distances change in relativity, we've talked about how time changes in relativity.

So I thought I'd talk about something, which combines those two things together, which is how speed,

which is distance divided by time, changes in relativity.

In the previous videos, we made what we called a Gamma Trilogy because they all have this gamma factor in it.

Actually one of the interesting things, about the way speed transforms

is that the gammas all disappear, they all cancel out so there is no gamma in this gamma video.

So, we need to think about how you made your speed, of course it's distance divided by time,

but particularly how it changes depending on what reference frame you're in.

So we need to think about how different people measure speed in different reference frames.

And some of the classic kind of thought experiment you might do is you've got somebody on a train

I'll call this reference frame S'. and they're gonna roll a bowling ball, along the train

at some speed or other,

and the train itself is moving so this whole reference frame is moving in this direction at some other speed v.

The way you're on the train you'd measure the speed is that you'd see how far the bowling ball had gone in your reference frame,

What should we call this?

Δx: how far it's gone, Δx' because it's in your reference frame.

and at what time you've measured that at, some time Δt' after you'd let go of the ball,

and then you could actually measure the speed in your reference frame of that bowling ball is just how

far it's gone divided by the time it took to get there.

That's the definition of speed, it's how far something's gone divided by the amount of time it took to get there.

So that's in one reference frame, so then the question is what does the person in this reference frame, the other reference frame see?

And let's deal with the non-relativistic case first:

What are they gonna measure?

The distance that they'll measure that the ball has gone is going to be how far the bowling ball has rolled away from you plus how far you've moved down the line.

That extra distance, how far the train has moved, depends on and is basically how fast the train was going.

You're sort of adding it all on top of each other?

Yeah, exactly! That's all you're doing.

So, that distance is vΔt.

If the train is going at some speed v, and it has traveled for some time Δt, then the distance it has covered is just the speed times the time it has taken to do it.

So, now we can put all this stuff together . . .

That says that Δx is equal to (how far you see the ball as having moved is) 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

Let's just divide through: 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 whether it's Δt or Δt', we can call it the same thing. It's all the same thing.

Or, we can rewrite this again: it just says that

the speed you see the balls moving is equal to v.

So that's Δx divided by Δt.

Plus the speed that the person on the train sees the ball moving.

And that, as you just said, is you basically just add the speed. The speed at which the bowling ball is moving is

the speed that the bowling ball is moving relative to the train plus the speed that the train is moving relative to you.

That's the simple Galilean transformation, no relativity. Everything works in the way that we're used to it working.

One of the things that get's people interested in relativity is that people say, you can't ever travel faster than light.

So one obvious question you could ask is: why can't I just travel faster than light by just doing these kinds of additions?

Supposing the person on the train, instead of rolling a bowling ball, was actually shining a torch along the train

Then the torch beam would be traveling at the speed of light relative to them and then if you ask how fast the torch beam was travelling relative to you

Well, it would be the speed of light, plus the speed that the train is moving away from you, and that's faster than the speed of light.

Unfortunately, relativity takes care of that, and it's not really the way that things work.

So, we need to start again, but we need to do the problem properly with relativity.

Unfortunately, my drawing is not quite up to the job, I suspect.

And then we've got the other chaps in here.

Moving along at some speed v.

Previously the formula we had said that the distance that the bowling ball is down the line

is equal to the distance relative to the train plus

how far the train has moved.

Now, the extra factor that we haven't put in yet is special relativity.

And, what we saw before was that what special relativity does is enter these factors of γ (gamma).

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 the distance depend on the reference frame you're in, but actually the time depends on which reference frame you're in.

And so we need the equivalent transformation for that.

So, there are these things called Lorentz transformations . . .

That's the first of them

and this is the other transformation that we need.

Turns out, these are the two Lorentz transformations that you need.

Now we can say, what's the speed as measured from the person who is beside the track watching the train go by.

And we just do the same thing that we did before.

And the speed is just the distance divided by the time.

So, we can just 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 on the top and a γ on the bottom.

So, we can just cancel them out.

I'll do a little bit of tidying up:

I'll divide top and bottom through by Δt'

So, I can just write this as, Δx' over Δt'.

This is kinda the answer we wanted because Δx over Δt is the speed of the bowling ball or whatever it is being thrown along the train

As seen from the perspective of the person standing beside 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 "u'"s we have before

This just says that u is equal to u' plus v divided by 1 plus, u'v divided by c squared.

Which 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 just 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, things all kind of come out right if things are going faster than the speed of light.

Now let's do the more interesting case of where the relativity actually matters.

And in particular let's go to the really extreme case of instead of rolling a bowling ball along the train, let's shine a light beam along the train.

So, in that case, u', the speed of whatever it is 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 . . .

Then 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 just do a little playing around with this.

So, I will rewrite the top here as . . .

c into 1 (so, I will pull that factor out) plus v over c.

Because if I multiply this out, I just 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 the c over c squared so I end up with 1 plus v over c.

Which you know is just the same as the term there.

So this is the same as that which just basically means that this whole thing is equal to the speed of light still.

So, there's the bizarre thing:

That we've taken

Remember, what we've done here is we've said, okay

So, there's a light beam that's moving relative to the train at the speed of light

Now, if you're watching from beside the train, what speed do you see that light beam going at?

The answer is still the speed of light.

You haven't actually added to its speed at all.

And I guess the physics behind it is that you have to worry about both space and time being changed by what reference frame you're in.

Which means that not surprisingly,

the speed that comes out is going to be change in a slightly strange way as well.

And it turns out that relativity takes care of this invariance in the speed of light.

That no matter what reference frame you're watching the light beam travel from, you'll always see it travel at the speed of light.

Is the universe wanting to keep things at the speed of light and everything changes to cater for that?

Or, does everything change all the time and the speed of light just falls out of that?

Is it chicken or egg?

It's a very good question.

And I actually really like your first explanation that

the universe arranges in things in such a way that

the speed of light always comes out as the speed of light.

and no matter how much you try to mess around with things by trying to run away from the light beam or run towards it . . .

When you come to measure it's speed, you'll always find that it's speed comes out at the speed of light.

and everything else kind of adjusts

the distances and the times adjust in just such a way

so that when you come to combine those distances and times to measure a speed of a light beam, it will always come out the speed of light.

Is there something that

is important about the speed of light being constant and unchanging?

So, what motivated Einstein to come up with all this in the first place is . . .

he had this idea that the laws of physics should the same no matter what reference frame you're in.

So that is you're in sealed box, there should be no experiment you can do that will tell you if that sealed box is moving at a constant speed or is stationary.

And in fact, in his view of things, it is a kind of meaningless question.

And what he also knew is that the speed of light comes out from the laws of electromagnetism.

And so

then the question is, in what reference frame do those laws of physics work

And his argument is that those laws of physics should work in whatever reference frame you're in.

which means that the speed of light has to be invariant

if you believe the laws of physics are the same in all the different reference frames.

What if it hadn't been that way?

What if the universe said, "nah, I can be different for different reference frames"?

Like, would you and I be ripped to pieces in some cosmic rip?

Or, would the universe just be a bit different and quirky?

Like, does the universe benefit from the fact that this is what happens?

It's very hard to construct a universe in which this wasn't true.

(In retrospect it's hard to construct a universe in which this isn't true.)

Bear in mind that for example, you know,

all the other things that come from relativity flow from that.

So, think that famous things like E=mc squared all flow 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 into energy

wouldn't work.

And if we weren't living in a relativistic universe

Now, you could imagine that the laws of physics in such a universe would come up with a way of generating energy from fusion.

But, the picture we currently have of it really would be completely different.

And the quantity of the speed of light (the speed that it actually is)

Does that matter?

Would the universe be the same if the speed of light had been, like halved?

Or it was quite, you know, if it was something close to the speed we walk at?

Does it need to be as fast as it is to our human brains?

So, it comes out of the laws of electromagnetism

it's to do with how strong magnetic fields are, how strong electric fields are.

which are really just arbitrary constants of nature as far as we know.

which means that the speed of light, which is some combination of these things, is just an arbitrary law of nature.

The universe would be a very strange place if the speed of light were very much slower.

Because of course all of these relativistic effects

that lead to all the weird things that come out of relativity

we don't have to worry about them in every day life.

But, if it turned out the speed of light was walking pace

then all sorts of strange relativistic effects would happen every time you walked to the post box.

You'd have to worry about all the relativistic time dilation and length contraction effects

So, it would be like, if I go to post this letter, am I going to die before I get there?

Or, will we all have relatives that have died of old age before you get home again.