E=mc² is Einstein's famous formula which states that energy (E) of a body is equivalent to the square of the speed of light (c²) times the mass (m) of that body. It is a statement that purports to relate all matter to energy. In fact, no theory has successfully unified the laws governing mass (i.e., gravity) with the laws governing light (i.e., electromagnetism), and numerous attempts to derive E=mc² in general from first principles have failed, though there are many derivations for special cases and experimental verifications.
In the USA, the popular Twilight Zone series featured E=mc² prominently, giving the equation greater currency with the public. The song Einstein A Go-Go by the band Landscape had a similar effect in the UK in the 1980s.
Mass is a measure of an object's inertia, in other words its resistance to acceleration. As shown by atomic bombs, matter can in certain circumstances be converted into energy through nuclear fusion or nuclear fission. (This is how the Sun generates energy.) In such limited cases E=mc² accurately calculates the amount of energy released by the amount of energy is completely converted into energy. This equation does not apply in situations where matter is conserved, for example chemical reations or electrostatic interactions.
The claim that E=mc² has never yielded anything of value and it has often been used as a redefinition of "energy" for pseudo-scientific purposes by non-scientific journals. At virtually all colleges and universities physicists explain how the equation is used in nuclear power generation, nuclear weapons, (nuclear fusion, nuclear fission, and speculation about antimatter).
The Theory of Relativity has never been able to mathematically derive E=mc² from first principles, and a physicist observed in a peer-reviewed paper published in 2011 that "Leaving aside that it continues to be affirmed experimentally, a rigorous proof of the mass-energy equivalence is probably beyond the purview of the special theory."
It has been known for a long time that radiation has a mass equivalence, which was correctly derived by Henri Poincare in 1904, but the equation E=mc² makes a claim far beyond that limited circumstance:
|“||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
Description for the layman
Ten top physicists were asked to describe in laymen's terms E=mc²:
|“||Things that seem incredibly different can really be manifestations of the same underlying phenomena.||”|
—Nima Arkani-Hamed, Theoretical Physicist, Harvard University
|“||You can get access to parts of nature you have never been able to get access to before.||”|
—Lene Hau, Experimental Physicist, Harvard University
|“||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
History of E=mc²
|“||Over time, physicists became used to multiplying an object's mass by the square of its velocity (mv²) to come up with a useful indicator of its energy. If the velocity of a ball or rock was 100 mph, then they knew that the energy it carried would be proportional to its mass times 100 squared. If the velocity is raised as high as it could go, to 670 million mph, it's almost as if the ultimate energy an object will contain should be revealed when you look at its mass times c squared, or its mc².||”|
The first experimental verification for the equation was performed 1932 by a team of an English and an Irish physicist, John Cockcroft and Ernest Walton, as a byproduct of "their pioneer work on the transmutation of atomic nuclei by artificially accelerated atomic particles" for which they were honored with the Nobel Prize in physics in 1951. The idea of the mass defect - and its calculation using E=mc² can be found on page 169-170 of his Nobel lecture.
The mass of the particles on the left hand side is 8.0263 amus, the mass on the right hand side only 8.0077 amu. The difference between this masses is .00186 amu, which results in the following back-of-an-envelope calculation:
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:
Some claim that the best empirical verification of E=mc2 was done in 2005 by Simon Rainville et al., as published in Nature (which is not a leading physics journal). The authors state in their article in Nature magazine that "Einstein's relationship is separately confirmed in two tests, which yield a combined result of 1−Δmc²/E=(−1.4±4.4)×10−7, 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."
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. Here is the most famous example of the mass change.
The decay path of Uranium that figured in the Hahn-Strassmann experiment may have been this:
- 235U → 140Xe + 91Sr + 4n
The masses of the particles are:
|Number of protons||92||54||38||0|
|Number of neutrons||235||140||91||4|
|Number of electrons||92||54||38||0|
The mass of the Uranium atom is 235.04393, and the sum of the masses of the products is 234.866503. The difference is .177427 amu, or, using the E=mc² equation, 165 million electron volts. (The generally accepted value for the total energy released by Uranium fission, including secondary decays, is about 200 million electron volts.)
The insight that the conversion from Uranium to Barium was caused by complete fission of the atom was made by Lise Meitner in December, 1938. She had the approximate "mass defect" quantities memorized, and so she worked out in her head, using the E=mc² equation, that there would be this enormous release of energy. This release was observed shortly thereafter, and the result is nuclear power and nuclear weapons.
A Topical Example: Speed of Extremely Energetic Neutrinos
Here is another example of the use of this formula in physics calculations. Recently there has been quite a controversy over whether neutrinos were observed traveling at a speed faster than light. 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=mc2.
The mass of a neutrino is about 0.44x10-36kilograms. (Normally all of these things are measured in more convenient units such as Giga-electron-Volts, but that makes implicit use of E=mc2. 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 .27x10-8Joules. Using the classical formula , we get v=110x1012meters per second. This is about 370,000 times the speed of light. However, the classical formula breaks down at speeds close to , and indeed, as the speed of a massive object approaches , the object's kinetic energy approaches .
Several scientists have gone on record stating that the neutrinos, which have mass, travel at precisely the speed of light. If true, this disproves the Theory of Relativity and the claim that E=mc2. However, it is more likely that those scientists are using language inaccurately. It is impossible to measure the speed of neutrinos precisely. What is meant is the difference between the speed of light and the speed of the neutrinos is too small to measure.
Deducing the Equation From Empirical Observation
While the equation was historically developed on theoretical grounds as an inevitable consequence of special relativity, it is possible to deduce it purely 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's, 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.
For example, the mass of a TNT molecule is 227.1311 Daltons, or 227.1311 g/mol, which is, for all practical purposes, the same as the mass of its constituent Carbon, Hydrogen, Nitrogen, and Oxygen atoms. It is essentially impossible to measure the difference. The principle of conservation of mass is upheld.
But when nuclear phenomena are discovered, we notice something different. The masses of the result particles after an event (e.g. alpha decay, nuclear fission, or artificial transmutation) is measurably less than the masses of the original particle(s). With the invention of the mass spectrometer around 1920, it became possible to measure atomic weights of various isotopes with great precision.
Radium-226 decays into Radon-222 by emission of an alpha particle with an energy of 4.78 MeV.
1 kg of Radium-226 = atoms. (The numerator is Avogadro's number, and the denominator is the atomic weight of Radium-226.) This is 2.6643647 * 1024 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 * 10-18 Joules. The total alpha energy from the decay of 1 kg of radium is 2.04 * 1012 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 = atoms. This is 2.7124611 * 1024 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 * 1012 Joules.
It looks as thought we have to rewrite the law of conservation of mass:
- In all "ordinary" interactions, mass is precisely conserved.
- In nuclear interactions, there is a small but measurable loss of mass.
- By the way, we can clearly see that atomic weights of pure isotopes are not integers, and that it has something to do with the energy released by nuclear disintegration. In retrospect, the formula E=mc² explains the non-integer character of atomic weights.
Making special cases like this is unsatisfactory, of course.
We do this for a few other interactions, including the explosion of TNT. This would include the Lithium-plus-Hydrogen and Uranium fission phenomena described above. We won't bother with the details. As observational scientists, we look for patterns in the behavior of nature. We make a table:
|interaction||energy released per kg, Joules||mass lost per kg of original substance, kg|
|explosion of TNT||4.184 * 106||seems to be zero|
|alpha decay of Ra-226||2.04 * 1012||.00002301 kg|
|alpha decay of Rn-222||2.39 * 1012||.00002703 kg|
We plot these, and a few others, not shown, on graph paper, and find to our amazement that the relationship is linear.
For Ra, m/E = .112794118 E-16 For Po, m/E = .113096234 E-16
If this is linear, the mass defect for TNT would have been .47 * 10-10. We couldn't possibly have measured this.
So we can rewrite the rule for conservation of mass in a more satisfactory way:
- In all interactions, there is a loss of mass, equal to about .113 * 10-16 kg per Joule of energy released.
What we thought was exact conservation is just very nearly exact, and we hadn't been able to measure it before.
But maybe there's more. This constant has dimensions of kilograms per Joule. From high-school physics, we know that that is seconds squared divided by meters squared. That is, it is the reciprocal of the square of a velocity. We calculate that velocity. It is about 2.97 * 108 meters per second. Very close to the speed of light! Very interesting! (The calculations above were not extremely precise. The formula has been verified with great precision, but not here.)
We don't understand why (that will have to wait for the invention of relativity), but we can formulate a hypothesis:
- In all interactions, there is a loss of mass, equal to times the amount of energy released.
We don't have to give the units any more, since everything is now dimensionally correct.
- There is a very interesting analogy with the discovery of Maxwell's Equations. Maxwell found an interesting relationship involving the fundamental constants and appearing in his equations. Specifically, has the dimensions of seconds squared divided by meters squared, and that:
- where "c" was the known velocity of light. He also showed that his equations predict electromagnetic waves, propagating at that speed.
- "... Einstein proves that energy and matter 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.)" Einstein: Genius Among Geniuses - PBS's NOVA
- E=mc² passes tough MIT test, MITNews, Dec 21, 2005
- John D. Norton Einstein for everyone - E=mc², Department of History and Philosophy of Science University of Pittsburgh
- Rod Nave HpyerPhysics - Relativistic Energy, Georgia State University
- Peter Tyson The Legacy of E=mc² October 11, 2005. PBS NOVA.
- Eugene Hecht: How Einstein confirmed E0=mc², American Journal of Physics, Volume 79, Issue 6, pp. 591-600 (2011)
- Herbert E. Ives Derivation of the Mass-Energy Relation, JOSA, Vol. 42, Issue 8, pp. 540-543 (1952)
- Lexi Krock, David Levin (editors) E=mc² explained, June, 2005. PBS NOVA
- David Bodanis Ancestors of E=mc², Nov 10, 2005, NOVA
- Nobel Prize Organization
- John D. Cockroft Experiments on the interaction of high-speed nucleons with atomic nuclei, Nobel Lecture, Dec 11, 1951
- Gerard Piel The age of science: what scientists learned in the 20th century, Basic Books, 2001, p. 144-145
- Nature 438, 1096-1097 (22 December 2005) doi:10.1038/4381096a; Published online 21 December 2005