Scientists say that many distant galaxies are moving away from us at
very high speeds. They also say that Time slows down the faster you
travel.

If so, then Time must vary from galaxy to galaxy, depending on the
relative speed of each galaxy.

Imagine, then, a clock in each galaxy. If the hands on clock A are
circling faster than those on clock B, then we might conclude that
galaxy A is traveling SLOWER than galaxy B (relative to the entire
universe).

(1) Is that conclusion correct? In other words, could we tell which
galaxies are traveling faster or slower than others simply by
comparing their clocks (assuming, of course, that we had the power of
God to see the clocks)?

(2) If we sent thousands of clocks in all directions at various
speeds, would the clock with the fastest circling hands be the clock
that is traveling the slowest (perhaps not even traveling at all ? a
perfect ?dead stop?)?

This sounds perfectly reasonable explanation. As Einstein said
"everything is relative".
In fact what you propose has already been done on a small scale. They
have already taken two perfectly aligned atomic clocks and sent one up
to space on a rocket. I think it was a saturn rocket but not sure.
They then compared the clocks when it returned from orbit and found
that time had ran more slowly on the moving clock (relative to the
earth based clock). As you point out the main problem is to actually
compare the clocks and if your clock is zooming away from you at near
light speed then you have no way of telling the time on the clock
until it comes back.
Of course the concept of dead stop may be meaningless. Dead stop in
relation to what reference point? Position in the universe is also
relative hence there may be no such thing as dead stop as it implies
relative to something else.

To knickers-ga:
If we sent thousands of clocks in all directions at various speeds,
what should we conclude about the clock whose hands are circling the
fastest? Should we not conclude that that clock is traveling the
slowest? (I?m not sure relative to what. But, if the clock?s hands are
circling the fastest, then what else can we conclude?)

Hallo,

   Special Relativity deals with pairs of inertial reference systems.
For a given relative velocity between two systems, there will be a
ratio of the clock rate within an observed distant system to that
within the observer's system. That's all it is, a ratio.
fv
   In SR, there is no such thing, in principle, as a clock rate
relative to the "entire universe". By this I mean that you cannot
expect to observe various distant systems, such as speeding galaxies,
measure their clock rates, and thereby infer their various absolute
velocities. All the observer will determine is their various
velocities relative to the observer. The clocks with the slower
observed rates will always be the ones with the highest velocity
relative to the observer. The slowest clocks will always be the ones
with zero relative velocity to the observer.

"The slowest clocks will always be the ones
with zero relative velocity to the observer."

   Pardon me. That last sentence wa supposed to be "The *fastest*
clocks will always be the ones with zero relative velocity to the
observer." (My mighty proofreading powers were neutralised; the cat
was doing a keyboard inspection.)

Isn't the speed of light absolute throughout the universe?

The clock whose hands are circling the fastest
is the clock that is traveling the slowest,
relative to the absolute speed of light. No?

"Isn't the speed of light absolute throughout the universe?

The clock whose hands are circling the fastest
is the clock that is traveling the slowest,
relative to the absolute speed of light. No?"

   In special relativity, the speed of light, or more accurately, the
speed of any massless object, is invariant for every inertial frame of
reference. All this says is that, within a system traveling at uniform
speed, the measured value of c within that system will be the same as
within any other similarly inertial system. The speed c *does* vary.
This is due to time "dilation". But what happens is that any two
inertial systems will have a Lorentz transformation, and the magnitude
of that transformation will depend upon the two systems' relative
velocity. The Lorentz transformation itself is most notably a scaling
of time & lengths. The scaling factor, commonly called "gamma", is
given by the following:

gamma = 1/SQRT[1 - v^2/c^2]

   Time in system A as observed from within system B will be slower.
Similarly, the clocks in B as seen by A will be slower. This is the
time dilation I mentioned above. The speed of light will of course be
slower, but lengths contract by the factor gamma along the direction
of motion, exactly compensating the slower value of c. (And in fact,
the slower rates across the board of all dynamic processes within the
system.) Thus, in every inertial (uniformly moving) system, the value
of c will be the same as measured within the system.

To qed100-ga:

I do appreciate your comments (although they're getting over my head).

You said, "Time in system A as observed from within system B will be slower.
Similarly, the clocks in B as seen by A will be slower."

Well, now I'm confused about the "twins" tale, where one twin goes
traveling very fast while the other twin remains on Earth. When the
traveler returns, he hasn't aged very much, but his earthly twin is
very old.

If everything is relative, how come the effects weren't the same for
both twins? (Who's to say which twin did the traveling?)

That's a perfectly reasonable question. Indeed, if twins are in
perpetual uniform relative motion, then there is symmetry and each
finds itself getting older than the other. The twist happens when the
symmetry is broken. There are several equally valid solutions to the
twin paradox. In fact, in the long run they're all related to one
another. But rather than quote them all here, I'll direct you to a
good website that tackles them nicely:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

To rambler, this discussion is getting very high brow now. However, to
keep things simple. The speed of light is an aboslute constant for any
given observer. Even if you were travelling close to the speed of
light then you would still measure the speed of light as C. So you can
not tell that your clock is running slower or faster by measuring the
speed of light. At the core of Einsteins relativity was "an observer
travelling at a constant speed can not tell he is in motion or
stationary" So in your examples, everyone observing a clock in there
own galaxy, regardless of how fast they were travelling, would see the
clock rotate at the same speed. However, to observers outside of that
reference point i.e. stationary they would see the clocks with
different times to their clocks. The whole point is that light speed
is a constant and everything else varies i.e. distance and time in
relation to the observer (relative!)
For your tiwns, time as varied for one and not the other. So as I said
before your proposal is essentially correct but only if you could
bring those clocks back to the origin to compare them all, otherwise
there is no way of telling which is running fast or slow. the speed
they move around the face stays the same for all the observers in
their locality.

Just to add another example and comment. On earth we are spinning
round at 24000 miles / hour. We dont feel it and we can not measure it
by measuring the clock speed or the speed of light. The speed of light
from our flash light is still C (186,000 miles/s). If you now go up in
the space shuttle and orbit stationary your speed relative to the
earth surface is now zero but the speed of light is still the same C.
However, there will be minor time differences between you both as a
result of you spinning round and the spaceman being shot into orbit
and now becomming stationary. If you now look outside of our solar
system, lets say someone watching earth from a close glaxy, both the
earth and the spaceman will be zooming away at some phenomenal speed.
The man on the earth, the spaceman and the distant observer will all
see the speed of light as C but we will all have slightly different
times relative to each other.

As a lay-person, I sort of understand the idea of multiple observers
and how everything is relative to the individual motions of those observers.

However, let's say that there are NO observers.
Instead, let's just use our own thoughts.

Imagine an empty universe: no planets or galaxies, no gravitational fields.

In our minds, we can imagine a clock that is traveling very fast.
We are not observers: we are simply using our minds.
We 'know' that the internal workings of that clock are slowing down.
In fact, if the clock travels fast enough, it could travel 
clear across the universe before it ticks one second.

If we imagine a different clock that is not traveling at all,
we 'know' that its internal workings are normal.
We don't have to look at it: we simply 'know' that it is working normally.

Interestingly, we canNOT imagine a clock working faster than normal.
A clock can work normally, or slower than normal, but not faster.
Reversing direction or reversing speed won't make the clock work faster.
The range of the clock's workings are between near-zero and normal.

Does any of this make sense, or am I wrong to eliminate observers?
Do we need observers? Can't we just 'know' what is happening?

I think you are losing the point.
The mechanics of the clock dont change. As far as the observer with
any clock goes it functions absolutely perfectly in their own time
frame i.e. if they time the motion of the hand then it appears to be
keeping correct time. However, the reality is that time is flexing not
the motion of the clock.

Further to my previous comment. Its better to think about the
definition of time as we know it. The current definition of the second
is based on the number of ocillations of the ceasium atom i.e. nothing
to do with mechanical clocks. We just count the number of oscillations
of the specific atom and thats how much time 1 second occupies. This
is the principle of atomic clocks. So if you use these clocks instead
of your mechanical clock you get the same result. The atom still
vibrates and you still count the number of vibrations. But as you
approach light speed it takes longer and longer to get the same number
of vibrations i.e. 1 second becomes longer. However that is only
relative to a motionless observer. For the person with the moving
clock the passage of 1 second seems exactly the same.

Another way to look at it is back to our school days definition. Time
for a journey is given by Time = Distance / Speed

So what we are saying is speed is constant i.e. the speed of light C.
So if this is true then for Time to vary the distances of moving space
 (moving observer) must be different to stationary space (observer).
This is exactly where einstein started his great mental exercise. He
started thinking about what would happen if you could ride as an
observer on a wave of light. What would the observer see happen to
light from his flashlight. The answer is that he sees exactly the same
thing as if he was motionless, light still travels away from him at C.
So as you can see the stationary observer and the moving observer must
see the lightwave travel over different distances and hence they
observe different times. A really good book to read is Einstein for
Beginners. It covers all of this discussion and is a great starting
place.

To knickers-ga:

I sincerely thank you for your comments.  May I ask you about
the "stationary observer" that you mentioned in your last comment?

How do you know he's stationary?

(Note that the original subject of my question
is "Coming to a dead stop in the universe".)

"How do you know he's stationary?"

   In SR, the closest a system can come to being stationary is to be
inertial, i.e., not accelerating. Given the postulates of special
relativity, it's meaningless to define any system as absolutely
stationary. Every inertial system is equally stationary with respect
to all other relatively moving systems.

A few months ago, I saw a science program on TV.  It was actually
one in a series of programs hosted by an African-American scientist
(I can't remember his name).

He was explaining why things operate slowly when you travel at high speed.
He presented a visual that absolutely stunned me. It was one of those
"Aha!" moments. Suddenly, years of puzzlement vanished. I thought to myself,
"It's so simple. Of COURSE things operate more slowly at high speed!"

He showed a glass cylinder standing on end (let's say 3" tall).
A small ball bounced from the bottom to the top, and back down again.
It continued bouncing endlessly. It took one second to travel one way.
Thus, its speed was 3 inches per second.

After a while, the scientist slid the cylinder 4" to the right.
He did this while the ball was moving upward, from bottom to top.
This meant that the ball had to travel 5" at an angle to reach the top
(it was tracing the hypoteneuse of a right-angled triangle).
Since the ball could only travel at 3" per second, it took
much long to reach the top.

That was my "Aha!" moment.  What I inferred from all this was:
(1) The speed of light is absolutely absolute throughout the universe.
(2) In fact, all sub-atomic activity is likewise absolute.
(3) Any object that travels (at ANY speed) operates at a slower internal speed.
(4) The faster the object travels, the slower its internal operation.
(5) Time does not really slow down. There is no such thing as Time.
    It's just that the object's sub-atomic activity can't "keep up".

Maybe the scientist wasn't implying any of this at all, but it's what
I inferred. And it makes it so easy to understand the "twin paradox".
The twin who travels simply doesn't 'age' as fast as the twin who
remains on Earth.

Of course, I may have misunderstood the scientist. Have I?

Let's state Einstein's postulates on which he built SR.

[1]- The laws of electrodynamics in general are invariant with respect
to inertial frame of reference.

[2]- The value of c specifically is invariant with respect to inertial
frame of reference.

   The motivation for these was the already well established fact
that, in Maxwell's electrodynamics, the speed of light, c, is
independent of frame of reference; there is nothing in the mathematics
to distinguish between frames. Thus, Einstein found his postulates
reasonable. Given these, there follows the Lorentz transformation of
time & length.

   The demonstration of the bouncing ball is an illustration of how
relatively moving systems have relativistic transformations. As you
say, the ball in a relatively stationary frame travels up & down at,
say, 3 inches/s. If the frame is relatively moving laterally at some
uniform speed, then relativistically the ball traces a longer path. (5
inches) So here's how it appears, depending on reference frame:

[1]- In the stationary frame, the ball moves at 3 inches/s.

[2]- In the traveling frame, it moves at 5 inches/s.

   Thus, in the traveling frame time is observed to be different in
the relatively moving frame by an amount proportional to the ratio of
the two lengths. It's important to point out that the ball represents
a pulse of light scattering between two opposing mirrors, an ideal
clock. It has to. If time differed between frames simply by the ratio
of the lengths as we've already discussed, then the effect would be
tremendous just by going jogging. Instead, the effect is of this
magnitude between frames for objects which move at c, and the effect
for massive objects approaches this same magnitude as their speeds
approach c.

   So, many of your Aha! inferences are essentially correct, though
they are in need of some fine tuning:

(1) The speed of light is absolutely absolute throughout the universe.

   The speed c is absolutely reliable within all inertial frames.

(2) In fact, all sub-atomic activity is likewise absolute.

   Yes, in the sense that such dynamics will be similar within all inertial frame.

(3) Any object that travels (at ANY speed) operates at a slower internal speed.

   Yes.

(4) The faster the object travels, the slower its internal operation.
(5) Time does not really slow down. There is no such thing as Time.
    It's just that the object's sub-atomic activity can't "keep up".

   Time does slow. I know a lot of people tend to say, "Time doesn't
slow down. It's all the clockwork dynamics which become slower." One
can put it this way, but as it turns out the only way in which the
measurement of time is at all meaningful is by way of such dynamical
processes. A clock is a dynamic system; a dynamic system is a clock.
If the dynamics within a system all display a uniform transformation
as a function of relative velocity, then there's no difference between
the dynamics slowing down and time itself doing so.

I am glad you had your Aha moment. It feels so good when that happens.
Its rather funny but I was just about to try and give you a similar
picture. Mine was going to be the flashlight beam bouncing across a
train. However the principle is the same. You have grapsed the concept
that the moving observer sees the ball travel a different distance to
the statinary observer so you now understand that, relatively, there
is a D1 and a D2 (where D = distance) for each observer. If the ball
were a light photon then we know the speed in both cases would be C.
So the time take for the ball to travel the distance is given by D1/C
for one observer and D2/C for the other observer i.e. you end up with
a T1 and T2 for the same object. So time does really change for each
observer. The hardest concept to grasp is the fact that Time is
relative and can flex. Its because we are so programmed to think of
time as a constant tick of a clock. It is really only our undertanding
of what time is that is difficult to grapple with. I find that if you
think of time as the distance speed concept if becomes far more easy.
In my simple universe anyway!

To knickers-ga and qed100-ga:
  Thank you both for your very helpful comments.
  If you were GA researchers,
  able to provide official answers,
  I would give you 5 stars!

Thank you. I appreciate the compliment.

"A few months ago, I saw a science program on TV.  It was actually
one in a series of programs hosted by an African-American scientist
(I can't remember his name)."

   I'm just curious, was it this gentleman?:

http://www.math.buffalo.edu/mad/physics/tyson_neildg.html

   If so, that's Neil deGrasse Tyson, a very well respected
astrophysicist who also does a lot of public outreach. I've seen him
on TV many times, explaining physics & astronomy.

Yes, that's him.  Thanks again!

Cheers, thanks for the compliments. Its been an interesting
discussion. We all learn a little more from these sorts of things as
it helps all of us to clarify our thinking whether right or wrong.

The "aha moment":
As another layman, I have to respect a noted astrophysist's
demonstration as being scientifically justified, but I have trouble
buying the explanation that it took longer than one second for the
ball to rise 3" when he moved the cylinder 4" during that period
because the ball then had to move 5".  Moving the cylinder imparted
additional motion, horizontal motion of 4".  Seems to me that in a
frictionless environment the ball would still rise 3" in that second.

(I don't believe the 5" is significant; seems more a vector
calculation, ie, swimming across a river and finding out how far
downstream the swimmer is when he reaches the opposite shore, not
having "swim" faster or further to get there.) Or looking at it
another way, if the observer of the ball also moved 4" during that
part of the demonstration (having the impression that the tube
remained still and that the room moved), to him the ball would seem to
be continuing to move only vertically.

I have to also wonder how the ball was activated in a vertical
position so that it's up and down motion took the same time despite
the effect of gravity - slowing its rise and accelerating its fall.

As to the clocks  - remember, I am just another layman -  as Knickers
initially pointed out:  "everything is relative".  We and our
earth-bound clock are not in motion relative to each other, regardless
of the rotation and other movements of the earth.  If the earth came
to a dead stop, we would still see it ticking at the same pace.
If we could send clocks everywhere in the universe, even send one to
the "dead" center while we on earth keep moving, that clock will be
moving relative to earth and slowing down from our point of view. 
Should the clock stop there, and we are in circular orbit around "dead
center", then it would then appear to keep time again with our
earth-bound clock, but now running a little late.

Does this add anything?  Hope so.

Keep in mind that the ball in the tube was only a handy way to
*illustrate* the difference between frames of reference: In one frame
it moves only 3 inches. In another it moves 5 inches. Keep also in
mind that, as I mentioned earlier, the ball represents a pulse of
light in flat spacetime (Flat spacetime is what puts the "special" in
special relativity.), so we needn't worry that the ball in the
demonstration changes its vertical speed as it responds to gravity.
It's just an aid to visualisation.

   Now, as for the 5 inches, it is relevant. You have to start out
with the proposition "The value of c is invariant for all inertial
frames." Given this, then a logical consequence is that the moving
frame (relative to an observer), in which the light pulse travels
farther per second than does a similar pulse scattering vertically
within the observer's relatively motionless frame, will have slowed
time, which is to say that all processes within that frame will
uniformly proceed more slowly than within the observer's frame. Time
will be scaled by the factor gamma. There will be a ratio of clock
ticks between the two frames such that, for each tick in the oberver's
frame, a similar clock in the observed moving frame will tick less
than once.

   This means, of course, that the speed of the light pulse in the
moving frame will be less than it's measured to be within the
observer's frame. But isn't c invariant? No. This is an opportunity to
reiterate: The value of c always measures out to the same value
*within* an inertial frame. A frame moving uniformly with respect to
the observer will, by the observer's reckoning, have a slower speed of
light. But due to the relativistic Lorentz contraction, anyone within
the other frame will reliably measure the speed of light as c. The
contraction will be exactly in inverse proportion to the time
dilation.