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Consider the following thought experiment:
* A watcher inhabits a planet which we'll call "Planet X".
* There is a space station, with a captain whose name is Homer, in
orbit around Planet X.
* And inside the space station's bay is a space ship, with a captain
whose name is Ulysses.
* Both space station and space ship are capable of moving swiftly in a
space environment.
1) Now, the space station decides on leaving Planet X and it
accelerates to a velocity "v" heading away from the planet. The space
station decides to coast at this velocity.
2) Then, the bay doors of the space station open allowing the space
ship to leave. It accelerates away from both Planet X and the space
station until it acquires a velocity "v" relative to the station. The
watcher on Planet X will observe the space ship to be travelling at a
speed of "[v+v]".
*where "[v+v]" equals "2v/(1+(v/c)²)"
*let "tW" be a period of time, as observed by the watcher
"tH" be a corresponding time, as observed on the space station
"tU" be a corresponding time, as observed on the space ship
Thus, when the watcher observes a time "tW" to have elasped, then
Homer will observe a time "tH" to have elasped and Ulysses will
observe a time "tU" to have elasped.
Now, it is obvious that
(1) "tH = y*tW"
and
(2) "tU = z*tW"
*where "y" equals "1/(1-(v/c)²)^½"
"z" equals "1/(1-([v+v]/c)²)^½"
Also, since the ship is travelling at velocity of "v" relative to the
station, we can also say that
(3) "tU = y*tH"
Now, we can solve these three equations for "v"! We find that "v"
equals "0"! But we shouldn't be able to know "v"!; the thought
experiment *should* work for any velocity, not just one particular
velocity. The problem here, specifically, is with the third equation,
which is wrong. It is wrong because, despite what Relativity
dictates, there is an *absolute* frame of reference (some may call it
a *preffered* or *unique* frame of reference).
We will get back to this thought experiment later. For now, you may
be asking, if there is an absolute frame of reference, how can we find
out where it is? It is actually surprisingly simple. Consider the
following experiment:
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* There is a watcher who is at rest with the entire Universe, that is,
he is at rest with an absolute frame of reference.
* The watcher watches two space stations (Station A and Station B)
which are both in the same frame of reference, that is, both stations
observe each other to be at rest. The distance between the two space
stations is "s". The watcher views both stations to be moving in the
right direction at a velocity "n".
* Homer remains on Station A for the entire duration of the
experiment.
* Meanwhile, Ulysses will travel from Station A to Station B, and then
he will promptly return back to Station A.
Let the period of the trip where the space ship is leaving Station A
and approaching Station B be called the "first phase" of the
experiment.
Let the period of the trip where the space ship is leaving Station B
and approaching Station A be called the "second phase" of the
experiment.
This is Ulysses flight plan for the experiment:
(1) Start on Station A with Homer.
(2) FIRST PHASE: Board space ship, and fire *right* thurster *once*
thus attaining a velocity "p" in the *right* direction relative to the
stations.
(3) SECOND PHASE: When Station B is reached, fire *left* thruster
*twice*, thus attaining a velocity "p" in the *left* direction
relative to the stations.
(4) When Station A is reached, fire *right* thruster *once*, thus
coming to rest relative to the stations.
(5) Dock on Station A and reunite with Homer.
--------------
In the following diagrams, the symbols mean..
* "-->" means the object is travelling at a velocity, "n",
in the right direction, relative to the watcher.
* "«" means the object is travelling at a velocity "p",
in the left direction, relative to the stations.
* "»" means the object is travelling at a velocity "p",
in the right direction, relative to the stations.
[to view the diagrams, one must use a fixed-width font]
----------------------------------------------
(1) Homer and Ulysses are on Station A which is travelling at a
velocity of "n" in the *right* direction as viewed by the watcher.
_______________________________________
| | | | |
ship *--> ship
Station A O--> O--> Station B
----------------------------------------------
----------------------------------------------
(2) FIRST PHASE: In his space ship, Ulysses fires his *right* thruster
*once*, causing him to travel in the right direction at a velocity "p"
relative to the stations.
_______________________________________
| | | | |
ship *-->» ship
Station A O--> O--> Station B
----------------------------------------------
----------------------------------------------
(3) SECOND PHASE: When Ulysses reaches station B he fires his *left*
thruster *twice*, causing him to travel in the left direction at a
velocity "p" relative to the stations.
_______________________________________
| | | | |
ship «*--> ship
Station A O--> O--> Station B
---------------------------------------------
----------------------------------------------
(4) When Station A is reached, Ulysses fires his *right* thruster
*once*, thus coming to rest relative to the stations.
_______________________________________
| | | | |
ship *--> ship
Station A O--> O--> Station B
----------------------------------------------
----------------------------------------------
(5) Ulysses' space ship docks on Station A. Experiment is over.
_______________________________________
| | | | |
ship *--> ship
Station A O--> O--> Station B
----------------------------------------------
--------------
Now, let's see how everybody aged:
---
*let "tW1" be the amount of time passed during the first phase,
as observed by the watcher
*let "tH1" be the corresponding time,
as observed on the space station
*let "tU1" be the corresponding time,
as observed on the space ship
---
*let "tW2" be the amount of time passed during the second phase,
as observed by the watcher
*let "tH2" be the corresponding time,
as observed on the space station
*let "tU2" be the corresponding time,
as observed on the space ship
---
Notice that the watcher and Homer will have difficulty starting and
stopping their clocks to time the first phase and second phase because
their observations will be delayed by the speed of light. Let us
assume here that the light of the stations and space ship reaches both
the watcher and Homer *instantaneuosly* so we do not need to worry
about the problem of timing both phases.
Ulysses, on the other hand, happens to be present where the starting
and stopping of the first phase and second phase is, and so, he can
time both phases correctly with ease.
--------------
HOMER:
Homer, who is in the same frame of reference as Station A, was
travelling at a velocity "n" relative to the watcher for the entire
experiment. Thus,
(1) "tH1 = y*tW1"
and
(2) "tH2 = y*tW2"
*where "y" equals "1/(1-(v/c)²)^½"
--------------
ULYSSES:
During the first phase, Ulysses had a velocity of "[n+p]" relative to
the watcher.
*where "[n+p]" equals "(n+p)/(1+(np/c²))"
During the second phase, Ulysses had a velocity of "[n-p]" relative to
the watcher.
*where "[n-p]" equals "(n-p)/(1-(np/c²))"
Thus,
(1) "tU1 = w*tW1".
*where "w" equals "1/(1-([n+p]/c)²)^½"
and
(2) "tU2 = x*tW2".
*where "x" equals "1/(1-([n-p]/c)²)^½"
--------------
WATCHER:
During the first phase, the watcher observed that Ulysses traversed a
distance "s+n*tW1" at a speed "[n+p]" and for a time "tW1". So, with
a bit of algebra, we get:
(1) "tW1 = s/([n+p]-n)".
Then, during the second phase, the watcher observed that Homer
traversed a distance "s+[n-p]*tW2" at a speed "n" and for a time
"tW2". With a bit of algebra, we get:
(2) "tW2 = s/(n-[n-p])"
--------------
We can solve the equations for Homer and the watcher to find the
velocity of the stations. But, in reality, Homer has trouble timing
both phases of the experiment correctly. So, instead, we will solve
the equations for Ullysses and the watcher. With a bit of algebra, we
can find:
"n = c²/p * (sqrt(B)-1)/(sqrt(B)+1)"
where "B" equals "tU1/tU2"
With all the equations for Homer, Ulysses and the watcher, you might
be asking why I described "n" as above. Two simple reasons: First,
with the above equation, we have no need for a watcher. That is, we
can calculate the velocity of the stations relative to the watcher
("n") without having the watcher make any measurements! Thus, we can
calculate the velocity of the stations relative to an absolute frame
of reference without prior knowledge of the absolute frame of
reference! Secondly, the equation relies only on the observations of
Ulysses, which is ideal considering that both the watcher and Homer
have trouble starting and stopping their clocks to time both phases of
the experiment.
--------------
Now, we have only considered the velocity of the stations in the right
direction. When the stations have a velocity in the left direction,
the value of "tW1" switches with "tW2", the value of "tH1" switches
with "tH2" and the value of "tU1" switches with "tU2". Suffice it to
say that the above formula for "n" still works.
Thus, if we were to do the above experiment, the equation for "n"
would only tell us the "speed" of the stations, and that they are
either travelling in the right direction or the left direction at that
speed.
To figure out which way the stations are travelling (left or right),
we need only accelerate both stations in either the left direction or
the right direction and redo the experiment. If the speed has
greatened, then we know for sure that the stations are travelling in
the direction of the acceleration. Else, if the speed is reduced, the
stations must be travelling in the opposite direction.
--------------
Still, the above experiment only figures out the velocity of the
stations relative to the Universe in one-dimension. One would need to
do the experiment in three directions, all perpendicular to each
other, to figure out the *true* velocity (in all directions) of the
stations. (This is assuming that the space near the stations is
Euclidean.)
--------------
Now, let's return to the thought experiment at the beginning of this
section.
The thought experiment yeilded a velocity when no velocity should be
found. This is because of the third equation. The third equation is
wrong because it assumed that the station was at rest, when in fact,
it wasn't. It was travelling at a velocity "v" observed by a watcher.
(We unknowingly assumed that the watcher was at rest with the entire
Universe.)
It is somewhat odd that I used parts of Special Relativity to disprove
other parts of Special Relativity. Thus, Special Relativity has gone
somewhat awry. To amend it, we should observe that the equation for
time dilations works only when we consider the object at rest to be an
object that is at rest with the Universe. Also, when you are "adding
velocities", one of the velocities needs to be relative to the rest
state of the Universe. If you follow those two rules, you will not
make mistakes like the one made in the above thought experiment.
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-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-
by Raheman Velji
December 27, 2004
you can also view this paper (and updated versions) at...
....http://www.angelfire.com/un/rv
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