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Re: The Necessity Alternate Theories

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Natalina Nazvarova wrote:

> Thomas 'PointedEars' Lahn wrote:
>> Metric of a manifold:  ds² = g_µν dx^µ dx^ν        (1)
> 
> How come, explain in details, so morons like use can understand.

Follow the references I give (not only in this thread), and STFW; read 
carefully; ask *specific* questions *where they are on-topic*.  I am neither 
able nor willing to give you a course in multi-dimensional differential 
geometry here.

Just a hint: “ds” is the line element of the manifold; it determines how 
distances in the manifold are measured.

>>     g_µν – (components of the) metric tensor
> 
> This must be the tensor, not its components.

No, unfortunately you still do not understand Einstein summation convention.
Actually, “g” is the metric tensor, the “g_µν” are its components, and “µ” 
and “ν” are variables for the indices of its components.

Connecting the dots for you again (now using “_00” instead of Unicode 
subscript “₀₀” etc.), for ν from 0 to 3, and for µ from 0 to 3 (since 
spacetime is a four-dimensional manifold):

  ds² =   g_µν dx^µ dx^ν

      =   g_00 dx^0 dx^0 + g_01 dx^0 dx^1 + g_02 dx^0 dx^2 + g_03 dx^0 dx^3
        + g_10 dx^1 dx^0 + g_11 dx^1 dx^1 + g_12 dx^1 dx^2 + g_13 dx^1 dx^3
        + g_20 dx^2 dx^0 + g_21 dx^2 dx^1 + g_22 dx^2 dx^2 + g_23 dx^2 dx^3
        + g_30 dx^3 dx^0 + g_31 dx^3 dx^1 + g_32 dx^3 dx^2 + g_33 dx^3 dx^3

> I am not the only moron around here.

That might be true, but different than you imply.

>>          (g₀₀   g₀₁   g₀₂  g₀₃)    (c²  0   0   0)
>>   g_µν = (g₁₀  g₁₁  g₁₂  g₁₃) = (0  −1   0   0)             (2)
>>          (g₂₀  g₁₁  g₂₂  g₂₃)   (0   0  −1   0)
                  ^^^
Typo; has to be g₂₁.

>>          (g₃₀  g₃₁  g₃₂  g₃₃)   (0   0   0  −1)
> 
> These must are the components.

On the near right-hand side of the equation you can see the tensor with its 
components and on the far right-hand side their corresponding values.  The 
far left-hand side is therefore maybe better written just “g” here to avoid 
confusion, but you can find both variants in the literature.  (The variable 
indexes of “g” indicate that it is a tensor, and how it transforms; they are 
required for expressing operations with the tensor exactly.)

> But does not imply that so many have to be zeroed.

But it does; that is how you write the metric tensor for Minkowski 
spacetime.  Otherwise the spacetime would not be a Minkowski one, it would 
not be flat (measuring distances would depend on the four-position).

And if you replace the components with their values in the expansion above, 
you can see that only the summands where µ = ν remain as the coefficients of 
those where µ ≠ ν are all 0.  (That is what multiplying with a diagonal 
matrix left-hand side does.)

Leaving only

  ds² =   g_µν dx^µ dx^ν

      =   g_00 dx^0 dx^0
                         + g_11 dx^1 dx^1
                                          + g_22 dx^2 dx^2
                                                           + g_33 dx^3 dx^3

Effectively, *exactly* as I have written in my previous follow-up.

> You could just use a diagonal matrix,

Obviously I already have.  The metric tensor of Minkowski spacetime *is* 
that matrix.

> in stead of the so many insignificant zeroes,

[I learned just now: The plural of the English numeral “zero” is _zeros_.  
“zeroes” is the third person singular of the verb “(to) zero” (to make sth. 
zero) instead:

<http://www.oxforddictionaries.com/definition/english/zero>
<http://www.oxforddictionaries.com/definition/american_english/zero>]

A diagonal matrix *has* zeros except in the main diagonal; therefore, too, 
the zeros are _not_ insignificant.  Which you could have seen if you had 
quoted me properly so that the columns of the matrices were still aligned.

<http://mathworld.wolfram.com/DiagonalMatrix.html>
<https://en.wikipedia.org/wiki/Diagonal_matrix>

> and represent it simpler by a "vector" notation.

Utter nonsense.

First of all, vectors are tensors of rank 1 (one index suffices to address a 
component), and matrices are tensors of rank 2 (two indexes are required for 
addressing a component).

<http://mathworld.wolfram.com/Tensor.html>
<https://en.wikipedia.org/wiki/tensor>

Second, tensor notation and Einstein summation notation are ways – so far 
the best ways, therefore *the* ways in relativity – of *avoiding* the 
cumbersome/tedious/impossible task of having to write vectors and matrices, 
and tensors of higher rank, with their components, either in rows and 
columns, or as sums (see below), explicitly.  As you can see above (count 
the lines or characters if you do not trust your eyes).

<http://mathworld.wolfram.com/EinsteinSummation.html>
<https://en.wikipedia.org/wiki/Einstein_notation>
 
>> Applying (2) in (1):
>> ds² = c² dx^0 dx^0 + (−1) dx^1 dx^1 + (−1) dx^2 dx^2 + (−1) dx^3 dx^3
>> ds² = c² dx^0 dx^0 − dx^1 dx^1 − dx^2 dx^2 − dx^3 dx^3    (3)
> 
> This must give a vector,

How good then that it does.

> not what you have shere.

You can write each vector as the sum of the products of its components and 
the corresponding orthonormal basis vectors of the coordinate system.

Homework assignment: Determine the orthonormal basis vectors of Minkowski 
spacetime for events given by the coordinates (t, x₁, x₂, x₃).

> You have (multiply) two vectors and a Matrice, which you call "metric".

The proper term is _matrix_.  Its plural is “matrices” or “matrixes”.

<http://mathworld.wolfram.com/Matrix.html>
<http://en.wikipedia.org/wiki/Matrix>
<http://en.wiktionary.org/wiki/matrix>
<http://www.oxforddictionaries.com/definition/english/matrix>

BTW, “metric” is not a term that I coined or introduced, and it is sensible 
one: it means “measurement” (of the distance between two points in a space); 
from Ancient Greek μέτρον ‎(métron) “measure”).

<http://mathworld.wolfram.com/Metric.html>
<https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)>

(Where there are differences, consider the first source the correct one, 
respectively.  Wikipedia articles may have been written/edited by interested 
laymen only.)

>> (from: WolframAlpha)
> 
> I see.

No, you don’t.


PointedEars
-- 
Q: What did the nuclear physicist order for lunch?  
A: Fission chips.

(from: WolframAlpha)

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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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