Why you can't go faster than light (with equations) - Sixty Symbols

So, we made some videos about relativity and we talked about how distances change in relativity, we talked about how time changes in relativity. So I thought I'd talk about something that combines those two things, which is how velocity, which is distance divided by time, changes in relativity. In the previous videos, we did what we called a Gamma Trilogy because they all have this gamma factor. Actually, one of the interesting things about the way the velocity transforms is that all the gammas disappear, they all cancel out, so there's no gamma in this gamma video. So we need to think about how you achieved your speed, of course it's distance divided by time, but particularly how it changes depending on what frame of reference you're in.

So we need to think about how different people measure speed in different benchmarks. frames. And one of the classic thought experiments you could do is have someone on a train. I will call this frame of reference S'. and they're going to roll a bowling ball along the train at one speed or another, and the train itself is moving, so this entire frame of reference is moving in this direction at some other speed v. The way you are on the train You would measure the speed to see how far the bowling ball had gone in your frame of reference.

What should we call this? Δx: how far it has come, Δx' because it is in your frame of reference. and at what point you have measured it, at some point Δt' after you release the ball, and then you could measure the speed in your reference frame of that bowling ball is how far it has gone divided by the time it took to get there. That's the definition of speed, it's how far something has gone divided by the amount of time it took to get there. So that's in one frame of reference, so the question is what does the person see in this frame of reference, the other frame of reference?

And let's address the non-relativistic case first: what are they going to measure? The distance they will measure the ball has traveled will be how far it has moved away from you plus how far you have moved down the line. That extra distance, how far the train has moved, depends and is basically how fast the train was going. Are you adding everything on top of each other? Yes, exactly! That's all you're doing. So, that distance is vΔt. If the train is going at a speed v and has traveled for some time Δt, then the distance it has traveled is simply the speed multiplied by the time it took to do so.

Now we can put all this together. . . That says that Δx is equal to (how far you see the ball has moved) how far the train has moved, plus how far the ball has moved relative to the train. Just the sum of the two. And we can rearrange this in the non-relativistic case to say that we simply divide: Δx divided by Δt is equal to v plus Δx' divided by Δt. And in this non-relativistic case we don't have to worry about different people seeing different times. . . So it doesn't matter if it is Δt or Δt', we can call it the same.

It's all the same. Or we can rewrite this again: it simply says that the speed with which you see the balls moving is equal to v. So that's Δx divided by Δt. Plus the speed at which the person on the train sees the ball moving. And that, as you just said, is basically adding speed. The speed at which the bowling ball is moving is the speed at which the bowling ball is moving relative to the train plus the speed at which the train is moving relative to you. That is the simple Galilean transformation, without relativity. Everything works as we are used to.

One of the things that gets people interested in relativity is that people say that you can never travel

faster

than

light

. So an obvious question you might ask is: why can't I travel

faster

than

light

just by making these kinds of additions? Suppose the person on the train, instead of rolling a bowling ball, was actually shining a torch along the train. So the torch beam would travel at the speed of light relative to them and then if you ask how fast the torch beam was traveling relative to you Well, it would be the speed of light, plus the speed at which The train is moving away from you, and that's faster than the speed of light.

Unfortunately, relativity takes care of that, and that's not really how things work. Therefore, we must start again, but we must solve the problem correctly with relativity. Unfortunately, I suspect my drawing is not up to the job. And then we have the other guys here. Moving at a certain speed v. Previously, the formula we had said was that the distance the bowling ball is along the line is equal to the distance relative to the train plus the distance the train has moved. Now, the additional factor that we haven't included yet is special relativity. And what we saw before was that what special relativity does is go into these γ (gamma) factors.

And so this Galilean transformation that we had before is a little bit modified by an extra factor of γ. γ is 1 over the square root of 1 minus v squared over c squared. One of the things we know about relativity is that not only does distance depend on the frame of reference you are in, but time actually depends on the frame of reference you are in. So we need the equivalent transformation for that. So there are these things called Lorentz transformations. . . That is the first of them and this is the other transformation we need. It turns out that these are the two Lorentz transformations you need.

Now we can say what the speed is measured by the person next to the track watching the train pass. And we just do the same thing we did before. And speed is just distance divided by time. So we can simply divide them with Δx divided by Δt. is equal to . . . Now the γs (gammas) are going to cancel because if we divide that by the other we end up with a γ above and a γ below. So, we can just cancel them. I'll tidy up a bit: I'll divide the top and bottom by Δt'. So, I can write this as, Δx' over Δt'.

This is more or less the answer we wanted because Δx over Δt is the speed of the bowling ball or whatever being thrown along the train, seen from the perspective of the person standing next to the train track. And Δx' over Δt' is the speed as seen from a person standing on the train. So, I can write that in terms of the "u's" and the "u'" that we have before. This just says that u is equal to u' plus v divided by 1 plus, u'v divided by c squared. What is the final answer in the non-relativistic limit.

So what we were looking at before, the Galilean transformation, both u' and v are small compared to the speed of light. That means this term is small. That's just 1 basically. . . And that simply says that the speed you see is just the sum of the ball relative to the train and the train relative to you. Which is the answer we had before. So, everything works out well if things go faster than the speed of light. Now let's look at the most interesting case where relativity really matters. And in particular let's go to the really extreme case that instead of rolling a bowling ball along the train, we shine a beam of light along the train.

So in that case, u', the speed of whatever is equal to the speed of light. And let's see what happens when we put that in there. So in that case we end up with, if u' is equal to the speed of light. . . So u is equal to c (the speed of light) plus v. . . divided by 1 plus (u' is the speed of light) cv over c squared. Now let me know, play with this a little bit. So, I'll rewrite the top part here as . . . c over 1 (so I'll take that factor out) plus v over c. Because if I multiply this, I end up with c plus cv over c, which is just v, so c plus v. divided by And then all I'm going to do is cancel c over c squared to end up with 1 plus v over c.

Which you know is the exact same term there. So this is the same thing that basically means that all of this is still equal to the speed of light. So, there's the strange thing: we've taken Remember, what we've done here is we've said, okay. So, there is a ray of light moving relative to the train at the speed of light. Now, if you are looking from the side of the train, how fast do you see that beam of light going? The answer remains the speed of light. In reality, you haven't increased its speed at all. And I guess the physics behind this is that you have to worry about both space and time changing depending on what frame of reference you're in.

Which means that, not surprisingly, the speed that emerges will change in a slightly strange way. also. And it turns out that relativity takes care of this invariance in the speed of light. That no matter what frame of reference you are watching the light beam travel from, you will always see it travel at the speed of light. Does the universe want to keep things at the speed of light and everything changes to satisfy that? Or does everything change all the time and the speed of light just decreases? Is it chicken or egg? It's a very good question.

And I actually really like your first explanation that the universe organizes things in such a way that the speed of light always turns out to be the speed of light. and no matter how much you try to mess around with things by trying to run away from the beam of light or run towards it. . . When you come to measure its speed, you will always find that its speed is the speed of light. and everything else adjusts the distances and times in such a way that when you combine those distances and times to measure the speed of a beam of light, it will always result in the speed of light.

Is there anything important about the fact that the speed of light is constant and unchanging? So what motivated Einstein to come up with all this in the first place is... . . he had the idea that the laws of physics should be the same no matter what frame of reference you are in. So if you're in a sealed box, there shouldn't be any experiments you can do that will tell you whether that sealed box is moving. at a constant speed or is stationary. And in fact, from his point of view, it's kind of a meaningless question. And what he also knew is that the speed of light arises from the laws of electromagnetism.

So the question is, in what frame of reference do those laws of physics work? And his argument is that those laws of physics should work in whatever frame of reference you're in. Which means that the speed of light has to be invariant if I believe that the laws of physics are the same in all different reference frames. What if it hadn't been like that? What if the universe said: "no, I can be different for different frames of reference"? Would you and I be torn apart in some cosmic tear? Or would the universe be a little different and peculiar?

Does the universe benefit from the fact that this is what happens? It is very difficult to construct a universe in which this would not be true. (In retrospect, it's hard to construct a universe in which this isn't true.) Keep in mind that, for example, all the other things that come from relativity flow from there. So, think that famous things like E = mc squared arise from the invariance of the speed of light. And that means that the equivalence of mass to energy is a natural consequence of relativity and so, for example, what powers the sun (fusion), converting mass to energy would not work.

What if we weren't living in a relativistic universe. Now, you could imagine that the laws of physics in such a universe would find a way to generate energy from fusion. But the image we currently have would be completely different. And the amount of speed of light (what speed it really is) does that matter? Would the universe be the same if the speed of light had been cut in half? Or was it pretty much, you know, kind of like the speed we walked at? Does it need to be as fast as it is for our human brain?

So, it arises from the laws of electromagnetism and it has to do with how strong the magnetic fields are, how strong the electric fields are. which are actually arbitrary constants of nature as far as we know. meaning that the speed of light, which is a combination of these things, is simply an arbitrary law of nature. The universe would be a very strange place if the speed of light were much slower. Because, of course, all these relativistic effects that lead to all the weird things that arise from relativity we don't have to worry about in everyday life.

But, if it turns out that the speed of light is the same as your walking pace, then all kinds of strange relativistic effects would occur every time you walk to the mailbox. You would have to worry about all the relativistic effects of time dilation and length contraction. So it would be like, if I'm going to publish this letter, am I going to die before I get there? Or will we all have relatives who will have died of old age before you return home? So yeah, it would be a very, very strange universe.