E=mc²
Einstein had no sooner despatched his paper On the Electrodynamics of Moving Bodies, than he realised that the same principles applied also to energy and mass as well. So just a day after the publication of his first paper, he submitted another, only three pages long, entitled Does the Inertia of a Body Depend Upon Its Energy Content?. Einstein discovered the relevant principles in an attempt to reconcile Maxwell's electromagnetic theory with the principles governing the conservation of energy and momentum.
According to Maxwell, a light wave carries an amount of energy and momentum, but according to the principles governing the conservation of momentum, in a closed system the sum of mass and its energy equivalent always remains the same. So if a body emits (kinetic) energy in the form of radiation, where does the energy come from? The answer Einstein arrived at was that the body must lose an equivalent amount of mass in accordance with E/c² . So the mass-energy relation tells us that when a body has converted mass to energy, the amount of mass converted can be computed according to the formula:
m = E … and vice versa!
c²
Einstein’s “most famous equation” flows as a result[1]. It carried the principles of Special Relativity another step further, and it was also found necessary to incorporate the Lorentz-Fitzgerald contraction factor, especially for objects moving at speeds close to the speed of light. This is because when observers from another vantage point, say in a laboratory, see a body, eg. an electron, moving at close to the speed of light, its rest mass appears to shrink, and the equation therefore needs to incorporate a correction factor to take this into account. When the electron is at rest, it always has the same mass, but when it is moving, if E=mc² and c is a constant, then m and c have to vary in exactly the same proportion as the energy increases. The electron’s inertial mass – its mass at its own rest frame – does not change, but its mass when moving as the observer in the laboratory measures it, does. One has to know how to calculate the electron’s rest mass, since it has appeared to contract. So to account for the apparent reduction in length of such objects, the familiar E=mc² is divided by the same factor for shrinkage as we noted when considering special relativity: √(1 - v²/c². The full equation therefore appears as:
_mc²__
E = √1 - v²/c² [2]
The reality here is that objects moving close to the speed of light gain mass, as well as, from the point of view of an observer at relative rest, becoming shorter and living longer. The reason, as explained by Philip Ball[3]: the electrons closer to the nucleus become more massive and move closer to the nucleus, lessening their influence on the outer electrons whose orbits expand and whose energies are lowered. As a case in point, experiments have shown that the decay times of muons slow significantly when they get closer to the speed of light.
The amount by which distances and times in one reference frame appear to shrink when perceived from another is governed by what is known as the Lorentz-FitzGerald “shrinkage” or “contraction” factor, which is capable of mathematical resolution as S = √(1 - v²/c², where S = shrinkage, v = velocity and c the speed of light[4]. So viewed from B, if someone on A is at rest, there is no apparent shrinkage. If someone on A is moving very slowly, the shrinkage factor is reduced by a very small amount, but for someone moving very fast, and say close to the speed of light, the amount by which distance and time appear to shrink is considerable.
So E= mc2 is actually a simplified version of what takes place.[5] However (and this should again be emphasised), in the realm of everyday life, Einstein’s various relativistic aberrations are virtually undetectable and space and time and mass and energy appear to behave normally[6].
Elaboration on the background to the L-F contraction and the M-M experiment
The Lorentz-Fitzgerald contraction is named after the physicists who first elaborated the relevant principles. They were trying to preserve the concept of the luminiferous ether which was said to propagate light through space, whereas Einstein, following hard upon the heels of the Michelson-Morley experiment which repeatedly failed to detect the existence of an ether, showed that the it was irrelevant, in fact non-existent.
The Michelson-Morley experiment had been predicated on a number of erroneous assumptions: "If light travelled through the ether, the Earth did as well. And if Earth were moving through the ether, the speed of light would change. When the Earth moved in the same direction as the ether, it would gain some ground on the light beam, and you would measure a smaller value. When the Earth moved in the opposite direction in its orbit around the sun, you’d measure a larger value for the speed of light, because you’d be losing ground". [4A]
The Michelson-Morley experiment (1887): A light source from the left is split in two and deflected up and down and left and right, and then refracted by mirrors back again. If there is an ether moving to the right (imagine the feeling of a wind as you put your hand out of the car while it is moving in a non-windy environment), one may have expected the light beam going up and down to beat the one going left and right, but the viewing screen at bottom detected no discernible difference (interference pattern) was found. In other words, no such hypothesized ether was found.
Source: https://www.google.com.au/search?q=michelson+morley+experiment and http://physics.about.com/od/relativisticmechanics/f/MichelsonMorleyExperiment.htm for explanation.
Michelson explained the original experiment to his children this way: Imagine a race between "two swimmers, one struggling upstream and back, while the other, covering the same distance, just crosses the river and returns ... The second swimmer will always win if there is any current in the river"[6A].
To return to our main theme, Einstein’s dismissal of the ether as a medium through which light travelled meant that electromagnetic waves are able in some mysterious way to snake their way through long stretches of thoroughly empty space entirely on their own without any material medium. They alone are waves of pure massless energy, so every time one looks at the light from a star or a flame, what one sees is pure incandescent energy – “as fantastic, in its own way, as beholding a disembodied soul”.[7]
Hitherto scientists had considered mass and energy as organically different, but in Einstein’s view, they were indistinguishable and interchangeable - rather like a person wearing different clothes or different hairstyles, or like different forms of monetary exchange such as US dollars and British pounds[8]. This “androgenous view of mass and energy” was reminiscent of science’s (at the time) recent discovery about the close connection between electricity and magnetism, as the result of which science’s picture of the world had become more unified but also more ambiguous and therefore less intuitive.
In 1927 Paul Dirac, then from Cambridge University[9] modified Einstein’s equation to include the momentum and spin of a particle along with its mass among relevant factors to be taken into account in determining a particle’s energy[10]. In so doing, he successfully predicted the existence of a positively charged electron, which he called an anti-electron (now referred to as a positron) with the capacity to self-annihilate along with its sister electron when they met, and thus pop out of existence having emerged from nowhere in the first place. The positron is the fundamental ingredient of positron emission tomography (PET scans). Dirac also totally changed our conception of so-called “empty” space into something teeming with energy with particles popping into and out of existence at will. Along the way, his work in this area also reconciled two of the most important but conflicting theories in physics: Einstein’s special theory of relativity encompassed in E=mc² and the bizarre world of the quantum. This startling development will be considered later, following some elaboration of the basic principles lurking within the quantum world itself.
[1] http://www.thebigview.com/spacetime/relativity.html See also Dana Mackenzie, The Universe in Zero Words – The story of Mathematics, Elwin Street Productions, Sydney, 2012op cit, 163.
[2] This account is drawn from Crease, Robert P. Crease, The Great Equations – Breakthroughs in Science from Pythagoras to Heisenberg, Norton, New York, 2008, 170. According to Dana Mackenzie, op cit, 163, E=mc2 was only ever an approximation by that “lazy dog” Einstein (as his teacher Hermann Minkowski - referred to below in the context of General Relativity - once called him), subject to correction at a later stage. All that Einstein proved initially was that energy is approximately equivalent to matter. Michael Guillen paints a somewhat different picture to that of Crease, namely that as a person’s speed increases, his mass and energy expand by the reciprocal of the shrinking factor with the result that as he approaches the speed of light, space and time shrink down to nothing , and his own mass and energy expand up to infinity. Since this is extremely improbable, Guillen says Einstein, in a thought experiment, interpreted the result to signify the impossibility of surpassing the speed of light and, as it were, catching up with his famous light beam: Guillen, op cit, 249-250.
[3] Claiming Einstein for Chemistry, http://www.rsc.org/chemistryworld/Issues/2005/September/einstein.asp
[4] Crease, op cit, 165-166. Crease’s more fulsome explanation and mathematics are laid out at 166-7. Guillen (op cit, 248-249) appears to say that a person’s perception of times and distances shrinks as he speeds up without obvious reference to different inertial rest frames as a factor.
[4A] Carlos I. Calle, Einstein For Dummies (Kindle Location 2102-2104). Wiley. Kindle Edition.
[5] See Robert P. Crease, The Great Equations – Breakthroughs in Science from Pythagoras to Heisenberg, Norton, New York, 2008, op cit, Ch 7, “Celebrity Equation”, esp, pp 164- 179; Michael Guillen, Five Equations that changed the world – The power and poetry of Mathematics, pp 215 ff.
[6] Guillen, 254.
[6A] Robert P Crease, op cit, 148.
[7] Ibid, 249.
[8] Ibid, 253.
[9] He was so taciturn that it was said of him that “if he said two words in a conversation, it meant he was in a talkative mood”: Dana Mackenzie, The Universe in Zero Words – The story of Mathematics, Elwin Street Productions, Sydney, 2012, 173.
[10] Ibid, 164 ff; see also page 93.