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We have been exploring spacetime diagrams over the last number

of video lectures and seeing how they can help us visualize.

And gain some insight into some of the key things about the special

theory of relativity such time dilation, length in contractions.

The Lorenz transformation and so on and so forth.

And so, in this video lecture we want to

consider this idea of Regions of Spacetime.

And so called light cone.

We'll get to that in a second.

But before we do that I want to go back to something we were doing the last video

in terms of representing two sets of coordinates,

two reference frames really on the same plots.

So remember how we did this, we didn't do this one, we'll get to that in a second.

But we did this, where we had again Alice observing Bob

going behind the spaceship to the right, in the positive.

This direction at velocity v, and

clearly we could have two separate coordinate systems, we could have Alice's.

We could have Bob's just as we normally do at right angles.

But we showed last time that we can actually, or the last couple times,

show that we can actually put both coordinate systems on the same plot.

And we did that by using the Lorenz transformation to take points from

Bob's coordinate axis.

And plot those transformed onto Alice's coordinates,

leaving Alice's the normal right angle coordinates we have there, the axis,

the X and T axes for Alice.

And showed that you end up with something at an angle here

We also made the point that you have of course, lines of same location and

lines of simultaneity, lines of same time.

And so on Alice's plot, the lines of same location are the vertical lines

represented by the tick marks.

And the lines of same time,

lines of simultaneity are the horizontal lines, parallel to the x sub a.

Axis, represented again by the tick marks.

Meaning of course anything on a horizontal line in Alice's frame of reference,

anything on that line any space time event that occurs on that line

is simultaneous to Alice.

And then any point on a vertical or any setup points on vertical line

are things that occur same location though at different times as time.

Goes on there.

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Are these lines here.

Here's his piece of the axis.

So this is the line of same location for X of B equals zero, of course.

And then the line for X sub b equal one, two, three and then -1, -2, -3 and so

on and so forth, and then the lines of same time,

simultaneity are Oriented parallel to the x of b axis.

Because this is the line for t's of b equals zero.

Anything that occurs at t's of b equals zero is along there.

This would be negative one, negative two, or one, two, three, and so

on and so forth.

So we can actually take a given space time event,

represented by this red dot here, and in Alice's frame of reference I've drawn it

such that It looks to be you know, about 2 and a half over, and then about 4 up.

Maybe not quite 4 up, and so that would be the location about 2 and

a half, someplace in there x sub a two and a half.

And then, 2 sub a 4 and if we're as we started doing,

we're using units of c being 1.

So light years per year, light seconds per second, so

this could be maybe, in terms of the location.

Might be two and a half light years over and time of one, two, three,

about four years or two and a half light seconds over time, about four seconds.

And that would be in Alice's frame, but in Bob's frame, according to his coordinate

system, We'd see that here's his line of same location.

So we go up there from that tick mark and we see it's approximately one.

And then the parallel line parallel to the xb axis for

the line of simultaneity looks like about three.

Just roughly, this isn't quite drawn exactly to The scale there but

you can see that.

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So in Bob's reference frame, he would measure that spacetime event as

XB equals about 1 it looks like,

and t, b equals about 3.

Okay, again these are approximations there.

And then for Alice, she would measure that same space time point, as we noted.

X of A she would have X of A Equal, we'll just call

it 2.5 and we'll just call it 4, t sub A = 4.

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So we see they measure different coordinates, of course.

And we could transform between those coordinates using Lorenz transformation.

Now my numbers might not quite work out, because I was just eyeballing.

Therefore, an actual transformation, but that's the idea,

a single space-time event.

The event's not changing.

It's the same thing.

It's a flash of light that occurs someplace in space and time, someplace

along the x-axis, because we're doing this in one dimension for the x-axis.

And Bob measures it to occur at xB = 1,

Time equals 3 on his clock, on his lattice of clocks.

Whereas Alice measures it at X sub A 2.5, and time for her lattice of

clocks at 4 seconds or years or whatever units we happen to be using there.

And we did a little bit more of that to show how we could

see time dilation on there.

And a little bit of lengthen traction, and so on and so forth.

So.

As just a reminder of how we can put both coordinate systems,

both frames of reference, really, on the same plot and get some use out of it in

terms of analyzing things with the special theory of relativity.

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You might ask what if we'd done it where we did it sort of from Bob's perspective

And then Alice was moving, which Alice then would be moving to the left.

So, if we consider Bob to be stationary in his frame of reference, and

Alice moving negative v to the left, what would our combination plot look like?

We're not going to try to prove this, we'll just say, here's what it looks like,

and that's what this is here.

So notice we've drawn the green coordinates, Bob's system here As x sub B,

t sub B in a normal 90 degree right angle configuration.

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And when you have the reference stream moving to the left, negative v,

then it turns out and you do the Lorentz transformation equations.

It turns out Alice's axes would be at these angles here,

sort of tilted away from Bob's axes.

But the concept is the same here.

In other words, here is the same event on this plot, and note that it is roughly,

in Bob's coordinate system, his reference frame is at 1, 3.

So 1, 2, 3, there's the red dot, 1, 3, same thing we got up here Okay.

And on Alice's, for Alice's plot, very roughly here, we can say okay,

what's x sub a?

Well, 1, 2, it's about 2 and half there.

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There is a parallel.

That's the line of same location on Alice's plot there.

And again, roughly at 4 because this has to be parallel.

Do this, so when you're using these skewed axises as we've

done a couple times before when we introduced them just very basically as

hey we could something like this.

Remember that you don't just measure over it right angles you have to use the angles

of the axes to indicate the location of a given event in this case.

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So again, same event from either Alice's or Bob's perspective.

And in either way of plotting there, you should,

you will get the same results assuming you draw in your diagram roughly.

Correctly there.

So that's just a brief reminder and

a few new things about if you did it from Alice's frame of reference, or

really if you had a situation where the moving frame was going to the left.

Then you'd get something like this.

Although typically, for whatever reason, this is more standard.

We like to have things going the right and deal with this system there.

One of the, note, here they'll be come important in our light cone is remember

when we take the units to be light years per year or light seconds per second.

Then c has the value one, therefore if we were to plot on either of these plots, if

we were to plot a light beam traveling out from the origin at velocity c, of course.

Then it's slope would be one.

It would go one light year in a year.

It's rise over run would be one.

Or it's velocity, the run over rise, would also be one.

So it would be, you could do two light years travelling in two years.

So two light years per year, would be a point.

Now, let's just put in here.

Can get it in here roughly speaking right there.

Three light years even though let's do one light year.

And one year is going to be about right there.

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Just put it over here.

You got a line at a 40 degree angle for

the speed of light traveling in that direction.

And actually going down that way as well.

So that is a light beam traveling in the positive x direction.

Remember, it's easy to get confused with all the angles here,

everything is happening just along the x axis.

And we're just plotting the motion through time here.

So this represents a light beam going along the X axis,

traveling at the velocity C.

And this tells us how far essentially it goes in a certain amount of time at

the speed of life 45 degree angle.

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So it splits not only Alice's axis here but it splits Bob's axis as well.

They're symmetrical across that 45 degree line representing the speed of light or

an object traveling at the speed of light.

Really not an object,

a photon a light being traveling at the speed of light there.

You can also, of course, have the light beam going in the negative v direction.

And it looks something like this.

At a 45 degree angle, and also you can go that way as well.

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The world line for a light beam or a photon will

always be a 45 degree angle either in that direction or the opposite direction here.

So having said that and sort of messed up our diagrams over there now

let's turn to this diagram and you should have already guessed.

You probably already have that, really talking about the same thing here.

so this is we're going back just a single plot,

single diagram for our reference frame.

Time up here, X axis here.

We will again assume that the speed of light is one.

We are representing it as one,

like one light year per year, one light second per second.

Sometimes you'll see this.

Diagram in certain books with, and

we mentioned this before I believe, with ct here.

c times t and that also essentially puts things in units of one.

But we'll just assume that t as in years, and

x has to be in light years or seconds, light seconds etc., etc.

So what's going on with this diagram?

Well, here's the idea, you can see all the dots we've got and

we've drawn in what's known as the light cone.

And we'll see why it's called that, so it's a cone like feature so it represents.

From the origin here where we have dot number 1, point number 1.

A light beam traveling in the positive x direction would go like that.

That's how we represent it.

It wouldn't go like that.

That's the world line.

It literally would be going along the x axis, and

this is its world line through time.

If they happened to start some place over here, if this is 0.

If it start over here some place.

And was going that way then, which we have it starting down here maybe there's

a flashlight or laser right here shooting off the light beam in the x direction.

And therefore, the world line would be going up like that and for

shooting the other direction maybe start over here.

And shoot it this way then, world line starts here and goes up [UNKNOWN.

So, the red lines represent the world lines a light beam.

Now, the key question here that we wan to ask, and

this gets into one of the themes for this week that we've been working toward and

that is what happens with cause and effect?

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And what happens with the speed of light?

If you can go faster than the speed of light, we've talked about how you can't.

We did some analysis of what happens if you had a light clock say

moving to get up to the speed of light,

what would happen then gamma becomes infinite and so on and so forth.

So that hints strongly that you can't get up to the speed of light.

But we want to consider some situations now in terms of specific question her is,

if I have an event happening at 0.1 here, say anything at all,

okay, can that event affect any of these other events here?

Okay, and one way to think about it is let's just go from one to two,

okay, so Note that from 1 to 2 here, if I.

Think about a message or just a space ship travelling.

Okay? Really, literally p\speaking 2 is

happening right here on the x axis, and 1 is over here.

So could I, in a spaceship, say get from 1 to 2.

In time to affect something happening at that point up here.

So it's really at position two and time, whatever time we have over here.

Well, the key thing here is note then,

if I were to draw the world line here from one to two, Something like that.

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That world line has a velocity less than the speed of light.

Remember how this works?

Velocity is proportional to the inverse of the slope of our world line,

so faster velocities are like this.

Slower velocities are up here.

This is the speed of light here.

So if I'm going less than the speed of light,

I've got to be over here some place in terms of my slope.

So from one to two here

represents of something that is traveling slower than the speed of light.

And we know that's possible.

So in theory, yes.

If I have a spaceship here and I need to get to position two over here

in this amount of time here, I can make it assuming my spaceship is fast enough.

It doesn't have to go to the speed of light.

It just has to go at some speed less than the speed of light.

Pretty fast, probably, but it would get there.

Okay?

And so, we say, yes, an event at 1 can affect an event at 2.

And in fact, if you think about it a minute, any events within this cone here,

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are able to be affected Affected by an event at one.

Because again I could get in spaceship that was fast enough and

I could travel to point 13 here, which is actually just on the X axis.

I really wouldn't have to do any traveling I'd just.

I'm sorry, not the X axis, the time axis.

I wouldn't have to do any traveling, I'd just sit there and

wait until 13 came along.

And then, ten seconds later, ten years later, or whatever it happened to be.

So I can get the places like that.

Note where I can't get however.

Look at number three over here, okay.

If I say okay, I'm going to get in my spaceship, go as fast as I can, and

I'm going to try to get to point three.

Which is out here at some position, Not position 12, that's just another dot.

But some position out here at some later time.

To do that, I'd have to have a world line, something like that.

Well I missed the dot there but at least hit the three.

That would be the world line to get there.

And note that that world line has a slope less than.

Then the speed of light slope, and

therefore it's a velocity faster than the speed of light.

In other words to get from one to three through space time,

which means traveling both through space and time, as it were.

Because time just ticks on,

I'd have to have a space ship that goes faster than the speed of light.

So I cannot get there.

And similarly, for For 12 here,

note for point 12, it's right on the x axis, and

that's a line of simultaneity if you're in this reference frame right here.

In other words, no matter where you are in here, if it's any distance away from you,

to affect something you'd have to have instantaneous communication.

Right, and that's going to be impossible.

Same thing over here, for number eight.

If I'm traveling in the negative x-direction,

I can't get to number eight in time to affect anything.

I don't have a spaceship faster.

I'd have to have a spaceship faster than the speed of light or

be able to send a signal faster than the speed of light.

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Anything outside the light cone, any point up there, is too far away from me.

Note now x8 here is farther away than 9.

We'll get to that in a second.

Certainly 3 here is farther away than 2.

So I don't have anything fast enough to get there.

What about these points on the bottom?

Well, the question there is not if something

at point 1 can effect these points, these are earlier in time than point 1.

So here the question is can an earlier event affect what's going on at 0.1 here?

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Well, again, same idea.

If a spacetime event at 0.6 happens, and

I want to affect something at 1 here, maybe I'm going to get in my spaceship and

travel really in the x direction over to here.

And also through time to that point.

And deliver a message or something.

Maybe there's a big vote in congress or

senate or parliament or something like that.

And I want to deliver a message to say vote this way.

Well, I can make it there in time to deliver that message and

affect the event there if I'm at six here.

Because where a line to get from there to there has a slope,

that tells me the speed is less than the speed of light.

Same thing for five here.

If I'm over here I'm travelling in a x direction to get there I can make

it in time to get there.

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13, you just sort sit there and wait for 13 to occur in time.

because this again is x equal 0 here.

11 over here, again just like 10 really, it can't get to point 1 in time,

it has to go faster than the speed of light.

What about 9 and 4?

And also 7 down here, note that they are right on the light cone.

Okay. What that means is if I'm at point one,

and I want to effect something going on at point four,

if I travel at the speed of light, I can just get there in time.

And the same thing over here going up, the negative x-direction,

I'll be able to get to point nine in time.

And then down here from point seven,

if I want to influence something at 0.1, I have to travel at the speed of light.

So, I would have to send a laser beam or

something or a light signal with a message that would get there in time.

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So, I got you all messed up there, but again, the idea is the light cone.

And you have different regions of spacetime, that it defines.

Just by these 45 degree axes representing the world lines of a light beam.

And we have some other names for this as well, and also it allows us to go back and

revisit the invariant interval.

We don't need this here.

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Okay, so again, remember the questions are, I'm at point one here, and can.

If I'm at point one here, can I influence something later on in time, and

at another point in space.

Or if I'm in time before point one, can where I end, can I get a message there or

travel there in time to influence something at point one.

And so, we have three names for these situations.

Because if you think about,

I can either get there in time with ongoing 2.1 or to another point.

So that's something and we call that a time like

interval a time like interval.

And really, the key question here is, we'll write it as

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And this should be a little intuitive if you think about it,

because Everyday lies we think about this a lot perhaps if you drive a car or

even you're taking a bus or train or something you think.

I have to travel a certain distance and I have the certain amount of time to do it.

And do I have enough time to get there given the distance and

the speed that I can go.

So that's why we were asking you.

Do I have enough time to make that.

That distance given the parameters I'm dealing with, whether car or walking or

however, I'm going from one space to another point

through time as well because as I'm walking, time is ticking on too.

So I'm traveling through both space and time that way.

And a timelike interval Is such that yes, you do have time to get there, okay.

And I'll write the.

Remember our invariant interval equation?

C squared t squared- x squared = a constant.

And it turns out we're not going to try to prove this rigorously or anything.

We're just going to state it and that is if you have a timelike interval,

what that really means is that this invariant interval c squared

t squared- x squared, turns out to be greater than 0.

And you might say well, why?

Well, we can motivate this a little bit actually,

because this means that, do a little bit of the algebra here.

We do, this means c squared t squared is greater than x squared.

Just moving the x squared over.

Or if we take the square root and everything is positive here,

we can say that c t in general is greater than x.

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Now what does that mean?

This is essentially c is of course the speed of light, c the speed of light

times a certain time, this is the distance that light can travel.

Okay, and I should say, that light does travel in a given time.

If I say, okay, I've got ten seconds,

I've got ten minutes, how far does light travel in that time?

Just use c times t.

That will tell me how far light travels.

And if that's the ultimate speed limit, if that's the fastest I can get some place,

I know if I have a certain amount of time to get some place,

ct is the fastest I can go.

Well, I, of course, can't quite go to ct, but I can get pretty close to it.

If that distance the light can travel is greater than the distance I need to go,

then, yes, I can get there, or at least the light beam can get there in that time.

So this is essentially saying, the distance light can travel in that certain

amount of time I have is greater than the distance, and yes, it's reachable then.

I have enough time to make the distance in that case.

And so that's called a timelike interval, and

again is represented in a diagram when you have points within the light cone,

that you can travel either at the speed of light to get to point 4 from point 1,

or point 9 from point 1, or less than speed of light to get from 1 to 2,

or down here from 5 to 1, or 6 to 1, or 7 to 1 at the speed of light.

Well, let me back up there.

We have a special name for the speed of light cases here.

So the timelike interval ct greater than x is,

for example, for 5 and 6, and 1 and 2 there.

Now, so that's the timelike interval, and

next one is actually called a lightlike interval.

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A lightlike interval, and as you might guess just from the name,

this is when you do have things going at the speed of light.

In that case you have the invariant interval c-squared,

t-squared- x squared = 0.

And pull the x-squared on the other side,

take the square root, and from this you get that ct = x.

And as we were just talking about, if ct is maximum distance anything can go in

that given time, because this represents the speed of light traveling over a given

time, that means that distance x is exactly equal to that maximum amount of

distance I can cover, okay?

In other words, do I have enough time to make the distance?

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If that distance is reachable, just barely,

by something going the speed of light there, we call it a lightlike interval.

And so in our diagram, that'd be like going from 1 to 4.

The only way to get from 1 to 4 in space and

time is to travel at the speed of light.

And again, you're really going out here to position x, but

you only have this much time to do it.

And you have to travel that much time or that much distance, and

to get there you have to travel at the speed of light.

And that's called the lightlike interval.

Or from 1 to 9, or from 7 to 1, going that way, those are lightlike intervals.

So a given point here, if it's within this cone this way, or

down here, it's a timelike interval from this point to that point.

If it's on the light cone, it's a lightlike interval.

And our third case is a spacelike interval.

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And the same analysis then tells us that ct,

the maximum distance we can cover in terms of what light can cover,

is going to be less than the distance x, in terms of the location of where x is.

So in other words, this is saying that, back to the question,

do we have enough time to make the distance?

No, because the maximum speed we can go is the speed of light.

And that means we can get a distance ct away,

or a light beam can get a distance ct away from where we are.

But if that's less than the distance that needs to be traveled,

then you can't get there.

And that's called a spacelike interval because space,

the space coordinate there, dominates over the time coordinate.

Lightlike is when space and time coordinates are equal, in a sense.

And timelike means the time coordinate dominates over the space coordinate.

And so a spacelike interval would be represented by, from 1 to 3 here.

That is a spacelike interval.

I can't get from 1 to 3 in my spaceship or even a beam of light,

cannot get from 1 to 3.

It's too far out here for the amount of time I have.

Or, again, 1 to 8 over in this direction.

To get over here in the amount of time I have, don't have enough time.

Same thing from 11 to 1, starting at 11, can't get to 1 in time, or

10 to 1, going that way.

Okay, so timelike intervals, lightlike intervals, spacelike intervals,

this is another way to see sort of the limitations that the special theory

of relativity puts on us as we travel, in a sense, through space and time.

And in the next few video clips,

we're going to explore some other aspects of this.