Re: Special Relativity is Dead! (second proof)


<jan.verheul@xxxxxxxxxx> wrote in message news:1163774298.220827.8260@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
There IS NO theory in which the universe appears the same, regardless
of inertial frame (considering the properties of light). S.R. is a good
try, but it won't work because you will get into trouble as soon as you
start accelerating.

There is a twin-experiment possible with only one acceleration, namely
that of the traveling clock at the far destination, to reverse speed
for the back journey. As soon as the traveling clock arrives back at
the stationary clock, it is discovered that the stationary clock is
time T ahead instead of time T behind, contrary to what was to be
expected, based upon the principles of S.R.

Ha, "contrary to what was to be expected, based upon the principles
of S.R.", you add.

Let's see how SR describes this experiment in practice.

We have a stationary clock K.
We have this clock K' that whizzes by at some speed v when
both clocks are set to show 0.

Phase 1:
Clock K' keeps coasting at this speed during some
time T according to clock K.
(sorry for hijacking your variable T)
Phase 2:
Clock K' fires its engines in the opposite direction of
K, during a time A according to clock K until it is at
rest w.r.t. clock K.
Phase 3:
Clock K' keeps firing its engines in the opposite direction
of K, during the same time A according to clock K, so
K' regains the same speed v w.r.t. K, but now towards K
Phase 4:
Clock K' keeps coasting at speed v during the same
time T according to clock K until it passes by K so they
can compare times.

Now, according to SR, knowing that the clock K remains
inertial (stationary), the total accumulated proper time
of clock K' will be given by the integral
int{ sqrt( 1 - [v(t)/c]^2 ) dt }
where v(t) is the velocity of clock K' at time t according to
clock K.

This integral can easily be calculated for the 4 phases:
Phase 1 ==> T / sqrt( 1 - v^2/c^2 )
Phase 2 ==> c / a argsinh( a A/c )
Phase 3 ==> c / a argsinh( a A/c )
Phase 4 ==> T / sqrt( 1 - v^2/c^2 )
where a is the proper acceleration, felt by clock K' during
the acceleration phase(s), and we have the following relations
between v, a and A:
v = a A / sqrt( 1 + (a A/c)^2 )
and solving for v,
a = v / ( A sqrt( 1 - v^2/c^2 ) )
A = v / ( a sqrt( 1 - v^2/c^2 ) )

So the travelling clock K' will show an elapsed time of
2 T sqrt( 1 - v^2/c^2 ) + 2 c / a argsinh( a A/c )
whereas the stationary clock K shows an elapsed time of
2 T + 2 A
which is, for every possible value of a, A, T and v, ahead
of the reading of clock K'.

Some remarks:
[1] Note that usually we let clock K' start at rest and
accelerate until it reaches its coasting speed, and that
we also let it decelerate to make it finally smoothly
rejoin clock K.
In that case we would have two additional phases and
the travelling clock K' will show an elapsed time of
2 T sqrt( 1 - v^2/c^2 ) + 4 c / a argsinh( a A/c )
whereas the stationary clock K shows an elapsed time of
2 T + 4 A.

[2] If we don't want the coasting phases, we put T = 0 and
the travelling clock K' will show an elapsed time of
4 c / a argsinh( a A/c )
whereas the stationary clock K shows an elapsed time of
4 A.

Note that in the limit for a --> 0 you have
c / a argsinh( a A/c ) --> A
and of course
v = a A / sqrt( 1 + (a A/c)^2 ) --> 0

[3] If you don't want acceleration phases we put A = 0 and
the travelling clock K' will show an elapsed time of
2 T sqrt( 1 - v^2/c^2 )
whereas the stationary clock K shows an elapsed time of
2 T.

As you surely know, this is all basic and elementary SR.

But let's stick to your experiment where the travelling clock
K' will show an elapsed time of
2 T sqrt( 1 - v^2/c^2 ) + 2 c / a argsinh( a A/c )
whereas the stationary clock K shows an elapsed time of
2 T + 2 A
which is, for every possible value of a, A, T and v, ahead
of the reading of clock K'.
You are of course free to imagine values for any of those
variables.

We see that like you say, "it is discovered that the stationary
clock is time D ahead", but can you - in terms of this - carefully
explain in detail what you mean with the part
"instead of time D behind, contrary to what was to be expected,
based upon the principles of S.R."?
(Sorry for having swapped your T for a D here).

Dirk Vdm


.

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