Re: (SR) Lorentz t', x' = Intervals

From: Dirk Van de moortel (dirkvandemoortel_at_ThankS-NO-SperM.hotmail.com)
Date: 03/20/05 

Date: Sun, 20 Mar 2005 11:50:07 GMT

"Mich" <mich@efni.com> wrote in message news:113plr9afrm1t2f@corp.supernews.com...
>
> I just realized that I made an error.
>
> Mich <mich@efni.com> wrote in message
> news:113pjerdapuj90c@corp.supernews.com...
>
> > T'A1B1= 2gL/v2
> > TA1B1 = 2 L/v2
> > XA1B1 = 2 L
> > X'A1B1 = 2gL
> >
> > If this correct, then, I see a problem.Here we have the observer on the
> > ground who sees the ball travel the distance 2L in the duration of 2L/v2.
>
> I just realized that g takes care of the problem.However, if g =
> 1/sqrt(1-v1^2/c^2)
> wouldn't X'A1B1 = 2gL represent a length dilation,since, as v increases, the
> denominator gets closer to 0, increasing X'A1B1 relative to XA1B1?Should it
> not be X'A1B1 = 2L/g instead?
>
> Andre

Be very careful with the real meaning of length contraction
and time dilation. Express them in terms of time differences
and distance differences between precisely defined events...

In the unprimed (ground) frame:
        A1: ( x, t ) = ( -L, 0 )
        B1: ( x, t ) = ( L, 2 L/v2 )
        A2: ( x, t ) = ( 0, L/c )
        B2: ( x, t ) = ( 0, L/c + 2 L/v2 )
The distance between A1 and B1 is
        XA1B1 = 2 L
The time interval between A1 and B1 is
        TA1B1 = 2 L/v2
The distance between A2 and B2 is
        XA2B2 = 0
The time interval between A2 and B2 is
        TA2B2 = 2 L/v2

In the primed frame (train):
        A1: ( x', t' ) = ( -g L, g L v1/c^2 )
        B1: ( x', t' ) = ( g L (1 - 2 v1/v2), g L (2/v2 - v1/c^2) )
        A2: ( x', t' ) = ( - g L v1/c, g L/c )
        B2: ( x', t' ) = ( - g L v1 (1/c + 2/v2) , g L (1/c + 2/v2) )
The distance between A1 and B1 is
        X'A1B1 = 2 g L (v1/v2 - 1)
The time interval between A1 and B1 is
        T'A1B1 = 2 g L (1/v2 - v1/c^2)
The distance between A2 and B2 is
        X'A2B2 = - 2 g L v1/v2
The time interval between A2 and B2 is
        T'A2B2 = 2 g L/v2

There is only one instance of standard time dilation:
        T'A2B2 = g TA2B2 (std time dilation!)
but:
        T'A1B1 # g TA1B1 (no std time dilation)
and there are no instances of length contraction:
        X'A1B1 # 1/g XA1B1 (no std length contraction)
        X'A2B2 # 1/g XA2B2 (no std length contraction)

Next exercise: find an event S1 that is simultaneous
with A1 in the train frame and calculate the distance
between A and B in the train frame.

Dirk Vdm

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