21 Aug 2005 [From Steuard Jensen]

> Thank you for the kind and timely response. And the endorsing manner.

You're welcome.  As I said, I enjoy this sort of thing.

[...]

> If the emission of light is an event that has specific space-time 
> coordinates independent of the motion/velocity of the emitter, then...

> 1.    Let's assume (if we can) that the light pulse from the light 
> clock is a laser beam a mere one photon wide.

Ok, but we'll have to be careful!  Sometimes it's easier to start by 
picturing a "light clock" whose pulses flash out in all directions at 
once.  (That way, you don't need to worry about aiming your laser 
beam.)

In the end, either approach will work the same way: only the "properly 
aimed" bit of a broad flash like that will actually bounce off the 
mirror and come back as expected, and different observers would agree 
on which photons were the ones that made it back successfully.  But 
figuring out exactly how the aiming works is a bit tricky (and I think 
that's the essence of your concerns below).

> 2.    Let's further assume that the beam is exactly perpendicular to 
> the direction of travel of the source of the beam (the base of the 
> light clock, I suppose).

And here's where that trickiness comes in.  _Who_ measures the beam to 
be exactly perpendicular to the direction of travel?  If an observer 
sitting on the clock aligns the beam to be exactly "vertical" (relative 
to the direction of her relative motion with Earth), then an observer 
on Earth will see the beam moving at an angle.  Conversely, if an 
observer on Earth is the one who decides what "perpendicular" means, 
then the observer on the clock will see the beam shoot off at an angle 
(and, as you've noticed, it will miss the clock).

I'll say more about this below.  It's _not_ actually a contradiction 
with what I told you last time, but it can certainly seem like it at 
first!

> 3.    And let's further assume that the reflecting mirror at the top 
> of the light clock is very, very, very tiny.

That's fine.

> 4.    THEN...If the light clock is not in motion relative to the 
> time/space coordinates of the event of the pulse, the photon will go 
> up to the mirror at the top and bounce back to the receptor at the 
> bottom of the light clock. And we can measure the elapsed time.

I should probably take this opportunity to explain a bit more about 
"motion relative to an 'event'".  (I'd actually started to write about 
this a bit in my last email, but it got too long and convoluted in that 
context, so I left it out.  Oops. :) )

It's easy to talk about "motion relative to a point (in space)": you 
just measure your position relative to the point at each time, and 
compute your relative velocity.

But an "event" isn't a point in space, it's a single point in 
spacetime.  In other words, you could only talk about your (spatial) 
position relative to the event at a single instant in time: when the 
time on your clock matched the time that you measure for the event.  
With only one data point like that, you could never measure relative 
motion: velocity only makes sense when you compare multiple positions 
over time.  You _could_ measure your velocity relative to "the point 
where the event happened", but different observers will disagree on the 
location of that point at any time except at the specific moment of the 
event itself.

So in the end, there's no reliable way to ask whether an object is "in 
motion relative to an event".  An object can really only be in motion 
relative to another object (or at least, relative to a path in space 
that could in principle be followed by another object).


With that caveat in mind, I think you'll start to see the ambiguity 
involved in this step in your reasoning.  The light clock follows a 
path through time that "passes through" the spacetime point in question 
just as the pulse happens.  But there's no good way to say whether it 
is in motion relative to the spacetime coordinates of the event (except 
in the obvious sense: the clock is moving through time, while the event 
is just a point in time).  The event itself is "agnostic" when it comes 
to motion through space.


I don't know if the "ASCII art" diagrams below will look right on your 
screen, but I'll give it a shot.  Imagine that time goes from bottom to 
top, so that objects move like a bubble rising through beer.  The 
left-right direction corresponds to space: to see the motion of the 
bubble in space, just imagine looking down on the beer from straight 
above (so you can't see that the bubble is rising).

If a bubble doesn't move from left to right as it rises, then from 
above it will look like it's standing still, and from the side its path 
will look like the left-hand picture below.  If it moves from left to 
right (maybe the beer is leaking out the side or something), then from 
the side it will look like the right-hand picture.  The faster the 
bubble moves from left to right, the sharper the angle if we look from 
the side.

     |               /
     |              /
     *             *
     |            /
     |           /

The beer-bubble example is just an analogy to explain what these 
pictures mean, of course.  There's no way that we could ever "look from 
the side" at an object moving through time, but drawing time as a 
direction like this is still remarkably useful.

In this case, the "event" is identified by that asterisk in the middle 
of each line.  Think of the asterisk as just a dot (so ignore the 
little points sticking out).  It doesn't know anything about which way 
is up; in fact, if you tilt your head, the second diagram will look 
just like the first one used to (or at least, it would if I were 
drawing pictures for real instead of using typewriter symbols).  The 
event itself doesn't pick out one picture or the other as "standing 
still".

> 5.    HOWEVER, if the light clock is moving fast enough relative to 
> the time/space coordinates of the pulse event, it seems to me that the 
> photon will miss the mirror at the top because the mirror will no 
> longer be perpendicular to the event coordinates by the time the 
> photon reaches the top of the light clock.

So in the spirit of my comments above, you may be able to see that that 
the distinction you've drawn here isn't really well defined.  Even 
here, the only good sense in which to talk about the clock being "in 
motion" relative to the event is that one is an object moving through 
time and the other isn't: the clock's path through time passes through 
the event point.  The point itself doesn't know anything about what 
direction that path is going or how fast it's moving through space.  So 
what you're _really_ describing here is the clock moving relative to 
some other observer, and the question is what that observer would see 
when the light was emitted.

To talk about this, let me actually refer back to a throwaway comment 
that I made in my previous email:

>> And an "event" is just a point; it knows nothing about directions or 
>> velocities.  It's the same way that a dot on a piece of paper doesn't 
>> care whether the paper is held horizontally or vertically (unlike an 
>> arrow, for instance, which would look very different if you rotated 
>> the paper).

In the scenario with the light clock that we're discussing here, the 
"event" is the emission of the light and it happens at a specific point 
in space and time.  Just as a dot on a piece of paper still looks like 
a dot (and still sits on the same physical place on the paper) if you 
rotate the paper, the "event" always corresponds to the same physical 
occurrence (the flash of light) no matter what angle you see it from or 
how fast you're moving.

But the _direction_ in which the flash of light is emitted is like the 
arrow on the paper!  If you rotate the paper, the arrow ends up 
pointing in a different direction (even though its physical 
relationship to the underlying paper hasn't actually changed).  
Similarly, observers moving at different speeds will see the flash of 
light moving in different directions, even though they'll all agree on 
how the flash was related to the underlying physical apparatus.


It may seem miraculous at first, but the difference in what direction 
different observers see the pulse move will always be _exactly_ the 
right difference to make them agree about true "physical" questions, 
like whether the light hit the mirror or not.  If the person sitting on 
the clock aims the beam straight up so that it hits the mirror, then 
when you work through the mathematics, an observer on Earth would see 
the beam aimed to the side with just enough of a lead so that it will 
reach the mirror's future position just as the mirror gets there.

As I said, it seems pretty miraculous: the equations that you'd use to 
figure out that angle are messy, and just from looking at them there 
would be no reason to expect that things would line up so nicely.  But 
when you study relativity on a deeper level, you'll discover that this 
"miracle" is really a reflection of simple geometry.  It's actually no 
more miraculous than the observation that if you draw an arrow from 
point A to point B on a piece of paper, the arrow still goes from point 
A to point B if you turn the paper on its side.  But sadly, too many 
explanation of special relativity never manage to explain this more 
elegant way of thinking of it.  (I could try to explain it, but it 
might be a bit of a challenge.)

> If my presumption is true, then I do not understand the light clock 
> diagrams that show the diagonal path of the light when the light clock 
> itself is moving rapidly through space.
>
> Any help you can give me on this sticking point will be greatly 
> appreciated.

Well, let me know if that explanation helped at all, and what parts of 
the issue remain confusing.  This is fun!  (Although it's feeding my 
longstanding temptation to write up a tutorial on special relativity 
for my web site!  That may or may not be a good thing right now. :) )