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Re: Inescapable (symmetric) twins paradox

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  Re: Inescapable (symmetric) twins paradox Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-08-09 22:24 +0200
    Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-09 17:59 -0700
      Re: Inescapable (symmetric) twins paradox Felipe Delgado <fd@spreadspectrum.org> - 2015-08-10 18:24 +0000
        Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-10 11:33 -0700
          Re: Inescapable (symmetric) twins paradox Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-08-19 22:16 +0200
            Re: Inescapable (symmetric) twins paradox Tom Roberts <tjroberts137@sbcglobal.net> - 2015-08-19 18:43 -0500
              Re: Inescapable (symmetric) twins paradox Odd Bodkin <bodkinodd@gmail.com> - 2015-08-24 09:52 -0500
              Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-08-25 00:01 -0700
                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-25 00:05 -0700
                  Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-08-25 00:29 -0700
                    Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-25 11:16 -0700
                      Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-08-25 22:36 -0700
                        Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-25 23:06 -0700
                          Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-08-26 00:19 -0700
                            Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-26 10:54 -0700
                              Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2015-08-26 20:32 +0200
                                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-26 15:58 -0700
                Re: Inescapable (symmetric) twins paradox Tom Roberts <tjroberts137@sbcglobal.net> - 2015-08-27 11:44 -0500
                  Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-08-27 15:24 -0700
                    Re: Inescapable (symmetric) twins paradox Tom Roberts <tjroberts137@sbcglobal.net> - 2015-08-27 22:15 -0500
                      Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-08-27 23:49 -0700
                        Re: Inescapable (symmetric) twins paradox Odd Bodkin <bodkinodd@gmail.com> - 2015-08-28 07:19 -0500
                        Re: Inescapable (symmetric) twins paradox Tom Roberts <tjroberts137@sbcglobal.net> - 2015-08-28 09:55 -0500
                          Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-08-28 16:52 -0700
                            Re: Inescapable (symmetric) twins paradox kefischer <emoneyjoe@iglou.com> - 2015-08-28 20:49 -0400
                            Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-28 21:45 -0700
                      Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-09-07 11:13 -0700
                        Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-07 12:21 -0700
                          Re: Inescapable (symmetric) twins paradox "Lleyton H. Bellucci" <lleyb@stratospheree.org> - 2015-09-07 19:36 +0000
                          Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-09-07 12:38 -0700
                            Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-07 15:49 -0700
                              Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-09-07 20:45 -0700
                                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 02:15 -0700
                                  Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-09-08 10:10 -0700
                                    Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 11:35 -0700
                                      Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-09-08 12:46 -0700
                                        Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 13:15 -0700
                                          Re: Inescapable (symmetric) twins paradox Maciej Woźniak <mlwozniak@wp.pl> - 2015-09-08 22:47 +0200
                                            Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 15:09 -0700
                                            Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 15:11 -0700
                              Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-09-08 10:17 -0700
                                Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 11:25 -0700
                                  Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-09-08 13:18 -0700
                                    Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 13:30 -0700
                                      Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-09-08 15:34 -0700
                                        Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-09-08 18:22 -0700
                                          Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-09-09 10:25 -0700
                                            Re: Inescapable (symmetric) twins paradox paparios <paparios@gmail.com> - 2015-09-09 10:54 -0700
                                              Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-09-09 12:40 -0700
            Re: Inescapable (symmetric) twins paradox JanPB <filmart@gmail.com> - 2015-08-20 22:24 -0700
              Re: Inescapable (symmetric) twins paradox kefischer <emoneyjoe@iglou.com> - 2015-08-21 02:44 -0400
              Re: Inescapable (symmetric) twins paradox Koobee Wublee <koobee.wublee@gmail.com> - 2015-08-20 23:58 -0700
              Re: Inescapable (symmetric) twins paradox alsor@interia.pl - 2015-08-21 08:14 -0700
      Re: Inescapable (symmetric) twins paradox Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-08-19 22:10 +0200
    Re: Inescapable (symmetric) twins paradox Felipe Delgado <fd@spreadspectrum.org> - 2015-08-11 18:19 +0000

Page 3 of 3 — ← Prev page 1 2 [3]

#363478

W dniu wtorek, 8 września 2015 00:49:44 UTC+2 użytkownik JanPB napisał:
> On Monday, September 7, 2015 at 12:38:47 PM UTC-7, al...@interia.pl wrote:
> > W dniu poniedziałek, 7 września 2015 21:21:07 UTC+2 użytkownik JanPB napisał:
> > > On Monday, September 7, 2015 at 11:13:25 AM UTC-7, al...@interia.pl wrote:
> > > > W dniu piątek, 28 sierpnia 2015 05:15:53 UTC+2 użytkownik tjrob137 napisał:
> > > > > On 8/27/15 8/27/15   5:24 PM, alsor@interia.pl wrote:
> > > > > > You are a Christoffel symbol itself, idiot...
> > > > > 
> > > > > You merely display yourself to be the idiot. It is QUITE CLEAR that you do not 
> > > > > know what a Christoffel symbol actually is. Your bluster and insults cannot hide 
> > > > > that fact.
> > > > > 
> > > > > 
> > > > > > BTW. What is the Christoffel symbol?
> > > > > > You don't know that... I bet 1000000$ !
> > > > > 
> > > > > You lose!
> > > > > 
> > > > > There are two types of Christoffel symbols, known as the first kind and the 
> > > > > second kind. They apply to the geometry of a manifold in specified coordinates: 
> > > > > the second kind is just the affine connection compatible with the metric, while 
> > > > > the first kind is just the second kind with an index lowered using the metric 
> > > > > components (this makes some of its symmetry properties more obvious, and is also 
> > > > > the one Christoffel found first).
> > > > > 
> > > > > It's remarkable how idiotic you behave -- the answer to your question is 
> > > > > available easily on the internet. And your silly bet merely displays how 
> > > > > out-of-touch you are -- I answered without referencing the internet or any book, 
> > > > > as this is part of my general knowledge about physics and math.
> > > > > 
> > > > > 	Send your $1,000,000 to the American Cancer Society
> > > > > 	https://donate.cancer.org/index
> > > > > 
> > > > > 
> > > > > Tom Roberts
> > > > 
> > > > That is wrong answer!
> > > 
> > > The question was: are Christoffel symbols tensors? The answer is no.
> > > 
> > > So it's the correct answer.
> > > 
> > > --
> > > Jan
> > 
> > Baby: Daddy, why the Sun is so bright?
> > Daddy: what you are talking about, babe?! the apples are not bright, at all!
> 
> So are you saying Christoffel symbols are tensors? I can prove they are not,
> easily.
> 
> --
> Jan

Symbols? HAHA!

Your Christoffel simbols are just final results,
ie. sloutions of... something.

These formulas are used by the stupid amateurs,
which never learn the classical geometry,
therefore they can't compute itself...
even the most primitive formulas are an enigma for them -
just like the magicial keys, casts for mistics or other dogmatic imbeciles.

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#363483

On Tuesday, September 8, 2015 at 10:17:55 AM UTC-7, al...@interia.pl wrote:
> W dniu wtorek, 8 września 2015 00:49:44 UTC+2 użytkownik JanPB napisał:
> > On Monday, September 7, 2015 at 12:38:47 PM UTC-7, al...@interia.pl wrote:
> > > W dniu poniedziałek, 7 września 2015 21:21:07 UTC+2 użytkownik JanPB napisał:
> > > > On Monday, September 7, 2015 at 11:13:25 AM UTC-7, al...@interia.pl wrote:
> > > > > W dniu piątek, 28 sierpnia 2015 05:15:53 UTC+2 użytkownik tjrob137 napisał:
> > > > > > On 8/27/15 8/27/15   5:24 PM, alsor@interia.pl wrote:
> > > > > > > You are a Christoffel symbol itself, idiot...
> > > > > > 
> > > > > > You merely display yourself to be the idiot. It is QUITE CLEAR that you do not 
> > > > > > know what a Christoffel symbol actually is. Your bluster and insults cannot hide 
> > > > > > that fact.
> > > > > > 
> > > > > > 
> > > > > > > BTW. What is the Christoffel symbol?
> > > > > > > You don't know that... I bet 1000000$ !
> > > > > > 
> > > > > > You lose!
> > > > > > 
> > > > > > There are two types of Christoffel symbols, known as the first kind and the 
> > > > > > second kind. They apply to the geometry of a manifold in specified coordinates: 
> > > > > > the second kind is just the affine connection compatible with the metric, while 
> > > > > > the first kind is just the second kind with an index lowered using the metric 
> > > > > > components (this makes some of its symmetry properties more obvious, and is also 
> > > > > > the one Christoffel found first).
> > > > > > 
> > > > > > It's remarkable how idiotic you behave -- the answer to your question is 
> > > > > > available easily on the internet. And your silly bet merely displays how 
> > > > > > out-of-touch you are -- I answered without referencing the internet or any book, 
> > > > > > as this is part of my general knowledge about physics and math.
> > > > > > 
> > > > > > 	Send your $1,000,000 to the American Cancer Society
> > > > > > 	https://donate.cancer.org/index
> > > > > > 
> > > > > > 
> > > > > > Tom Roberts
> > > > > 
> > > > > That is wrong answer!
> > > > 
> > > > The question was: are Christoffel symbols tensors? The answer is no.
> > > > 
> > > > So it's the correct answer.
> > > > 
> > > > --
> > > > Jan
> > > 
> > > Baby: Daddy, why the Sun is so bright?
> > > Daddy: what you are talking about, babe?! the apples are not bright, at all!
> > 
> > So are you saying Christoffel symbols are tensors? I can prove they are not,
> > easily.
> > 
> > --
> > Jan
> 
> Symbols? HAHA!
> 
> Your Christoffel simbols are just final results,
> ie. sloutions of... something.

This is not under discussion. Reread the thread. The question was whether 
Christoffel symbols were tensors, not how they were originally derived.

> These formulas are used by the stupid amateurs,
> which never learn the classical geometry,

It's simply the transformation formula for them, therefore it's used
by everyone who happens to need it.

> therefore they can't compute itself...

There is nothing to "compute" here. The discussion was whether the
transformation rule for those symbols was tensorial or not.

> even the most primitive formulas are an enigma for them -
> just like the magicial keys, casts for mistics or other dogmatic imbeciles.

As usual, lots of empty posturing, no content. If you are saying that
Christoffel symbols are tensors, then I've just provided a disproof.

--
Jan

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#363505

W dniu wtorek, 8 września 2015 20:25:53 UTC+2 użytkownik JanPB napisał:

> This is not under discussion. Reread the thread. The question was whether 
> Christoffel symbols were tensors, not how they were originally derived.

Are you sure... it isn't an pussensor? :)

You just utilise some primitive things, like the so-called black box,
ie. totally unconsciously... just like the prehistoric humans used a fire...
spontaneously only, because only when they found it... usually after a fortunate hit of a thunderbolt...

Do you know what is a gibbon with a computer?
It's a relativistic physicists... personally,
but hysically it is a fantastic quantum phenomen - faster than time. :))))
 
> As usual, lots of empty posturing, no content. If you are saying that
> Christoffel symbols are tensors, then I've just provided a disproof.

Prepared, calculated earlier formulas.
(studenci nazywają to zwkle: wzorami... formułkami, do zaklepania dla głąbów).

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#363508

On Tuesday, September 8, 2015 at 1:18:51 PM UTC-7, al...@interia.pl wrote:
> W dniu wtorek, 8 września 2015 20:25:53 UTC+2 użytkownik JanPB napisał:
> 
> > This is not under discussion. Reread the thread. The question was whether 
> > Christoffel symbols were tensors, not how they were originally derived.
> 
> Are you sure... it isn't an pussensor? :)
> 
> You just utilise some primitive things, like the so-called black box,
> ie. totally unconsciously... just like the prehistoric humans used a fire...
> spontaneously only, because only when they found it... usually after a fortunate hit of a thunderbolt...
> Do you know what is a gibbon with a computer?
> It's a relativistic physicists... personally,
> but hysically it is a fantastic quantum phenomen - faster than time. :))))

Are you on drugs or something?

> > As usual, lots of empty posturing, no content. If you are saying that
> > Christoffel symbols are tensors, then I've just provided a disproof.
> 
> Prepared, calculated earlier formulas.
> (studenci nazywają to zwkle: wzorami... formułkami, do zaklepania dla głąbów).

I'm sorry to disappoint you but the entire edifice of mathematics happens
to consist of formulas.

--
Jan

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#363522

W dniu wtorek, 8 września 2015 22:30:46 UTC+2 użytkownik JanPB napisał:

> I'm sorry to disappoint you but the entire edifice of mathematics happens
> to consist of formulas.

Yes.
The key is just to recognise the sense of these formulas...
not only remembering their names - symbols.

You simply never know what are these... Kartoffellos dupperelless. :)

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#363526

On Tuesday, September 8, 2015 at 3:34:54 PM UTC-7, al...@interia.pl wrote:
> W dniu wtorek, 8 września 2015 22:30:46 UTC+2 użytkownik JanPB napisał:
> 
> > I'm sorry to disappoint you but the entire edifice of mathematics happens
> > to consist of formulas.
> 
> Yes.
> The key is just to recognise the sense of these formulas...
> not only remembering their names - symbols.

Who told you I only "remembered their names"? Have we met?

> You simply never know what are these... Kartoffellos dupperelless. :)

No, I am the Queen of England, you keep forgetting. In other words: stupid word salad
on your part, sorry.

--
Jan

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#363571

W dniu środa, 9 września 2015 03:22:47 UTC+2 użytkownik JanPB napisał:
> On Tuesday, September 8, 2015 at 3:34:54 PM UTC-7, al...@interia.pl wrote:
> > W dniu wtorek, 8 września 2015 22:30:46 UTC+2 użytkownik JanPB napisał:
> > 
> > > I'm sorry to disappoint you but the entire edifice of mathematics happens
> > > to consist of formulas.
> > 
> > Yes.
> > The key is just to recognise the sense of these formulas...
> > not only remembering their names - symbols.
> 
> Who told you I only "remembered their names"? Have we met?
> 
> > You simply never know what are these... Kartoffellos dupperelless. :)
> 
> No, I am the Queen of England, you keep forgetting. In other words: stupid word salad on your part, sorry.
 
Indeed, You can to be the queen...
after all she, without any doubt,
has no clue how to calculate the geodesic lines.

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#363581

On Wednesday, September 9, 2015 at 2:25:14 PM UTC-3, al...@interia.pl wrote:
> W dniu środa, 9 września 2015 03:22:47 UTC+2 użytkownik JanPB napisał:
 
> > No, I am the Queen of England, you keep forgetting. In other words: stupid word salad on your part, sorry.
>  
> Indeed, You can to be the queen...
> after all she, without any doubt,
> has no clue how to calculate the geodesic lines.

It is obvious that neither the nym-shifter troll nor the two angry polish lovers are here to learn (or teach) anything. They just love to troll writing nonsense which they do know is total nonsense. The three of them appear to be some sort of lousy low level programmers with a huge ego. Best thing to do is just to ignore them, as they are not even good for a laugh. Amazing there are so many people willing to waste other's time, instead of studying.

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#363593

W dniu środa, 9 września 2015 19:54:03 UTC+2 użytkownik paparios napisał:
> On Wednesday, September 9, 2015 at 2:25:14 PM UTC-3, al...@interia.pl wrote:
> > W dniu środa, 9 września 2015 03:22:47 UTC+2 użytkownik JanPB napisał:
>  
> > > No, I am the Queen of England, you keep forgetting. In other words: stupid word salad on your part, sorry.
> >  
> > Indeed, You can to be the queen...
> > after all she, without any doubt,
> > has no clue how to calculate the geodesic lines.
> 
> It is obvious that neither the nym-shifter troll nor the two angry polish lovers are here to learn (or teach) anything. They just love to troll writing nonsense which they do know is total nonsense. The three of them appear to be some sort of lousy low level programmers with a huge ego. Best thing to do is just to ignore them, as they are not even good for a laugh. Amazing there are so many people willing to waste other's time, instead of studying.

You should try to learn to walk earlier,
as any babe do, instead try to fly directly. :)

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#361421

On Wednesday, August 19, 2015 at 1:20:12 PM UTC-7, Thomas 'PointedEars' Lahn wrote:
> JanPB wrote on 10.08.2015 20:33:
> > On Monday, August 10, 2015 at 11:24:13 AM UTC-7, Felipe Delgado wrote:
> >> JanPB wrote:
> >>> It's not only that, he writes a 4-tuple of symmetric tensor squares of
> >>> covectors and equates it to g. This simply has no meaning unless he
> >>> defines what this notation is supposed to mean exactly.
> >>
> >> Lol, good you say it. People might wrongfully assert you were a 
> >> mathematician. 
> >>
> >> Come on, you are supposed to be at least a programmer, familiar with 
> >> stuff, recipes and algorithms. No wonder Koobee Woblee bets you in math 
> >> anywhere any time. You can't even read tensors equations, why am I talking 
> >> to you.
> > 
> > Talk substance. Leave the poetry out. The point is that the notation:
> > 
> >     g = [-(cdt)², (dx)², (dy)², (dz)²] 
> > 
> > ...is nonsense. A metric is not a _quadruple_ of symmetric tensor powers,
> > it's a _linear combination_ of them.
> 
> I am pretty sure that "dx", "dy" and "dz" are _not_ tensors.  So how can
> "(dx)²", "(dy)²", and "(dz)²" be "tensor powers", let alone "symmetric"
> ones?

Good grief, this is standard stuff (meaning, basic differential geometry), see e.g.:

Wald - "General Relativity", pp. 18-23
Abraham, Marsden, Ratiu - "Manifolds, Tensor Analysis, and Applications"
Spivak - "Comprehensive Introduction to Differential Geometry" vol. I
...and tons of others.

A Reader's Digest version:

1. symbols like "dx", "dq^i", etc. are coordinate covector fields dual to the coordinate
vector fields denoted by symbols "d/dx", "d/dq^i", etc. (dual in the standard linear
algebra sense),

2. it can also be shown that these symbols are exterior derivatives of the corresponding
coordinate functions, e.g. "dx" is the exterior derivative of the x-coordinate function, etc.,

3. metrics are symmetric covariant 2-tensors(*) of rank 2, hence they can be expanded
in the usual linear basis of the relevant tensor space as sums of the form:

    A_ij dx^i dx^j

...where it is _traditional_ NOT to write the symmetric tensor product sign explicitly.
It would normally sit between the covectors, like so:

    A_ij dx^i (x) dx^j

It stands for the symmetric tensor product, like so:

    dx^i dx^j =[by definition]= 1/2*(dx^i X dx^j + dx^j X dx^i)

...where by "X" I denoted the standard tensor product (running out of ASCII here).

(*)meaning tensor _fields_ - the word "field" is frequently omitted, it's another
example of slight terminological abuse to prevent too much pedantry.

> And there is no interpretation in which "-(cdt)²" could be a
> "(symmetric) tensor power" either, as the exponent is one of the expression
> "cdt" only.

Yeah yeah, sure.

--
Jan

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#361423

On Thu, 20 Aug 2015 22:24:37 -0700 (PDT), JanPB <filmart@gmail.com>
wrote:

>On Wednesday, August 19, 2015 at 1:20:12 PM UTC-7, Thomas 'PointedEars' Lahn wrote:
>> JanPB wrote on 10.08.2015 20:33:
>> > On Monday, August 10, 2015 at 11:24:13 AM UTC-7, Felipe Delgado wrote:
>> >> JanPB wrote:
>> >>> It's not only that, he writes a 4-tuple of symmetric tensor squares of
>> >>> covectors and equates it to g. This simply has no meaning unless he
>> >>> defines what this notation is supposed to mean exactly.
>> >>
>> >> Lol, good you say it. People might wrongfully assert you were a 
>> >> mathematician. 
>> >>
>> >> Come on, you are supposed to be at least a programmer, familiar with 
>> >> stuff, recipes and algorithms. No wonder Koobee Woblee bets you in math 
>> >> anywhere any time. You can't even read tensors equations, why am I talking 
>> >> to you.
>> > 
>> > Talk substance. Leave the poetry out. The point is that the notation:
>> > 
>> >     g = [-(cdt)², (dx)², (dy)², (dz)²] 
>> > 
>> > ...is nonsense. A metric is not a _quadruple_ of symmetric tensor powers,
>> > it's a _linear combination_ of them.
>> 
>> I am pretty sure that "dx", "dy" and "dz" are _not_ tensors.  So how can
>> "(dx)²", "(dy)²", and "(dz)²" be "tensor powers", let alone "symmetric"
>> ones?
>
>Good grief, this is standard stuff (meaning, basic differential geometry), see e.g.:
>
>Wald - "General Relativity", pp. 18-23
>Abraham, Marsden, Ratiu - "Manifolds, Tensor Analysis, and Applications"
>Spivak - "Comprehensive Introduction to Differential Geometry" vol. I
>...and tons of others.
>
>A Reader's Digest version:
>
>1. symbols like "dx", "dq^i", etc. are coordinate covector fields dual to the coordinate
>vector fields denoted by symbols "d/dx", "d/dq^i", etc. (dual in the standard linear
>algebra sense),
>
>2. it can also be shown that these symbols are exterior derivatives of the corresponding
>coordinate functions, e.g. "dx" is the exterior derivative of the x-coordinate function, etc.,
>
>3. metrics are symmetric covariant 2-tensors(*) of rank 2, hence they can be expanded
>in the usual linear basis of the relevant tensor space as sums of the form:
>
>    A_ij dx^i dx^j
>
>...where it is _traditional_ NOT to write the symmetric tensor product sign explicitly.
>It would normally sit between the covectors, like so:
>
>    A_ij dx^i (x) dx^j
>
>It stands for the symmetric tensor product, like so:
>
>    dx^i dx^j =[by definition]= 1/2*(dx^i X dx^j + dx^j X dx^i)
>
>...where by "X" I denoted the standard tensor product (running out of ASCII here).

 
        Here is a few more for ya;



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>
>(*)meaning tensor _fields_ - the word "field" is frequently omitted, it's another
>example of slight terminological abuse to prevent too much pedantry.
>
>> And there is no interpretation in which "-(cdt)²" could be a
>> "(symmetric) tensor power" either, as the exponent is one of the expression
>> "cdt" only.
>
>Yeah yeah, sure.

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#361425

On Thursday, August 20, 2015 at 10:24:39 PM UTC-7, JanPB wrote:

> [GR Bible passages snipped]

Not important.  <shrug>

> 1. symbols like "dx", "dq^i", etc. are coordinate covector fields dual
> to the coordinate vector fields denoted by symbols "d/dx", "d/dq^i",
> etc. (dual in the standard linear algebra sense),

They are simply elements to the observation (coordinate system) represented by a matrix.  So, it all depends on if you want to use a column or a row matrix to describe your observation.  <shrug>

> 2. it can also be shown that these symbols are exterior derivatives of
> the corresponding coordinate functions, e.g. "dx" is the exterior
> derivative of the x-coordinate function, etc.,

This should be partial derivatives.  This stuff should have been well understand up to Maxwell's time.  However, it was very nice that Gibbs was able to define a few mathematical operators describing just that.  <shrug>

https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs

> 3. metrics are symmetric covariant 2-tensors(*) of rank 2, hence they
> can be expanded in the usual linear basis of the relevant tensor space
> as sums of the form:

Since the observation (coordinate system) can be represented by a one-dimensional matrix, the metric can also be represented by a matrix --- a square matrix --- in this case.  <shrug>

>     A_ij dx^i dx^j

To be more precise,

**  ds^2 = [g]_ij d[q]^i d[q]^j

Where

**  ds^2 = Invariant geometry regardless whichever [q]
**  [g]_ij = Elements of the matrix that represents the metric
**  [q]^i, [q]^j = Elements of the matrix that presents the coordinate system

Following the footsteps of Gibbs, Koobee Wublee sees the above equation can be written into the following by introducing another math operator.  <shrug>

**  ds^2 = [g] * [dq^2] = [g]_ij d[q]^i d[q]^j

Where

**  [dq^2] = Matrix with elements (d[q]^i d[q]^j)
**  [] * [] = Dot product of 2 matrices

Thus, we have:

**  GEOMETRY = METRIC * COORDINATE SYSTEM

Where

**  GEOMETRY = ds^2
**  METRIC = [g]
**  COORDINATE SYSTEM = [dq^2]

There is no fvcking way that the metric can be invariant since the geometry must be invariant.  <shrug>

> ...where it is _traditional_ NOT to write the symmetric tensor product sign
> explicitly.

Bullshit!  <shrug>

> [rest of nonsense snipped]

This is all Newtonian stuff.  <shrug>

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#361452

W dniu piątek, 21 sierpnia 2015 07:24:39 UTC+2 użytkownik JanPB napisał:
> On Wednesday, August 19, 2015 at 1:20:12 PM UTC-7, Thomas 'PointedEars' Lahn wrote:
> > JanPB wrote on 10.08.2015 20:33:
> > > On Monday, August 10, 2015 at 11:24:13 AM UTC-7, Felipe Delgado wrote:
> > >> JanPB wrote:
> > >>> It's not only that, he writes a 4-tuple of symmetric tensor squares of
> > >>> covectors and equates it to g. This simply has no meaning unless he
> > >>> defines what this notation is supposed to mean exactly.
> > >>
> > >> Lol, good you say it. People might wrongfully assert you were a 
> > >> mathematician. 
> > >>
> > >> Come on, you are supposed to be at least a programmer, familiar with 
> > >> stuff, recipes and algorithms. No wonder Koobee Woblee bets you in math 
> > >> anywhere any time. You can't even read tensors equations, why am I talking 
> > >> to you.
> > > 
> > > Talk substance. Leave the poetry out. The point is that the notation:
> > > 
> > >     g = [-(cdt)², (dx)², (dy)², (dz)²] 
> > > 
> > > ...is nonsense. A metric is not a _quadruple_ of symmetric tensor powers,
> > > it's a _linear combination_ of them.
> > 
> > I am pretty sure that "dx", "dy" and "dz" are _not_ tensors.  So how can
> > "(dx)²", "(dy)²", and "(dz)²" be "tensor powers", let alone "symmetric"
> > ones?
> 
> Good grief, this is standard stuff (meaning, basic differential geometry), see e.g.:
> 
> Wald - "General Relativity", pp. 18-23
> Abraham, Marsden, Ratiu - "Manifolds, Tensor Analysis, and Applications"
> Spivak - "Comprehensive Introduction to Differential Geometry" vol. I
> ...and tons of others.
> 
> A Reader's Digest version:
> 
> 1. symbols like "dx", "dq^i", etc. are coordinate covector fields dual to the coordinate
> vector fields denoted by symbols "d/dx", "d/dq^i", etc. (dual in the standard linear
> algebra sense),
> 
> 2. it can also be shown that these symbols are exterior derivatives of the corresponding
> coordinate functions, e.g. "dx" is the exterior derivative of the x-coordinate function, etc.,
> 
> 3. metrics are symmetric covariant 2-tensors(*) of rank 2, hence they can be expanded
> in the usual linear basis of the relevant tensor space as sums of the form:
> 
>     A_ij dx^i dx^j
> 
> ...where it is _traditional_ NOT to write the symmetric tensor product sign explicitly.
> It would normally sit between the covectors, like so:
> 
>     A_ij dx^i (x) dx^j
> 
> It stands for the symmetric tensor product, like so:
> 
>     dx^i dx^j =[by definition]= 1/2*(dx^i X dx^j + dx^j X dx^i)

Indeed... very impressive. :)
But why do you use still the primitive jargon,
to communicate something, I presuppose in a fly... but what in fact?

Just nothing... and that's the your private big problem. :)

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#361319

JanPB wrote on 10.08.2015 02:59:

> On Sunday, August 9, 2015 at 1:28:36 PM UTC-7, Thomas 'PointedEars' Lahn wrote:
>> wobbly wrote:
>>> JanPB wrote:
>>>> On Saturday, August 1, 2015 at 11:48:00 AM UTC-7, wobbly wrote:
>>>>> JanPB wrote:
>>>>>> This is very basic, it simply follows from the general formula for
>>>>>> arc length which is sqrt(g_ij qdot^i qdot^j) dt when the coordinates
>>>>>> ARE standard inertial, i.e. g = -c^2 dt^2 + dx^2 + dy^2 + dz^2
>>>>>
>>>>> Is this the same as
>>>>>
>>>>>     g = [-(cdt)², (dx)², (dy)², (dz)²]
>>>>
>>>> I don't know what you wrote on the right-hand side. It's not correct
>>>> mathematics, so if it's some private notation that you use, explain it
>>>> first.
>>>
>>> Are you serious, not familiar with vector notation of the same?
>>> https://en.wikipedia.org/wiki/Vector_notation
>>
>> In your own reference, the notation that uses commas (ordered set notation) 
>> uses (round) parentheses or *angular* brackets (like the bra-ket notation in 
>> QM), and the notation that does not (matrix notation) uses rectangular or 
>> round brackets.
>>
>> Your notation uses rectangular brackets *and* commas.
> 
> It's not only that, he writes a 4-tuple of symmetric tensor squares of covectors
> and equates it to g. This simply has no meaning unless he defines what this
> notation is supposed to mean exactly.

I do not think that

  [-(cdt)², (dx)², (dy)², (dz)²]

is “a 4-tuple of symmetric tensor squares of covectors”.  I do not think 
your composition of mathematical terms even has a sensible mathematical 
meaning.

See also:

<http://mathworld.wolfram.com/Tensor.html>
<http://mathworld.wolfram.com/SymmetricTensor.html>
<https://en.wikipedia.org/wiki/Tensor_product#Tensor_powers_and_braiding>
<http://www.wolframalpha.com/input/?i=covector>
<http://mathworld.wolfram.com/n-Tuple.html>


PointedEars
-- 
“Science is empirical: knowing the answer means nothing;
 testing your knowledge means everything.”
   —Dr. Lawrence M. Krauss, theoretical physicist,
    in “A Universe from Nothing” (2009)

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#360362

Thomas 'PointedEars' Lahn wrote:

>>>>     g = [-(cdt)², (dx)², (dy)², (dz)²]
>>> 
>>> I don't know what you wrote on the right-hand side. It's not correct
>>> mathematics, so if it's some private notation that you use, explain it
>>> first.
>> 
>> Are you serious, not familiar with vector notation of the same?
>> https://en.wikipedia.org/wiki/Vector_notation
> 
> In your own reference, the notation that uses commas (ordered set
> notation) uses (round) parentheses or *angular* brackets (like the
> bra-ket notation in QM), and the notation that does not (matrix
> notation) uses rectangular or round brackets.
> 
> Your notation uses rectangular brackets *and* commas.

Cretin! Vastly Multilateral and Ample.

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