Post by Richard Livingston
Post by Luigi Fortunati
The graduated belt of my animation is directly *bound* to the Earth and
This makes its measurement (direct and immediate) more reliable than
the measurement at distance with light.
I'll try one last time to make this point. The time "now" for events outside
the light cone, e.g. events that are spatially separated, are not uniquely
defined for different observers.
I must ask what meaning you attach to a time "now" back on earth? It
is not the time you can see, via light, from earth. In fact there is nothing
you can do to interact with earth "now" in any way. The only way to
know what is happening "now" is to wait for that event to enter your
past light cone, at which time you can see it. At that later time you can
know what happened back when you identified a time as "now".
If you want to define some time as "now" by some mechanism like you
describe, that is OK. But the conventional way to define "now" is to
synchronize clocks as described in most any book on special relativity.
That process allows you to define a "now" that conforms with what
we all normally think of as "now". It gives a different result for different
inertial observers, but it gives a result that each observer can make
sense of as "now". That is, if they do some experiment such as send
a time message via light, the result is consistent with special relativity.
There is another factor that you are not taking into account: The
Lorentz contraction of each side of the belt will be different for
moving observers. This may seem as it should be negligible for
such a slow moving belt, but note that magnetic fields are the result
of just such a difference in Lorentz contraction even though the
electrons in a wire are moving at less than 1 mm/second. In the
case of your belt model, a rapidly moving observer will see that
one side of the belt has shrunk compared to the other, and thus
the earth is rotated relative to what the "stationary" observer
[[Mod. note -- That's a very clear (and completely correct) exposition.
It sounds nice, but it is yet wrong. There is no 'rotation' of the two
Earths relative to a traveller along their axis of symmetry.
What *is* true and correct, is that the belts are observed (Lorentz-wise
as well as 'looking'-wise) to advance at different speeds 'back' and
'fro', by the traveller.
Take a look at the graph I proposed before:
For an example, choose b=0.43 (traveller velocity) and d=0.75 (belt
velocity). Choose "Show voyager's POV" (and unchoose "Show Earths'
POV"). Now let the traveller go from Earth1 (left) to Earth2 (right)
with parameter s. Look at the following events:
O. s=0 :
traveller leaves Earth1. Belt check points B (in Earths' POV) leave
their respective Earth, say, B1 and B2. But in traveller's POV, B'1
coincides with B1, but B'2 is already underway, as traveller's x-axis
points to Earths' future.
A. s=0.24 :
Checkpoints B'1 and B'2 cross each other midways (same event for both
traveller and Earths)
B. s=0.364 :
Traveller crosses checkpoint B'2 (heading to the left).
C. s=0.47 :
Checkpoint B'1 reaches Earth2 at the right.
! Immediately a "new" checkpoint B'2bis starts heading to the left! [not
Checkpoint B'2 is still on its way left.
! In the upper part there is no checkpoint at present!
D. s=0.7 :
Checkpoint B'2 reaches Earth1.
! Immediately, but only just now, a new checkpoint B'1bis starts heading
to the right! [not shown]
E. s=1 :
Voyager reaches Earth2.
From this we can conclude:
The belt running in voyager's direction seems shorter and running at
higher velocity (at times, without checkpoint on it).
The belt running in voyager's opposite direction seems longer and
running at lower velocity (at times, with two checkpoints on it).