Five-minute Explainer: What Is Gravity?

This essay continues from the previous one in this series, “Five-minute Explainer: Why Is Mass Equivalent to Energy?”

An old story relates that Newton figured out gravity when an apple fell on his head. Newton himself doesn’t mention the apple falling on his head—this appears to be a later embellishment—but he does mention the apple anecdote a couple of times in his dotage. John Conduitt remembered,

In the year 1666 [Newton] retired again from Cambridge to his mother in Lincolnshire. Whilst he was pensively meandering in a garden it came into his thought that the power of gravity (which brought an apple from a tree to the ground) was not limited to a certain distance from Earth, but that this power must extend much further than was usually thought.

Why not as high as the Moon said he to himself & if so, that must influence her motion & perhaps retain her orbit, whereupon he fell a calculating what would be the effect of that supposition.

This anecdote describes a key quality of gravity as understood then: its nature as an occult force—something working mysteriously and unseen across space.

Before Newton, it was known that the planets moved according to well known laws (Kepler’s laws) which allowed their motions to be predictable. It was not understood, however, why they should move in that way. Kepler’s laws merely came from generalizations after many observations.

Philosophers at the time were troubled that the planets appeared to have no reason to move as they did. Aristotelian thought required that something must drive the planets in their motions. If concentric spheres of quintessence did not, what could this be? For a while, we believed space might be full of a kind of fluid which moved in vortices which propelled the planets like clockwork. This explanation was unexpectedly successful for decades precisely because it did not require belief in occult forces—which is to say, it didn’t require something invisible to reach magically over distances and cause a thing to happen without touching it. It pushed instead of pulled.

Newton had looked at the apple and realized nothing had pushed it to the ground. It seemed to have fallen of its own accord. Newton then extrapolated this idea out beyond the garden into the stars. Once he did, a compact set of laws allowed him to explain all the motions of the heavens very tidily. His explanation, eventually known as the Principia, laid the groundwork for fundamental physics for centuries to come. It was a feat on par with Euclid’s Elements and fully completed the Scientific Revolution which Galileo had inaugurated.

From Hypotheses to Theories

In the second edition of the Principia, Newton tacked on some notes by popular demand. In this General Scholium, he explained that he was in no position to explain what gravity could tangibly be. Famously, he said, “Hypotheses non fingo” (“I do not feign hypotheses [of what gravity could be]”). He described nature as he found it, and the explanation worked. That’s how the matter lay for centuries.

One problem is that, over time, we observed that Newton’s explanations were not perfect after all. There were subtle but galling errors which cropped up in very rare circumstances (such as predicting where Mercury would be over time). Another problem was more metaphysical—Newton’s laws only explained how gravity worked, not what it was.

Einstein solved both problems in a single stroke with general relativity. His theory of general relativity followed in the decade after special relativity as a consequence of the latter. The general theory extended the special one to more situations and provided a more fundamental explanation of universal phenomena, particularly gravity.

Equivalence All the Way Down

If you’ve made it this far, you’ve read how energy is an impetus to change over time. Motion can be a form of energy because it can impart motion on another object, accelerating it. Energy is also equivalent to mass, and mass to energy—even at rest. Finally, you’ve seen how motion itself changes energy, space, and time relative to someone observing the motion.

Now we add a new equivalence—one so incredible in its implications that Einstein called it his “happiest thought.” It’s now simply known as the equivalence principle, special enough to stand alone by that name. It states that it’s impossible to distinguish between acceleration and gravity in any real, physical way.

That is to say, if you were trapped in some enclosed box and unable to see outside, you could not devise any instrument which would be able to tell you whether that box were accelerating in some direction steadily (and therefore drawing you toward the floor) or within a gravitational field (which would accomplish the same effect). Therefore, experiencing acceleration is equivalent to experiencing a gravitational field.

Einstein realized this in November 1907. From that point, he realized that energy, mass, space, time, and gravity were all inseparably linked, and he spent the next several years feverishly working toward a general theory of relativity to explain how it all works. The explanation he came up with in 1915 works so well that its predictive power overturned Newton and has held up even to this day.

Motion in a Bottle

As a result of special relativity, we saw that motion warps space and time. We also know that motion relative to an observer represents kinetic energy, which is equivalent to any other form of energy. Finally, we know that energy is equivalent to mass and vice versa. The final piece of the puzzle to put into place here is that, since motion—and therefore energy—warps time and space, so does mass.

Think of mass as bottled motion. Mass–energy equivalence lets us treat mass as energy which has congealed, more or less, into one place. As I said in the last essay, it’s not enough to think of mass and energy as distinct things sharing some properties—they are a single substance. Therefore, all the same properties and consequences which apply to one form also apply to the other. That means that all the warping effects which apply to energy—to motion—also apply to mass.

So mass warps time and space, but what does this actually mean in reality? The result is gravity! Gravity is an emergent consequence of how mass warps time and space, exactly the same way motion warps time and space due to special relativity. Gravity is in fact not a force reaching mysteriously across distances but instead a bending of space and time which changes the paths of objects traveling through that space and time, leading them inexorably closer to one another.

The Conservative Appeal of Gravity

Let’s dispense with the tired bowling-ball-on-a-rubber-sheet imagery and talk about what that last paragraph actually means. We can begin with the classic assumptions about how objects behave. Newton’s laws state that objects in motion tend to stay in motion, or at rest, unless acted on. They also state that there’s always an opposite and equal reaction for every action.

These are, at their heart, conservation laws. For things to behave otherwise would mean creating or destroying energy. An action must impart an opposite and equal reaction, or energy would go missing. An object at rest must stay at rest, or energy would spontaneously appear. An object in motion must stay in motion, or energy would vanish.

In flat space, therefore, moving objects tend to stay the course in order to conserve energy. You can trace the line of how the object moves geometrically as a straight line. Now if we introduce a mass nearby, space and time contract and stretch, respectively, in the vicinity of that mass. The object’s path still needs to conserve energy, and in order to do so, the line we trace now curves closer to the mass. It appears as if the object “falls” inwards toward the mass—exactly as you’d expect from a gravitational field.

Occult Forces and Fictitious Forces

We no longer need an “occult force” to explain the mechanism of gravity. General relativity—which geometrically describes space and time as it bends under the influence of mass and energy—provides the complete picture.

As it turns out, gravity is not a force at all in the ordinary sense. It only appears to exert a force in the way that a merry-go-round in motion appears to make a ball curve through the air when you throw it from one side to the other. Gravity plays a similar trick on us: we’re constantly on a path through time and space which, were it not for the gigantic rock beneath us, would cause us to curve inexorably toward the center of the Earth. Since the Earth itself interrupts our course, we press against it, and it against us, which imparts the force we’re familiar with.

Making Waves

By uniting conservation laws and a handful of postulates, we can fully explain the substance and behavior of gravity. When we combine this knowledge with the speed limit of the universe, we see that even gravity takes time to travel, which means that changes in gravity take time to travel. This allows gravity to ripple across space and time. We’ll now be prepared to look at these waves in the next explainer.

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.