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E=mc²
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'''E=mc²''' is [[Einstein]]'s famous formula which asserts that the energy ('''E''') which makes up the [[matter]] in any body an unmoving particle is equal to the square of the [[speed of light]] ('''c²''') times the [[mass]] ('''m''') of that bodyparticle.<ref>"... Einstein proves that energy Energy and matter mass are linked in the most famous relationship in physics: E = mc². (The energy content of a body is equal to the mass of the body times the speed of light squared.)" [httphttps://www.pbs.org/wgbh/nova/physics/einstein-genius-among-geniuses.html Einstein: Genius Among Geniuses] - PBS's NOVA</ref> The complete form, when applied to moving objects, is [[E^2=(mc^2)^2+(pc)^2|E²=(mc²)²+(pc)²]], where '''p''' represents momentum,<ref>https://www.youtube.com/watch?v=NnMIhxWRGNw</ref> It is an equation a statement that purports to relate all [[matter]] to [[energy]]. In it's most practical example it is fact, no [[theory]] has successfully unified the [[law]] that explains how nuclear fission creates energy s governing [[mass]] (as seen in nuclear reactors and bombs''i.e.'', [[gravity]])with the laws governing light (''i. Because it is experimentally e.'', [[electromagnetism]]), and mathematically tested on a daily basis with numerous attempts to derive ''E=mc²'' from first principles have failed.<ref name="wvarticles">Five lectures at Wikiversity. The 4th one derives the same positive results formula, using the assumptions in the "What the Equation Means" section.*[https://en.wikiversity.org/wiki/Special_relativity/space,_time,_and_the_Lorentz_transform Lecture 1]*[https://en.wikiversity.org/wiki/Special_relativity/momentum Lecture 2]*[https://en.wikiversity.org/wiki/Special_relativity/energy Lecture 3]*[https://en.wikiversity.org/wiki/Special_relativity/E_%3D_mc%C2%B2 Lecture 4]*[https://en.wikiversity.org/wiki/Special_relativity/spacetime_diagrams_and_vectors Lecture 5]</ref> [[Politics|Political]] pressure, however, has since made it is impossible for anyone pursuing an academic career in [[science]] to even question the validity of this nonsensical [[equation]] without being able to generate any evidence to the contrary since there extraordinary claims must be proven by extraordinary evidence. Simply put, '''''Simply put, E=mc²is [[liberal claptrap]]''''' is accepted as scientific fact.
The formula asserts that the mass of an object, at constant energy, magically varies precisely in inverse proportion to the square of a change in the speed of light over time,<ref>http://www.livescience.com/29111-speed-of-light-not-constant.html</ref> which violates [[Biblical Scientific Foreknowledgeconservation of mass]] predicts and disagrees with commonsense.<ref>The formula asserts that a unified theory of all the laws mass of physics an object has energy associated with it, even when it is impossiblenot moving (p=0). The formula asserts a relationship between the rest mass of an object, because light its energy and matter were created at different timesthe speed of light. According to the formula, in different ways, as described in the apparent mass of an object depends on its energy and so [[Book conservation of Genesismass]]is not satisfied. Instead, relativity proposes that the total energy of a [[closed system]] is conserved, when we "convert" the masses into energies using this formula.</ref>
The [[Theory of Relativity]] has never been able to mathematically derive claim that '''E=mc²''' from first principles, has never yielded anything of value and it has often been used as a physicist observed in a peer-reviewed paper published in 2011 that "a rigorous proof redefinition of the mass-"[[energy equivalence is probably beyond the purview ]]" for pseudo-scientific purposes by non-scientific journals. Claims can be found not only on liberal, second-tier college websites but at those of [[Baylor]] and the special theory[[MIT]] that the equation is used in [[nuclear power]] generation and [[nuclear weapon]]s ([[nuclear fusion]] and [[nuclear fission]]) and about [[antimatter]]."<ref>[http://adsabswww.harvardpitt.edu/abs~jdnorton/2011AmJPhteaching/HPS_0410/chapters/E=mcsquared/index.html John D.79..591H Eugene Hecht: Norton ''How Einstein confirmed for everyone - E=mc²''], Department of History and Philosophy of Science University of Pittsburgh<sub/ref>0<ref>[http:/sub/hyperphysics.phy-astr.gsu.edu/hbase/relativ/releng.html Rod Nave ''HpyerPhysics - Relativistic Energy''], Georgia State University</ref><ref>[https://www.pbs.org/wgbh/nova/physics/legacy-of-e-equals-mc2.html Peter Tyson ''The Legacy of E=mc²''] October 11, American Journal of Physics, Volume 79, Issue 6, pp2005. PBS ''NOVA''. 591-600 (2011)]</ref>
It has been known for a long time that radiation has a mass equivalence, which was correctly derived by [[Henri Poincaré]] in 1904,<ref>[http://www.opticsinfobase.org/josa/abstract.cfm?uri=josa-42-8-540 Herbert E. Ives ''Derivation of the Mass-Energy Relation'', JOSA, Vol. 42, Issue 8, pp. 540-543 (1952)]</ref> but the equation '''E=mc²''' makes a claim far beyond that limited circumstance: {{cquote|The equality of the mass equivalent of radiation to the mass lost by a radiating body is derivable from Poincaré’s momentum of radiation (1900) and his principle of relativity (1904).|||[[Herbert Ives]], 1952}}<!--In 1907, [[Max Planck]] proved in his fundamental paper that the formula ''E=mc²'' is not a general law and any system submitted to an external pressure will obey a different law: its mass will be proportional to its enthalpy ''H=E+PV'', that is, ''m=H/c²''<ref name="Capria">{{cite book |author=Marco M. Capria, Aubert Daigneaut et al. |title=Physics Before and After Einstein |publisher=IOS Press |year=2005 |chapter=2.Mechanics and Electromagnetism... |pages=43|isbn=1-58603-462-6 |url=http://www.dmi.unipg.it/~mamone/pubb/PBAE.pdf |quote=}}</ref>.-->
== Description for the layman ==
cquote|''It certainly is not an equation that reveals all its subtlety in the few symbols that it takes to write down.''|||Brian Greene Theoretical Physicist Columbia University}}
The principle of ''conservation of energy'', universally accepted for well over 100 years, says ::Total energy (kinetic + potential) is always conserved. Hundreds of years of research by chemists (and, before that, the alchemists) worked out the potential energies that are characteristic of various substances, and that the potential and kinetic energies are accurately converted from one to the other, leading to the principle of conservation of total energy. An interesting fact is that, normally, one considers only ''changes'' in potential energy; one doesn't need an absolute scale. A rock at the top of a hill has more potential energy than after it rolls to the bottom of the hill, but the energy at the bottom isn't necessarily zero. We could dig a hole and let it roll down farther, with its energy going negative. Only changes matter. Now it turns out that, once one accepts the implications of E=mc², one ''could'' assign an absolute potential energy to something—its mass times c², and changes in potential energy would work out correctly because of the mass changes. But that isn't necessary, and, in any case, it would require accepting E=mc² and would therefore be getting ahead of the story. With those preliminaries out of the way, it is possible to give a concise explanation of what the equation means: ::'''Potential energy has mass.''' That is, it weighs something. Whenever anything has potential energy of any kind in it, improbable as this may sound, it weighs more. The proportionality constant is 1/c<sup>2</sup>, or 1.11 x 10<sup>−17</sup> kilograms per joule. A fresh battery weighs more than a spent one, a wound-up alarm clock weighs more than a run-down one, etc. Now that's way too small to measure for anything other than nuclear reactions, which is why it escaped everyone's notice for so long. But it has been measured and experimentally verified for nuclear transformations all across the periodic table. :There's an interesting parallel with heat. Before the rise of thermodynamics, it was believed that heat was a "substance". That substance was called "caloric". When heat travels from one body to another, what was really happening was presumed to be a transfer of caloric. Much effort was put into measuring the mass of this mysterious "substance". It was always found to be zero, and we now know that what is actually being transferred is thermal energy. So it is not unheard-of to assign mass to intangible properties. The nonzero mass of potential energy, and the equation E=mc², were determined on theoretical grounds, before any experimental observations were made. The logic of this follows from these assumptions: #Galilean and Newtonian mechanics.#Galilean relativity, that is, the notion that there is no absolute frame of reference.#Conservation of energy.#Conservation of momentum. (So far this is just classical physics.)#The universality of the speed of light. (That is, special relativity.) Keep in mind that, under special relativity, it's not just space and time that need to be redefined. The definitions of momentum and energy need to change also. This is necessary so that the '''conservation of energy and of momentum will be absolutely precise in all circumstances.''' Under classical Newtonian mechanics, the momentum and kinetic energy of a moving mass are:<math>p = mv\,</math>and:<math>E = \frac{1}{2}mv^2\,</math>respectively. But under special relativity they are:<math>p = \frac{mv}{\sqrt{1 - v^2/c^2}}\,</math>and:<math>E = mc^2\left(\frac{1}{1 - v^2/c^2} - 1\right)\,</math>One can verify that, in the non-relativistic limit, the relativistic values converge to the classical ones. It is this requirement, and some "''gedanken experiments''"<ref>https://www.britannica.com/science/Gedankenexperiment</ref> involving conversion between potential and kinetic energy, that lead to E=mc².<ref name="wvarticles"/> These experiments involve some kind of object that isn't moving (though there might be internal motion that doesn't figure in the experiment) and therefore has no kinetic energy and only potential energy, turning into some things that have kinetic energy. The requirements of strict conservation of total momentum and total energy prove the equation. Einstein's famous derivation<ref name="einstein1905b">[http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ "Does the Inertia of a Body Depend its Energy Content?" Albert Einstein, Sept 1905]</ref> involved light instead of tangible objects, but the result is the same. ==History of Experimental Verification==Because the change in mass arising from a given release of energy is so small (<math>1/c^2</math>, which 1.11 x 10<sup>−17</sup> kilograms per joule), it is essentially impossible to check this equation for normal processes. For example, a flashlight battery loses about 1 picogram of mass when it discharges, and the resultant atoms from the detonation of 1 kilogram of TNT weigh 47 nanograms less than the TNT. Even if all the particles of smoke and gas could be collected reliably, the difference couldn't be detected. Measuring the effect requires process that release vastly more energy than ordinary chemical processes. The discovery of Radium and Polonium around 1898 gave a tantalizing hint that there were processes that released far more energy than chemical processes could account for. These elements continuously released measurable heat, and also glowed in the dark. Einstein touched on this possibility in his original 1905 paper.<ref name=einstein1905b/>{{cquote|''It is not impossible that with bodies whose energy content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.''}}It would take more than a decade to develop an understanding of the nuclear process involved. The first thing that was required was accurate knowledge of atomic weights. Atomic weights of the various elements were first measured, with accuracy of a few decimal places, by J. J. Berzelius in the late 1820s. This required extremely painstaking (for the time) measurements. The figures were refined to even more accuracy by J. A. R. Newlands in the 1860s. The values were accurate enough to clearly show the rather interesting property that the atomic weights were nearly integers, but not exactly so. The reason for this would turn out to be partly because of different isotopes (discovered by Frederick Soddy in 1913) and partly because of E=mc<sup>2</sup>. In 1907 Rutherford determined that the "alpha" radiation from Radium was Helium. In 1911 he formulated the theory of the nucleus. In 1919 he demonstrated that nuclear transmutations could take place, such as ::<math>{}_7^{14}\mathrm{N}\, +\, {}_2^4\mathrm{He}\,\rightarrow\, {}_8^{17}\mathrm{O}\, +\, {}_1^1\mathrm{H}</math> The discovery of Radon, and much further investigation, revealed that the behavior of Radium was ::<math>{}_{88}^{226}\mathrm{Ra}\,\rightarrow\, {}_{86}^{222}\mathrm{Rn}\, +\, {}_2^4\mathrm{He}</math>and::<math>{}_{86}^{222}\mathrm{Rn}\,\rightarrow\, {}_{84}^{218}\mathrm{Po}\, +\, {}_2^4\mathrm{He}</math> Accurate ways of measuring speed of a charged particle, by deflecting it in a magnetic field, had been developed by then, so that, by very painstaking observation and measurement, it was determined that the first alpha particle (Helium nucleus) had an energy of 4.78 MeV and the second an energy of 5.49 Mev. This confirmed E=mc<sup>2</sup> up to the accuracy of the measurements. The equation, along with knowledge of isotope mixes, now explained why the atomic weights appearing in the periodic table were nearly integers, but not exactly so. Around 1925, the development of the mass spectrograph, by Francis Aston, made it possible to measure atomic weights to extreme precision. The 1932 Cockcroft-Walton experiment, described in more detail below, started to make the equation famous by confirming it, with reasonable accuracy, for an artificially induced nuclear reaction. (Confirming E=mc<sup>2</sup> was not a goal of the experiment; it was an incidental consequence. The equation had already been known and understood for many years.) In the decades since, nuclear transmutations have been performed, in particle accelerators, all over the periodic table, observing in detail the properties of various isotopes. These have confirmed E=mc<sup>2</sup> with great precision. See [[Quantitative Analysis of Alpha Decay]]. ==The Rainville test==Perhaps the most precise direct empirical verification of E=mc<sup>2</sup> was done in 2005 by Simon Rainville ''et. al.''<ref>[http://www.nature.com/nature/journal/v438/n7071/full/4381096a.html Nature 438, 1096-1097 (22 December 2005)] doi:10.1038/4381096a; Published online 21 December 2005</ref> The article states that "Einstein's relationship is separately confirmed in two tests, which yield a combined result of 1−Δmc²/E=(−1.4±4.4)×10<sup>−7</sup>, indicating that it holds to a level of at least 0.00004%. To our knowledge, this is the most precise direct test of the famous equation yet described." ==The [[Cockcroft and Walton Experiment|Cockcroft/Walton experiment]]==This experiment is not one usually cited as validating E=mc². That was not its goal. The generally accepted important tests of this equation are the measurements of alpha decay energies, described above. In 1932 English physicist John Cockcroft and Irish physicist Ernest Walton performed the first artificial nuclear transmutation of nuclei, for which they were awarded the 1951 [[Nobel Prize]] in physics.<ref>[https://www.nobelprize.org/nobel_prizes/physics/laureates/1951/cockcroft-lecture.pdf John D. Cockroft ''Experiments on the interaction of high-speed nucleons with atomic nuclei''], Nobel Lecture, Dec 11, 1951</ref> The award was for ''"their pioneer work on the transmutation of atomic nuclei by artificially accelerated atomic particles."''<ref>[https://www.nobelprize.org/nobel_prizes/physics/laureates/1951/# Nobel Prize Organization]</ref> Verifying E=mc² was not the goal of the experiment, and the Nobel prize was awarded for the transmutation itself, not any verification of the equation. This experiment could not have proved any general truth to the equation, since it was a test of just one specific reaction. But data from this experiment was consistent with the equation for the particular transmutation involved. They bombarded [[Lithium]] atoms with [[protons]] having a [[kinetic energy]] less than 1 [[Electron-Volts|MeV]]. The result were two (slightly less heavy) [[α-particle]]s, for which the [[kinetic energy]] was measured as 17.3 MeV
:::::<math>{}_3^7\mathrm{Li}\, +\, {}_1^1\mathrm{H}\,\rightarrow\,2\, {}_2^4\mathrm{He}</math>
The mass of the particles on the left hand side is 8.0263 [[atomic mass units|amu]]s, the mass on the right hand side ''only'' 8.0077 amu.<ref>Gerard Piel ''The age of science: what scientists learned in the 20th century'', Basic Books, 2001, p. 144-145</ref> The difference between this masses is .00186 amu, which results in the following back-of-an-envelope calculation:
::::<math>0.00186\,\mathrm{amu} \cdot c^2 = 0.0186 \cdot 1.66 \cdot 10^{-27}\,\mathrm{kg}\cdot\left(3\cdot10^8\,\mathrm{\frac{m}{s}}\right)^2</math>
Accurate measurements and detailed calculations allowed for verifying the theoretical values with an accuracy of ±0.5%. This was the first time a nucleus was artificially split, and thereby the first transmutation of elements using accelerated particles:
==A Famous Example -- Nuclear Fission of Uranium==
For most types of physical interactions, the masses of the initial reactants and of the final products match so closely that it is essentially impossible to measure any difference. But for nuclear reactions, the difference is measurable. That difference is related to the energy absorbed or released, described by the equation E=mc². (The equation applies to '''all''' interactions; the fact that nuclear interactions are the only ones for which the mass difference is measurable has led people to believe, wrongly, that E=mc² applies only to nuclear interactions.)
The [[Theory of Relativity]] played no role in this work, but proponents later tried to retrofit the theory to the data in order to explain the explain the observed mass changes. <ref>Actually, the formula E=mc<sup>2</sup> was published in 1905, and has not changed since then. Fission of Uranium was discovered in 1938. It is not possible that the equation was retrofitted to explain this discovery.</ref> Here is the most famous example of the mass change.
Nuclear fission, which is the basis for nuclear energy, was discovered in experiments by [[Otto Hahn]] and [[Fritz Strassman]], and analyzed by [[Lise Meitner]], in 1938.
:<sup>235</sup>U → <sup>140</sup>Xe + <sup>91</sup>Sr + 4n
(The [[Xenon]] decayed within about a minute to <sup>140</sup>Ba. There are a large number of fission paths and fission products, but they They were searching for the chemical signature of [[Barium]].)
The masses of the particles are:
==A Topical Example: Speed of Extremely Energetic Neutrinos==
Here is another example of the use of this formula in physics calculations. Recently In 2011 there has been quite a controversy over whether neutrinos were [https://www.theguardian.com/science/2011/sep/22/faster-than-light-particles-neutrinos?newsfeed=true reports] that high-energy neutrinos had been observed traveling at a speed faster than the speed of lightin an experiment at the Gran Sasso laboratory in Italy. Specifically, they seemed to have arrived at the detector 60 nanoseconds faster than light would have. Relativity doesn't allow that, and, since neutrinos have nonzero (but incredibly tiny) mass, they aren't even supposed to travel ''at'' the speed of light. This very issue came up on the [[Talk:Main_Page#Neutrinos]]. The speeds under discussion were calculated by the use of E=mc<sup>2</sup>.
The mass of a neutrino is about 0.44x10<sup>-36−36</sup>kilograms. (Normally all of these things are measured in more convenient units such as Giga-electron-Volts, but that makes implicit use of E=mc<sup>2</sup>. If we don't accept that, we have to do the calculations under classical physics, using SI (meter/kilogram/second) units.) The neutrinos were accelerated to an energy of about 17GeV, or .27x10<sup>-8−8</sup>Joules. Using If one did not accept relativity and had to use classical physics and the classical formula <math>\mathrm{E} = \frac{1}{2}mv^2</math>, we one would get v=110x10<sup>12</sup>meters per second. This is about 370,000 times the speed of light, something that scientists would certainly have noticed.However In fact, the classical formula breaks down at speeds close to <math>c</math>with special relativity, and indeed, as the speed is just under the speed of a massive object approaches <math>c</math>light, such that the object's kinetic energy approaches neutrinos should be received at the detector about .26x10<mathsup>+\infty−24</mathsup>seconds (.26 yoctoseconds) later than the speed of light itself. This is far too small to measure—15 orders of magnitude smaller than the resolution of the GPS signals in the experiment.
==Deducing the Equation From Empirical Observation==
So, for the purposes of this section, imagine that one is in the era of "classical physics"; prior to 1900 or so. Relativity has not been invented, but, inexplicably, nuclear physics has. Imagine that the phenomena of radioactivity and nuclear fission have been observed, without any knowledge of relativity.
A well-accepted physical law of classical physics was the law of conservation of mass. This was not easy to deduce. It required careful analysis of such phenomena as combustion, in the 1700's1700s, to eliminate the various confounding sub-phenomena that made the law difficult to see. But, by 1900, the law was well established:
:::*'''In all interactions, mass is precisely conserved.'''
Radium-226 decays into Radon-222 by emission of an alpha particle with an energy of 4.78 MeV.
1 kg of Radium-226 = <math>\frac{6.622 022 \times 10^{2723}}{226.0254}</math> atoms. (The numerator is [[Avogadro's number]], and the denominator is the atomic weight of Radium-226.) This is 2.6643647 * 10<sup>24</sup> atoms.
That number of Radon-222 atoms has mass .98226836 kg. That number of alpha particles has mass .01770863 kg.The mass lost is .00002301 kg.
Each emitted alpha particle has energy of 4.78 MeV, or 4.78 * .1602 * 10<sup>-18−18</sup> Joules. The total alpha energy from the decay of 1 kg of radium is 2.04 * 10<sup>12</sup> Joules.
Also, Radon-222 decays into Polonium-218 by emission of an alpha particle with an energy of 5.49 MeV.
1 kg of Radon-222 = <math>\frac{6.622 022 \times 10^{2723}}{222.0176}</math> atoms. This is 2.7124611 * 10<sup>24</sup> atoms.
That number of Polonium-218 atoms has mass .98194467 kg. That number of alpha particles has mass .01802830 kg.
The mass lost is .00002703 kg.
Each emitted alpha particle has energy of 5.49 MeV. The total alpha energy from the decay of 1 kg of polonium is 2.39 * 10<sup>12</sup> Joules.
It looks as thought we have to rewrite the law of conservation of mass:
| alpha decay of Ra-226
| 2.04 * 10<sup>12</sup>
| .00002301 kg
|-
| alpha decay of Rn-222
| 2.39 * 10<sup>12</sup>
| .00002703 kg
|}
For Po, m/E = .113096234 E-16
If this is linear, the mass defect for TNT would have been .47 * 10<sup>-10−10</sup>. We couldn't possibly have measured this.
So we can rewrite the rule for conservation of mass in a more satisfactory way:
::where "c" was the known velocity of light. He also showed that his equations predict electromagnetic waves, propagating at that speed.
==See also==
*[[Attempts to prove E=mc²]]
*[[Counterexamples to Relativity]]
*[[essayEssay:Rebuttal to Counterexamples to Relativity]]*[[Logical Flaws in E=mc²]]*[[Essay:Rebuttal to Logical Flaws in E=mc²]]*[[Quantitative Analysis of Alpha Decay]]*[[E^2=(mc^2)^2+(pc)^2]]
== References ==
<references />
[[Category:relativityRelativity]][[Category:physicsLaws of Physics]][[Category:science]][[Category:BiblePhysics]]