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.

Five-minute Explainer: Special Relativity

This is going to be the the briefest history of time ever. When I’m done, my goal is for you to understand not only that motion changes time and space but how and why.

Einstein created the theory of special relativity to answer these questions, and it does so in a very satisfying and complete way which physicists still haven’t improved on. It may surprise you to know that the paper in which he originally described it was only thirty-one pages long.

Postulates

Before we start, we need just the slightest background here on what Einstein had to work with when he came up with special relativity. Namely, he used two assumptions.

• The laws of physics are the same for any point of view which seems stationary to the observer. This is known as the principle of relativity. It means that it’s physically impossible to distinguish between whether I’m moving or the world is moving if I were to jump up and down. Both are true at the same time. You can take whichever point of view you like. That stationary point of view is called an inertial frame of reference.
• The speed of light is always the same in all inertial frames of reference, regardless of the speed of whatever is emitting the light.

There are some additional assumptions you have to bake in, like conservation laws and so on, but none of those would strike you as terribly strange, and they’re subtle enough points that they don’t bear discussing here.

From those two assumptions, which are today called postulates, the rest of the theory emerges. Remember that carefully—every strange part of special relativity is an emergent consequence of those two postulates. No alternative to the theory could work without some change to the postulates, and at the time Einstein was working, he was almost certain they were true. Einstein merely carried the assumptions to their logical conclusion: special relativity.

The Principle of Relativity

Einstein loved to visualize things, and he used trains to illustrate his theory originally because of the train station in Bern, Switzerland. We’ll use trains, too. To make things even easier to visualize, we’ll not deal with light directly but instead use a thought experiment involving a thrown ball.

Imagine a person standing still on the ground can throw a ball at 100 kilometers per hour. The ball crosses a distance of 27.78 meters in 1 second. The thrower can always throw at that speed, so over the course of 1 second, the distance traversed is always 27.78 meters.

Now imagine this same person is standing on a train car. The train is moving at 30 kilometers per hour. They throw the ball in the direction of travel at 100 kilometers per hour. How far does the ball travel?

For an ordinary ball, the distance traveled depends on where you stand. The person throwing the ball on the train sees nothing out of the ordinary because they too are moving along. They see the ball travel at 100 kilometers per hour, and so they likewise see the ball cross 27.78 meters in 1 second. This accords with the first postulate, the principle of relativity. No matter that the train is moving. To the person on the train, it might as well be still while the rest of the world moves backward.

However, a person standing stationary on the ground next to the train would see the ball thrown at 130 kilometers per hour because the train’s motion adds onto the ball’s motion. Therefore, the ball travels 36.11 meters over 1 second. The velocities add together.

A Constant Speed of Light

So far, I have described the ordinary, intuitive behavior you would expect in this situation. Now let’s change things up: The ball can never travel at any other speed than 100 kilometers per hour in any direction, regardless of who is standing where or who is in motion compared to whom. This is similar to how light behaves, according to the second postulate.

The person on the train throws the ball, and it travels at 100 kilometers per hour relative to them, crossing 27.78 meters in 1 second. So far, so good! But the person on the ground also sees the ball travel 100 kilometers per hour, despite the velocity of the train being 30 kilometers per hour. The velocity of the train no longer adds onto the velocity of the ball, yet the ball is still in motion when it is thrown.

How far does the ball travel now?

Here, the universe encounters some very awkward bookkeeping problems. If the person on the ground also saw it travel 27.78 meters in 1 second at 100 kilometers per hour, that ball would land somewhere other than where the person on the train sees it land. This train is in motion, along with the ball. We expect it to cover a distance in 1 second equivalent to the motion of the train in addition to the 27.78 meters which the person on the train sees. That’s 36.11 meters. Yet it cannot cross that distance in 1 second because the ball cannot go faster than 100 kilometers per hour.

For the ball to go different places for different people violates causality itself—it would mean cause and effect were broken. Einstein assumed cause and effect would work out because without them, science wouldn’t do us much good for describing the universe. At the same time, however, he had this very annoying issue of how to solve this problem of reconciling time and distance under these conditions. He thought about it until he understood that some of the assumptions he held about what the universe would keep constant were not, in fact, fixed.

The way to resolve the problem above is that the ball does travel only 27.78 meters. How can that be? Because the train itself gets shorter. The person on the ground, looking up at the train, would see the train squished in the direction of travel. The universe solves the bookkeeping problem around the speed of light by altering space itself! In point of fact, those 27.78 meters over which the ball travels would look different to a person standing on the ground compared to someone on the train.

This is not just a coincidence. It is the way the universe must work because of the finite and invariant speed of light—or, in this case, the invariant speed of the ball thrown. It simply cannot be any other way without breaking causality or changing one of the postulates.

There is one additional problem though. If a ball is moving at 100 kilometers per hour, and the distance it’s moving over has just shortened, wouldn’t it arrive in less than a second now? This problem really bothered Einstein until he let go of the assumption that the universe kept an absolute time clock somewhere—in other words, that there’s no one real, absolute time. This let the universe bend time in order to make the math come out right.

This means that the solution to the problem is for the ball to slow down, along with everything else, when it’s in motion relative to an observer. The sum total of these effects leads to the stationary person on the ground seeing the ball move through a shorter distance while also moving more slowly. There can be no distortion in space without a matching distortion in time, and it appears that space and time are so inextricably bound that it’s easier to deal with them as a single thing called spacetime.

For the person on the train, they see everything happening there normally, but due to the principle of relativity, they see things on the ground also squished and in slow motion.

Cause and Effect

By changing the rules the ball followed in our thought experiment, some very unintuitive consequences emerged. As it happens, light really does behave the way the ball does. Because time and space warp for light, they warp for anything moving at any speed, though the speed limit is so high that the effects aren’t obvious in ordinary life.

It was only necessary to change the behavior of the ball—and hold everything else equal—to see the effect of special relativity on time and space. It caused the everything in motion to squish (length contraction) and begin moving in slow motion (time dilation) when seen by the person on the ground next to the train.

In the next five-minute explainer, I’ll describe even more incredible effects which emerge from the invariant speed of light which Einstein and others found later.

I am grateful again to Zuzu O. for her thoughtful suggestions on improving the readability of this post.