Changes

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.'''

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.'''

For example, under classical Newtonian mechanics, the kinetic energy of a moving mass is

Under classical Newtonian mechanics, the momentum and kinetic energy of a moving mass are

:<math>E = \frac{1}{2}mv^2\,</math>

:<math>E = \frac{1}{2}mv^2\,</math>

But under special relativity is is

respectively.  But under special relativity they are

:<math>E = mc^2\left(\frac{1}{1 - v^2/c^2} - 1\right)\,</math>

:<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 second of those equations converges to the first.

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" 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.

It is this requirement, and some "gedanken experiments" 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.

Around 1925, the development of the mass spectrograph, by Francis Aston, made it possible to measure atomic weights to extreme precision.

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=mc² was not a goal of the experiment; it was an incidental consequence.  The equation had already been known and understood for many years.

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=mc² 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=mc² with great precision.  Perhaps the most precise test, by Rainville ''et. al.''<ref name="rainville">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>, confirmed the equation to an accuracy of a few parts per million.

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=mc² with great precision.  Perhaps the most precise test, by Rainville ''et. al.''<ref name="rainville">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>, confirmed the equation to an accuracy of a few parts per million.

==Experimental verification==

==Experimental verification--the Cockcroft-Walton experiment==

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>[http://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>.  ''&quot;their pioneer work on the transmutation of atomic nuclei by artificially accelerated atomic particles&quot;''<ref>[http://www.nobelprize.org/nobel_prizes/physics/laureates/1951/# Nobel Prize Organization]</ref>  Like all of the very famous experiments of modern physics, this experiment has been replicated, with modern equipment, hundreds of times, all around the world.

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>[http://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>.  ''&quot;their pioneer work on the transmutation of atomic nuclei by artificially accelerated atomic particles&quot;''<ref>[http://www.nobelprize.org/nobel_prizes/physics/laureates/1951/# Nobel Prize Organization]</ref>  Like all of the very famous experiments of modern physics, this experiment has been replicated, with modern equipment, hundreds of times, all around the world.

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:

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:

Perhaps best empirical verification of '''E=mc²''' was done in 2005 by Simon Rainville et al., as published in ''[[Nature (journal)|Nature]]'' (which is not a leading physics journal).<ref name="rainville"/>  The authors state in their article in ''Nature'' magazine that &quot;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.&quot;

Perhaps best empirical verification of '''E=mc²''' was done in 2005 by Simon Rainville et al., as published in ''[[Nature (journal)|Nature]]'' (which is not a leading physics journal).<ref name="rainville"/>  The article states that &quot;Einstein's relationship is separately confirmed in two tests, which yield a combined result of 1−&Delta;mc²/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.&quot;

==A Famous Example -- Nuclear Fission of Uranium==

==A Famous Example -- Nuclear Fission of Uranium==