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Re: Inescapable (symmetric) twins paradox

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On Tuesday, March 29, 2016 at 12:50:54 PM UTC-7, Maciej Woźniak wrote:

> > > > ds²=(1+αx'/c²)²dct'²-dx'²-dy²-dz²
> > > > where α is a function of t'.
> >
> > > This is a light sphere equation only
> >
> 
> > No its not crackpot. Its
> > ds²=dct²-dx²-dy²-dz²
> > under the transformation 
> > ct=∫γdct'+γβx' 
> > x=γx'+∫γβdct' 
> > where γ and β are to be expressed as functions of t'. 
> > and 
> > α=γ²(dβ/dt')c 

> As we're talking about mathematics, let's remind that the oldest
> and one of most trustful pieces of it was rejected by Great Guru,
> because he found it's no fun anymore.

I know this is over your crackpot head, but I’ll suppose I’ll spell it out anyway. Lets say Al is an unaccelerated observer in the flat spacetime of special relativity. Coordinates appropriate for an inertial frame observer express the spacetime line element as
ds²=dct²-dx²-dy²-dz²
Al’s inertial frame coordinates will be t,x,y,z and that will be the expression for the line element according to his coordinates. 
Now consider another observer Bert, who may even be accelerated. 
Local to him using locally rectilinear coordinates he will find that the line element reduces to
ds²→ dct’²-dx’²-dy’²-dz’²
He doesn’t move with respect to himself so using this as a description of his own path through spacetime the spatial differentials vanish and the path describes his wristwatch time, proper time.
ds²→ dcτ²
Since ds² is an invariant we see that the path of Bert as described by Al’s inertial frame coordinates is
dct²-dx²-dy²-dz²= dcτ²
Algebra
dct²{1-[(dx/dct)²+(dy/dct)²+(dz/dct)²]}= dcτ²
dt²{1-[(dx/dct)²+(dy/dct)²+(dz/dct)²]}= dτ²
dt²[1-(v²/c²)]= dτ²
dt²= dτ²/[1-(v²/c²)]
dt= dτ/sqrt[1-(v²/c²)]
So this time dilation equation is how the inertial frame observer Al, observes the other observer Bert to be time dilated, no matter whether Bert is accelerated or not. 
Now Julio is making the crackpot assertion that even when accelerated Bert would get the result finding Al time dilated. Relativity says he doesn’t and relativity yields no contradiction, so claiming otherwise is a strawman fallacy. To find how things go according to the accelerated spacetime standards of Bert, you must first assign Bert accelerated coordinates appropriate to his accelerated state. So lets say Bert accelerates along the x-axis. We’ll assign him accelerated coordinate of t’ and x’ according to 
ct=∫γdct'+γβx' 
x=γx'+∫γβdct'
where γ and β are to be expressed as functions of t’. Now he accelerates with arbitrary time dependence but two things can clearly be seen before we even go any further. First of all, these reduce to Lorentz transformation only when he is not accelerating, and only then will he observe Al to be time dilated, however, second of all, since he is always at x’=0, he has to agree with Al that in any round trip when the watches are compared side be side that it will be his own watch that will have recorded less time in accordance with special relativistic time dilation of
Δct=∫γdct'
as that comes directly from the first transformation equation for the coordinates assigned him.
Now under this transformation the line element expressed in accelerated Bert’s coordinates is
ds²=(1+αx'/c²)²dct'²-dx'²-dy²-dz²
α=γ²(dβ/dt')c
From a general relativity perspective, what Bert describes as going on is he describes inertial Al’s motion as a geodesic in what he perceives as a gravitational acceleration field which yields a gravitational red or blue shift or gravitational time dilation effect on clocks far along the x’ direction corresponding to the (1+αx'/c²)² element of the metric. According to the accelerated spacetime standards appropriate for Bert, he finds that when Al is far away, and he accelerates toward Al, Al’s watch runs fast corresponding to a blue shift. In the end both observers agree that after a round trip, Burt aged less and by how much. 
Now Julio just introduces another accelerated frame. It doesn’t matter. You still have to treat that frame as accelerated, the same way we had to treat Bert’s frame as accelerated and assigned him appropriate accelerated coordinates. The kind of acceleration that Bert experiences is absolute. It is not relative and is not treated as relative in relativity. Saying otherwise is a strawman fallacy. You can tell who accelerates and in the end all observers can agree on that. If the spacetime you observe is 
ds²=dct²-dx²-dy²-dz²
then you’re not accelerating.
If the spacetime you observe is something like
ds²=(1+αx'/c²)²dct'²-dx'²-dy²-dz²
then you absolutely ARE accelerating. 
The spacetime length of the acceleration 4-vector on you in that case is nonzero, and that is an invariant.

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Re: Inescapable (symmetric) twins paradox Julio Di Egidio <julio@diegidio.name> - 2016-03-29 04:40 -0700
  Re: Inescapable (symmetric) twins paradox "Paul B. Andersen" <relativity@paulba.no> - 2016-03-29 14:54 +0200
    Re: Inescapable (symmetric) twins paradox Julio Di Egidio <julio@diegidio.name> - 2016-04-03 15:13 -0700
      Re: Inescapable (symmetric) twins paradox moroney@world.std.spaamtrap.com (Michael Moroney) - 2016-04-04 01:03 +0000
      Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-03 20:57 -0700
        Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2016-04-03 22:14 -0700
          Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-04 10:53 -0700
          Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-04 11:24 -0700
            Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2016-04-04 23:44 -0700
              Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-05 02:05 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-05 02:51 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-05 02:58 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-05 04:08 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-05 04:25 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-05 04:41 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-05 05:14 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-05 05:46 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-05 06:46 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-04-05 20:54 +0200
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-05 12:45 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-05 23:17 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-06 06:41 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-06 07:56 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-06 09:00 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-04-06 09:54 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-06 11:51 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-06 11:29 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-06 12:01 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-06 14:33 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-09 10:17 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-09 15:45 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-10 12:08 -0700
        Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-03 23:48 -0700
      Re: Inescapable (symmetric) twins paradox "Paul B. Andersen" <relativity@paulba.no> - 2016-04-04 20:22 +0200
  Re: Inescapable (symmetric) twins paradox "Dono," <sa_ge@comcast.net> - 2016-03-29 06:45 -0700
  Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-29 08:18 -0700
    Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-03-29 11:17 -0700
      Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-29 11:54 -0700
        Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-29 21:50 +0200
          Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-29 22:09 +0200
          Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-29 13:11 -0700
            Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-29 22:43 +0200
              Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-29 14:24 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-29 15:58 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-29 23:15 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 00:07 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 00:51 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 01:45 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 02:06 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 02:35 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 03:00 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 03:11 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 03:21 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 03:43 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 03:53 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 04:34 -0700
                Re: Inescapable (symmetric) twins paradox Julio Di Egidio <julio@diegidio.name> - 2016-03-30 04:43 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 05:31 -0700
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 06:08 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 12:21 -0700
                Re: Inescapable (symmetric) twins paradox Odd Bodkin <bodkinodd@gmail.com> - 2016-03-30 14:48 -0500
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 23:39 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-30 22:34 +0200
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 14:36 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 23:17 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-31 00:04 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-31 00:15 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-31 00:30 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 05:29 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 12:13 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-30 22:30 +0200
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 13:35 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-30 22:53 +0200
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 14:39 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-30 23:23 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-31 00:06 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-31 00:23 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-31 00:42 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-31 00:56 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-31 11:11 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-31 22:10 +0200
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-31 13:50 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-31 23:03 +0200
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-31 15:01 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-31 23:09 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-01 00:11 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-01 00:34 -0700
                Re: Inescapable (symmetric) twins paradox moroney@world.std.spaamtrap.com (Michael Moroney) - 2016-04-01 14:53 +0000
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-04-01 17:55 +0200
                Re: Inescapable (symmetric) twins paradox moroney@world.std.spaamtrap.com (Michael Moroney) - 2016-04-01 18:44 +0000
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-04-01 21:02 +0200
                Re: Inescapable (symmetric) twins paradox moroney@world.std.spaamtrap.com (Michael Moroney) - 2016-04-04 01:10 +0000
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-01 10:30 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-01 10:38 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-01 13:36 -0700
                Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-04-04 11:41 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-04 15:02 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-04 23:01 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-05 00:27 -0700
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-04-05 00:52 -0700
                Re: Inescapable (symmetric) twins paradox Gary Harnagel <hitlong@yahoo.com> - 2016-04-01 15:37 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-04-01 21:10 +0200
                Re: Inescapable (symmetric) twins paradox Bimpy <pimp@pimp.com> - 2016-04-01 19:15 +0000
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-04-01 13:38 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-04-02 09:05 +0200
                Re: Inescapable (symmetric) twins paradox moroney@world.std.spaamtrap.com (Michael Moroney) - 2016-03-31 14:42 +0000
                Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 02:45 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 12:10 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2016-03-30 12:07 -0700
                Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-30 22:26 +0200
                Re: Inescapable (symmetric) twins paradox mlwozniak@wp.pl - 2016-03-29 23:25 -0700
        Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-03-30 10:30 -0700
          Re: Inescapable (symmetric) twins paradox David Waite <waitedavid1618@yahoo.com> - 2016-03-30 10:48 -0700
            Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2016-03-31 07:20 -0700

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