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Re: The Necessity Alternate Theories

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The ’nym-shifting troll wrote as "Natalina Nazvarova":

> Thomas 'PointedEars' Lahn wrote:
>> Thomas 'PointedEars' Lahn wrote:
>>> […]  (The proper time interval for uniform motion in spacetime is
>>> 
>>>    ∆τ = √((∆t²) – (∆x_i)²∕c²).
>> 
>> Correction:
>> 
>>   ∆τ = √((∆t)² – (∆x_i)²∕c²).
> 
> Why should it be like that?

Because the metric of a space is in general

   ds² = g_µν dx^µ dx^ν

and in a flat (Minkowski) spacetime (3+1-space) the metric tensor can be 
written as (the matrix)

         (c²  0   0   0)
     g = (0  −1   0   0)
         (0   0  −1   0)
         (0   0   0  −1)

where one can define the indexes starting from 0, so that

  dx^0 = dt

from which follows

   ds² =  c² dt² − (dx^i)² =   c² dt² −  (dx^1)² − (dx^2)² − (dx^3)²

                           =   c² dt² − ((dx^1)² + (dx^2)² + (dx^3)²)

   ds                      = √(c² dt² − ((dx^1)² + (dx^2)² + (dx^3)²)))

And

  ∆τ = ∫ dτ = ∫ ds/c
       W
            = ∫ 1∕c ds

            = ∫ 1∕c √(      c² dt² − ((dx^1)²    + (dx^2)² + (dx^3)²)).

            = ∫     √(1∕c² (c² dt² − ((dx^1)²    + (dx^2)² + (dx^3)²))).

            = ∫     √(         dt² − ((dx^1)²∕c² + (dx^2)²∕c²
                                                 + (dx^3)²∕c²)).

along the world line W (“the proper time interval ∆τ between two events on a 
world line W equals the sum of infinitesimal proper time intervals dτ along 
W”).  And if the motion is uniform, then

         ∆τ = ∫  √(dt² − ((dx^1)²∕c² + (dx^2)²∕c² + (dx^3)²∕c²))

            =    √(∆t² − ((∆x_1)²∕c² + (∆x_2)²∕c² + (∆x_3)²∕c²))

            =    √(∆t² − (∆x_i)²∕c²). ∎

See also:

<http://mathworld.wolfram.com/MetricTensor.html>

and

<https://en.wikipedia.org/wiki/Proper_time#In_special_relativity> pp.

> I bet you dont even know what that /i/ stands for.
            ^^^^ YSCIB

In this notation, “i” is the 1-based index of the spatial coordinate.

You lose.

> Real life you MUST always specify that /i/ (plus all the others)

No, real life includes Einstein notation, also known as Einstein summation 
convention, where repeated indexes (here: i) are summed over:

  a_i b^i := ∑ a_i b^i := a₁ b^1 + a₂ b^2 + …
             i

(b^2 here does not mean „b-squared“ – that would be and is above 
b² accordingly –, but the second component of the contravariant tensor b, 
which in this example is a coordinate vector).  And so,

  (∆x_i)² := ∆x_i ∆x_i

          := ∆x₁ ∆x₁ + ∆x₂ ∆x₂ + ∆x₃ ∆x₃

           = (∆x₁)²  + (∆x₂)²  + (∆x₃)²

for 3 spatial dimensions.

Granted, this is not the trivial real *troll* life that you are living, 
where you are blathering about tensors without having the slightest clue 
about them; but this is *sci*.physics.relativity, so you will have to get 
used to it.

[1] cf. <https://youtu.be/h96SW0PfQcg?list=PLQrxduI9Pds1fm91Dmn8x1lo-O_kpZGk8&t=1016>


PointedEars
-- 
Q: Who's on the case when the electricity goes out?  
A: Sherlock Ohms.

(from: WolframAlpha)

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Thread

The Necessity Alternate Theories kefischer <emoneyjoe@iglou.com> - 2015-10-08 13:01 -0400
  Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-08 17:09 +0000
  Re: The Necessity Alternate Theories Odd Bodkin <bodkinodd@gmail.com> - 2015-10-08 13:30 -0500
    Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-08 22:45 +0200
      Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-08 22:58 +0200
        Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-08 21:56 +0000
          Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-09 06:39 +0200
            Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-09 10:59 +0000
              Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-09 20:29 +0200
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-09 18:44 +0000
                Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-10 09:40 +0200
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-10 12:08 +0000
                Re: The Necessity Alternate Theories paparios <paparios@gmail.com> - 2015-10-10 05:25 -0700
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-10 12:37 +0000
                Re: The Necessity Alternate Theories Maciej Woźniak <mlwozniak@wp.pl> - 2015-10-10 14:54 +0200
                Re: The Necessity Alternate Theories paparios <paparios@gmail.com> - 2015-10-10 06:02 -0700
                Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-10 17:49 +0200
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-10 16:08 +0000
                Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-10 18:13 +0200
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-10 16:16 +0000
                Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-10 19:08 +0200
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-10 18:02 +0000
                Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-10 20:17 +0200
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-10 18:37 +0000
                Re: The Necessity Alternate Theories Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-10-10 20:54 +0200
                Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-10 19:00 +0000
                Re: The Necessity Alternate Theories benj <nobody@gmail.com> - 2015-10-10 16:40 -0400
                Re: The Necessity Alternate Theories underante <underante@yahoo.com> - 2015-10-09 15:31 -0700
  Re: The Necessity Alternate Theories kefischer <emoneyjoe@iglou.com> - 2015-10-09 02:42 -0400
    Re: The Necessity Alternate Theories Natalina Nazvarova <natana42@hotmail.com> - 2015-10-09 11:44 +0000
      Re: The Necessity Alternate Theories kefischer <emoneyjoe@iglou.com> - 2015-10-09 08:00 -0400

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