30 Aug 2005 [From Steuard Jensen]

> Your response of a week ago required some think time. Still does, as a 
> matter of fact.

That's to be expected.  Relativity is really strange!  The main reason 
that I understand it better than you do is that I've devoted rather a 
lot of think time to it over many years.

> Let's assume the same space ship with the same single-photon-wide 
> light beam. Also assume everything is from the point of view of an 
> observer on that space ship.
>
> The space ship is not in acceleration or deceleration. It is in 
> uniform motion.
>
> The observer tunes the light clock so the narrow beam bounces off the 
> very tiny mirror and returns to a very tiny receptor at the bottom of 
> the light clock.
>
> Okay so far??

Sounds good.

> Now, suppose the spaceship (including the light clock and the 
> observer/technician) accelerates rapidly in a direction that is 
> roughly perpendicular to the light beam.

This gets a little dangerous: although special relativity _can_ handle 
situations in which an observer watches other objects accelerate, it 
_isn't_ general enough to describe the point of view of an observer who 
is accelerating herself.  That is the province of general relativity 
(hence the name), which is the most beautiful theory in physics.  But 
it's also a good step more complicated than special relativity, and it 
is hard to understand the general theory before you have a pretty good 
grasp on the special one.

The most bare-bones description of the problem is this: it is 
impossible for an observer to distinguish between "accelerating faster 
and faster" and "sitting still in a gravitational field".  That is, if 
I stuck you in a small room with no windows and you felt yourself 
pulled to the floor, you would not be able to tell whether the room was 
sitting on the surface of the earth or flying ever faster on a rocket 
through empty space.

The math behind this is very unfamiliar to most people, and I won't try 
to explain it now.  To deal with the problem, I'll use that 
"acceleration = gravitation" intuition when speaking from the point of 
view of the accelerating observer.  But I'll also comment on what an 
_inertial_ observer would see, just to show that special relativity 
_can_ handle these questions.

> Will the light beam still bounce back to the receptor...
>
> 1.    During acceleratiion?

No.  From the point of view of an observer on the rocket, the light 
bends in the "gravitational field" that she feels.  Because it bends 
away from its previously straight path, it will miss the tiny mirror.

From the point of view of an _inertial_ observer outside the rocket 
ship, the answer is still no (remember, all observers must certainly 
agree on the answers to "physical" questions like "did the light reach 
the receptor").

Imagine that from his point of view, the rocket had been moving at some 
constant speed when the laser and mirror were aligned.  Then what he 
saw at that point was that the laser was carefully aligned to aim at 
where the mirror _would be_ when the light got there (much as a 
football quarterback throws a pass to where he expects the running 
receiver to end up).

But when the rocket starts accelerating, the mirror will keep moving 
faster _after_ the laser has already been aimed and fired.  Because the 
mirror ends up moving faster than the laser was set to expect, the beam 
of light will miss.  To continue the football analogy, imagine that the 
quarterback throws the ball, but then the receiver speeds up faster 
than the quarterback expected.  The ball will fall behind him, for an 
incomplete pass.

Does all that make sense?

> 2.    At the end of acceleration (once again in uniform motion but 
> without deceleration)?

Yes, the laser will hit the mirror and bounce back to the receptor.

From the point of view of the observer on the rocket, the 
"gravitational field" has now gone away.  Thus, the laser's previous 
alignment is no longer being disturbed, and the light will bounce off 
of the mirror just as it did before.

From the point of view of the observer who was in an inertial frame the 
whole time, the laser is once again aiming at just the right angle 
ahead so that it will hit the mirror.  That angle has to be different, 
of course, since the rocket is now moving at a different speed than it 
was before.

But if you give it a few minutes' thought, you'll see that it's not 
surprising that the angle looks different now: that doesn't even depend 
on relativity.  To use a different sports analogy, imagine a pair of 
basketball players inside a train car (one with lots of windows so that 
we can see in), passing the ball back and forth perpendicular to the 
direction the train is moving (perpendicular from their point of 
view!).  If we watch them doing this from the outside, we will see the 
ball moving in a zig-zag pattern: the ball "inherits" the motion of the 
train along the tracks even though the people inside feel like they're 
passing it straight back and forth.  The same thing works for our laser 
in the rocket ship (even if the math involved in "inheriting" the 
motion is a bit more complicated when relativity is involved).

This basketball player example can give some insight into the 
acceleration part of your scenario, too.  If one player passes the ball 
but the train accelerates before it reaches the other player, the pass 
will go off center (because it was aimed at where the receiver was 
supposed to be before he accelerated out of the way).  But once the 
train's speed stabilizes again, their passes will stay on target... and 
the observer on the platform will see a more "stretched out" zig-zag 
because the ball is now "inheriting" the train's faster sideways speed.

I hope those explanations aren't overly complicated!  But let me know 
if I've managed to make things more confusing instead of less. :)

> If the photon beam misses the mirror during or after acceleration, 
> that supports the previous idea of the light pulse from one space ship 
> reaching the other space ship even though the two ships are receding 
> from each other at 120% the speed of light. But then, it also seems to 
> support some kind of stationary spatial concept (similar to but not 
> identical to the ether idea), doesn't it?

The light pulse from one ship will most certainly reach the other one, 
but _without_ any need for a notion of "stationary space".  Let me 
explain what each of the three observers sees, given your premise that 
one observer (on Earth, say) sees each ship moving at 60% of the speed 
of light in opposite directions.

From the point of view of the emitting ship, the other ship is not 
moving away faster than the speed of light.  (In fact, when you work 
through the math, people on the emitting ship will see the other one 
moving at about 88% of the speed of light.)  So they certainly expect 
to see the light catch up with the other ship eventually.

From the point of view of the observer on Earth, the light pulse still 
moves at the speed of light, even though it was fired "backwards" from 
the emitting ship.  (That's where the weirdness of relativity comes 
in.)  So the Earth observer certainly expects to see the pulse of light 
catch up with the receiving ship eventually.

From the point of view of the observer on the receiving ship, the light 
pulse _still_ looks like it's moving at the speed of light.  Given that 
that observer thinks that he is sitting still, he will expect the pulse 
of light to reach him without trouble (it won't even have to "catch up" 
with anything!).


But for the record, the three observers won't agree on everything!  If 
the emitting ship emits a _series_ of pulses (say, one pulse per second 
as they measure it), the three observers will disagree on what the rate 
of pulses is.  The Earth observer will see one pulse every [1.25] 
seconds, and the receiving ship will see about one pulse every [2.1] 
seconds.  For that matter, since the frequency of light determines its 
color (and since frequency is a lot like a pulse rate), the Earth 
observer will see the pulses a bit more red than the people on the 
emitting ship do, and the people on the receiving ship will probably 
see the pulses all the way down in the infrared range (and hence 
invisible to the naked eye).

> If the photon beam hits the mirror during or after acceleration, 
> doesn't that support the idea that the travel of the light is somehow 
> related to or influenced by the acceleration or deceleration of the 
> emitter?

That's at least somewhat accurate, too.  But only in the sense that I 
discussed with the basketball player example: if I see the emitting 
device moving, then I will see that motion "inherited" to some extent 
in the motion of the emitted light.  If another observer were in the 
rest frame of the emitter, she wouldn't see any motion of the emitter 
that the light could "inherit".  So the motion of the light is 
"influenced" by the motion of the emitter only in a fairly weak sense: 
things that travel together, travel together, no matter who's watching.

> Thanks for your interest in this. It seems to be a sticking point that 
> keeps me from understanding other stuff.

It's really complicated stuff, totally at odds with our usual intuition 
for how motion works in some ways.  But I've found that by thinking 
about motion in relativity carefully, I've actually come to understand 
_familiar_ motion better, too.  (As I've tried to explain above, some 
of the confusing bits in relativity can also be a bit confusing in our 
familiar experience, too, but we generally aren't forced to think about 
them in that case!)