Five-minute Explainer: Why Is Mass Equivalent to Energy?

This essay continues from the previous one in this series, “Five-minute Explainer: Special Relativity.”

The most recognizable equation of the twentieth century equates mass to energy.

\(E = mc^2\)

Specifically, this equation relates a very small quantity of mass to a huge quantity of energy. Why should that be true? What does it imply?

It follows as a consequence of special relativity—one which emerged only after Einstein and his friends worked out the initial theory when they considered how energy is conserved.

Energy: Always Transforming Yet Never Changing

The conservation of energy means that it is never created or destroyed; it only changes form. This cosmic bookkeeping of energy suggests that its different forms are in fact the same underlying phenomenon which is conserved in quantity through each transformation. We should look at this idea more closely, though.

What is energy? This is actually a hard question to answer in an univocal way, but we should adopt a definition useful for our purposes. I will describe energy as that quantitative property of an object which provides an impetus to change over time.

For example, kinetic energy is capable of accelerating an object. Chemical energy is capable of inducing a chemical reaction and changing one substance into another. Nuclear energy is capable of driving a nuclear reaction, thereby disassembling an atomic nucleus. All these forms of energy are equivalent in their respective quantities.

Being equivalent, they may endlessly transform from one form to another. Friction is an example of a phenomenon by which kinetic energy becomes heat energy. We can turn heat back into motion (assuming we could capture it with perfect efficiency, though we can’t) by using the heat to drive a turbine. We can turn the rotational motion of a turbine into electrical energy using induction, and we can transform that into light, sound, motion, magnetism, or heat all over again.

Everyday life relies on hundreds of examples of energy transformations. Even if we can’t capture and reuse energy with total efficiency, we always can always measure it and account for it. Since energy is always perfectly conserved, it makes sense to think of energy as a single phenomenon which changes form endlessly.

Energy in Motion

Now we need to understand how special relativity agrees with conservation of energy. In our thought experiment for special relativity, we watched a train pass by at 30 kilometers per hour. Let’s revisit that train.

While the train is in motion, it has energy—kinetic energy—relative to an observer standing on the ground watching it pass. The faster the train goes, the more velocity it has, the more kinetic energy it acquires, and so on.

However, our observer standing on the ground has already noticed odd effects due to the cosmic bookkeeping which makes special relativity work. That train is getting shorter, and the time on the train is getting slower. Our thought experiment has significantly exaggerated the effects of special relativity because we’ve lowered the speed limit of the universe to 100 km/h. Otherwise, everything we see obeys the actual laws of physics.

Kinetic energy is only dependent on the mass and velocity of an object: as both increase, so does the kinetic energy. This fact remains true whether you consider special relativity or not. However, instead of being half the product of the mass and the square of the speed, as in classical mechanics, the kinetic energy instead tends toward infinity as we approach the speed limit in relativistic physics. As motion begins to warp time and space the closer we come to the speed limit, it must make similar adjustments to kinetic energy.

Points of View in Collision

If kinetic energy didn’t consider the speed limit of the universe, energy would not be conserved properly. Stranger yet, these consequences affect not only energy but mass. We can show this with an example.

Imagine watching from the train as someone throws a ball at 100 km/h to us standing still on the ground watching the train pass at 30 km/h. The person who threw it—who is moving along at the same speed with the ball—doesn’t see anything out of the ordinary with the ball’s motion, energy, or momentum. It moves at 100 km/h relative to them because that’s the speed at which they threw it.

From our vantage point on the ground, we also see the ball arrive at 100 km/h because that’s the speed limit of the universe. The train has been moving at 30 km/h, and so the train imparted some kinetic energy onto the ball, even if the train could not in fact make the ball go any faster than the speed limit. Although the ball is stuck moving at the speed limit, it took some kinetic energy from the train regardless. We know this because the ball imparted an opposite and equal reaction to the train as it was thrown. This means the train lost some energy, and that energy has to go somewhere—so it went into the ball.

We catch the ball at 100 km/h, but the ball somehow has more kinetic energy—and more momentum—than it should because it was thrown from a 30 km/h train. We feel that additional energy in the impact when we catch it. It makes a louder thud in our catcher’s mitt, too. Yet it’s not going any faster than a 100 km/h ball thrown from the ground.

What could be happening here? Here’s more cosmic bookkeeping: since we know the ball cannot move any faster than 100 km/h in our thought experiment, some other quantity has to increase to make up the difference. We also know that kinetic energy relates velocity and mass to one another. The only two things which impart more energy to an impact is adding a heavier object or making it go faster. Therefore, if velocity must stay constant, then mass must increase as a result. We are forced to conclude that the energy imparted onto the ball has added to the mass content of the ball instead of the velocity.

The amount of mass added isn’t much, to be sure—just enough to make up the difference between one ball thrown at 100 km/h and another ball thrown at 130 km/h. Remember also that we’ve lowered the speed limit of our imaginary universe, which exaggerates all the effects. In reality, the speed limit is actually about 1,079,252,848.8 km/h, so differences in speed impart vanishingly tiny bits of mass because ordinary, everyday speeds are tiny in proportion to the universal speed limit, \(c\). The difference in mass to “make up” the missing velocity is usually quite small.

Velocity isn’t the only quantity which “turns into” mass, due to the way energy transforms. All forms of energy are equivalent, so they all represent some amount of mass which can be quantified and calculated.

Once we take this idea to its logical conclusion, we hit upon the unavoidable consequence that the relationship works in reverse as well—that all forms of mass also are equivalent to energy and are quantifiable as such. Even mass at rest has some energy content, the amount of which grows as the mass is set into motion. Motion merely increases the mass–energy.

The Implications of Mass–energy Equivalence

As we just worked out, the math works out such that any tiny bit of mass adds up to an enormous amount of energy, thanks to the fact that the speed of light is so fast. For this reason, it took us a very long time to notice or test this phenomenon.

For example, one kilogram is equivalent to almost ninety quadrillion joules of energy. That’s the same energy output as a twenty-one megaton bomb, or four-fifths the energy output of the 2004 Indian Ocean earthquake and tsunami. In the other direction, the output of a sixty-watt incandescent bulb over an hour—both its light and heat—weighs only 2.4 nanograms, or about the mass of thirty red blood cells.

Special relativity implies that mass and energy are in actuality a single underlying phenomenon, called mass–energy, which we encounter in two familiar forms. In other words, they’re not just similar on some level—they literally are the same thing. Consider, for example, that the Earth weighs approximately 2.38 billion metric tons more due just to the rotational energy of spinning than it would if we changed nothing at all except to cause it not to spin. To stop the world from spinning would be the equivalent of shedding over thirteen million blue whales of mass.

Generalizing Relativity

From the seemingly contradictory postulates of the principle of relativity and the invariance of the speed of light, we have been able to learn new things about the very substance of the universe. If we add in one more principle, we generalize special relativity into a much broader and much more powerful theory which overturned Newton’s theory of gravity. I’ll cover that in the next five-minute explainer.