0:03

In the last two video clips, we have been considering this case of the light clock

and through our analysis of it, we came to this whole concept of time dilations.

So what we want to do is, pull those pieces together a little bit here, and

look at where it leads us, in terms of a conclusion about time dilation,

how we might think about it.

And then also consider, is this really real or is this just some special,

almost philosophical, case that doesn't apply in the real world.

So, remember what we were doing here.

We had identical clocks, one moving and one at rest.

So here's the clock at rest, perhaps we're Bob, right, so

we're Bob sitting here, we've got our clock,

our light clock ticking away with the light beams going up and down.

And then we have Alice with an identical clock,

who is moving with some velocity v with respect to us.

Now remember, to Alice, she's in her frame of reference.

And so as far as she's concerned, her light clock is just bouncing up and

down normally, but to Bob, observing her light clock.

And by the way, occasionally, I use the term seeing as well,

Bob seeing her light clock, and

remember we're using that in a very special sense of technically observing.

We're not taking in time delays of light traveling from some place to your eye, and

so on and so forth.

So, if you catch me saying seeing sometimes,

just insert observing there, okay?

In our precise definition of what it means to observe using a lattice of synchronized

clocks and so on and so forth, as we talked about before.

So, clock at rest, Bob has his clock at rest.

Alice, moving clock, so Bob is observing her moving clock and our conclusion

was that the duration of one clock tick, Bob observing Alice's moving clock.

So, this is one clock tick, as far as Bob is concerned observing Alice's clock,

equals some factor of gamma times one clock tick for his clock at rest,

which again is identical, you've got two identical clocks here.

And yet, by putting one clock into motion,

all the sudden from Bob's perspective, the clocks are running differently,

that the moving clock is running more slowly than his clock at rest.

Because there's a gamma factor here, remember, it's 1 over the square root of

1- v squared over c squared, and that's always greater than or equal to 1.

If v is 0, we just get 1 there, but if v is any positive value,

any value at all really because it's v squared,

then this number down here will be, the whole thing will be less than 1,

it'll have 1 divided by a number less than 1, which will be greater than 1.

And in the next video clip, we'll explore some

values of gamma a little bit further here to get a feel for what that's like and

what values of v give us what values of gamma, but for now, we'll just

say that clearly gamma is greater than or equal to 1, so that's where we ended up.

Let's draw a little picture here of actually what's going on with this,

because really, we're not so interested in clock ticks when we're comparing things.

We want to really talk about elapsed time.

What's the elapsed time on Bob's clock right here,

versus the elapsed time as he observes it on Alice's moving clock

as it goes by versus counting clock ticks and things like that.

So, we need to convert this equation into an elapsed time equation.

So let's look at this, we'll just say that (gamma = 3), that's actually fairly high

velocity, but so just say (gamma = 3), Alice is going by at a fairly high speed.

3:39

Here, if I sort of create a diagram of the clock ticks, here's what it'd look like.

So let's look at the rest clock first, the rest clock ticks, okay, so

that's Bob here with his clock just ticking away nicely.

So here are the ticks, each line represents a tick of his clock.

And the distance between the two ticks, that's this (delta t) rest here.

That's the elapsed time, the time duration for one tick, the light goes up and

comes back down.

And then, the moving clock ticks,

according to our analysis here, with (gamma = 3),

you're going to get a tick every third tick for the at rest clock, right?

Because the duration of a tick for

the moving clock is three times the duration of the tick for the rest clock,

just again, assuming (gamma = 3) so we can see we've got one,

two, three rest clock ticks is going to equal one moving clock tick.

And so, again,

you get one tick of the moving clock every three ticks of the rest clock.

And all that's saying is sort of in pictorial form what this equation is

saying when gamma = 3.

And so, clearly we had the moving

clock ticking more slowly, from Bob's perspective observing,

ticking more slowly than his rest clock there, all right?

Well, again, we're not so much interested in number of ticks.

We want elapsed time, I'd like to say okay,

Bob measures some elapsed time on his clock.

What would be the corresponding elapsed time on Alice's clock

as he observes it going by?

And we can see this right here is that, the way I've drawn it here,

I've got 12 ticks of the rest clock, his rest clock here,

is going to equal four ticks of the moving clock, okay?

It's a 3 to 1 ratio, again, because we're choosing (gamma = 3) here.

So let's just say every tick is one second, so on the rest clock,

his clock sitting right next to him, he's going to have 12 seconds elapse,

the moving clock will only have four seconds elapse, okay?

And note that that means it's a slightly different equation than this and

it's really easy to get this switched around.

And so that's why we're trying to make a semi-big deal of it here,

but let's think about this now.

We want an equation for elapsed time.

I'll do it right here, so we'll do elapsed time.

6:51

Okay, so it's the same form as this except we spelled it out.

(Elapsed time) here = something times (Elapsed time) of the rest clock.

Here, these were clock ticks, right?

The duration of one clock tick versus the clock tick on the other one,

we've got the gamma factor in there.

Okay, clearly it's going to involved gamma, but

note what it is here just in our example when (gamma = 3),

the elapse time of the rest clock is 12, right?

And so, and the elapse time of the moving clock was four.

So, we need an equation essentially in that specific example that

would say 4 equals something times 12.

Well, what is that?

It's one-third, right, 12 times one-third, 12 divided by 3 is 4.

So, this number right here is one-third, in other words,

in general, it's 1 over gamma, okay?

You see that, again, the duration of each tick for the moving clock,

as Bob observes it going by, is longer than one tick on the rest clock

by a factor of gamma, in this case gamma equals three.

But, that means the laps time of the rest clock is

greater than the laps time on the moving clock.

8:14

Moving clock ticks are longer in duration, that means they are fewer,

then therefore you flip it around.

Elapse time is going to be greater, going to get more ticks on the rest clock

For the number of text on the moving clock.

And so this factor right here is one over gamma,

not very good gamma there, a little bit better

anyway one over gamma where gamma is as we've defined it before, okay?

So let's rewrite that so we can see this a little bit better here,

and note that this is what we

derive using our light clock analysis this equation here that's for the text.

We're going to move away from that now we said okay we derived it

we have this form here, but

the new form we're going to write on the board here Involving elapsed time.

That's the form we want to memorize.

And so we can use it in various situations and and so on and so forth.

Okay, so this was our example.

Let's erase our example here and say

10:15

but if gamma is, v is in the other value gamma is greater than one.

So we see that the lapse time on the moving clock is always going to be less

than the elapsed time on the clock at rest as Bob is observing it here.

Very important point, it's Bob's observation of the moving clock or

whoever is observing that moving clock.

So you use Elapsed time moving clock,

1 over gamma (Elapsed time) at their clock, identical clock at rest.

So this is where we get time dilation, as we mentioned previously.

The moving clock here runs more slowly, moving clocks run slow,

or run slowly, in compared to an identical clock at rest.

So sometimes just to write this in a little better form,

you might write like this, so Delta t where we used capital T for elapsed time,

at least for the moment because we used lowercase t here.

In the future, we may just start using lowercase t's for this equation as well.

But for now, we'll say, delta T of the moving clock.

So elapse time on moving clock again equals 1 over gamma Times

delta T of an identical clock at rest.

11:33

So that is our time relation equation and

it going to shows the moving clock runs slow.

Now it have to be precise about this.

In fact awhile back, I got an inquiry from a television game show

about a question they wanted to ask their contestants.

Or they were thinking about asking their contestants, and, essentially,

without going into all the details.

They wanted a question along the lines of,

if you could travel near the speed of light Would time slow down?

Or time would slow down.

Would it slow down or speed up or go to infinity or go to zero?

There are a number of different choices.

The answer they thought was correct was time would slow down, okay.

And you can look at our elapsed time equation and

the moving clock time slows down.

Yeah, that makes sense.

Think about it carefully however.

Because this is Bob observing the moving clock.

Alice, sitting on her spaceship say as she's traveling along there

at some high velocity v, her clock is just running normally.

She does not perceive any difference in time at all.

For her, time does not slow down, so that's often when we say things like that,

if you could travel near the speed of light,

the special theory of relativity says, Einstein says that time would slow down.

Not for the person who's actually traveling at that speed, it's the person

outside Observing the clock that's moving past that sees time slowing down.

And does to some things that we'll talk about a little later on in the course

especially, the so called twin paradox of two people actually age differently and

so on and so forth.

But, we're not there yet but, that would be coming up in a In a few weeks here.

So time dilation equations.

Delta t, elapsed time on a moving clock, from an observer watching it go by,

is one over gamma delta t on a clock at rest next to them, okay.

So again, time dilation, we could write it in different form,

we could say delta t rest equals gamma delta t moving.

But usually it's best to pick one form,

this is the form that I think shows the time dilation affect most effectively.

Because the gamma is always greater than or equal to one, so this will

be a fraction like one third, if we did a guess of three or something like that.

So the moving clock left time less than the rest time on identical rest clock.

Okay, so that's time dilation and again just to

emphasize this does not mean time slows down for the person on their spaceship.

Alice in this case it means the person deserving them sees their clock

running more slowly than the identical clock that they have next to them.

And of course, you can actually reverse analysis and have Alice look at Bob,

and to Alice, Bob is moving away from her.

She will also see her clock running normally and

Bob's clock ticking more slowly.

And that's where you get into some of these paradoxes in terms of the twin

paradox, which we'll consider later on, figure out how that works.

One more thing to do here though for time dilation, and that is might say,

well this is all very nice and I sort of see where it comes from.

But these light clocks they're a little weird, they're strange.

How do we know regular clocks work like that?

Maybe this is just a special case with these light clocks.

And this could also go back to one, further point where these all coming from.

Remember, it's because Einstein's Led to the fact

that the speed of light is the same for all observers no matter how you're moving.

And therefore, for Bob here with his clock at rest,

he just sees the speed of light going up and down at c.

He sees the light on Alice's clock go taking a diagonal path, also at c.

It's clearly a longer path, therefore it takes longer to get up and down.

And our three snapshots here,

remember these are the snapshots, one, two, and three as we watched the light.

As Bob watches Alice's light go up and down.

Therefore, it ticks more slowly.

But, is this just a special case.

Well, let's consider that a minute and earlier in one of the videos.

I promised that we would make a little argument here to say no,

if a light clock acts like this.

All clocks must act like this If the special, if,

not the special theory of relativity.

It's true, but if the principle of relativity is true.

So let's consider that here.

17:22

And so you might say well, again, maybe that's just the light clock.

Maybe this regular clock over here, whether it's battery driven or

spring driven or whatever, maybe it doesn't obey this type of stuff,

maybe it just runs normally, all right.

So, what would happen if that's the case?

Let's assume that that's true, the light clock is a really special type of thing,

this special theory of relativity,

all we've been talking about here really only applies to the light clock,

quote unquote, normal clocks will act differently, will act normally say.

So Bob takes a series of photographs, he sees Alice's clock ticking more slowly,

right, keeping time, there's a digital readout on that clock which we compare

to this clock, and takes a series of photographs.

Therefore, as Alice goes by,

Bob will have photographs that show these two clocks are getting out of sync with

each other because this one is running slow as the analysis says.

But we're assuming this clock runs normally, okay.

So now Bob has a series of photographs that show that.

That, as Alice goes by here, this runs slow, therefore they get out of sync,

we're assuming they start off in sync with each other, they fly by.

And then Bob sees this one, quote unquote, running normally, well,

running faster than this one because of the slowing down effect that we get

with our light clock analysis.

So he has those series of photographs.

Alice can also take her own photographs at the same time Bob takes his, all right?

Now think about this, in that,

think about the principle of relativity that we emphasized at the beginning.

18:57

We were going to do a train car, I forgot to put the wheels on the train car here,

so whatever you want there, or a spaceship or whatever.

So whatever it is, going by velocity v there, think about the principle

of relativity and our initial thought experiment we did that we emphasized

saying that for Alice here inside that train car, assuming she can't look out.

There are no reference points which she can tell whether she's moving or not.

If she ends up having photographs where these two clocks are running differently,

that's evidence that she is moving with some velocity.

because if she's at rest with respect to Bob here,

Bob just sees the light clock running normally.

That clock's running normally as well.

And therefore, no difference.

But put her in motion then Bob sees a discrepancy there.

If Alice also has photographic evidence of that discrepancy, or

really, just looking at the clocks as she's going along,

her photographs would have to agree with Bob's photographs.

If every time Bob took a flash photograph as he goes by,

Alice also took one at the exact same instant and location and

looked at the comparison of her two clocks, if she saw a difference in those

two clocks, that'd be clear evidence that she was moving.

Because if she wasn't moving, there'd be no difference between the two clocks,

they'd both be running normally.

Ergo, it would be a violation of the principle of relativity.

So you could say, well, maybe the principle of relativity is violating

a case like that, but Einstein's assertion was, and assumption was,

that no, it's a universally valid principle.

And, of course, based on that, and this principle of light constancy,

he built the foundations of his special theory of relativity.

So, in this case we're going with Einstein, we're saying yes,

the principle of relativity is true, a lot of good evidence that it's true.

And therefore, a regular clock, no matter what it's constructed of,

has to run the same as a light clock, like we've got right here.

And therefore, any moving clock runs slowly, as Bob observes it.

Okay, not just a special light clock.

Now in principle, you could do a very difficult and

complicated analysis of any clock to analyze what's going on,

the various motions in the clock, and get the same result.

But that's, I suppose in certain cases you might be able to do that.

But much easier to just say hey, if the principle of relativity is true,

these two clocks have to run the same, and in this case, as Bob is whizzing on by,

it means they both have to run more slowly than Bob here.

We'll give him his identical light clock here and give him also a regular clock.

21:38

Okay, and so his clocks are running normally,

perfectly in sync with each other, Alice goes by at some velocity v.

Those clocks also have to keep in sync with each other, otherwise a violation of

the principle of relativity, Alice could tell whether she's moving or not.

And they're going to move more slowly by the gamma factor

compared to Bob's two clocks here.

So whether we're talking about light clocks or just, quote unquote,

regular clocks, we get this time dilation effect.

And again, remember the basic equation, delta T,

capital T, moving as 1 over gamma delta T at rest.

Again, the time dilation effect.

And as we move along here in the videos we'll be exploring more of

the consequences of this.

And also asking some more questions about how some of this actually works.