E = MC2 AND THE WEIGHT OF SUNSHINE - BIG THINGS - Quantum Theory Cannot Hurt You - Marcus Chown

Quantum Theory Cannot Hurt You - Marcus Chown (2007)




Photons have mass?!? I didn’t even know they were Catholic.

Woody Allen

It’s the biggest set of bathroom scales imaginable. And, oh, yes, it’s heat resistant too. It’s so big in fact that it can weigh a whole star. And today it’s weighing the nearest star of all: our Sun. The digital display has just come to rest and it’s registering 2 × 1027 tons. That’s 2 followed by 27 zeroes—2,000 million million million million tons. But wait a minute, something’s wrong. The scales are superaccurate. That’s another remarkable thing about them, in addition to their size and heat resistance! But every second, when the display is refreshed, it reads 4 million tons less than it did the previous second. What’s going on? Surely the Sun isn’t really getting lighter—by the weight of a good-sized supertanker—every single second?

Ah, but it is! The Sun is losing heat-energy, radiating it into space as sunlight. And energy actually weighs something.1 So the more sunlight the Sun gives out, the lighter it gets. Mind you, the Sun is big and we’re only talking about it losing about a 10-million-million-millionth of a per cent of its mass per second. That’s hardly more than 0.1 per cent of its mass since its birth.

The fact that energy does indeed weigh something can be seen vividly from the behaviour of a comet. The tail of a comet always points away from the Sun just like a windsock points away from the gathering storm.2 What do the two have in common? Both are being pushed by a powerful wind. In the case of the windsock, it’s a wind of air; in the case of the comet tail, a wind of light streaming outward from the Sun.

The windsock is being hit by air molecules in their countless trillions. It is this relentless bombardment that is pushing the fabric and causing it to billow outward. The story is pretty much the same out in deep space. The comet tail is being battered by countless tiny particles of light. It is the machine-gun bombardment of these photons that is causing the glowing cometary gases to billow across tens of millions of kilometres of empty space.3

But there is an important difference between the windsock being struck by air molecules and the comet’s tail being hit by photons. The air molecules are solid grains of matter. They thud into the material of the windsock like tiny bullets, and this is why the windsock recoils. But photons are not solid matter. They actually have no mass. How then can they be having a similar effect to air molecules, which do?

Well, one thing photons certainly do have is energy. Think of the heat that sunlight deposits on your skin when you sunbathe on a summer’s day. The inescapable conclusion is that the energy must actually weigh something.4

This turns out to be a direct consequence of the uncatchability of light. Because the speed of light is terminally out of reach, no material body can ever be accelerated to the speed of light, no matter how hard it is pushed. The speed of light, recall, plays the role of infinite speed in our Universe. Just as it would take an infinite amount of energy to accelerate a body to infinite speed, it would take an infinite amount of energy to push one to light speed. In other words, the reason that getting to the speed of light is impossible is because it would take more energy than is contained in the Universe.

What would happen, however, if you were to push a mass closer and closer to the speed of light? Well, since the ultimate speed is unattainable, the body would have to become harder and harder to push as you get it closer and closer to the ultimate speed.

Being hard to push is the same as having a big mass. In fact, the mass of a body is defined by precisely this property—how hard it is to push it. A loaded refrigerator which is difficult to budge, is said to have a large mass, whereas a toaster, which is easy to budge, is said to have a small mass. It follows therefore that, if a body gets harder to push as it approaches the speed of light, it must get more massive. In fact, if a material body was ever to attain the speed of light itself, it would acquire an infinite mass, which is just another way of saying its acceleration would take an infinite amount of energy. Whatever way you look at it, it’s an impossibility.

Now, it is a fundamental law of nature that energy can neither be created or destroyed, only transformed from one guise into another. For instance, electrical energy changes into light energy in a lightbulb; sound energy changes into the energy of motion of a vibrating diaphragm in a microphone. What, then, happens to the energy put into pushing a body that is moving close to the speed of light? Hardly any of the energy can go into increasing the body’s speed since a body moving at close to the speed of light is already travelling within a whisker of the ultimate speed limit.

The only thing that increases as the body is pushed harder and harder is its mass. This, then, must be where all the energy goes. But, recall, energy can only be changed from one form into another. The inescapable conclusion, discovered by Einstein, is therefore that mass itself is a form of energy. The formula for the energy locked up in a chunk of matter of mass, m, is given by perhaps the most famous equation in all of science: E = mc2, where c is the scientists’ shorthand for the speed of light.

The connection between energy and mass is perhaps the most remarkable of all the consequences of Einstein’s special theory of relativity. And like the connection between space and time, it is a two-way thing. Not only is mass a form of energy, but energy has an effective mass. Put crudely, energy weighs something.

Sound energy, light energy, electrical energy—any form of energy you can think of—they all weigh something. When you warm up a pot of coffee, you add heat-energy to it. But heat-energy weighs something. Consequently, a cup of coffee weighs slightly more when hot than when cold. The operative word here is slightly. The difference in weight is far too small to measure. In fact, it is far from obvious that energy has a weight, which is of course why it took the genius of Einstein to first notice it. Nevertheless, one form of energy at least—the energy of sunlight—does reveal its mass when it interacts with a comet.

Light can push the tail of a comet because light energy weighs something. Photons have an effective mass by virtue of their energy. Another familiar form of energy is energy of motion. If you step into the path of a speeding cyclist, you will be left in no doubt that such a thing exists. Energy of motion, like all other forms of energy, weighs something. So you weigh marginally more when you are running than when you are walking.

It is energy of motion that explains why the photons of sunlight can push a comet tail. An explanation is needed because they actually have no intrinsic mass. If they did, after all, they would be unable to travel at the speed of light, a speed that is forbidden to all bodies with mass. What light has instead is an effective mass—a mass by virtue of the fact that it has energy of motion.

The existence of energy of motion also explains why a cup of coffee is heavier when hot than when cold. Heat is microscopic motion. The atoms in a liquid or solid jiggle about, while the atoms in a gas fly hither and thither. Because the atoms in a cup of hot coffee are jiggling faster than the atoms in a stone-cold cup, they possess more energy of motion. Consequently, the coffee weighs more.


So much for energy having an equivalent mass, or weighing something. The fact that mass is a form of energy also has profound implications. Since one form of energy can be converted into another, mass-energy can be transformed into other forms of energy and, conversely, other forms of energy can be changed into mass-energy.

Take the latter process. If mass-energy can be made out of other forms of energy, it follows that mass can pop into existence where formerly no mass existed. This is exactly what happens in giant particle accelerators, or atom smashers. At CERN, the European centre for particle physics near Geneva, for instance, subatomic particles—the building blocks of atoms—are whirled around a subterranean racetrack and slammed together at speeds approaching that of light. In the violent smash-up, the tremendous energy of motion of the particles is converted into mass-energy—the mass of new particles that physicists wish to study. At the collision point, these particles appear apparently out of nothing, like rabbits out of a hat.

This phenomenon is an instance of one type of energy changing into mass-energy. But what about mass-energy changing into another type of energy? Does that happen? Yes, all the time.


Think of a piece of burning coal. Because the heat it gives out weighs something, the coal gradually loses mass. So if it were possible to collect and weigh all the products of burning—the ash, the gases given off, and so on—they would turn out to weigh less than the original lump of coal.

The amount of mass-energy turned into heat-energy when coal burns is so small as to be essentially unmeasurable. Nevertheless, there is a place in nature where a significant mass is converted into other forms of energy. It was identified by the English physicist Francis Aston in 1919 while he was “weighing” atoms.

Recall that each of the 92 naturally occurring atoms contains a nucleus made from two distinct subatomic particles—the proton and neutron.5 Since the masses of these two nucleons are essentially the same, the nucleus, at least as far as its weight is concerned, can be thought of as being made from a single building block. Think of it as a Lego brick. Hydrogen, the lightest nucleus, is therefore made from one Lego brick; uranium, the heaviest, is made from 238 Lego bricks.

Now, there had been a suspicion since the beginning of the 19th century that perhaps the Universe had started out with only one kind of atom—the simplest, hydrogen. Since that time, all the other atoms have somehow been built up from hydrogen, by the process of sticking together hydrogen Lego bricks. The evidence for the idea, which had been proposed by a London physician named William Prout in 1815, was that an atom like lithium appeared to weigh exactly six times as much as hydrogen, an atom like carbon exactly 12 times as much, and so on.

However, when Aston compared the masses of different kinds of atoms more precisely with an instrument he invented called a mass spectrograph, he discovered something different. Lithium in fact weighed a shade less than six hydrogen atoms; carbon weighed a shade less than 12 hydrogen atoms. The biggest discrepancy was helium, the second lightest atom. Since a helium nucleus was assembled from four Lego bricks, by rights it should weigh four times as much as a hydrogen atom. Instead, it weighed 0.8 per cent less than four hydrogen atoms. It was like putting four 1-kilogram bags of sugar on a set of scales and finding that they weighed almost 1 per cent less than 4 kilograms!

If all atoms had indeed been assembled out of hydrogen atom Lego bricks, as Prout strongly suspected, Aston’s discovery revealed something remarkable about atom building. During the process, a significant amount of mass-energy went AWOL.

Mass-energy, like all forms of energy, cannot be destroyed. It can only be changed from one form into another, ultimately the lowest form of energy—heat-energy. Consequently, if 1 kilogram of hydrogen was converted into 1 kilogram of helium, 8 grams of mass-energy would be converted into heat-energy. Amazingly, this is a million times more energy than would be liberated by burning 1 kilogram of coal!

This factor of a million did not go unnoticed by astronomers. For millennia, people had wondered what kept the Sun burning. In the 5th century BC, the Greek philosopher Anaxagoras had speculated that the Sun was “a red-hot ball of iron not much bigger than Greece.” Later, in the 19th century, the age of coal, physicists had naturally wondered whether the Sun was a giant lump of coal. It would have to be the mother of all lumps of coal! They found, however, that if it was a lump of coal, it would burn out in about 5,000 years. The trouble is that the evidence from geology and biology is that Earth—and by implication the Sun—is at least a million times older. The inescapable conclusion is that the Sun is drawing on an energy source a million times more concentrated than coal.

The man who put two and two together was English astronomer Arthur Eddington. The Sun, he guessed, was powered by atomic, or nuclear, energy. Deep in its interior it was sticking together the atoms of the lightest substance, hydrogen, to make atoms of the second lightest, helium. In the process, mass-energy was being turned into heat and light energy. To maintain the Sun’s prodigious output, 4 million tons of mass—the equivalent of a million elephants—was being destroyed every second. Here, at last, was the ultimate source of sunlight.

This discussion conveniently skirts over the matter of why making a heavy atom out of a light atom converts so much mass-energy into other forms of energy. A digression may help.

Imagine you are walking past a house and a slate falls from the roof and hits you on the head. Energy is released in this process. For instance, there is the whack as the slate hits your head—sound energy. Maybe it even knocks you over. Then there is heat energy. If you could measure the temperature of the slate and your head very accurately, you would find they were slightly warmer than before.

Where did all this energy come from? The answer is from gravity. Gravity is a force of attraction between any two massive bodies. In this case, the gravity between Earth and the slate pulls them closer together.

Now, what would happen if gravity was twice as strong as it is? Clearly, the slate would be pulled towards Earth faster. It would make a bigger noise when it hit, create more heat, and so on. In short, more energy would be released. What if gravity were 10 times stronger? Well, even more energy would be unleashed. Now, what if gravity was 10,000 trillion trillion trillion times stronger? Obviously, a mind-bogglingly huge amount of energy would be released by the crashing slate (and the combination of Earth and slate would be lighter, like the helium atom).

But isn’t this just fantasy? Surely, there is no force that is 10 trillion trillion trillion times stronger than gravity? Well, there is, and it is operating in each and every one of us at this very moment! It is called the nuclear force, and it is the glue that holds together the nuclei of atoms.

Imagine what would happen if you took the nuclei of two light atoms and let them fall together under the nuclear force rather like the slate and Earth falling together under gravity. The collision would be tremendously violent and an enormous amount of energy would be liberated—a million times more energy than would be released by burning the same weight of coal.

Atom building is not only the source of the Sun’s energy. It is also the source of the energy of the hydrogen bomb. For that’s all H-bombs do—slam together hydrogen nuclei (normally, a heavy cousin of hydrogen, but that’s another story) to make nuclei of helium. The helium nuclei are lighter than the combined weight of their hydrogen building blocks, and the missing mass reappears as the tremendous heat energy of the nuclear fireball. The destructive power of a 1-megaton hydrogen bomb—about 50 times greater than the one that devastated Hiroshima—comes from the destruction of little more than a kilogram of mass. “If only I had known, I should have become a watchmaker!” said Einstein, reflecting on his role in the development of the nuclear bomb.


Even though Einstein downgraded mass, showing that it was merely one among countless other forms of energy, it is special in one way: It is the most concentrated form of energy known. In fact, the equation E = mc2 encapsulates this fact. The physicists’ symbol for the speed of light, c, is a big number—300 million metres per second. Squaring it—multiplying it by itself—creates an even bigger number. Applying the formula to 1 kilogram of matter shows that it contains 9 × 1016 joules of energy—enough to lift the entire population of the world into space!

Of course, to get this kind of energy out of a kilogram of matter, it would be necessary to convert it entirely into another form of energy—that is, to destroy all of its mass. The nuclear processes in the Sun and a hydrogen bomb liberate barely 1 per cent of the energy locked up in matter. However, it turns out that nature can do far better than this.

Black holes are regions of space where gravity is so strong that light itself cannot escape—hence their blackness. They are the remnant left behind when a massive star dies, shrinking catastrophically in size until they literally wink out of existence. As matter swirls down into a black hole, like water down a plug hole, it rubs against itself, heating itself to incandescence. Energy is unleashed as both light and heat. In the special case when a black hole is spinning at its maximum possible rate, the liberated energy is equivalent to 43 per cent of the mass of the matter swirling in. This means that, pound for pound, the in-fall of matter onto a black hole is 43 times more efficient at generating energy than the nuclear processes powering the Sun or an H-bomb.

And this isn’t just theory. The Universe contains objects called quasars, the superbright cores of newborn galaxies. Even our own Milky Way galaxy may have had a quasar in its heart in its wayward youth 10 billion years ago. The puzzling thing about quasars is that they often pump out the light energy of 100 normal galaxies—that’s 10 million million suns—and from a tiny region smaller than our solar system. All that energy cannot be coming from stars; it would be impossible to squeeze 10 million million suns into such a small volume of space. It can only come from a giant black hole sucking in matter. Astronomers, therefore, firmly believe that quasars contain “supermassive” black holes—up to 3 billion times the mass of the Sun—that are steadily gobbling whole stars. But even black holes can convert barely half of the mass of matter into other forms of energy.

Is there a process that can convert all of the mass into energy? The answer is yes. Matter actually comes in two types—matter and antimatter. It is not necessary to know anything about antimatter other than the fact that, when matter and antimatter meet, the two destroy, or annihilate each other, with 100 per cent of their mass-energy flashing instantly into other forms of energy.

Now, our Universe, for a reason nobody knows, appears to be made almost entirely of matter. This is a deep puzzle because, when tiny amounts of antimatter are made in the laboratory, their birth is always accompanied by an equal amount of matter. Because there is essentially no antimatter in the Universe, if we want any we have to make it. It’s difficult. Not only do you have to put in a lot of energy to make it—as much energy as you are likely to get out!—but it tends to annihilate as soon as it meets ordinary matter, so it’s difficult to accumulate a lot of it. So far scientists have managed to collect less than a billionth of a gram.

Nevertheless, if the problem of making antimatter in quantities could be cracked, we would have at our disposal the most powerful energy source imaginable. The problem with all spacecraft is that they have to take their fuel along with them. But that fuel weighs a lot. So fuel is needed to lift the fuel into space. The Saturn V rocket, for instance, weighed 3,000 tons and all that weight—mostly fuel—was needed to take two men to the surface of the Moon and return them safely to Earth. Antimatter offers a way out. A spacecraft would require hardly any antimatter to fuel it because antimatter contains such a tremendous amount of energy pound for pound. If we ever manage to travel to the stars, we will have to squeeze every last drop of energy out of matter. Just as in Star Trek, we will have to build starships powered by antimatter.

1 I am using the word weight here the way it is used in everyday life as synonymous with mass. Strictly speaking, weight is equivalent to the force of gravity.

2 A comet is a giant interplanetary snowball. Billions of such bodies are believed to orbit in the deep freeze beyond the outermost planet. Occasionally, one is nudged by the gravity of a passing star and falls toward the Sun. As it heats up, its surface cracks, and buckles, and boils off into the vacuum to form a long, glowing tail of gas.

3 Actually, the tail of a comet is pushed by a combination of the light from the Sun and the solar wind, the million-mile-an-hour hurricane of subatomic particles—mostly hydrogen nuclei—that streams out from the Sun.

4 Strictly speaking, the thing photons possess is momentum. In other words, it takes an effort to stop them. This effort is provided by the comet’s tail, which recoils as a result.

5 Except, of course, the most common isotope of hydrogen, the nucleus of which consists simply of one proton and no neutrons.