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How to specify the time for the x, y, and z coordinates.

We assume, of course,

we have some measuring system and units of length and so on and so forth.

But, and certainly for the time too, we'd have clocks.

But the question is if an event happens out there someplace, a flash of light or

a classic standard event, and we can specify the x, y, and z coordinates, or

just the x coordinate along the x axis, how do we specify the time?

Well, we could say I've got a nice master clock

right next to me here if I'm the observer.

I see the flash occur, record the coordinates out there, x, y, and z.

And then, when I see the flash of light,

if it is a flash of light, I record the time on my clock.

Of course,

there's a delay from when the event occurs, and the time the flash gets to me.

In most of our real-life situations, the delay is imperceptible, so we don't worry

about it, but if we want to be precise we have to take that into account.

Now we certainly can do that, so that we could sort of

have a flow of form for the delay it's not that difficult.

But we'd rather not have to deal with that.

So, alternatively, what we've imagined is that we have a latus or a grid of clocks.

In other words, we have imaginary clocks at every single point along the X-axis or

in three dimensions, every single point in space that we want.

And when we want to specify the time of an event,

we just look at the clock at that location.

So the flash of light, say it occurs right here, and the x coordinate of it,

or x, y, z coordinates.

And we look at the clock at that instant of time when the flash occurs,

the clock right next to it, so there's no delay.

We'll get the clock as close as we want to that point, the imaginary clock.

And then that will record the time.

So we talked about the so-called photo clock principle.

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The problem is, as we will learn a little bit later on here in the course, is

once you start moving clocks with respect to each other, they get out of synch.

And that was one of the results of Einstein's special theory of relativity.

Now, again, it's a question of precision.

So if you say, well, I'm only interested in precision to a tenth of a second.

If you move them slowly enough to their positions, they won't get out of sync very

much, and the amount they get out of sync is very, very small.

If you want to be very precise about it, that can have an effect.

In principle, yes, we can move them out very slowly, slowly enough so

that there would be slight.

Desynchronization as you could say, but

not enough to affect whatever measurements we are doing.

So that could certainly work, we'd rather avoid that however.

So we talked about a second method, and

that second method was essentially have one master clock again.

Put all the clocks out, say along the x axis,

the clocks that we want in our lattice.

But they're not running yet.

Then, send a signal out from our master clock.

Say when the mater clock is 12 noon,

it sends a signal out in both directions along our line of clocks.

As each clock receives that signal, it starts running.

Now of course the signal has a slight delay to reach any given clock.

And so we'd have to take that into account.

And the way we imagine doing that is let's say the signal takes five minutes to

reach this clock right here, so it's pretty far away, at the speed of light.

And so we would say okay, we're going to put this clock at this position.

We know it's going to take like five minutes to reach that exactly, and so

we'd set this clock at 12:05.

And so the master clock sends out the light pulse, light signal at 12 noon,

five minutes later it reaches that clock and turns it on.

So it's reading 12:05, the master clock is reading 12:05, and so on and so

forth through all the other clocks.

So, that is certainly another way we could synchronize all our clocks and

avoids this little problem with, once you start.

Moving clocks with respect to each other, they get out of sync.

So, we consider the problem of how to synchronize.

We've talked about those two methods, then we moved on to space-time diagrams.

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So here is our x-axis, some origin,

some tick marks representing how close we want our measurements to be.

Tens of meters, meters, billions of meters or whatever.

Everything happens along the x-axis.

And again, we imagine our lad as a grid of clocks along that x-axis.

And so, we could record a series of events.

If something happens here and then here, and then here, and then here.

Maybe four flashes.

To represent that in a spacetime diagram, we add, of course, a time axis.

And then we say, okay if something happens at time equals 0, we plot it along here.

If it's at position 4, time equals 0, it'd be right here.

Just put a little dot there.

So position 4, x equals 4.

Time equals zero, it happens sort of right there.

Don't want to mess up my marker too much there.

Let's get some more tick marks up here.

And of course, if it happens, anything that happens at time t equals 1,

assuming it's one seconds, two seconds, three seconds, four seconds,

that'd be anywhere along here.

So if I have another flash, maybe at position two that happens at one there.

It'd be there and so on, and so forth.

So any given event that occurs at a certain x value and

a certain time value, I just plot the x value and

then the time value and then we expanded more with our

the idea of a spacetime diagram and talk about world lines.

So that if we have an object moving along the x-axis here, constant velocity.

We start off with that case, then what we get is a straight line.

Something like this at a certain slope or angle there and

this represents the world line of a object.

Remember, it is moving along the x-axis through time.

So we can see at time equals 1, it was about position 1.

At time equals 2, roughly it was in position 2.

At times equals 3, it was position maybe 3.25 and so on and so forth.

Actually, that wouldn't be constant velocity motion.

But hopefully, you get the idea that constant velocity,

straight line motion like that.

We also did some other examples of world lines, just remind you of a few of them.

So, that the world line of an object moving to the positive x direction.

If you have something moving in the negative direction,

it's going to be look something like that for constant velocity motion.

If you have something that's just like this, so we're aligned like that.

That is something that is sitting at x equals 2, as time goes on.

Time is going up here and so this object is just sitting at x equals 2 there.

That's a stationary object on this spacetime diagram.

Another example is something like a curved line.

You have a curved line, that's something that is accelerating or

decelerating as the case may be.

In this case, we can tell it's actually accelerating.

Because you can see for a given amount of time here,

it's traveling a farther distance than it is down here.

So when it bends over, it's accelerating.

If it was going something like that, let's not quite do that.

Let's get the infinite acceleration there or deceleration there, I mean.

Let's just do something like that.

So this is something that's decelerating, it's actually slowing down and

another reason we can tell that is we talked about velocity.

And let's not do the accelerating or decelerating so much here.

Because as we've mentioned, we're just dealing in this

course with constant velocity motion, but it's important to know that a curve line

is something where the velocity is essentially changing.

It's not a constant velocity, because if we talked about,

if you have a straight line like this,

velocity is just distance covered by elapsed time.

So distance covered say,

if it goes to this point right here maybe, something like that.

One, two, three, four.

It's covered four units in three units of time.

Distance covered 4 divided by 3.

So, it would be four-thirds meters per second or

whatever units we would be using there.

So we talked about velocity related to the slope of our world line and

we talked about velocity being run, the run divided by the rise.

So that if we have a world line like that,

that's a big run divided by a relatively small rise.

That's a high velocity object.

It's a object that's moving at a higher velocity, it covers a lot of distance in

a relatively short amount of time versus a world line for an object like this,

lower velocity.

It covers less distance in a bigger amount of time.

So we noted that the slope of a line, this is a line with bigger slope than this,

because slope, remember is rise over run.

Velocity on our diagram is run over rise, the inverse of the slope.

So low slopes,

smaller slope on our spacetime diagrams means something that's going faster versus

bigger slope means something that's actually going slower there.

So, that's how velocity works on our diagrams.

And so then if we, let's do a green one here.

Go back just to the acceleration case very briefly.

If you have something like this,

you can see the slope of this line is gradually bending over more horizontal.

That means it's getting faster over here.

It's accelerating versus if you have something going the other direction,

the slope of the line is getting steeper again on our spacetime diagram.

That means it is slowing down, it's decelerating.

And it's still moving to the right, but it's decelerating as it does that.

Remember once again, I repeated this a number of times that this is

not the actual motion of the object.

It's not shooting off at some angle or curving this way, or that way.

It's just going along the x-axis.

So this green line would be something that start off fast along the x-axis and

now is gradually slowing down, still moving in that direction.

This one here is an object moving fairly, quickly in that direction.

This object here is something that's moving relatively slower in that

direction.

This world line here represents something moving in the negative x direction

at a constant loss velocity.

So in our assessment quizzes, we did a lot of examples with world lines.

So hopefully, if you haven't already, you'll be working through those and

you'll see more of them on the week two review quiz as

well based on the things that you did in the assessment quizzes.

So, that was spacetime diagrams and world lines.

Then from there, so I think we covered everything, again, the main points there.

A lot of examples, again, in assessment quizzes to The work on,

okay, let's look at next topic which was I think we're up to number five now.

Frames of reference [SOUND] spent several video

lectures looking at frames of reference,

did examples with Alice and

Bob and maybe Earth is involved as well.

In assessment quizzes we introduced drifts into some of our examples.

What's the basic idea here?

Well the basic idea goes back to our lattice of clocks idea.

And so, in fact we've had some props here.

Here's Alice, say in her space ship.

This is her lattice of clocks in the X direction.

And we can imagine that as far as she's concerned, she's just sitting there.

And she's watching Bob fly by in his spaceship.

She can measure his progress by taking photographs as he goes by each of her

clocks, there are more clocks here too, just not shown.

So every single point has a clock where if she wanted to take a photograph at that

point as Bob flies or whatever happens at that point,

it records the time and the location of Bob.

And therefore, she can track his progress, but

her frame of reference then is her lattice of synchronized clocks.

And as far as she's concerned, with respect to your frame of reference,

she's just sitting there.

She's not going anyplace at all.

But of course, Bob you can say, is also in the same situation.

Now, he's moving by here like this.

But as far as he's concerned, he could be sitting there and it's Alice who's moving.

In this case, she'd be moving backwards.

I would say hey Alice, I'm not moving, you're moving backwards.

And Alice would say I'm not moving, you're moving in that direction,

in the positive X direction.

So, Bob also has his frame of reference which consists of his lattice of clocks.

And his frame of reference is, as far as he's concerned, he's at rest.

He's got his lattice of clocks.

he can measure anything, any spacetime event.

According to his frame of reference.

Alice can measure any space time event with vector her frame of reference.

And a little bit we get to the Galilean translation that allows us to

convert between the two different measurements.

What Alice might measure versus what Bob might measure

in terms any given space time event.

But, before we got to that,

we had to make sure the concept was clear of a frame of reference again.

Imagine, if these props help, or if these images help, imagine it as, let's see,

for Alice here, just her lattice of clocks, or any observer.

It doesn't have to be Alice and Bob and their spaceships.

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And we did it with Alice and Bob.

In the assessment quiz on one of them,

we actually just introduced just person one and person two,

because it's good not to get too hung up on certain, Alice is always doing this.

Bob is always doing that, because this is one of those things that we can really get

confused with plus and minus signs.

So, let me introduce sort of another way to look at it that can be helpful.

The idea here is you memorize one basic situation.

And if you are sure about that,

you can usually very easily apply it to second situation.

So we're going to have two frames of reference here and

we're going to label them, the Lab frame.

So this actually isn't officially, it's a summary of what we've done, but

slightly different terminology here.

So, we're introducing something a little new and

we're going to talk about the Rocket frame.

Two frames, Lab and Rocket frame.

So the advantage of this is versus say Alice and Bob, you say, okay,

in general, usage the Lab is stationary.

The rocket is what's moving here.

And so, here is our rocket going by with some velocity v and

we're going to assume that the rocket is going to the right.

So that's going to be our standard situation, Lab frame,

observing a Rocket or whatever object it is moving to the right there.

And of course, if this is Bob or Alice whoever in the rocket frame.

The rocket has it's own frame of reference.

So as far as this is concerned, whoever is in the rocket is stationary

has their imaginary clocks going in both directions there and

they could measure any event on their clocks.

But the question is if they measure something on their clocks,

what is the measurement going to be?

What are the coordinates going to be in the Lab frame?

And again, without going into sort of how we semi-derived it, let's just

write down the basic equation here is assuming something has been measured.

Again, let's say, it's Bob here.

He's going along here and he sees an event out here some place, and

he records it on his clock there.

The question is what is that on, let's say, Alice is in the Lab here.

So, Alice is stationary watching Bob goes by.

Bob records something way out there.

The question is what is the location in time on Alice's clock?

So, it'd be Alice's clock something like that at that instance of time.

So, here is the basic equation in Lab and Rocket terms.

XLab = xRocket and

then here's the key thing.

Is it plus or is it minus?

Because essentially, we know the distance between the Rocket and

the Lab in terms of the origin.

And we always assume that if t equals 0, I should've mentioned this,

reminded you of it.

At t equals 0, their origins were the same.

So, they're right next to each other.

And then at a later time, the Rocket goes in that direction to the right.

That's what we're assuming.

So, note that the Lab position here is going to be greater.

So, we're imagining it was right out there that last clock say on our prop here.

And clearly, if the rock was moving that direction and

measure something right there this isn't a time.

Then for the Lab frame, it's going to be at a greater position,

a greater x location than in this frame.

So, it's a plus sign.

So it goes plus vt where the velocity times the time-lapse is

just how far the Rocket has moved in that given amount of time.

So if time t equals 0, they're right here.

And if Bob at that instant in time saw something here, if I can get this.

Saw something right there,

then Alice would see it at the same clock in position.

But a little bit later, velocity times time.

Bob has moved a certain distance.

And therefore, if it's the same clock,

it's that clock distance plus the time he's moved or the distance he's moved.

Velocity times time.

So this is a basic equation xLab = xRocket + vt.

So again, Lab frame, Rocket frame.

Rocket moving to the right, they start at t equals 0 right next to each other.

Rockets coming by and then right at t equals 0, it pass,

it passes by the origin there and then continues on that direction.

So, the Lab value here for the location of whatever event occurs has been measured

in a Rocket frame is going to be greater by vt than what it is in the Rocket frame,

because the Rocket is going off there to the right in a positive x direction.

So if you remember this basic situation say, okay, Lab frame,

Rocket moving to the right.

Here's the Galilean transformation.

If I measure something in the Rocket frame and I need to find out what it is,

what the location is in the Lab frame.

The stationary frame in our example, you just add vt to it.

And then given that you can say, well, what about other situations?

Well, what if the Rocket was moving the other direction to the left,

you just change the sign here.

It just becomes -vt and

then you can say that's really the same thing,

so you can say xLab=xRocket-vt.

So, this is motion to right.

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And this is motion to the left, negative direction.

And then you say, well, what about the Rocket itself?

Remember, the person in the Rocket?

They have their own frame of reference.

With respect to them, the Lab is moving to the left.

They're moving to the right, but if they're motionless,

everything else is moving to the left.

And therefore, you use the left equation, the left motion equation.

So we have a left motion equation and a right motion equation and

you just have to say, okay, if we're really talking about the Rockets,

the person in the Rocket, the observer in the Rocket, then they are the Lab now and

whatever they're watching outside of them is the Rocket or the moving object.

So we could say, the stationary observer and

the moving object if you wanted to look at it that way.

But again, I recommend these were get one situation visualized in your mind and

know how it works and then you should know how to figure out the other situation,

depending on whether you're moving left or right.

It's just a difference in minus sign there.

And again, it's very easy to mix it up and get the wrong minus sign there.

So it does usually require, so you've done this a lot.

Requires a little bit of thinking.

Make sure you understand the situation which way the motion is,

who is the stationary observer?

Could be either the Rocket person or person in the Lab?

It just depends whose frame of reference we're talking about and

then the correct equation based on, either motion to the right or motion to the left.

So that's maybe a little bit longer summary than anticipated here,

but I did want to go into a few of those details and

it gives you a nice overview of the things we've done in week two.

And as we move into week three where we talk about things like light waves,

and the Michelson-Morley experiment, and stellar aberration, and

Einstein's two key principles or postulates that were the foundation for

his special theory of relativity.

So, all that is coming up.