Hey, everybody. Welcome to this next episode of Your Daily Equation. Today I’m going to focus upon the relativistic mass equation, the relativistic mass formula. Some people love this equation. Some people despise it. I’ll describe why that is, but let me just give you a quick sense of why I think it’s important for us to cover.
Many people ask me why is it that the speed of light is the maximum possible speed? Why is it a barrier? And the relativistic mass formula at least gives you some intuition for an answer to that important question. It gives you some understanding of why it is that if you try to push an object and speed it up to the speed of light, you will always fail. You can get close to the speed of light, but you can’t actually reach the speed of light and certainly you can’t exceed the speed of light.
Okay, so what is the relativistic mass formula? Let me begin by even just writing it down for you. Then we’ll explain it. So it says that the relativistic mass is equal to the mass of an object with a little zero on the bottom. That means the mass of the object at rest. This is called the rest mass.
And there’s an additional factor, which is one over the square root of one minus the velocity squared of the object divided by C squared. And for those of you who’ve been following along in previous discussions, you will know that this is the gamma factor that crops up all over the place in the special theory of relativity. And the key part of this equation is that you see that the relativistic mass depends on V, on the speed of an object.
So the first thing that I want to do is try to give you some understanding of why in the world you’d ever suspect that there’s a useful notion of mass or heft that depends not just on the stuff that makes up the object but also the speed from any given perspective that that stuff is executing. Why would speed come into the story?
And two, to give you a little intuition for that, I’m going to tell you a brief little story that I think helps you gain that rough understanding, that intuition for speed affecting heft. And here’s the story. I call it the parable of the two jousters. So cast your mind back to medieval times and imagine that there are two opponents in a stadium that are engaged in a joust, but I’m going to modify the joust from probably the image you have in mind in two important ways.
Number one, the lance that each of these two opponents carries doesn’t have a sharp blade at the top. Rather it has a metallic sphere at the top. Second change, rather than taking their metallic spheres and trying to like knock the opponent in the head or in the body to try to knock them off their horse, in this particular version of the joust, what the opponents do is they slam their spheres together as they pass and in that way try to knock the other one off the horse.
Okay, so let me show you animation of this. And in this animation, before I show it, they’re going to be two opponents that I call Brian and Evil Brian. They kind of look like me a little bit and the stipulation, and it will be clear why I’m saying this in the outcome of the joust, is that Brian and evil Bryant are completely equally matched in every way.
So when they engage in this joust, they go toward each other on their horses. They thrust their respective lances at each other and because they’re equally matched, neither falls off the horse. It’s a draw. It’s a tie.
Okay. Now all I want to do is a simple change of perspective. In that animation, we were looking at the joust, say from the standpoint of somebody in the bleachers looking down at the competition. Now I want you and me to take my perspective in this competition and view the unfolding from my perspective.
Now from my perspective, I am an observer moving at fixed speed in a fixed direction so I can claim to be at rest. So from my view, I’m just sort of sitting there, as Evil Brian is coming toward me. Now imagine that the horses involved, they’re like really fast horses, relativistic horses, so their speed is like really large. It means the effects of relativity are more pronounced, right?
Now from my perspective, if I carefully think through what happens to Evil Brian, if I observe what happens and then really follow through my understanding of the special theory of relativity that we’ve already discussed, I recognize that because Evil Brian is in motion, Evil Brian’s watch must be ticking off time slower than my watch. And look, when we talk about that effect, that time dilation effect, bear in mind that we’re not like referring to some weird physicists abstract notion of time. I’m really referring to time itself, the rate at which processes unfold. So when Evil Brian is experiencing this time dilation from my perspective, that applies to everything. All of Evil Brian’s motions slow down, right? The blinking of the eyes are slow, the turning around is all slow and in particular, I conclude from that thinking through the situation that Evil Brian’s thrust of the lance is going to be really slow too.
And so naively at first blush, I come to the conclusion that this is going to be an easy victory, an easy win, a piece of cake, because Evil Brian is thrusting the lance at me in slow motion. But in reality, of course, we know that it can’t be a victory for me, because we already saw from the perspective of the bleachers that it is a draw.
So indeed, if we now look at this situation, Evil Brian thrust it slowly. I thrust it quickly, but it is still a draw. Now at first, I’m a little confused by the fact that I didn’t win, but then I think things through a little bit more carefully and I realized that the impact that the thrust that I experienced, the force that I experienced from Evil Brian actually depends not on one, but on two things, right?
One of those things is indeed the speed of the thrust. So we’ve got actually two speeds in this story. You’ve got the speed of Evil Brian’s horse. You’ve got the speed of the thrust. So to distinguish them, I’ll call it the speed of the thrust. I’ll just write it underneath there.
So the speed of the thrust from my perspective is indeed decreased by a factor of gamma. I’m actually put a gamma of V in there with that V and let me just give some colors here. Is this V right here. That’s the V of the horse. Okay. The speed of Evil Brian approaching me from my perspective. So the speed of a thrust is decreased by this factor of gamma, but I realize that there’s an additional factor that affects the impact. And that factor is, of course, the mass of the object that is hitting me, right?
I mean we all know this in everyday life. If, if a mosquito slammed into you, even at high speed, are you afraid of that? I don’t think so. Right? Because even if it’s relatively high speed, I’m not talking relativistic speeds here, but even if it’s relatively high speed, the mass of the mosquito is so minuscule that the impact is tiny. But if a Mack truck is slamming into you, even if it has low speed, even if it’s going slowly, because the Mack truck has such a huge mass that can really cause significant damage. So it’s the product of these two factors, not just the speed, but the mass comes into that effect.
And therefore, if I want to explain how it is that I didn’t win in this competition, I said to myself, “Look, it is the case that Evil Brian is thrusting that lance at me in slow motion, but it must be the case that the mass of Evil Brian’s sphere must compensate for that slow down of the thrust. How would it compensate? Well, if it picks up a factor of gamma of V, then the gamma of the upstairs and the gamut of the downstairs.
Oops, sorry about that little ringing of the telephone. That happens on occasion here, but let’s just ignore it and keep on going.
The gamma that we get from the slowdown in the thrust and the gamma that we get-
Oh, be quiet, telephone, already over there. All right. I’m going to have to answer this phone if I can find it. Just going to let it go.
So the slow down in the thrust… Ah, it stopped ringing. Thank God. So the slowdown of the thrust is compensated for by an increase in the mass. And there you have basically our formula. If I just scroll down over here, relativistic mass is the mass at rest and that’s really what I mean by this term over here multiplied by the factor of gamma.
So this little parable of the jousters at least gives you some sense of where we would be led to think about a mass that would be velocity dependent, that would increase as a factor of the velocity. And when we now write this out in a little bit more detail and analyze it, we see that it yields this wonderful intuition as to why the speed of light is a speed limit. So if you write M relativistic is M not times one over the square root of one minus V squared over C squared and ask yourselves what happens to the relativistic mass as V approaches C?
Well, it gets larger and larger. In fact, let me show you that. Bring up this little graph here and notice that when the speed of small, the relativistic mass hardly differs from the rest mass, but as V approaches the speed of light, the curve zips up, goes arbitrarily large, zips up toward infinity, and that’s a very useful realization, because if you have an object, whatever, even if it’s a ping pong ball and you’re trying to speed it up ever more quickly, you apply a force. But if the mass of the ping pong ball gets ever bigger, as its speed gets ever larger, then you have to give an even larger force to speed it up further still. And as the ping pong ball or any object approaches the speed of light, it’s heft. It’s relativistic mass sourced toward infinity, which means you need an infinite push today to get it to go faster.
Still, there’s no such thing as an infinite push and that’s why you can get close to the speed of light, but you can’t push an object up to the speed of light. That’s why the speed of light is indeed a limiting speed for any material object.
Final point that I want to make before I’m done is that when you think about Einstein’s E equals MC squared, you should now ask yourself which M is it in E equals MC squared, right? Is it the relativistic mass or is it the rest mass? And the answer is it’s actually the relativistic mass, because when we talk about energy on the left-hand side, we’re talking about the total energy, right? The energy from motion has to be included in that expression and you only include it if you have a V on the right hand side and indeed therefore, the real way of writing Einstein’s famous equation is equals M not one over the square root of one minus V squared over C squared times C squared.
Now I trust you will agree that saying equals M not one of the squared, one minus V squared over C squared times C squared does not have the same ring as E equals MC squared. And that then motivates you to introduce the definition that we started with, call this the relativistic mass and then you can write E equals M relativistic and that should be an L, not a V there, M relativistic times C squared and that’s the full version of Einstein’s E equals MC squared.
And it’s also useful to write this in one other equivalent way, making use of what’s known as a Maclaurin series or a Taylor series expansion, which is valid for those of you who are familiar with this little additional detail. When V over C is a good deal less than one, V is a good deal less than C, you can do, if you know a little bit of calculus, an expansion of that one over the square root of one minus V squared over C squared empowers of V over C squared.
And if you do that, and maybe at some point, I don’t know how long we’re going to go on with the series, but if we do some calculus and some expansions, I’ll show you how this goes, but for the time being, let me just write down the answer that you get if you expand the one of it squared of one minus V squared over a C squared, multiply it by the M not C squared, what will you get? Well, you will get M not C squared plus one half M not times V squared plus three-eighths times M not V to the fourth over C squared. And I think the next term, if I’m doing this in my head, which is always dangerous, so do correct me if I’m wrong in this, I think it would be five-sixteenths V to the six over C to the fourth and blah, blah, blah, dot, dot, dot.
Now this is a kind of wonderful little expression here, because one of these terms is familiar to anybody who took high school physics, which I hope is all of you. This is just ordinary kinetic energy that you learned from Isaac Newton in your course on classical physics.
This term over here is the new term that Einstein gives us, and it tells us that the total energy of an object actually is non-zero even when the object is at rest, right? This term does not have a V in it. And it says, and that’s why we call it frozen energy, not the best terminology, but it’s energy that the particle has, even when it’s not moving, when it’s sitting still. And that’s it’s rest mass times C squared. And then you have all this other stuff, which are relativistic corrections that Newton didn’t know about, that emerged from this more complete understanding.
So it’s a nice formula that brings together Newtonian physics, Einsteinian physics, relativistic physics in one complete package. Okay? So that’s all I had to say today about the relativistic mass formula. And we’ll carry on next time. But for today, that is Your Daily Equation. Looking forward to seeing you next time. Until then, take care.