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 Lorentz 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 sees Bob pass or go by in a spaceship at

velocity v according to her lattice of clocks and measurements.

And then Bob could be in his spaceship,

could imagine really he could be stationary.

He could see Alice receding behind him at negative velocity v

according to his lattice of clocks and measuring system.

So v is the relative velocity between the two reference frames,

c is 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,

and Alice measures Bob at the speed of light, look what happens, of course.

We got c squared on the top, c squared at the bottom.

This is one, one minus one is zero.

Square root of zero is zero.

One divided by zero is infinity.

Gamma becomes infinite.

So that's a hint right there, a very strong hint that something either

can't go at the speed of light, or we have problems with our theory.

And 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 the amount of energy needed to take a massive object,

anything with mass, and get it up to the speed of light.

And you find that essentially infinite energy is required.

So there are other arguments too for why you can't get there.

They're actually or the last, it's about 30 years or so ago,

there's the idea that maybe you can't get to the speed of the light but

perhaps there going to be things that actually were faster than the speed of

light, served 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 would actually work out.

The theory ran into some insurmountable problems along the way eventually and

therefore was discarded.

But for our purposes certainly the gamma factor,

the Lorentz factor is indicating we can't get up to c.

Can't 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.

Now, let's see how that factors in with our light clock example.

Remember from the light clock we found that the elapsed time on a moving clock,

if I'm observing here I've two identical clocks, maybe 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 should really say ticks 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 our identical stationary clock.

So this is us observing Bob's clock going by.

Remember for Bob, he's just sitting there with his clock and

it looks fine to him and he's viewing our clock as running more slowly than his.

And again it all goes back to the two basic principles that Einstein enunciated

and the fact that simultaneity in clock synchronization is relative.

But just remind ourselves time dilation, elapsed time on a moving clock

is less than the elapsed time on an identical stationary clock.

Here is the formula.

So, again delta T, the elapsed time that we would measure in the lab,

is gamma times the laps time on the rocket clock.

So the number of ticks on the rocket clock,

maybe we get ten ticks on the rocket clock is received, the number of ticks on our

clock again might be 20 so the gamma was two in that case.

The moving clock would run more slowly, or just reverse the equation here.

The rocket clock, the lapse on the rocket clock, is going to be less,

than the 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 an infinite quantity.

And so that's why even the math might say, if we say we can't have that physically.

But just imagine if we could here, this would say that

the elapsed time of the lab equals infinity compared to the rocket clock.

Isn't it sort of weird?

That how do you have an infinite elapsed 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 of the rocket clock, 1 over gamma times the lab clock.

So whatever the lab clock is saying, 1 over infinity,

if this was 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 here,

it's ticking away very nicely.

If Bob was 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 just sit there, it would be frozen, time would be frozen as far as we would

see his clock going by according to our lattice of clock.

So now 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

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 c.

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 three snapshots of the light clock.

Snapshot 1, snapshot 2, snapshot 3.

It's 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 beam on his clock would go up and then back down again,

bounce off the upper mirror here, and then down to the lower mirror.

That would 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, let's do this even in orange here.

So we'll say v = c.

What happens in that case, sort of geometrically here?

Well, that means from 1 to 2 here, okay?

His light clock is moving at the velocity c.

In that amount of time, okay, to move say this distance here,

the 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?

Well it's moved 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 the light clock gets to position 2,

presuming it's moving at velocity c, speed of light, it's moved to here.

The light beam heading up to the upper mirror can only go that far.

And so what happens then, when it gets up,

presuming that mirror, that mirror is not there anymore.

Because, a little bit later on, it's moved now over here to this position here.

So it's right there.

And the lightbeam can't hit the mirror there.

In fact, the lightbeam can never hit the upper mirror

because the upper mirror is always just out of its reach.

And another analysis we'll do later on, we can actually do this more quantitatively,

to show that in this instance the lightbeam will never hit the upper mirror,

it'll never get there.

This is another way of saying the rocket clock is frozen.

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 ticks at all.

And it actually becomes just a horizontal line there.

It never reaches the upper mirror.

Like a clock it's frozen.

Again, we'd say that's an unphysical situation for

a moving object like our light clock here.

And therefore, we don't expect it to actually happen.

But again the math here gives us some hints that,

it's a 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 towards the upper mirror will

never be able to reach it because the mirror is moving it.

Velocity c is supposedly this way.

It'll always be just out of its reach.

So interesting to see

how the velocity of 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.

The ultimate speed limit

Understanding Einstein: The Special Theory of Relativity

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