What is Mass? Part I

This is the second installment of our “What is…?” series. 

During these months of quarantine, a lot of us have probably thought once or twice that we have been putting on some quarantine-weight. And while that is true (or not, for you, athletic reader) what is actually happening, is that you body is acquiring mass. A somewhat trivial difference, you may say, as weight is just the force inflicted by gravity on to some body. And it strikes me as a funny thing that in our everyday physics, everything comes intuitively to us, force, speed, acceleration, rotation stuff. Everything -but mass. It feels intuitive, but the more you think about it, the more you will end up asking yourself… What is mass?

As a disclaimer here, for the philosophically inclined.: I think I should say that I may be able to scratch only the surface of this huge topic by trying to give a small clarification to the main question by stating what we mean with mass as physicists. In this first part, I will talk about what mass and matter meant up to Newton and how Einstein started changing the ideas around it. In contrast, on the second installment, I will try to explain more in the realm of the question “How is mass?”. How is it created, what have energy and Quantum Mechanics to do with it, and how our view of it was altered even further.  As well as for the remaining question, the infinitely problematic “Why is mass?”, well, this is metaphysics. While it is a relevant and interesting question, I don’t think I can tackle it in the course of this article, or my lifetime.

So let’s start.

Classical Mechanics (a.k.a every day life)

Whenever you wake up walk to kitchen and make yourself breakfast and a coffee, that is classical mechanics. The physics you learned in high-school and the one that governs the egg falling into the pan, or supports your whole house, or keeps your bike steady, are one and the same. All of this is classical mechanics. Mass, in this theory, and honestly in most of physics is somewhat ad hoc, an arbitrary concept. I say that because mass was originally defined as the resistance of a body to changes in its acceleration, also called inertia. This means that although it seems to be a fundamental property, it is one that explicitly determined upon measurable quantities, such as position, velocity and acceleration (or a gravitational field) [1].

Untitled_Artwork 6.png
In everyday life everything is Newtonian mechanics. Classical means non-quantum and non-relativistic.

And yet, it seems to us to be something so obvious, right? I personally think that it is because we want to hold on to the notion of perceptual tangibility of objects. This happens at a more subconscious level, where ascribe this apparent obviousness of “what mass is” to a combination of the ideas of the material substance of objects and practical intuition over inertia. Don’t believe me?  It is easy to think about the mass of something you push, but think a bit about the mass of the atmosphere, or any gas, and you will see that all that intuition and obviousness suddenly goes away.

It is in this realm of intuition, where ideas like how matter is immutable, become apparent. Recall from high-school/university that

Matter is neither created nor destroyed, it can only be transformed
-The Conservation of Mass Law, a scientific (fact?)

This is just our intuition from the world, right? It seems so, even as early as in the (greek) pre-socratic philosophy we can find such notions being formed. Even though they talked in general about it in a more abstract way. We definitely don’t want to stay talking about these philosophers for too long, but unfortunately, no talk about mass and matter can be remotely complete without the pre-socratic gang. So let’s summarize: Basically all of them were wrong [2]. But some got  parts slightly well, and those good bits tend to be more interesting.

Let’s start with Empedocles [3], who basically believed that things cannot just appear out of nothing, and that things cannot just disappear into nothingness, so that the total amount of fundamental substance (whatever that was for him) couldn’t possibly vary. About this substance, there were some disagreements.

The Mileto school of philosophy believed originally in one fundamental substance, permanent, all encompassing. Then Anaximander (one of the Mileto boys) stated that contraries rise from it, hot and cold, light and darkness, ice and fire [4]. Anaximenes, his student, on the other hand, declared directly the substance to be air. The interesting part is that his principle of everything, was thickening -and thinning- the air. He stated everything can be obtained by changing the density of the fundamental substance, air. This is very similar to how we know physics works, but you know, with the wrong building blocks.

Untitled_Artwork 7.png
The Presocratic Gang (P.S.A). From left to right: Anaximander, Heraclitus, Empedocles, Democritus  and Anaximenes.

Now, the truly revolutionary ideas came with Democritus and Leucippus, who believed in the atomic principle, which we know is right now. So they were right on the money on that, right? Well, definitely not. In some sense they did get right that there are some fundamental building blocks which make everything up. However, they suggested atoms were shaped in different shapes, and from there it gets weird. Some were balls, some were like coins, some were sticks, some were hooks, and some were shaped like human eyes. Yes, human eyes. But wait, there is more! If you would, for example, eat sweet stuff, that was a specific type of sweet “atom”, caressing your tongue, while bitter stuff would be pointy atoms scratching it.

 

From this point onward, the permanence of mass was more or less assumed, but very rarely checked or questioned. Finally, it was Antoine Lavoisier who performed the experiments that allowed to state that in chemical reactions mass is conserved (in reality this is slightly violated, but more about that later). Let us remember that in the 19th century, chemistry was the only matter-changing science.Physics was about motion, biology was mostly about cataloging life, and was barely starting. And nobody knew they were all connected!

At this point we can stop for a second to relax and breath in the fact that throughout most of history nobody had one clue of what mass is, but most agreed it was not possible to destroy it. Finally, we discovered that up to measurement errors, that seemed to be true.

Nice. But, as it turns out, it is not.

Mass can be fluctuated in and out of existence if help by exchanging it with a little something called energy. How is that possible, you ask? Well…

Relativity

Einstein’s Theory of Relativity is not one, but two theories. There is the so-called Special one, describing the relation between motion and the speed of light, and the General one, also related to that, but on the context of Gravity. We will focus on Special Relativity, because it was the first one introduced the brave new concept that matter could be “destroyed”.

A crash course into relativity (more like a violent crash into it): Relativity is a set of ideas put forward by Albert Einstein -pictured below- which in fact collected a bunch of  results and conjectures going up-and-down physics during the late 19th and early 20th centuries [5]. Before I even go into it, it is important that you know that for us physicists, it is of vital importance that

A physical system, looked from two different inertial frames of observation will yield the same physics.

Imagine there are two cars moving towards each other, you are in the first, and I am in the second. If at a random place along the way a someone drops a ball, we both should be able to correctly assess when and where it hits the floor, after intuitively correcting for our relative speeds. As I hinted with the italics, this applies for inertial frames. Here, inertial frame means a frame which is moving at a constant speed, without any acceleration. Non-inertial frames are a bit more fun, but not relevant here and I explain them a bit in footnote [6]. Now, classical mechanics relies in Newton’s Three Commandments. I mean… Laws. If you change your frame, you would transform it by just shifting your position by the amount of distance introduced by constant speed would actually give, say velocity x time. Time flows at the same rate in both frames.

An actual, 100% real picture of the man himself. Source: Trust me bro

This will leave the governing laws of movement (force=mass x acceleration) exactly the same. The problem was that these transformations, also called Galilean, are not satisfied by electromagnetic waves (light), even for inertial frames! That means that if you were standing light would look differently than if you were in your car. And I mean very different, not just shifting a frequency, but like light suddenly moving in very different ways, and disappearing. We know this is not the case.

So to try to fix this, Einstein assumed two ideas:

A) The change of  inertial frames does not affect the evolution of a system. 
B) The speed of light is invariant of the frame [2].

If you go and play -mathematically- with these postulates, the result is very counter-intuitive. To explain it, let me exploit a bit more the last analogy. Now you are still in the car, but I am the one dropping the ball. And you are going a bit faster, almost at the speed of light. What you see, apart from the normal movement of the ball falling, is that it falls, in fact, slower. Quite slower. And not only that! You suddenly realize that the ball is not spherical, but it looks oblong, more like a football. It seems now to be flattened. We do the same again, and you see that the closer you go to the speed of light the flatter and flatter the ball looks, while getting slower and slower.  When you are very close to the speed of light, say 99.99%, you will see the ball basically as a pancake!

Ok, all of that is very interesting indeed, but why and how is that related to mass? And most importantly, how is that related to mass not being conserved?

IMG_0591
You are in three different scenarios. In one, you are still, watching the ball fall. In the next two, you are travelling at 0.75 and 0.9 times the speed of light. This is what you would see: The ball falling slower, and looking squeezed along the direction of motion. The stopwatches symbolize the amount of time you would see the ball needing to hit the bottom. 

It turns out that those transformations, and mostly the conceptual inclusion of space contraction and time dilation make it very difficult for momentum (the product of mass and velocity) to be conserved. In fact, it was found by Einstein that kinetic energy, the energy of movement not only changes differently in relativity with the change of mass, but now it is completely differently defined. The biggest difference here is that Kinetic Energy in our realm depends on the mass and the speed so that the faster or heavier you are, the more energy you possess. But without movement, you can always say there is zero Energy. In relativity, this is not true anymore. Now, Energy and Mass are tied forever together, so that even when there is no movement, there is always some residual energy.

There is always Energy, the sheer energy of a resting mass. And this is what gives the very well known formula:

Untitled_Artwork 8

One of the most fascinating parts of Relativity is that the definitions match when you have very small velocities. That is brilliant, so it is not that Newton was wrong, it is that this just works for small speeds. He was just incomplete!

The equivalence of mass and energy is in fact a very big deal. It allowed for particles to be able to join into heavier particles, if antity you should care about. Once again, Newton was not wrong, he was incomplete!

So… in the what is mass?

Mass is the measure of resistance to change. A manifestation of the reticence objects to move around, or heat up, or just, plainly, change. Mass is also Energy, higher masses being directly related to higher intrinsic energies. For now, these are enough, but there is so much more we can ask. Many more questions that I will address in the next installment. Questions like what is mass in the context of another successful theory, Quantum Mechanics? How does mass change when you put it into quantum systems? What makes the mass of an electron or a molecule? Where does it come from, fundamentally? And of course, the very popular question, What is the Higgs Boson?

But for now, let’s just stop for a moment and bask on the realization that from the pre-socratic boys, Heraclitus was perhaps the least wrong. He believed the fundamental substance to be fire, in which every change was possible. Bear with me as I will give myself a bit of an artistic interpretation here, by saying that in our modern eyes, that is the closest to the manifestation of energy as they had.

And he was right. Everything is energy, and energy is change.

Notes and References

[1] Through the history of physics, inertial mass and gravitational mass have been distinguished, because it was not clear if they were different. We have performed countless experiments and not one time we have had significant differences. With the current understanding of gravity as acceleration, this distinction is even less prominent, so I will omit it for the sake of clarity.
[2] This list is not in chronological order.
[3] It is really funny that in at thousand years -if humankind has not annihilated itself by then- some outreach-loving scientist will most probably say the same about us: “Ah, yes, the physicists from the 20th, they all were dead wrong”.
[4] Yes, you got the reference.
[5] The contraction of space and time dilation was found mathematically by Hendrik Lorenz in 1892, by investigating the fact the equations that describe electromagnetic waves didn’t transform with the normal transformations of classical mechanics. Henri Poincaré made it very mathematically formal. What was really cool about this was that Einstein derived it from physical statements. He assumed what I explained before, and bing-bang-boom, Lorentz result was there.
[6] Non-inertial frames are a bit more fun, because forces appear if you move between them. Take for example, a rollercoaster. While riding it, you feel the sinking sensation, or pressure in your chest (apart from the regret of eating that burrito). Those forces are not “real” forces, like gravity, or the repulsion -or attraction-  felt by electric charges. So… what are those? Those are forces coming from being in an accelerated frame.
[7] In the vacuum, it is the maximal velocity of things in the Universe, approximately c=  300 000 km/s. That is roughly going around the earth 7.5 times in one second!

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