What did Einstein conclude about the relationship between energy and mass

Einstein's equation E=mc² pops up on everything from baseball caps to bumper stickers. It's even the title of a 2008 Mariah Carey album. But what does Albert Einstein's famous equation really mean?

For starters, the E stands for energy and the m stands for mass, a measurement of the quantity of matter. Energy and matter are interchangeable. Furthermore, it's essential to remember that there's a set amount of energy/matter in the universe.

If you've ever read Dr. Seuss's children's book "The Sneetches," you probably remember how the yellow, birdlike characters in the story go through a machine to change back and forth between "star-bellied sneetches" and "plain-bellied sneetches." The number of sneetches remains constant throughout the story, but the ratio between plain- and star-bellied ones changes. It's the same way with energy and matter. The grand total remains constant, but energy regularly changes form into matter and matter into energy.

Now we're getting to the c² part of the equation, which serves the same purpose as the star-on and star-off machines in "The Sneetches." The c stands for the speed of light, a universal constant, so the whole equation breaks down to this: Energy is equal to matter multiplied by the speed of light squared.

Why would you need to multiply matter by the speed of light to produce energy? The reason is that energy, be it light waves or radiation, travels at the speed of light. That breaks down to 186,000 miles per second (300,000 kilometers per second). When we split an atom inside a nuclear power plant or an atomic bomb, the resulting energy releases at the speed of light.

But why is the speed of light squared? The reason is that kinetic energy, or the energy of motion, is proportional to mass. When you accelerate an object, the kinetic energy increases to the tune of the speed squared. You'll find an excellent example of this in any driver's education manual: If you double your speed, the braking distance is four times longer, so the braking distance is equal to the speed squared [source: UNSW Physics: Einsteinlight].

The speed of light squared is a colossal number, illustrating just how much energy there is in even tiny amounts of matter. A common example of this is that 1 gram of water -- if its whole mass were converted into pure energy via E=mc² -- contains as much energy as 20,000 tons (18,143 metric tons) of TNT exploding. That's why such a small amount of uranium or plutonium can produce such a massive atomic explosion.

Einstein's equation opened the door for numerous technological advances, from nuclear power and nuclear medicine to the inner workings of the sun. It shows us that matter and energy are one.

Explore the links on the next page to learn even more about Einstein's theories.

Originally Published: Sep 14, 2010

By the time Einstein came along with his special theory of relativity in 1905, it had been over 200 years since Newton wrote down his laws of motion. His followers had developed his notions further, elaborating and clarifying them. And two centuries of scientific experiments, of engineering and of technology had been based on these ideas, confirming them directly and indirectly to a very great degree. In all of the life of the 18th and 19th century, Newton’s equations worked. They weren’t likely to stop working in the 20th century, and indeed they didn’t: 19th century clocks,  engines, ships, mills, refrigerators, gyroscopes and cannons worked just fine after Einstein’s theory of special relativity appeared in 1905. So how could Einstein “overthrow” Newton? How could Newton have been “wrong”?

It is easy to write a new set of equations and say “I think the conventional equations are wrong and I think my new equations are right.” Anyone can do that. Indeed I get dozens of papers sent to me or mentioned to me every year (by amateur physicists) that say exactly that about Einstein’s equations. But they mostly can be seen to be wrong on the first page, because there’s some experiment or technology that wouldn’t work at all if the new equations were correct. It is extraordinarily difficult to invent equations that are consistent with all previous well-established experiments and technology.  Such is the high standard of science, and of nature.

Einstein — and a few of his colleagues — had some clues that some significant changes to theoretical physics might be necessary.  In the late 19th century there were two sets of equations (“theories”) being used in physics, for different phenomena. These were Newton’s 17th century equations for how forces cause objects to change their motion (accelerate) and the 19th century equations that govern electricity, magnetism and light, whose complete form was first given by Maxwell, following on the work of Faraday, Ampere, and many others.   For phenomena where both sets of equations were needed, one could find situations where the equations led to  inconsistent predictions.   Knowing that there is an inconsistency is one of the things that allows theoretical physicists to have confidence that they should modify their equations, and can give them a hint as to where that modification might be necessary. But that doesn’t mean finding an acceptable modification is easy.

The trick — and if you think about for a bit, the only possible trick in coming up with new physical theories (and one which, in my experience, very few amateur physicists really appreciate) — is to find equations that extend previous equations, so that it can be proven that in the realms that experiments and technology have already tested, the new equations equal the old equations up to tiny deviations that generally are too small to have yet been noticed in any existing technology and in all (or almost all) previous scientific experiments.

What I’m going to do now is show you how that works for some of Einstein’s equations.

For an object moving freely through empty space, and as viewed by an observer who sees the object moving at a speed v, Einstein proposed relations between energy E, momentum p, mass m and speed v that can be represented through a triangle, as described in detail in this article on mass and energy, and as reproduced more briefly here by the triangle in Figure 1. The energy E is a sum of mass-energy  mc2 and motion-energy (which in this article I will call “K”, because the technical term for motion-energy is “kinetic energy”).  Einstein‘s equation relating energy and momentum and mass is the Pythagorean relationship

• E2 = ( p c )2 + ( m c2 )2

while the speed v is given by a trigonometric relation

(Here and always on this website I define “mass” as a quantity on which all observers agree, sometimes called “invariant mass” to distinguish it from the archaic “relativistic mass”. That is, as in Einstein’s original paper and in his later-in-life viewpoint, mass should be defined so that E = m c2 only for a particle at rest, that is, for a particle with p=0; this invariant mass, unlike relativistic mass and unlike energy, does not increase with speed. I’ll explain why particle physicists take this point of view in an upcoming article.)

Newton’s equations (and their elaboration by his scientific descendants) have a different set of relations, given by the three line segments shown in Figure 2. Newton and followers did not know about mass-energy — that was something that Einstein proposed (as did at least one other of his colleagues, but only Einstein got the details right). So the only energy they talked about (for an object moving freely) was its motion-energy, (which I’ll call “K”, because the technical term for motion-energy is “kinetic energy”.) Newton and friends also didn’t know about the speed of light. So the pre-Einstein relation between motion-energy K and momentum and mass was

while the relation involving speed is

(When you combine these two relations you get the formula you may be more used to: K = 1/2 m v2.) Now this set of relations is not given by a triangle, but by the three line segments given in Figure 2.

Now how did Einstein manage to get a set of formulas that could agree with all previous experiments? (With all of them!) What’s the trick?

Einstein’s relations are almost identical to Newton’s in the case that an object’s speed is small compared to the speed of light! You can see that in Figure 3. Suppose the triangle has a small angle α, which corresponds to having a speed much less than the speed of light.  For Einstein, a particle traveling on its own has motion-energy K equal to its total energy E minus its mass-energy  m c2 , as shown in the figure.  Notice K is much less than E.  Then, as you can see by eye in Figure 3, Newton’s prediction for K and Einstein’s prediction for K are almost the same size; the smaller is the momentum p, the closer the two predictions become.  Meanwhile, the formula relating the speed to momentum and mass is almost the same too, because Newton says

(in the second equation I put an extra “c” in both the top and bottom of the fraction), while Einstein says

but these two equations are almost the same when v is small, because mc2 and E are almost the same when v is small. (You can also see this in terms of the angle of the triangle, because tan α and sin α, which are related to Newton’s and Einstein’s formulas for the speed, are almost identical for a small angle.)  So Einstein’s Pythagorean relationship, and Newton’s three line segments, are experimentally indistinguishable as long as the speed of an object is very, very small compared to c.

In short, Einstein’s equations are very, very cleverly formed; whenever velocities are small, Einstein’s relations and Newton’s relations are almost indistinguishable. This is why — as Einstein checked himself, before presenting his equations to his colleagues (amateurs, please take note) — all existing scientific experiments and all existing 18th and 19th century technologies were consistent with Newton’s equations. The predictions obtained from Newton’s theoretical framework were and remain correct, for all practical purposes, as long as all velocities of massive objects are slow compared with that of light.

That’s for stuff on the slow side.  What about fast stuff?

In the 19th century the only objects known that traveled anywhere near the speed of light were electromagnetic waves (including but not limited to visible light.) Those waves were known to have v = c, at least to the accuracy that anyone could measure. But Einstein’s grand and extremely radical proposal was that these waves were actually made from massless quanta (i.e. `particles’ of a sort, called “photons”), which would also satisfy his relations between E, p, m and v. If you put m=0 in Einstein’s equations, the equations still make sense: try it, you’ll find E = p c and v / c = 1 . (See Figure 4.) This is what you would expect for photons — and thus Einstein’s relations work both for ordinary massive objects of daily life and for the massless photons that make up light!

No one could have made Einstein’s proposal before him. Because to propose light was made from massless particles wouldn’t have made any sense. Look at Newton’s equations. If you have an object whose momentum p isn’t zero, and you put m = 0, you get K = infinity and v = infinity! That’s something people already knew experimentally wasn’t true for light.  And if instead you take p=0 and m=0, Newton’s equations for K and v are both “0 divided by 0” and thus don’t make any prediction at all.

To summarize, Einstein slipped his proposal in between two things people really knew at the turn of his century. They knew that massive objects obeyed Newton’s equations to very high accuracy up to the speeds of bullets and cannonballs, which though fast are still tiny, tiny fractions of c. They knew that light was made from electromagnetic waves moving at about the speed c. Einstein, in suggesting that light was made from massless photons and that Newton’s laws for massive objects needed to be extended to a set of equations that could govern both massive and massless particles, was a great unifier. And through this unification he made a prediction: that massive objects moving close to the speed of light would not obey Newton’s equations, but his own.

Since experimental techniques weren’t instantly available to check what Einstein was proposing, and the proposal was so radical, there was a lot of pushback initially from people who thought what he was proposing made no sense. But already some parts of his relations were tested between 1908 and 1920.  And it didn’t stop there.  Technology and scientific technique have come a long, long, long way since then. Many 20th and 21st century experiments and technologies access much higher velocities than were possible in 1905; indeed particle physics experiments and their medical applications depend crucially on particles with speed v comparable to c, as does our detailed understanding of atoms. And for such particles, Newton and Einstein’s equations disagree sharply! This is shown in Figure 5. All measurements done on fast particles agree with Einstein’s formulas, and not Newton’s.   The debate is long over (except at the inevitable fringes.)  Even the Global Positioning Satellite (GPS) system depends on using Einstein’s equations, instead of Newton’s.

Someday we may find situations in which Einstein’s equations don’t work, and themselves need to be extended. Maybe our first hints will come from an experiment. Or maybe they will arise through the recognition of a theoretical inconsistency. But so far, the special relativity equations that Einstein proposed to govern E, p, m and v, for objects moving freely on their own — and the speed limit that those equations encode, that no object can be measured by any observer to be traveling at a speed faster than c — are still operating without any conflicts.