Let's say a few words about the speed of light as an absolute speed limit using our light clock as an example. Remember we derived our expression for the Lorens factor gamma in this form, 1 over the square root of 1 minus v squared over c squared. Clearly there is a problem here if, remember V is the relative velocity between two reference frames, two inertial reference frames. we've been using Alice and Bob. So if Alice is stationary she see's Bob perhaps go by in the spaceship. At velocity v according to her lattice of, of clocks and measurements. And then Bob could be in his spaceship, could image. Really, he could be stationary. You could see Alice receding behind him at, negative velocity V according to his lattice of clocks, and. And and measuring systems. So, v is the relative velocity between the two reference frames c the speed of light of course, if the relative velocity between two reference frames is c, if Bob is traveling by at the speed of light compared to Alice. Alice is, Alice measures Bob at the speed of light, look what happens of course. We've got C squared on the top, C squared on the bottom. This is 1. 1 minus 1 is 0. Square root of 0 is zero. 1 divided by 0 is infinity. Gamma becomes infinite. So that's a hint right there, a very strong hint that something either can't go to, go at the speed of light or we have problems with our theory. Since the theory has been well verified we say okay. This must be an absolute speed limit that nothing can actually get up to the speed of light. You can do further calculations with this that really go a little bit beyond our course in terms of energy needed to take a mass of object, anything with mass to get up to speed of light and you will find it essentially infinite. energy is required. So, there are other arguments too for why you can't get there. There are actually, over the last, it was about 30 years or so ago, there is the idea that maybe you can't get to the speed of light.

But, perhaps there could be things that actually were faster than the speed of light. You know, sort of on the other side as it were. And they were given the name Tachyons. there was some theoretical work done on them. some development to figure out whether this was actually, would actually work out the problem. The theory ran into some, insurmountable problems along the way eventually and therefore, was, was, was discarded. But. For our purposes certainly, the gamma factor, the Lorentz factor, is indicating you can't get up to C. You can get close to C, in which case gamma becomes very, very, very large. But you can never quite get to C, because then it becomes infinite. And now let's see how that factors in with our light clock example. Remember from the light clock we found that the elapsed time in a moving clock, if I'm observing here, we've got two identical clocks, maybe I'm, I'm Alice is here with me, and we see Bob going by with his light clock and we compare the two. The light that we see in his clock has to travel a longer distance than the light in our clock. And therefore his clock ticks longer I shouldn't really say slower, each tick takes a longer time to go from here in this diagonal path versus our just up and down path with our identical light clock. And therefore the elapsed time on a moving clock is less than the elapsed time on identical stationary clocks. So this is us observing Bob's clock going by. Remember for Bob, his clock, he's just seeing it with his clock, it looks fine to him. And he's viewing our clock as, as running more slowly than, than his. And again it all comes back to the two basic principles that Einstein enunciated and the fact that simultaneity in clock synchronization is, is relative. But just to remind ourselves time dilation elapsed time in a moving clock is less than the lapsed time in the identical stationary clock. It was the formula. So again delta T the elapsed time that we measure in the lab, is gamma times

elapsed time on the rocket clock. So the number of ticks on the rocket clock, maybe we get ten ticks on the rocket clock, as we see. The number of ticks on our clock, depending on gamma, might be 20, or so, if gamma was, was two in that case. the moving clock would run more slowly or just reverse the equation here. The rocket clock the elapsed time of the clock is going to be less than elapsed time on the lab clock by the 1 over gamma factor. Well what if gamma then is infinite. What if we put C in here and make gamma infinite look what that indicates with our time dilation equation. Well first of all, we haven't encountered anything in physical reality that is in infinite quantity, and so that's why even the math might say it's infinite, we say we can't have that physically. But just imagine if we could here, this would say that the, the elapsed time of the lab equals infinity compared to the rocket clock. that's sort of weird, that, How do you have an infinite elapse time on our lab clock ticking away right here. You get a little more insight, perhaps, with the second equation. That this says the elapsed time on the rocket block 1 over gamma times the lab clock. So whether the lab clock is saying 1 over infinity. If, if this really is infinity up here. That's just 0. That's saying no matter what the lab clock says here, the rocket clock is 0. In other words, the rocket clock never moves. As the lab clock is ticking away very nicely, we're looking at it here, it's ticking away nicely. If Bob is moving by in his rocket, with his light clock there, and he was moving actually at the speed of light. We would see his clock as never ticking at all. It'd just sit there. It'd be frozen. Time would be frozen as far as we would see his clock going by according to our lattice of clocks. In other words, we'd see him going by, and our lattice of clocks would be there. We could take flash photographs all the way along our lattice of clocks and can look at those photographs later and see

what Bob's clock said. Along the way in each of those cases and his clock would be the same thing for all of them if he's traveling at sea his clock would never tick at all even though ours were all ticking nicely. And we can see that geometrically a little bit with our light clock diagram here remember this is like 3 snapshots of the light clock snapshot 1, snapshot 2, snapshot 3. So moving at velocity V here, and so we had the light beam as we see it in the lab. As Bob goes by the light, light beam on his clock will go up and then back down again. Bounce off the upper mirror here and then down to the lower mirror. That'd be one tick while our clock would just be going up and down like that. Well think about what happens if V equals C. So if we say, and let's do this even in orange here, so we'll save V equals C. What happens in that case, sort of geometrically here? Well that means from one to two here, 'kay? This light clock is moving with the velocity C. In that amount of time, 'kay, to move say this distance here in green so, in that amount of time it's moved that distance there. So, to get to this position 2. So, if this light beam here is traveling up toward the upper mirror, how far has it moved? Let's move the same distance. So, if we sort of spin this up here, like that, that's where that light beam would be at that instant. When it, when the light clock gets to position 2, it's moving at velocity C, the speed of light. It's moved to here, the light beam heading up to the upper mirror can only go that far. So what happens then it gets up to that mirror that mirror's not there any more. Because later on it's moved to this position here. So it's right there. And the light beam can't hit the mirror there. In fact the light beam can never hit the upper mirror. Because the upper mirror is always just out of it's reach.

And another analysis we'll, we'll do later on, we can actually do this more quantitatively to show that, in this instance the light beam will never hit the upper mirror. Will never get there. This and now this thing of the rock clock is frozen. The, the light beam actually, in fact, as you elongate this out, the higher the velocity here is, in terms of this triangle, this triangle sort of gets elongated out this way. So at very high velocity, the light beam would go way over here and bounce, and then way back down. And when it gets up to C it never can catch up to that upper mirror. And so it never picks it all. And actually becomes just a horizontal line there, it never reaches the upper mirror like a clock that's frozen. again we say that's an unphysical situation for a moving object like our light clock here and therefore we don't expect it to to actually happen. But again, the math here gives us some hints that, very strong hint, that that's not possible. And that's backed up just by looking at the situation physically or in a geometrical sense that the light beam heading toward the upper mirror will never be able to, to reach it. Because the mirror is always, because the mirror is moving at velocity C as opposed to this way, it'll also be just out of its, out of it's reach. so interesting to see how the velocity's light then becomes an absolute, speed limit. Later on we'll also see that there's some cause and effect arguments that also say why this can't be possible.