E=mc² is perhaps the most famous, most immediately recognizable to the public, most widely quoted by the public, and least understood by the public, of all equations in science. Most people recognize it instantly, but when asked to explain it, might say something like "It says that matter and energy are the same/equivalent", or "It explains relativity", or "It tells where nuclear power comes from", or "It's about atomic bombs."
Mass refers to what is often called inertia (though that term is used much less now than it was 100 years ago). It is sometimes called inertial mass. Inertial mass is the resistance of an object to being accelerated when a force is applied. The m in E=mc² is the same as the m in Newton's formula of motion F=ma. Gravitational mass is what makes things heavy, that is, what is acted upon by gravity. It is the m1 and m2 in Newton's formula for gravity . The difference between inertial mass and gravitational mass used to be a burning issue among physicists, eventually being tested, and found to be the same, by the famous Eotvos experiments. (Note that general relativity does have an explanation for why this should be, but this is not relevant here.)
Energy refers to kinetic energy of motion, or radiated energy.
- E=mc² is not about gravity or gravitational mass.
- E=mc² is not about light, or other electromagnetic radiation, except insofar as it is a form of energy.
- E=mc² is not about nuclear power per se. It applies everywhere.
- E=mc² is not about where the power of the atomic bomb comes from, though the physicists developing atomic power and atomic bombs were completely familiar with it.
A little calculation will show that E=mc² is dimensionally correct. So, for example, in SI units (also called MKS units), E is in Joules, m in kilograms, and c in meters per second. In CGS units or units of the US Customary System (pounds and feet) it is also dimensionally correct, as long as the units are used consistently. 
In any normal units, the value of c, the speed of light, is enormous, its square even more so. This means that E=mc² equates an extremely tiny amount of mass with an extremely large amount of energy. This gives the equation some of its mystique.
E=mc² is a meaningless though working, almost nonsensical though often applied, e.g., in nuclear physics, statement that purports to relate all matter to energy and light. In fact, no theory has successfully unified the laws governing mass (i.e., gravity) with the laws governing light (i.e., electromagnetism). Simply put, E=mc² is liberal claptrap.
Biblical Scientific Foreknowledge predicts that a unified theory of all the laws of physics is impossible, because light and matter were created at different times, in different ways, as described in the Book of Genesis.
Mass is a measure of an object's inertia, and is directly related to the force of gravity. In contrast, the intrinsic energy of an object (such as an atom) is a function of electrostatic charge and other non-inertial forces, having nothing to do with gravity. Declaring the object's energy to be a function of inertia rather than electrostatics is an absurd and impossible attempt to unify the forces of nature, contrary to Biblical Scientific Foreknowledge.
For more than a century, the claim that E=mc² has never yielded anything of value. Often it seems to be used as a redefinition of "energy" for pseudo-scientific purposes, as by the lamestream media. There have been attempts to find some justification for the equation in already understood processes involved in nuclear power generation and nuclear weapons, and in the speculation about antimatter. The energy released in matter/antimatter annihilations corresponds to E=mc² for example the annihilation of an electron and a positron (each with rest energy of 0.511 mega electron volts according to E=mc²) results in the release of two gamma ray photons, which each have energy 0.511 mega electron volts.
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:
Probably the best empirical verification of E=mc² was done in 2005 by Simon Rainville et al., as published in Nature, as the authors write in their abstract:
|“||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 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 2.6643648 * 1024 atoms.
That number of Radon-222 atoms has mass .98226836 kg. That number of alpha particles has mass .01770864 kg. The mass lost is .00002299 kg.
Each emitted alpha particle has energy of 4.78 MeV, or 4.78 * .16021765 * 10-18 Joules. The total alpha energy from the decay of 1 kg of radium is 2.040 * 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 2.7124612 * 1024 atoms.
That number of Polonium-218 atoms has mass .98194455 kg. That number of alpha particles has mass .018028315 kg.
The mass lost is .00002713 kg.
Each emitted alpha particle has energy of 5.49 MeV. The total alpha energy from the decay of 1 kg of polonium is 2.386 * 1012 Joules.
It looks as though we have to rewrite the law of conservation of mass:
For the Cockcroft-Walton experiment, we us Avogadro's number of particles. The mass of that many of the various atoms, in kilograms, is just their atomic mass. Avogadro's number of Lithium-7 atoms weighs 7.01600455 kg. The Hydrogen atoms weigh 1.007825032 kg, and the Helium atoms weigh 4.00260325 kg each. The atoms weigh 8.023829582 kg before, and 8.0052065 kg after. The mass lost is .018623082 kg.
The energy released is 17.3 MeV per reaction, or 1669 * 10-12 Joules per Avogadro's number.
It looks as though 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 out of nuclear interaction versus non-nuclear ones is unsatisfactory, of course.
We do this for a few other interactions, including the explosion of TNT. This would include many other radioactive decays, and the 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||2.040 * 1012||.00002299 kg|
|alpha decay of Rn-222||2.386 * 1012||.00002713 kg|
|Cockcroft-Walton experiment, per Avogadro||1669 * 1012||.018623082 kg|
We plot these, and a few others, not shown, on graph paper, and find to our amazement that the relationship is linear.
For Radium decay, m/E = .1126 * 10-16 For Polonium decay, m/E = .1137 * 10-16 For the Cockcroft-Walton experiment, m/E = .1116 * 10-16 As a linear relationship, 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 .112 * 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.)
Since we are just making empirical observations, we don't understand why this is so (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.
This sort of inductive approach is a common way that scientific discoveries are made. For example, Newton's law of motion, F=ma, is empirical and backed up by enormous amounts of confirming data.
The formula E=mc² could have been discovered this way in our alternative universe, but, as stated above, that is not historically how it happened. It was formulated on theoretical grounds in order to obtain conservation of momentum in relativistic interactions, and then confirmed by experiment.
- Physicists, when they are really into theory, sometimes like to use "natural units", which are calibrated so that c is dimensionless and has the value 1. That is not the approach taken here, however.
- The equation claims that the total energy E of a body in all forms is equal to that body's mass m multiplied by the square of the speed of light c.
- Peter Tyson The Legacy of E=mc² October 11, 2005. PBS NOVA.
- Lawrence Berkeley National Laboratory The ABC of Nuclear Science
- 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, NOVA, Nov 10, 2005
- 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
- Simon Rainville, James K. Thompson, et. al World Year of Physics: A direct test of E=mc² Nature 438, 1096-1097 (22 December 2005)] doi:10.1038/4381096a; Published online 21 December 2005