In the last video lecture, we considered the case where we had

the spaceship moving from San Francisco to St. Louis to New York.

And the observer in New York,

where they were actually saw the spaceship seemingly coming toward them

at four times the speed of light or even nine times the speed of light.

In other words, is a case of apparent faster than light motion.

Even though the way we setup, clearly the ship was not moving faster than light

is moving at 0.6 or 0.8 or 0.90, or whatever it chose.

But putting the numbers together it seemed like,

just because of the way the flashes of light worked as it traveled across and.

The observer in New York would see the flashes of light coming right after each

other, such that the effect was,

seemed that the ship was moving faster than light.

What we want to do in this video clip is actually see what happens when you have

a ship that can move faster than light, or

just something in general moving faster than light.

Just like the last example, actually this is from Taylor and

Wheeler's book on space time physics, although it's a classic example,

you can find it in many other places.

I've modified it, just a little bit as well,

to get at the salient points, that we want to make here.

And so the theme is cause and effect, or vice versa.

And so, it gives the ending away just slightly but

let's set up the situation here.

We've got a diagram, but

here are the events that we're going to imagine occurring.

We've got the good guys and bad guys.

And they don't like each other very much.

But, at a certain point, they sign a peace treaty.

So in year zero, there's a peace treaty signed and the negotiating team for

the good guys then leaves at, leaves the home planet of the bad guys.

They actually went to the home planet of the bad guys and

negotiated the peace treaty and got it all set and everything signed and then they

left in their ship to go back home at a velocity of 0.6 times the speed of light.

And turns out that in the mean time, after they have left, over the next four years.

Okay, so that was, we're calling that year zero.

Over the next four years the bad guys are able to invent a faster than

light space ship.

And being bad guys, they say,

we're going to take off in pursuit of are the good guys, our enemies.

Because now we have a chance to really put it to them.

And so, they use their faster light ship and

take off at a velocity of three times the speed of light.

And that happens in year four.

And then in year five, they're able to catch up to the good guys,

and they launch a sneak attack against them.

So diagrammatically, on the space time diagram, this is what it looks like, okay?

So this is the frame of reference of the bad guys, for b here, for bad guys.

It's in light-years and years, so the velocity of light is one,

light-year per year here.

And the green line represents the negotiating team of the good guys,

taking off from the home planet of the bad guys, which is right here, and

traveling back towards their home wherever it might be.

Now, again, remember, they are actually not traveling at an angle.

They are traveling along the x axis here, going in this direction, but

the space-time diagram shows their world line progress through time as well.

So, space that way and then moving through time this way.

And if the speed of light, again,

is at a 45 degree angle here this roughly is 0.6c as we've drawn it.

In other words, in one year they go about 0.6 of a light year.

In two years they go about 1.2 of a light year and so on and so forth.

Not precise, but pretty close there in terms of the units involved, and

the scale involved.

So they're traveling along here, going back home.

Really, traveling along here, of course, this is their world line at 0.6 c, and

then in year four, the bad guys invent their faster than light spaceship.

At that point they take off from their home planet.

So, they've just been sitting on their home planet here as time goes on.

One year, two years, three years, four years, trying to make this faster than

light spaceship work and they finally got to work in year four.

So, they take off at a speed of three times the speed of light because

the other ship was going 0.6 times the speed of light you work it out.

They were able to catch up with them in one year.

They go, essentially, in one year from here to here.

They're able to go three light years because it's three times the speed

of light.

So, three light years per year is what their speed is.

Over five years, the other ship goes five

times 0.6c which again, is three light years.

So, they both end up three light years away from the bad guys planet there.

And that is where the sneak attack occurs.

And from the bad guys perspective that sneak

attack occurs at x of b equals three.

Three light years away and a time of five years.

Okay, taking off from zero here.

And what we'd like to figure out is, what is the time and

location of this attack in the good guys frame of reference?

Okay, where they assume their on their ship, they actually see their home planet

receding away from them, they consider themselves at rest or can consider that.

And let's just do the Lorentz transformation, see what we get here.

So that's what we're dong here,

where the difference in the two frames of reference is 0.6c.

And also note that from the good guy's perspective,

the bad guy's planet is moving to the left, in the negative x direction,

so we use the minus sign version of the Lorentz transformation.

So, the X location of the sneak attack here is going to be gamma X sub B,

minus vtB, where, we've got x sub B and t sub B here, three and five.

Gamma, for 0.6c turns out to be 1.25.

So we just plug in the numbers, 1.25 for gamma, 3 for X sub B,

the location of the bad guy frame where the sneak attack occurred and

the time where the sneak attack occurred in the bad guy frame and

we're transforming it now into the good guys' coordinates.

And know what happens here, X sub B is three,

V is 0.6c really, but we're using unit c as one.

So, we can just put 0.6 there, t sub B is five and therefore, 5 times 0.6 is 3.

We get 3-3, we get 0 there.

You say, well, does that make sense?

Well, think about it a minute.

From the perspective of the good guy ship,

the sneak attack occurs right at their ship.

That's their zero point, that's the origin in their frame of reference.

So yes, it does make sense that they get X sub G equals zero.

It occurs right at their ship, which again is their origin point for

their frame of reference.

Now, what about the time, though, where they see that occur?

So again we plug in the numbers, equation gamma t sub B minus v over c squared x

sub B, of course again with the minus sign because from the good guys, the planet,

the bad guy frame of reference is moving to the left, negative x direction.

And plug in the numbers, you get a 5-0.6 over c squared, again c is one.

So, that just becomes 1 times 3.

So, this is 3 times 0.6 is 1.8.

5 minus 1.8 is, what, 3.2.

Multiply that by 1.25 and you get four years.

Okay?

So that in the good guys frame of reference which we're not showing

on this plot, we could put it on here but

we're not going to get it too complicated in terms of a plot that's going on.

But we see their clocks read four years at that point.

And really, that makes sense because you just have a time dilation effect going on

here, between the bad guys' clocks, which took off five years, but

in from the bad guys' perspective, the good guys' clocks are running slow.

And so they'll tick off four years, assuming they've both had their zero

point for their clocks when the good guys left the planet there.

Okay, so nothing too amazing there or confusing, just in case of how

confusing it might be in terms of time dilation and things like that.

But nothing that is overly surprising there.

Now however, let's ask the question what is the time

of the invention of the faster than light ship?

And the launch of that ship in the good guy frame.

So, that's this point right here.

In the bad guy frame, they launch their ship, put a black mark there

to indicate that's the launch of the ship the bad guys launch.

The invention of the ship, and the launch of the fast and

light ship, what time does that occur in the good guy's frame?

We know the sneak attack occurred four years in their frame.

Let's see what this does.

Same equation again.

Well actually, just do it over here.

So we have in this case, t sub G equals gamma

times t sub B minus v over c squared, x sub B.

And for the invention and launch right here, notice that,

well we could put in gamma in first.

So we'll do that, so it's 1.25.

What time does that occur in the bad guy's frame?

Well it occurs at t = four, right?

That's the time that occurs, so it's four.