Kamp reveals:
And then that hackneyed combination of Einstein and the “E = mc2,” endlessly bandied about in popular-scientific Western folklore! True, it can be deduced from the theory, but it does not prove STR
As for the origin of the formula, it wasn’t until five years before his death (1955) that Einstein publicly attributed the basis of E = mc2 to the 1862 charge-momentum field equations of James Clerk Maxwell Previous to Maxwell was the work of J. Soldner who assigned mass to light and thus could calculate its deflection in a gravitational field Michael Faraday’s 1831 experiments with electricity and induction coils had already introduced the energy/mass relationship, and Maxwell put this in the reciprocal m = E/c2 equation In fact, one can go back as far as Isaac Newton in 1704 for the theoretical relationship between mass and energy Samuel Tolver Preston used the formula in 1875 Julius Robert Mayer put the formula in terms of ether pressure
A curious twist in this saga occurs in 1881 with J. J. Thomson in his work with charged spherical conductors in motion, since he derived a slightly higher coefficient, E = 4/3mc2  The same E = 4/3mc2 was found by F. Hasenöhrl in 1904 when he published the first explicit statement that the heat energy of a body increases its “mechanical” mass The 1905 Nobel Prize winner, Philipp Lenard, a staunch opponent of Einstein, was one of the first to reveal this fact in his 1921 book Ether and Para-ether. In the book, Lenard demonstrated how simple it was to arrive at E = mc2 without any reference to Relativity theory – something Einstein would also admit a few years prior to his death. In his 1929 book Energy and Gravitation, Lenard honored Hasenöhrl as “the first to demonstrate that energy possesses mass (inertia).”
The history of the 4/3 coefficient is intriguing. Arthur Miller shows both its origin and how Einstein sought to remove it. Although Einstein purports to have legitimately removed it, Miller shows he did not succeed. Einstein had attributed the excess 1/3 to mechanical constraints, but Poincaré had demonstrated earlier that it was due to forces that avoid the explosion of the electron. Engrossed in his General Relativity theory, Einstein did not visit the problem again. Max Von Laue demonstrated that to obtain the final formula E = mc2 “one type of energy…the new physics must eliminate from its list…is kinetic energy.” The reason is that if mass is based on energy, as E = mc2shows, then there cannot be a kinetic energy, K = ½mv2, which, in turn, depends on the mass. In other words, to obtain E = mc2 one must abandon the most obvious and primary form of energy, kinetic energy.
Prior to this, in 1889 Oliver Heaviside used the E = mc2 principle in his work with capacitors. Henri Poincaré used the rudiments of the E = mc2 formula long before Einstein commandeered it for his Special and General Relativity theories. In 1903 the Italian scientist Olinto De Pretto had already published E = mc2 two years before Einstein did, but Einstein did not mention De Pretto in his 1905 paper on Special Relativity, which is odd considering that he spoke fluent Italian and, by his own admission, read all the Italian physics journals. In 1907, Max Planck, expanding the work of Hasenöhrl and using Poincaré’s momentum of radiation formula, gave the final derivation of the E = mc2 formula. All in all, E = mc2 is readily derivable apart from the theory of Relativity, as both Joseph Larmor in 1912; Wolfgang Pauli in 1920, Philipp Lenard in 1921, and M. Simhony in 1994, demonstrated independently.
 De Labore Solis, p. 51. Van der Kamp cites Carl A. Zappfe’s A Reminder on E = mc2 for the “three non-relativistic ways,” but there are actually a half dozen or more paths to the formula. See text and footnotes. Albert Einstein, Out of My Later Years, Philosophical Library, New York, viii, 282, 1950. Also Edward Schilpp’s, Albert Einstein, Philosopher Scientist, Library of Living Philosophers, 1949, p. 62, has Einstein quoted as saying: “The special theory of relativity owes its origin to Maxwell’s Equations of the electromagnetic field.” J. Soldner, Berliner Astronomisches Jahrbuch, 1804, p. 161. Also cited in Annalen der Physik, 65:593, 1921. The derivation of E = mc2 originates from Maxwell’s formula [ f = δE/cδt ] which equates the force exerted on an absorbing body at the rate energy is received by the body. Since force is also the rate of the change of momentum of the body, which, by the conservation of momentum, is also the rate of change in the momentum of the radiation, the momentum lost by the radiation is equal to 1/c times the energy delivered to the body, or M = E/c. If the momentum of the radiation of a mass is M times the velocity c of the radiation, the equation m = E/c2 is derived. In Newton’s Query 30 he writes: “Gross bodies and light are convertible into one another…” (Opticks, Dover Publications, Inc., New York, p. cxv). Newton’s Opticks also reveal that he believed gravity would bend light. This is further evidence that many of Einstein’s ideas are not original. Stephen Hawking adds that “a Cambridge don, John Michell, wrote a paper in 1783 in the Philosophical Transactions of the Royal Society of London in which he pointed out that a star that was sufficiently massive and compact would have such a strong gravitational field that light could not escape…A similar suggestion was made a few years later by the French scientist the Marquis de Laplace…” (A Brief History of Time, pp. 81-82). Preston’s purpose in the paper Physics of the Ether was to dispel Newton’s spiritualistic notion of “action-at-a-distance” and replace it with the mechanical concept of ether. The total force required in Preston’s following example is said to be equivalent to E = mc2.
To give an idea, first, of the enormous intensity of the store of energy attainable by means of that extensive state of subdivision of matter which renders a high normal speed practicable, it may be computed that a quantity of matter representing a total mass of only one grain, and possessing the normal velocity of the ether particles (that of a wave of light), encloses a store of energy represented by upwards of one thousand millions of foot-tons, or the mass of one single grain contains an energy not less than that possessed by a mass of forty thousand tons, moving at the speed of a cannon ball (1200 feet per second); or other wise, a quantity of matter representing a mass of one grain endued with the velocity of the ether particles, encloses an amount of energy which, if entirely utilized, would be competent to project a weight of one hundred thousand tons to a height of nearly two miles (1.9 miles).” (S. T. Preston, Physics of the Ether, E. & F. N. Spon, London, 1875, #165).
 “If a mass M, originally at rest, while traversing the effective space s, under the influence and in the direction of the pressure p, acquires the velocity c, we have ps = mc2. Since, however, every production of motion implies the existence of a pressure (or of a pull) and an effective space, and also the exhaustion of one at least of these factors, the effective space, it follows that motion can never come into existence except at the cost of this product, ps = mc2. And this it is which for shortness I call ‘force’” (J. R. Mayer, translated by J. C. Foster, “Remarks on the Mechanical Equivalent of Heat,” The Correlation and Conservation of Forces, 1867, pp. 331, 336). Thomson’s use of the formula has not escaped the notice of at least some modern physics textbooks. In Fundamentals of Physics by Halliday, et al, they state: “A decade before Einstein published his theory of relativity, J. J. Thomson proposed that the electron might be made up of small parts and that its mass is due to the electrical interaction of the parts. Furthermore, he suggested that the energy equals mc2” (John Wiley, fourth edition, p. 735). Cunningham, The Principle of Relativity, 1914, p. 189. N. M. Gwynne, Einstein and Modern Physics, p. 36; F. Hasenöhrl in Annalen der Physik, 4, 16, 589, 1905, and Wien. Sitzungen IIa, 113, 1039, 1904. Hasenöhrl’s original equation was 8E/3c3, which was then changed to 4E/2c3. Some sources have ¾ E=mc2; Kostro has E = ¾ mc2 (Einstein and the Ether, p. 135). Philip Lenard, Über Äther und Uräther, Leipzig, Verlag von S. Kirzel, 1921, cited in Kostro’s Einstein and the Ether, p. 135. Philip Lenard, Über Energie und Gravitation, Berlin/Leipzig, Walter de Gruyter und Co., 1929, cited in Kostro’s Einstein and the Ether, p. 136. Arthur I. Miller, The Special Theory of Relativity: Emergence and Early Interpretation, 1998, pp. 338-339. Miller writes: “But where is the 4/3-factor? It is reasonable to conjecture that by May 1907, when Einstein submitted…for publication, he knew full well that the electron’s mass occurred in kinematical quantities deduced from its self-fields as 4/3 times its electrostatic mass – for example…the role of Poincaré’s stress and very probably of Abraham’s (1905) which contained a detailed discussion of the necessity for an extra energy to correct the Lorentz-electron’s total energy. In fact, Einstein may well have avoided the particular example of Lorentz’s electron because of his having been unable to deduce the 4/3-factor from the relativistic kinematics.” Max von Laue in Albert Einstein: Philosopher Scientist, ed., P. A. Schlipp, 1988, p. 529. He continues: “…we must explain why Abraham’s model of the electron as well as cavity radiation yield the different relationship m = (4/3) (Eo/c2). The reason is the same in both cases. The electromagnetic field is not capable of existing by itself alone, it requires certain supports of a different nature. Cavity radiation can exist only within an envelope, and the charged sphere would fly apart if it were not for certain cohesive forces. In both cases, motion will give rise to an energy current within the material supports which is directed opposite to the motion. It contributes to the total momentum a negative amount and reduces the factor 4/3 to 1” (ibid., pp. 528-529). This discrepancy can be seen, for example, in the kinetic energy of the electron in the hydrogen atom compared to the speed of light. The ionization energy of the electron is 13.6 eV or 2.17 × 10-18 joules. Transposing K = ½mv2 to v = (2K/m)½, and then making the binding energy of the electron equal to the ionization energy, we have v = (2 x 2.17 × 10-18 J / 9.1 × 10-31 kg)½ = 2.18 × 106 meter/second as the velocity of the electron, but this value is 137.6 times slower than c, the speed of light. The Flash of the Cathode Rays: J. J. Thomson and His Contemporaries, 1998, by Per F. Dahl: “…not only did Thomson anticipate Einstein’s mass-energy equivalence by 24 years…the expression was also anticipated by Oliver Heaviside in 1889.” See also David Bodanis’ book, E=mc2: A Biography of the World’s Most Famous Equation. See a critique of Bodanis’ book by Hans Melberg, How Much Gossip is Required Before Science Becomes Interesting, Walker Publishing, 2000. In his 1900 paper “The Theory of Lorentz and the Principle of Reaction,” Poincaré derived the expression M = S/c2, representing M as the momentum of radiation, S as its flux, and c as the velocity of light. Poincaré reasoned that, since electromagnetic energy behaved like a fluid with inertia, if it is discharged from a source there must be a recoil, just as there is a recoil when a ball is shot from a cannon. Using μ for the mass of the recoiling body, and v for its velocity, the equation is μv = S/c2. Since S = Ec, we have μv = Ec/c2 = E/c2 times c, where the E/c2 represents the role of mass. When v = c, the equation reduces to E = mc2. Poincaré also developed the concepts of relativity and the limit of light’s velocity. Einstein makes no reference to Poincaré in his famous 1905 paper, or anyone else. This is all the more significant since Poincaré wrote 30 books and 500 papers, none of which Einstein claimed to have read. Perhaps Poincaré returned the favor to Einstein since, until his death in 1912, he only mentioned Einstein’s name in print once, and that was to register an objection (Holton, Thematic Orgins of Scientific Thought, p. 249). Regarding the 1905 paper, Clark, an admirer of Einstein, states: “…it was in many ways one of the most remarkable scientific papers that had ever been written. Even in form and style it was unusual, lacking the notes and references which give weight to most serious expositions and merely noting, in its closing paragraph, that the author was indebted for a number of valuable suggestions to his friend and colleague, M. Besso” (Einstein: The Life and Times, p. 101). Later, however, Einstein eliminated Besso’s name from a paper he submitted to the Berlin Academy in 1915 regarding the perihelion of Mercury, even though the equations were “simply to redo the calculation he had done with Besso in June 1913” (Michel Janssen, “The Einstein-Besso Manuscript,” p. 15). As for the 1905 paper, how it is that a 9,000 word paper on one of the most controversial ideas ever presented to mankind made it past the editor of Annalen der Physik, the world’s leading physics periodical, is anyone’s guess. The most likely reason is that Max Planck, the chief editor of Annalen in 1905, published it due to his total acceptance of Special Relativity, which he demonstrated by defending it against Kaufmann in 1906. In any case, an editor of a prestigious physics journal should want to know whether anyone prior to Einstein had written about the ideas being presented, especially since the editors themselves were very familiar with the work of Lorentz and Poincaré. When asked about whether his 1905 paper was guilty of plagiarism, Einstein retorted in his 1907 paper: “It appears to me that it is the nature of the business that what follows has already been partly solved by others. Despite that fact, since the issues of concern are here addressed from a new point of view, I am entitled to leave out a thoroughly pedantic survey of the literature…” (Über die vom Relativitätspringzip geforderte Trägheit der Energie,” Annalen der Physik 23 (4), p. 373). Yet in a 1935 paper Einstein admitted: “…because the Lorentz transformation, the real basis of special relativity theory…” (“Elementary Derivation of the Equivalence of Mass and Energy,” Bulletin of the American Mathematical Society 61:223-230; first delivered as The Eleventh Josiah Willard Gibbs Lecture at a joint meeting of the American Physical Society and Section A of the AAAS, Pittsburgh, December 28, 1934, emphasis Einstein’s). There was hardly any way to avoid this realization, since Lorentz’s Transformation equation is identical to the equation for Einstein’s Special Relativity. My thanks to Richard Moody in Nexus Magazine, vol. 11, no. 1, Dec.-Jan. 2004 for many of the above quotes. Against all this is Gerald Holton’s view that Einstein never read Lorentz and Poincaré before 1905; that Einstein showed “painful honesty,” and that “the so-called revolution which Einstein is commonly said to have introduced into the physics in 1905 turns out to be at bottom an effort to return to a classical purity” (Thematic Origins of Scientific Thought, pp. 199, 200, 195 in order of ellipses). Umberto Bartocci, Professor of Mathematics at the University of Perugia, Italy, in his book, Albert Einstein E Olinto De Pretto: la vera storia della formula piu’ famosa del mondo (translated: “Albert Einstein and Olinto De Pretto, the true history of the most famous formula in the world,” Societa Editrice Andromeda, via S. Allende1, 40139) provides documentation that De Pretto published an article in which he gave, in its final form, the equation E = mc2. This article was published on June 16, 1903, and published again in February 27, 1904, the second time in the Atti of the Reale Instituto Veneto di Scienze. De Pretto thereby preceded Einstein’s famous 1905 E = mc2 paper by at least a year and half. Could Einstein have copied from De Pretto? No one can prove definitively that Einstein saw De Pretto’s article, but Professor Bartocci offers some intriguing speculation. Professor Bartocci traced a link between De Pretto and Einstein, through Einstein’s best friend, Michele Besso. As we noted, Besso is the only person credited in the famous E = mc2paper of 1905. See also R. Carroll’s, “Einstein’s E = mc2 ‘was Italian’s idea,’” (The Guardian, Nov. 11, 1999, cited in Moody). Planck writes: “…through every absorption or emission of heat the inertial mass of a body alters, and the increment of mass is always equal to the quantity of heat…divided by the square of the velocity of light in vacuo” (M. Planck, Sitz. der preuss. Akademie der Wissenschaften (Berlin), Physik. Math. Klasse. 13 (June, 1907), p. 566. Regarding Einstein’s 1905 paper (Annalen der Physik 18, 639), Planck shows that, although Einstein came to “essentially the same conclusion by application of the relativity principle to a special radiation process,” he did so by assuming the existence of one of the mathematical components. Thus Planck continues, “however under the assumption permissible only as a first approximation, that the total energy of a body is composed additively of its kinetic energy and its energy referred to a system with which it is at rest” (cited in The Einstein Myth and the Ives Papers, Part II, p. 185). Larmor in “On the dynamics of radiation,” Proc. Intern. Congr. Math., Cambridge, 1912, p. 213; W. Pauli, Jr., “Relativitätstheorie,” Encyclopedia Math. Wiss. V-2, hft 4, 19, 679, 1920, as reported by Herbert Ives in Journal of the Optical Society of America 42: 540-543, 1952, and cited in The Einstein Myth, pp. 84, 109, 184.
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Einsteins Theory of Special Relativity
Elizabeth Howell, Space.com Contributor
The theory of special relativity explains how space and time are linked for objects that are moving at a consistent speed in a straight line. One of its most famous aspects concerns objects moving at the speed of light.
Simply put, as an object approaches the speed of light, its mass becomes infinite and it is unable to go any faster than light travels. This cosmic speed limit has been a subject of much discussion in physics, and even in science fiction, as people think about how to travel across vast distances.
The theory of special relativity was developed by Albert Einstein in 1905, and it forms part of the basis of modern physics. After finishing his work in special relativity, Einstein spent a decade pondering what would happen if one introduced acceleration. This formed the basis of his general relativity , published in 1915.
Before Einstein, astronomers (for the most part) understood the universe in terms of three laws of motion presented by Isaac Newton in 1686. These three laws are:
(1) Objects in motion (or at rest) remain in motion (or at rest) unless an external force imposes change.
(2) Force is equal to the change in momentum per change of time. For a constant mass, force equals mass times acceleration.
(3) For every action, there is an equal and opposite reaction.
But there were cracks in the theory for decades before Einstein’s arrival on the scene, according to Encyclopedia Britannica. In 1865, Scottish physicist James Clerk Maxwell demonstrated that light is a wave with both electrical and magnetic components, and established the speed of light (186,000 miles per second). Scientists supposed that the light had to be transmitted through some medium, which they called the ether. (We now know that no transmission medium is required, and that light in space moves in a vacuum.)
Twenty years later, an unexpected result threw this into question. Physicist A.A. Michelson and chemist Edward Morley (both Americans at the time) calculated how Earth’s motion through this "ether" affected how the speed of light is measured, and found that the speed of light is the same no matter what Earth’s motion is. This led to further musings on light’s behavior — and its incongruence with classical mechanics — by Austrian physicist Ernst Mach and French mathematician Henri Poincare.
Einstein began thinking of light’s behavior when he was just 16 years old, in 1895. He did a thought experiment, the encyclopedia said, where he rode on one light wave and looked at another light wave moving parallel to him.
Classical physics should say that the light wave Einstein was looking at would have a relative speed of zero, but this contradicted Maxwell’s equations that showed light always has the same speed: 186,000 miles a second. Another problem with relative speeds is they would show that the laws of electromagnetism change depending on your vantage point, which contradicted classical physics as well (which said the laws of physics were the same for everyone.)
This led to Einstein’s eventual musings on the theory of special relativity, which he broke down into the everyday example of a person standing beside a moving train, comparing observations with a person inside the train. He imagined the train being at a point in the track equally between two trees. If a bolt of lightning hit both trees at the same time, due to the motion of the train, the person on the train would see the bolt hit one tree before the other tree. But the person beside the track would see simultaneous strikes.
"Einstein concluded that simultaneity is relative; events that are simultaneous for one observer may not be for another," the encyclopedia stated. "This led him to the counterintuitive idea that time flows differently according to the state of motion, and to the conclusion that distance is also relative."
Einstein’s work led to some startling results, which today still seem counterintuitive at first glance even though his physics is usually introduced at the high school level.
One of the most famous equations in mathematics comes from special relativity. The equation — E = mc2 — means "energy equals mass times the speed of light squared." It shows that energy (E) and mass (m) are interchangeable; they are different forms of the same thing. If mass is somehow totally converted into energy, it also shows how much energy would reside inside that mass: quite a lot. (This equation is one of the demonstrations for why an atomic bomb is so powerful, once its mass is converted to an explosion.)
This equation also shows that mass increases with speed, which effectively puts a speed limit on how fast things can move in the universe. Simply put, the speed of light (c) is the fastest velocity at which an object can travel in a vacuum. As an object moves, its mass also increases. Near the speed of light, the mass is so high that it reaches infinity, and would require infinite energy to move it, thus capping how fast an object can move. The only reason light moves at the speed it does is because photons, the quantum particles that make up light, have a mass of zero.
A special situation in the universe of the small, called "quantum entanglement," is confusing because it seems to involve quantum particles interacting with each other at speeds faster than the speed of light. Specifically, measuring the property of one particle can instantly tell you the property of another particle, no matter how far away they are. Much has been written about this phenomenon, which is still not fully explained in terms of Einstein’s conclusions.
Another strange conclusion of Einstein’s work comes from the realization that time moves relative to the observer. An object in motion experiences time dilation, meaning that time moves more slowly when one is moving, than when one is standing still. Therefore, a person moving ages more slowly than a person at rest. So yes, when astronaut Scott Kelly spent nearly a year aboard the International Space Station in 2015-16, his twin astronaut brother Mark Kelly aged a little faster than Scott .
This becomes extremely apparent at speeds approaching the speed of light. Imagine a 15-year-old traveling at 99.5 percent the speed of light for five years (from the astronaut’s perspective). When the 15-year-old gets back to Earth, according to NASA, he would be only 20 years old . His classmates, however, would be 65 years old.
While this time dilation sounds very theoretical, it does have practical applications as well. If you have a Global Positioning Satellite (GPS) receiver in your car, the receiver attempts to find signals from at least three satellites to coordinate your position. The GPS satellites send out timed radio signals that the receiver listens to, triangulating (or more properly speaking, trilaterating) its position based on the travel time of the signals. The challenge is, the atomic clocks on the GPS are moving and would therefore run faster than atomic clocks on Earth, creating timing issues. So, engineers need to make the clocks on a GPS tick slower , according to Richard Pogge, an astronomer at Ohio State University.
The clocks in space tick faster, according to Physics Central , because the GPS satellites are above Earth and experience weaker gravity. So even though the GPS satellites are moving and experience a seven-microsecond slowing every day because of their movement, the result of the weaker gravity causes the clocks to tick about 45 microseconds faster than a ground-based clock. Adding the two together results in the GPS satellite clock ticking faster than a ground-based clock, by about 38 microseconds daily.
Special relativity and quantum mechanics
As our knowledge of physics has advanced, scientists have run into more counterintuitive situations. One is trying to reconcile general relativity — which describes well what’s going on with large objects — with quantum mechanics, which is best used for very small things (such as uranium atom decay). The two fields, which excellently describe their individual fields, are incompatible with one another — which frustrated Einstein and generations of scientists after him.
"Relativity gives nonsensical answers when you try to scale it down to quantum size, eventually descending to infinite values in its description of gravity. Likewise, quantum mechanics runs into serious trouble when you blow it up to cosmic dimensions," an article in The Guardian pointed out in 2015.
"Quantum fields carry a certain amount of energy, even in seemingly empty space, and the amount of energy gets bigger as the fields get bigger. According to Einstein, energy and mass are equivalent (that’s the message of E=mc2), so piling up energy is exactly like piling up mass. Go big enough, and the amount of energy in the quantum fields becomes so great that it creates a black hole that causes the universe to fold in on itself. Oops."
There are several ideas to overcome this (which are beyond the scope of this article), but one approach is to imagine a quantum theory of gravity that would have a massless particle (called the graviton) to generate the force. But as physicist Dave Goldberg pointed out in io9 in 2013, there are problems with that. At the smallest scales, gravitons would have infinite energy density, creating an unimaginably powerful gravity field. More study will be required to see if this is possible.
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Einstein Was Right! Scientists Confirm General Relativity Works With Distant GalaxySpace
- Elizabeth Howell, Space.com Contributor
Elizabeth Howell is a contributing writer for Space.com who is one of the few Canadian journalists to report regularly on space exploration. She is pursuing a Ph.D. part-time in aerospace sciences (University of North Dakota) after completing an M.Sc. (space studies) at the same institution. She also holds a bachelor of journalism degree from Carleton University. Besides writing, Elizabeth teaches communications at the university and community college level. To see her latest projects, follow Elizabeth on Twitter at @HowellSpace .
- Elizabeth Howell, Space.com Contributor