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both frames of reference into one diagram.

So both Bob and Alice where, as usual we had Bob traveling at velocity b,

although I guess sometimes we've had Alice doing that as well, but

Bob traveling at velocity b to the right.

Alice observing.

Here was the Lorentz transformation,

if we're given coordinates in Bob's frame of reference, we can transform them,

of course, into Alice's frame of reference into her coordinate system.

And we want to figure out how to plot that and see them on the same plot or

really two plots in one and by using the Lorentz transformation and

starting with Alice's normal plot, right angle plot,

we're able to get Bob's axes on there as well.

So of course we have Alice's x-axis and t-axis and,

with respect to that, then, Bob's x-axis is at an angle and

his t-axis is also at an angle there.

And the angle depends, actually, on the relative velocity

between the frames because that gives us a slope of those lines.

And we also noted that, of course, we have lines of simultaneity and

lines of same location.

So for Alice, the lines of same location are just the straight, vertical lines.

In other words, one, two, three, four, I've got four right here for Alice.

Everything that occurs on this vertical line

is at the same location going through time.

So if something is just sitting there, we've talked about this before, even

before this week, if something is sitting in Alice's frame at a position four,

it's world line is just going to be a straight vertical line there.

And for a horizontal line for Alice,

the world line there just indicate something or series of things that

are all simultaneous, in line with each other there.

And of course, you can just step through time that way.

For Bob on this plot, however, because the axes are skewed,

remember we have to draw the lines of simultaneity, parallel to the time axis.

And the lines of, actually, I said that backwards there.

Lines of same location parallel, I don't know which way I said it now so

I'll say it correctly this time.

Lines of the same location parallel to the time access.

Lines of simultaneity parallel to the x-axis.

There so you get this skewing effect, but for a given space on

the event represented by this say a flash of light or whatever, this red dot here.

It's the same event,

but Bob and Alice would just have different coordinates for that.

And so in this example here, we can see that flash of light for

Bob occurs at x sub b equals one and t sub b equals one, two, three.

Those are the coordinates for Bob.

X equals one, t equals three.

For Alice, those coordinates though, same spacetime event, roughly speaking,

that's not an exact diagram here but, I've got it about two and a half for

x there and one, two, three, three and a half, maybe a little bit more for

the t coordinate there.

So Alice and Bob same space on the event,

but they lead it in different coordinates or observe it with different coordinates.

We also noted that we can see time dilation on here because note that for

this event Alice is meeting it and again time about three and

a half something like that.

For Bob, the time is three.

Alice again observing Bob's clock, be observing it running slow, time dilation.

And same thing for length contraction here,

that if you look at the distance for Bob here, it's at x sub b equals one.

Alice is reading it at x sub a equals two and a half or something like that.

So, as she observes Bob going by, and remember that's the basic picture here,

she is observing Bob going by at some velocity v.

She sees objects in his frame of reference

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contract their length versus their rest length in Bob's frame of reference.

So, you can see both length contraction and time dilation,

if you look at the diagram carefully here, You can also see,

remember, the relativity of simultaneity because we did an example where

Alice had a whole series of flashes that went off simultaneously.

So that'd be on one of her lines of simultaneity here.

So let's just do it on this one here.

So a flash there, there, there, there and there.

Get a few more in here, roughly speaking there.

A little off, but those are all on a horizontal line

in Alice's diagram here meaning that to her, they're all simultaneous.

They all occurred at t equals one.

So it's like having a line of flash bulbs that, for Alice, they

all go off at the same time, they're all synchronized with her synchronized clocks.

And so,

if you were to take flash photographs at every flash there compared to the clocks

at the location, all Alice's clocks would read the same time for those flashes.

But look what happens in terms of Bob here that we can see.

Now, of course, we know from the Lorentz transformation there's going to be

a difference, but we can see this right off the diagram because Bob's lines of

simultaneity are you know like this.

Okay, so he sees you know if this is,

let's actually just try to draw this in here a little bit.

So here is the line of simultaneity,

one of them looks like that, another one right here.

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And that is sort of crooked there, but hopefully, you can get the idea.

So, all these are synchronized and simultaneous for Alice, but

clearly for Bob, let's just pick some in the middle here.

This one right here occurs between time t equals zero and one.

This one over here, though,

is occurring after time t equals one, this line of simultaneity.

And this one way over here occurs after time t equals two.

So in other words, to Bob, these flashes are occurring before these flashes.

In fact, he will see it go off as flash and

then flash and then flash and then flash.

So it'll be a series of flashes, not simultaneous, but in line with each other.

And, again, also, you may remember,

we can see leading clocks lag on this as well because to Bob.

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So we spent a fair amount of time developing this tool,

really as it is, a visual tool for

seeing some things about the implications of the special theory of relativity.

Then we did another version of it, where we didn't do too long on the same plot,

but we just said, okay, let's do a spacetime diagram here.

And let's note that when we put a light beam on the spacetime diagram,

If we can choose the units of c to be light years per year or

light seconds per second then c at the speed of light has the value of one.

It's one light year per year, one light second per second.

Which is nice for us because if we use sort of regular units of time and

distance like meters per second, then a light beam along here

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would be very close to the x-axis.

Almost indistinguishable from the x-axis because it travels a very far distance

in a a very short amount of time.

And so by choosing c equals one, one light year per year,

one light second per second, one light day per day, one light month per month,

whatever, then c becomes a world line on our diagram here that has a slope of one.

Or if it's going the opposite direction, a slope of negative one, v equals minus c.

And once again, you'll get tired of me saying this,

but just as a reminder, this does not mean the light beam's going at an angle.

Everything we're doing is going along the x-axis,

just shows its progression in time.

So time is flowing upward, it in this case is moving along the x-axis

at a velocity of c, which means it goes maybe one light year here

in one year of time, or one light day in one day of time, and so on and so forth.

So, we call this the light cone, because as we analyzed it further,

we said if this is an event right here, or we're standing right here,

can we influence events later in time?

And we analyze this and learn that,

as long as the event is within this cone here, this cone of light,

then we can influence it because we can get a signal to that or

travel to that spacetime event In time to have some influence over it.

Or, down here, if we want to influence an event occurring here,

anything in this cone here, would work.

But if we're outside the light cone,

then I can say we're over here in this section, what that means is, I'd have to

travel faster than the speed of light to get there in the amount of time I have.

The question was, given the time I have, can I make that distance?

Just like we often ask in real life if we're taking a trip.

You know, how long is it going to take,

do I have enough time to get there given the speed I can go?

So here, the ultimate speed is the speed of light and,

therefore, we actually analyzed it into three

intervals based on our invariant interval to give some more insight into it.

And we had the name timelike interval for,

really, those things that occur in the cone here or the cone here.

And what that meant, we didn't try to prove this, but we just stated it.

It means that c squared t squared minus x squared,

the value of the invariant interval between two points here,

say the point here and then a point up here, if that's greater than zero,

then it's possible to get there and have an influence on it.

And what that means is c t is greater than x.

If you think about it, c times t is the distance light can travel,

if that's the ultimate speed limit, we can't go faster than that, then if that,

the distance light can travel is greater than the distance we actually want to go,

then yes we can get there.

In principle, at least,

assuming we can travel up to some speed less than the speed of light.

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Then I have a light-like interval in between me.

And if I can go at the speed of light, I can't go, but if I shoot a laser beam,

the laser beam could get there in time to get to that spacetime point.

If, however, the inner variant interval between two points in space and

time is less than zero then that means ct is less than x,

it means the distance is farther than a light beam could get there even.

And therefore, I can't have any cause and effect relationship between them.

And it'd be like, if I'm here, it'd be like I'm out over here or out over here.

Or, if I'm over here and I want to influence this point,

I'd have to travel faster than the speed of light.

And remember again that world lines with slopes less than 45

degrees in our diagram would indicate a speed faster than the speed of light.

So we analyze that a little bit.

That was an interesting way to think about cause and

effect relationships with this light cone diagram.

And then we considered a couple of examples at the end.

Where we had a question of is faster than light travel possible?

And we did it in a thought experiment as it were of a spaceship

traveling from San Francisco to St. Louis to New York City and shooting off some

flashes along the way, so the flashes, the images would continue on in front of them,

go in front of them to an observer in New York City.

And to the observer in New York City, it seemed like that they saw the image

of the San Francisco photograph appear and then the image of the St. Louis photograph

appear pretty quickly after that such that it seemed like the spaceship went from

San Francisco to St. Louis at maybe four times the speed of light or even faster.

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And, but clearly this ship was not going faster than the speed of light.

It was just traveling, it's going very fast in our example, but

clearly less than the speed of light.

So, it shows there can be situations where you can get, observations that seem

to show a violation of the fact that the speed of light is an ultimate speed limit,

but when you analyze it further, it actually is not true.

And then finally we did an example with the quote unquote true

faster than light travel.

So we said okay what if we actually had a way,

a spaceship where we could travel faster than the speed of light.

What would be the implications of that?

We did a simple example with that and

the result was we saw a violation of cause and effect.

That the effect of something

actually did not happen after the cause, but happened before the cause.

And the example we gave was an invention of a spaceship that could travel faster

than light, and then catch up with another one, and do the sneak attack on it.

And what we showed in the frame of reference of the good guys who were being

attacked, that the invention of the spaceship and

their frame of reference actually happened after the sneak attack,

which is clearly a violation of cause and effect.

You can't have a sneak attack, if the spaceship doesn't even exist yet.

So it works backwards.

Another way you could say it is that the spaceship actually attacked them first and

then moved in their frame of reference back to its invention state, so it'd

be like, a little bit like time running backwards is one way to look at that.

So anyway true faster than light travel leads to true being it's not possible,

but we do an example where we say,

what if we could do it, leads to a violation of cause and effect.

So next week some of the things we've developed here will be useful.

And thinking about things like the twin paradox, which we'll get to, and

also things like the pull in the barn paradox.

And unlike this example, faster than light travel,

which it's almost an optical illusion that you get that.

These sort of strange paradoxes are actually true and

they come out of the special theory of relativity.

That's one of the things that we've been working toward and

we'll be getting to those next week.