Let's finish off the pole in the barn paradox by doing a space-time diagram,

as we've done several times before now.

And here's all the calculations we did before.

And here's a space-time diagram, where the black axes, 90 degree axes,

horizontal and vertical, are Bob's barn frame of reference.

And we've got meters and nanoseconds.

And put the units on there.

Here's where the barn is.

We're assuming the barn is at x sub b equals zero and extends to

x sub b equals eight, so that's eight meters there, pretty close to that.

And in nanoseconds, 10, 20, 30, 40 nanoseconds, and so on and so forth.

We'll get to the green stuff in a minute, but

the red is the essentially space time path, the world line

of Alice's pole as she moves through the barn from Bob's perspective.

So right here at this point, at t sub B equals zero,

the pole is just at the front door of the barn there and

goes backwards here eight meters, because that's how,

that's the length the Bob sees, the pole beam.

And then as time goes on, when it's at 10 seconds here, it's at, not 10 seconds,

10 nanoseconds, it's this position here, and then at 20 it's here, 30 here.

In other words, through, in terms of the space time world line it moves this way,

which of course physically means it's moving in the barn like that.

And then when it reaches this point right here,

it's reached the end of the barn there.

And of course it fills up the whole barn from one end to the other from Bob's

perspective.

And note, this is at 44 nanoseconds, just as we had over here,

44.4 nanoseconds, is when the photos are taken by Bob.

So he has one clock at each end, one clock here and here,

that take the photographs of each end and get the 44.4 nanoseconds.

Then what we've done here is put on Alice's frame of reference.

We haven't done the whole thing, we've just done the x sub A axis,

because we're most interested in the lines of simultaneity.

We could certainly put on the t sub B axis as well, but

then the diagram gets sort of messy and it's hard to see what's going on.

So x sub A, you may remember the slope as we calculated as v over c-squared.

And using the numbers up here, we get 2 nanoseconds per meter.

So that's where we get this line from, and then try to plot it correctly there.

So we did that calculation when we first derived how to put one frame

of reference on another frame of reference here using the Lorentz transformation.

So that's where that comes from.

If you're wondering,

how do I know what angle to put this at, that's the answer right there.

And there's a similar calculation we did, of course, for the t sub A axis,

which again we'll leave off for this diagram.

And so in this case we have the lines of simultaneity.

And I've just drawn parallel lines through the x sub A axis.

I haven't tried to figure out exactly the spacing in terms of precise numbers here.

But the key thing we can see is, as time ticks off here for

Alice, okay, so these are one tick, two ticks,

three ticks, four ticks, or whatever it happens to be.

Right here, at this time, we can see that the barn reaches the end of the pole,

the front of the pole from her perspective,

and that's when Bob would take the 44.4 nanosecond picture.

But then also, if we trace the line of simultaneity back here,

look where she sees the other end of the pole.

It's still well outside the barn.

Here's the barn right here, and so it's still back there, just as we did,

we erased the picture, but

just as we had with the picture of the pole still sticking out the front end.

Because from her perspective of course, the pole is 10 meters,

the barn is only 6.4 meters.

Now if you did a very precise diagram and put both of Alice's axes on here,

you could also show that you get the length contraction effect as well,

in terms of how long she would see the barn

being not 9 meters but actually 6.4 meters there.

But again, that gets a little messy in terms of the diagram.

But it does show that in our basic combined space-time diagram,

you can actually see these effects of the relativity of

simultaneity in terms of when the barn doors close here and

how far the pole fits in to the barn and so on and so forth.

So it essentially is consistent with our quantitative analysis we did before.