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Schwarzschild's paper and the uniqueness of the solution

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  Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 17:05 -0800
    Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 22:28 -0800
      Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 22:47 -0800
        Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 23:01 -0800
          Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 23:04 -0800
            Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 23:27 -0800
              Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 23:28 -0800
                Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 23:40 -0800
                  Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-24 00:21 -0800
                    Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-02-24 10:39 +0100
                    Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-24 12:59 -0800
                      Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-24 13:12 -0800
                  Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-24 08:46 -0600
                  Re: Schwarzschild's paper and the uniqueness of the solution "Paul B. Andersen" <relativity@paulba.no> - 2016-02-24 19:54 +0100
              Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-24 01:20 -0800
                Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-24 01:39 -0800
                  Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-02-24 11:12 +0100
              Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-24 08:37 -0600
    Re: Schwarzschild's paper and the uniqueness of the solution "Dono," <sa_ge@comcast.net> - 2016-02-24 07:16 -0800
      Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-24 22:31 -0800
        Re: Schwarzschild's paper and the uniqueness of the solution "Dono," <sa_ge@comcast.net> - 2016-02-25 06:30 -0800
        Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 09:11 -0800
          Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 16:13 -0800
        Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:38 -0800
          Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 17:44 +0000
            Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:50 -0800
              Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-25 11:51 -0600
                Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:57 -0800
                  Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-25 12:05 -0600
                    Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 10:19 -0800
                  Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 18:13 +0000
                    Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 10:21 -0800
                Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 18:05 +0000
        Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:48 -0800
        Re: Schwarzschild's paper and the uniqueness of the solution Jackpol11@hotmail.com - 2016-02-25 14:06 -0500
          Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 12:35 -0800
            Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 21:44 +0000
            Re: Schwarzschild's paper and the uniqueness of the solution Jackpol11@hotmail.com - 2016-02-25 17:37 -0500
              Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 15:11 -0800
                Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 12:44 +0000
                  Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-26 06:57 -0800
                    Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 14:59 +0000
                      Re: Schwarzschild's paper and the uniqueness of the solution Gary Harnagel <hitlong@yahoo.com> - 2016-02-26 07:36 -0800
                        Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 18:45 +0000
                      Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-26 07:37 -0800
                        Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 19:36 +0000
                    Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-26 07:50 -0800
                      Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 18:47 +0000
                      Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 11:44 -0800
            Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 15:47 -0800
              Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 16:00 -0800
                Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 16:09 -0800
                  Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-25 18:29 -0600
                    Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 18:43 -0800
                    Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-26 23:04 +0100
                      Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 14:59 -0800
                        Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-26 16:19 -0800
                        Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-27 10:27 +0100
                          Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-27 17:55 -0800
                            Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 12:13 +0100
                              Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 11:37 +0000
                                Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 12:56 +0100
                                  Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 08:03 -0800
                                    Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:13 +0100
                                      Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:26 +0100
                                        Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:36 -0800
                                          Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:50 +0100
                                            Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:54 -0800
                                              Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 20:40 +0100
                                                Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 19:44 +0000
                                                  Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 12:20 -0800
                                                    Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 20:58 +0000
                                                      Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 13:02 -0800
                                                Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 12:31 -0800
                                                Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-29 22:40 -0800
                                            Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 10:54 -0800
                              Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 07:30 -0800
                                Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 17:46 +0100
                                  Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 08:51 -0800
                                  Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 17:01 +0000
                              Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 07:34 -0800
                                Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 15:15 -0600
                                  Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-03-03 19:54 -0800
                              Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 08:48 -0800
                                Re: Schwarzschild's paper and the uniqueness of the solution David Fuller <fuller.david@hotmail.com> - 2016-02-28 09:13 -0800
                                  Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 10:09 -0800
                                    Re: Schwarzschild's paper and the uniqueness of the solution David Fuller <fuller.david@hotmail.com> - 2016-03-06 10:56 -0800
                                Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:19 +0100
                                  Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:24 -0800
                                    Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:34 +0100
                                      Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:44 -0800
                                        Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 19:12 +0100
                                          Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 10:19 -0800
                                          Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 11:17 -0800
                              Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-01 17:06 -0800
                                Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-01 17:54 -0800
                                  Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-01 19:41 -0800
                                Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-03-04 01:17 +0100
                                  Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-03 17:12 -0800
                                    Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-03 17:53 -0800
                                    Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 22:19 -0600
                                      Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-03 21:36 -0800
                                        Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-10 11:14 -0600
                                          Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-10 18:32 +0100
                                          Re: Schwarzschild's paper and the uniqueness of the solution Luigi Durante <luigdu@jaccappo.org> - 2016-03-10 20:22 +0000
                                            Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 13:02 -0800
                                              Re: Schwarzschild's paper and the uniqueness of the solution Luigi Durante <luigdu@jaccappo.org> - 2016-03-10 21:08 +0000
                                                Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 13:19 -0800
                                                  Re: Schwarzschild's paper and the uniqueness of the solution Luigi Durante <luigdu@jaccappo.org> - 2016-03-10 21:27 +0000
                                                    Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 14:04 -0800
                                                      Re: Schwarzschild's paper and the uniqueness of the solution "Paul B. Andersen" <relativity@paulba.no> - 2016-03-11 13:33 +0100
                                              Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-10 22:49 +0100
                                                Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 14:20 -0800
                                    Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-03-05 05:24 +0100
                                      Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-04 21:36 -0800
                                        Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-04 22:50 -0800
                                        Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-03-05 11:24 +0100
                                          Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-05 20:07 -0800
                              Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 14:49 -0600
                                Re: Schwarzschild's paper and the uniqueness of the solution Helénē Swanhild <helenes@gardenclosure.org> - 2016-03-03 21:40 +0000
                                  Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 18:58 -0600
                                Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-03 14:04 -0800
                                Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-03 17:46 -0800
                                  Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 22:21 -0600
                  Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 16:42 -0800
                  Re: Schwarzschild's paper and the uniqueness of the solution "Paul B. Andersen" <relativity@paulba.no> - 2016-02-26 20:00 +0100
                    Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 19:03 +0000
    Re: Schwarzschild's paper and the uniqueness of the solution John Gogo <jfgogo22@yahoo.com> - 2016-02-25 19:36 -0800
    Re: Schwarzschild's paper and the uniqueness of the solution alsor@interia.pl - 2016-02-26 09:36 -0800
      Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 11:33 -0800
        Re: Schwarzschild's paper and the uniqueness of the solution alsor@interia.pl - 2016-02-26 12:05 -0800
          Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 12:47 -0800
            Re: Schwarzschild's paper and the uniqueness of the solution alsor@interia.pl - 2016-02-26 14:16 -0800
              Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 15:02 -0800

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#377202

On Friday, February 26, 2016 at 5:44:55 AM UTC-7, PC wrote:
> JanPB wrote:
> 
> >> No, ds^2 is a scaler product formed from ds^2 = dX^i*dX_j delta ij,
> >> Einstein summation. The dX's are covariant and contravariant vectors
> >> (components of dX).
> > 
> > No, that's incorrect. Your ds^2 is equal to dx^i dx_i which is not true.
> > Also "The dX's are covariant and contravariant vectors (components of
> > dX)" is a nonsensical sequence of words.
> 
> That's what happen when you give same thing ten meaning, Janny. You must 
> admit Koobee Woblee is stronger than you when it comes to tensors.

Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu for
ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
if his life depended on it. He probably couldn't even write down a matrix representation of g_mu_nu for it.

[toc] | [prev] | [next] | [standalone]

#377204

David Waite wrote:

>> >> No, ds^2 is a scaler product formed from ds^2 = dX^i*dX_j delta ij,
>> >> Einstein summation. The dX's are covariant and contravariant vectors
>> >> (components of dX).
>> > 
>> > No, that's incorrect. Your ds^2 is equal to dx^i dx_i which is not
>> > true.
>> > Also "The dX's are covariant and contravariant vectors (components of
>> > dX)" is a nonsensical sequence of words.
>> 
>> That's what happen when you give same thing ten meaning, Janny. You must
>> admit Koobee Woblee is stronger than you when it comes to tensors.
> 
> Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu for
> ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
> if his life depended on it. He probably couldn't even write down a matrix
> representation of g_mu_nu for it.

Economic Update: Capitalism is the Problem
http://www.rdwolff.com/download/sites/default/files/Econ_Update_Truthout_2016.02.18.WEB_.mp3

[toc] | [prev] | [next] | [standalone]

#377207

On Friday, February 26, 2016 at 7:59:51 AM UTC-7, PC wrote:
>
> David Waite wrote:
> >
> > Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu for
> > ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
> > if his life depended on it. He probably couldn't even write down a matrix
> > representation of g_mu_nu for it.
> 
> Economic Update: Capitalism is the Problem
> http://www.rdwolff.com/download/sites/default/files/Econ_Update_Truthout_2016.02.18.WEB_.mp3

Non sequitur.

Dishonest person's scheme:  When out of his depth, change the subject.

[toc] | [prev] | [next] | [standalone]

#377239

Gary Harnagel wrote:

>> > Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu
>> > for ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
>> > if his life depended on it. He probably couldn't even write down a
>> > matrix representation of g_mu_nu for it.
>> 
>> Economic Update: Capitalism is the Problem
>> http://www.rdwolff.com/download/sites/default/files/
Econ_Update_Truthout_2016.02.18.WEB_.mp3
> 
> Non sequitur.
> 
> Dishonest person's scheme:  When out of his depth, change the subject.

Shut up pretender. You know nothing about anything.

[toc] | [prev] | [next] | [standalone]

#377208

On Friday, February 26, 2016 at 7:59:51 AM UTC-7, PC wrote:
> David Waite wrote:
> 
> >> >> No, ds^2 is a scaler product formed from ds^2 = dX^i*dX_j delta ij,
> >> >> Einstein summation. The dX's are covariant and contravariant vectors
> >> >> (components of dX).
> >> > 
> >> > No, that's incorrect. Your ds^2 is equal to dx^i dx_i which is not
> >> > true.
> >> > Also "The dX's are covariant and contravariant vectors (components of
> >> > dX)" is a nonsensical sequence of words.
> >> 
> >> That's what happen when you give same thing ten meaning, Janny. You must
> >> admit Koobee Woblee is stronger than you when it comes to tensors.
> > 
> > Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu for
> > ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
> > if his life depended on it. He probably couldn't even write down a matrix
> > representation of g_mu_nu for it.
> 
> Economic Update: Capitalism is the Problem
> http://www.rdwolff.com/download/sites/default/files/Econ_Update_Truthout_2016.02.18.WEB_.mp3

And what "ism" isn't in some way a problem? Not relevant anyway.

[toc] | [prev] | [next] | [standalone]

#377251

David Waite wrote:

>> > Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu
>> > for ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
>> > if his life depended on it. He probably couldn't even write down a
>> > matrix representation of g_mu_nu for it.
>> 
>> Economic Update: Capitalism is the Problem
>> http://www.rdwolff.com/download/sites/default/files/Econ_Update_Truthout_2016.02.18.WEB_.mp3
> 
> And what "ism" isn't in some way a problem? Not relevant anyway.

You are deluded. Try this then.

Michael Parenti - The U.S. War on Yugoslavia
https://www.youtube.com/watch?v=GEzOgpMWnVs

[toc] | [prev] | [next] | [standalone]

#377209

On Friday, February 26, 2016 at 7:57:30 AM UTC-7, David Waite wrote:
> On Friday, February 26, 2016 at 5:44:55 AM UTC-7, PC wrote:
> > JanPB wrote:
> > 
> > >> No, ds^2 is a scaler product formed from ds^2 = dX^i*dX_j delta ij,
> > >> Einstein summation. The dX's are covariant and contravariant vectors
> > >> (components of dX).
> > > 
> > > No, that's incorrect. Your ds^2 is equal to dx^i dx_i which is not true.
> > > Also "The dX's are covariant and contravariant vectors (components of
> > > dX)" is a nonsensical sequence of words.
> > 
> > That's what happen when you give same thing ten meaning, Janny. You must 
> > admit Koobee Woblee is stronger than you when it comes to tensors.
> 
> Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu for
> ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
> if his life depended on it. He probably couldn't even write down a matrix representation of g_mu_nu for it.

Hell, I'd bet money he doesn't even know what g^mu_nu is, which is a no brainer.

[toc] | [prev] | [next] | [standalone]

#377240

David Waite wrote:

>> > That's what happen when you give same thing ten meaning, Janny. You
>> > must admit Koobee Woblee is stronger than you when it comes to
>> > tensors.
>> 
>> Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu for
>> ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
>> if his life depended on it. He probably couldn't even write down a
>> matrix representation of g_mu_nu for it.
> 
> Hell, I'd bet money he doesn't even know what g^mu_nu is, which is a no
> brainer.

Shut up, David Waste. Or answer the question.

[toc] | [prev] | [next] | [standalone]

#377253

On Friday, February 26, 2016 at 7:50:23 AM UTC-8, David Waite wrote:
> On Friday, February 26, 2016 at 7:57:30 AM UTC-7, David Waite wrote:
> > On Friday, February 26, 2016 at 5:44:55 AM UTC-7, PC wrote:
> > > JanPB wrote:
> > > 
> > > >> No, ds^2 is a scaler product formed from ds^2 = dX^i*dX_j delta ij,
> > > >> Einstein summation. The dX's are covariant and contravariant vectors
> > > >> (components of dX).
> > > > 
> > > > No, that's incorrect. Your ds^2 is equal to dx^i dx_i which is not true.
> > > > Also "The dX's are covariant and contravariant vectors (components of
> > > > dX)" is a nonsensical sequence of words.
> > > 
> > > That's what happen when you give same thing ten meaning, Janny. You must 
> > > admit Koobee Woblee is stronger than you when it comes to tensors.
> > 
> > Oh please, I doubt koobee could even calculate g^mu^nu from g_mu_nu for
> > ds^2=dx0^2-dr^2-r^2dOmega^2-(R/r)[(dr-dx0)^2]
> > if his life depended on it. He probably couldn't even write down a matrix representation of g_mu_nu for it.
> 
> Hell, I'd bet money he doesn't even know what g^mu_nu is, which is a no brainer.

You are probably right.

--
Jan

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#377142

On Thursday, February 25, 2016, JanPB wrote:
> On February 25, 2016 at 11:06:18 AM, Jack...@hotmail.com wrote:
> > Jan Bielawski wrote:

> > > Two metrics ds1^2 and ds^2 determine the same geometry if and
> > > only if there exists a _diffeomorphism_ F (of the manifold)
> > > which induces a transformation of one metric tensor into the
> > > other, say ds1^2 into ds2^2:
>
> > >     F^*(ds1^2) = ds2^2
>
> >  ds1^2 and ds2^2 are not tensors, they are called "metrics" and
> > they are pure scalars.
> 
> No, they are tensors.

No, ds^2 is just a scalar that represents the geometry.  In the following description, you can find what the metric [g] and the coordinate system are.  <shrug>

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

Where

**  ds = spacetime
**  [g] = matrix that represents the metric
**  [q] = matrix that represents the coordinate system
**  [g]^i, [g]^j, [q]^i, [q]^j = elements to these matrices

> [matheMAGICS snipped]
>
> OTOH the "d" in "ds^2" is merely a convenient metaphor, it's neither a 
> derivative nor a square of anything. The reason for this metaphoric
> symbol is that the line element in path integrals (denoted by "ds"
> usually, that "d" again is metaphoric) is equal to the square root
> of the "ds^2" tensor evaluated at the tangent vector V to the path:
>
>     "ds in line integrals" = sqrt( ds^2(V,V) )

Not even wrong!  <shrug>

> [more nonsense snipped]

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#377143

On Thursday, February 25, 2016 at 3:47:09 PM UTC-8, Koobee Wublee wrote:
> On Thursday, February 25, 2016, JanPB wrote:
> > On February 25, 2016 at 11:06:18 AM, Jack...@hotmail.com wrote:
> > > Jan Bielawski wrote:
> 
> > > > Two metrics ds1^2 and ds^2 determine the same geometry if and
> > > > only if there exists a _diffeomorphism_ F (of the manifold)
> > > > which induces a transformation of one metric tensor into the
> > > > other, say ds1^2 into ds2^2:
> >
> > > >     F^*(ds1^2) = ds2^2
> >
> > >  ds1^2 and ds2^2 are not tensors, they are called "metrics" and
> > > they are pure scalars.
> > 
> > No, they are tensors.
> 
> No, ds^2 is just a scalar that represents the geometry.

Incorrect (also, the phrase "scalar that represents the geometry" is
a meaningless string of words).

You don't understand the rest at all, so I won't bother any further.

--
Jan

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#377144

On Thursday, February 25, 2016, Jan the relativistic moron wrote:
> Koobee Wublee wrote:

> > ds^2 is just a scalar that represents the geometry.  In the
> > following description, you can find what the metric [g] and
> > the coordinate system are.  <shrug>
>
> > **  ds^2 = [g]_ij d[q]^i d[q]^j
>
> > Where
>
> > **  ds = spacetime
> > **  [g] = matrix that represents the metric
> > **  [q] = matrix that represents the coordinate system
> > **  [g]^i, [g]^j, [q]^i, [q]^j = elements to these matrices

> Incorrect (also, the phrase "scalar that represents the geometry"
> is a meaningless string of words).

It is meaningless to relativistic morons.  <shrug>

> You don't understand the rest at all, so I won't bother any
> further.

For example, the Minkowski spacetime can be described as the following in the Cartesian coordinate system.  <shrug>

**  ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2

Notice that it is a number with a unit of length^2.  It is not a tensor nor a metric.  When you are that fvcking dumb, it is best not to broadcast your ignorance.   <shrug>

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#377146

On 2/25/2016 6:09 PM, Koobee Wublee wrote:
> On Thursday, February 25, 2016, Jan the relativistic moron wrote:
>> Koobee Wublee wrote:
>
>>> ds^2 is just a scalar that represents the geometry.  In the
>>> following description, you can find what the metric [g] and
>>> the coordinate system are.  <shrug>
>>
>>> **  ds^2 = [g]_ij d[q]^i d[q]^j
>>
>>> Where
>>
>>> **  ds = spacetime
>>> **  [g] = matrix that represents the metric
>>> **  [q] = matrix that represents the coordinate system
>>> **  [g]^i, [g]^j, [q]^i, [q]^j = elements to these matrices
>
>> Incorrect (also, the phrase "scalar that represents the geometry"
>> is a meaningless string of words).
>
> It is meaningless to relativistic morons.  <shrug>
>
>> You don't understand the rest at all, so I won't bother any
>> further.
>
> For example, the Minkowski spacetime can be described as the following in the Cartesian coordinate system.  <shrug>
>
> **  ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2
>
> Notice that it is a number with a unit of length^2.  It is not a tensor nor a metric.  When you are
> that fvcking dumb, it is best not to broadcast your ignorance.   <shrug>
>

s^2 = c^2 * t^2 - x^2 - y^2 - z^2 is a number with units of length^2. 
Notice the OPERATOR d in front in the expression above.

I'm not surprised that you don't know the difference. You obviously 
didn't know much about calculus anymore. Pity you've forgotten so much.

To illustrate your ignorance, consider a function y = x^2, where x has 
units meters. So y has a scalar VALUE of 4 m^2 when x=2 m. Now, Koobee, 
kindly tell me the VALUE of dy at that value of x.

PS. You can keep your CalTech diplomas, if you like. They are the only 
remaining evidence that you ever learned anything.


-- 
Odd Bodkin --- maker of fine toys, tools, tables

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#377151

On Thursday, February 25, 2016 at 4:30:07 PM UTC-8, Odd Bodkin wrote:
> On 2/25/2016, Koobee Wublee wrote:

> > ds^2 is just a scalar that represents the geometry.  In the
> > following description, you can find what the metric [g] and
> > the coordinate system are.  <shrug>
>
> > **  ds^2 = [g]_ij d[q]^i d[q]^j
>
> > Where
>
> > **  ds = spacetime
> > **  [g] = matrix that represents the metric
> > **  [q] = matrix that represents the coordinate system
> > **  [g]^i, [g]^j, [q]^i, [q]^j = elements to these matrices
>
> > For example, the Minkowski spacetime can be described as the
> > following in the Cartesian coordinate system.  <shrug>
> >
> > **  ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2
> >
> > Notice that it is a number with a unit of length^2.  It is not a
> > tensor or a metric.

> s^2 = c^2 * t^2 - x^2 - y^2 - z^2 is a number with units of length^2. 
> Notice the OPERATOR d in front in the expression above.

<shaking head>

> I'm not surprised that you don't know the difference. You obviously 
> didn't know much about calculus anymore. Pity you've forgotten so much.

The 'd' in ds^2 above is not an operator.  Technically, d of dANYTHING is not an operator.  Leibniz was the one who founded the set of rules to reduce it.  You are an idiot, PD.  <shrug>

> [rest of garbage snipped]

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#377279

Odd Bodkin wrote:

> On 2/25/2016 6:09 PM, Koobee Wublee wrote:
>> On Thursday, February 25, 2016, Jan the relativistic moron wrote:
>>> Koobee Wublee wrote:
>>>> ds^2 is just a scalar that represents the geometry.  In the
>>>> following description, you can find what the metric [g] and
>>>> the coordinate system are.  <shrug>
>>>
>>>> **  ds^2 = [g]_ij d[q]^i d[q]^j
>>>
>>>> Where
>>>
>>>> **  ds = spacetime
>>>> **  [g] = matrix that represents the metric
>>>> **  [q] = matrix that represents the coordinate system
>>>> **  [g]^i, [g]^j, [q]^i, [q]^j = elements to these matrices
>>>
>>> Incorrect (also, the phrase "scalar that represents the geometry"
>>> is a meaningless string of words).
>>
>> It is meaningless to relativistic morons.  <shrug>
>>
>>> You don't understand the rest at all, so I won't bother any
>>> further.
>>
>> For example, the Minkowski spacetime can be described as the following in
>> the Cartesian coordinate system.  <shrug>
>>
>> **  ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2
>>
>> Notice that it is a number with a unit of length^2.  It is not a tensor
>> nor a metric.  When you are
>> that fvcking dumb, it is best not to broadcast your ignorance.   <shrug>
> 
> s^2 = c^2 * t^2 - x^2 - y^2 - z^2 is a number with units of length^2.

_Dimension_, not “units”; _quantity_, not “number”.  Even then your 
statement is only true if c has the dimension length∕time, t has the 
dimension time, and x, y, and z have the dimension length.

> Notice the OPERATOR d in front in the expression above.

That does not change a lot regarding the question at hand.  To me there 
appears to be as much confusion on the one side as negligence on the others 
so far.

According even to Einstein himself, ds *is* “a scalar that represents the 
(spacetime) geometry” indeed; that is not at all “a meaningless string of 
words”.  It is a spacetime differential, the infinitesimal (infinitely 
small) change in spacetime position.  If ds < 0, then (using the time-like 
sign convention), if I do not confuse the first and the last ones, the 
spacetime interval is space-like (events thus connected lie outside of their 
light cones), if ds = 0 it is light-like (… on their light cones), and if ds 
> 0 it is time-like (… within their light cones).  The same is true for just 
s if t, x, y, and z represent *measured* *differences* in corresponding 
coordinates (but it is important that they do).

Compare the Pythagorean theorem in flat 2-space of which this is, in a 
manner of speaking, just the calculus version adapted for the peculiarities 
of Minkowski (flat)/pseudo-Riemannian (curved) space(time).  In the 
Pythagorean theorem in its common form, you calculate the length of the 
hypotenuse c from the square of the length of the cathetes a and b: c² = a² 
+ b²; you *actually* calculate the *distance* between the points on either 
cathete that the hypothenuse connects, based on their coordinates.  The 
distance between points is always a number, a scalar.  That does not change 
with a different metric or higher dimensions.

Now, by definition, a scalar is a 0th-order tensor (just as a vector is a 
1st-order tensor, and a matrix is a 2nd-order tensor), so it is not 
incorrect (of Jan) to say of that scalar that it is a tensor, and it is 
incorrect (of the other person) to say that it is not a tensor; but it also 
does not strike me as overly useful to insist on the term “tensor” for ds 
(and, IIUC, for dt, dx, dy, and dz).

It is true that ds is _not_ “spacetime” (that would be the entire manifold 
instead); it is the *line element* of the corresponding spacetime instead.  
But “g” *is* the spacetime *metric* (not something else that “represents the 
metric”).

It certainly *is* useful to know that the “g” and “dq” are also tensors in 
the aforementioned equation for *curved* spacetime, because “g” – therefore 
also called the metric *tensor* – occurs in the Einstein field equations, 
and it matters how tensors transform.

See also:
<http://www.britannica.com/topic/Albert-Einstein-on-Space-Time-1987141>

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

CMIIW.

> I'm not surprised that you don't know the difference. You obviously
> didn't know much about calculus anymore. Pity you've forgotten so much.
  ^^^^^^^^^^^
YGCIB.
 
> To illustrate your ignorance, consider a function y = x^2, where x has
> units meters.

I do not think it is helpful to introduce an physical unit of measurement 
here.  This problem can be solved without such (more easily).

> So y has a scalar VALUE of 4 m^2 when x=2 m. Now, Koobee,
> kindly tell me the VALUE of dy at that value of x.

I could tell you the value of ∂y(x)∕∂x for x, and at that value of x (it is, 
of course, equivalent to the [function of the] slope of the corresponding 
curve at a value of x, which is a number/scalar).  But “the value of dy (at 
that value of x)” – does that even have meaning in and of itself, 
considering that dy is a differential, an *infinitesimally small* value?
 

PointedEars
-- 
A neutron walks into a bar and inquires how much a drink costs.
The bartender replies, "For you? No charge."

(from: WolframAlpha)

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#377285

On Friday, February 26, 2016 at 2:05:01 PM UTC-8, Thomas 'PointedEars' Lahn wrote:
> 
> According even to Einstein himself, ds *is* "a scalar that represents the 
> (spacetime) geometry" indeed; that is not at all "a meaningless string of 
> words".

This is because professionals don't need to be pedantic and the difference
between ds^2 as a tensor (which is what it really is) and its value on
the relevant curve's tangent vector (which is a number), as well as the
square root of this number (the "ds") is presumed understood and obvious
and not worth elaborating. Everyone knows what's what.

> It is a spacetime differential, the infinitesimal (infinitely 
> small) change in spacetime position.

Well, yeah, except "infinitesimal" is an undefined notion (unless you
are explicitly using non-standard analysis which AFAICT no physicist
ever used or uses). "Infinitely small" is not a number: for any real x,
either x < 0 or x = 0 or x > 0. There is no room in this scheme for
"positive but smaller than all positive numbers" other than as a means
of torturing undergraduate students with it.

Again, professionals can use this hand-waving terminology because they
know what this shortcut refers to. It's like that saying about eating
fish: those who know that one ought not to use fork and knife for fish
can use fork and knife for fish.

Finally, keep in mind Einstein's notation is a bit old-fashioned in places,
none of the developments in global analysis due to Georges de Rham (Lausanne 
and Geneva!) and Élie Cartan seem to have made any impact on his writing.

> -- 
> A neutron walks into a bar and inquires how much a drink costs.
> The bartender replies, "For you? No charge."
> 
> (from: WolframAlpha)

"That's strange", said the kaon.

--
Jan

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#377298

On Friday, February 26, 2016 at 2:59:27 PM UTC-8, JanPB wrote:
> On Friday, February 26, 2016 at 2:05:01 PM UTC-8, Thomas 'PointedEars' Lahn wrote:
> > 
> > According even to Einstein himself, ds *is* "a scalar that represents the 
> > (spacetime) geometry" indeed; that is not at all "a meaningless string of 
> > words".
> 
> This is because professionals don't need to be pedantic and the difference
> between ds^2 as a tensor (which is what it really is) and its value on
> the relevant curve's tangent vector (which is a number), as well as the
> square root of this number (the "ds") is presumed understood and obvious
> and not worth elaborating. Everyone knows what's what.
> 
> > It is a spacetime differential, the infinitesimal (infinitely 
> > small) change in spacetime position.
> 
> Well, yeah, except "infinitesimal" is an undefined notion (unless you
> are explicitly using non-standard analysis which AFAICT no physicist
> ever used or uses). "Infinitely small" is not a number: for any real x,
> either x < 0 or x = 0 or x > 0. There is no room in this scheme for
> "positive but smaller than all positive numbers" other than as a means
> of torturing undergraduate students with it.
> 
> Again, professionals can use this hand-waving terminology because they
> know what this shortcut refers to. It's like that saying about eating
> fish: those who know that one ought not to use fork and knife for fish
> can use fork and knife for fish.
> 
> Finally, keep in mind Einstein's notation is a bit old-fashioned in places,
> none of the developments in global analysis due to Georges de Rham (Lausanne 
> and Geneva!) and Élie Cartan seem to have made any impact on his writing.
> 
> > -- 
> > A neutron walks into a bar and inquires how much a drink costs.
> > The bartender replies, "For you? No charge."
> > 
> > (from: WolframAlpha)
> 
> "That's strange", said the kaon.
> 
> --
> Jan

The ranting above from Jan the relativistic moron does not contain a single correct sentence.  <shrug>

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#377334

JanPB wrote:

> On Friday, February 26, 2016 at 2:05:01 PM UTC-8, Thomas 'PointedEars'
> Lahn wrote:
>> According even to Einstein himself, ds *is* "a scalar that represents the
>> (spacetime) geometry" indeed; that is not at all "a meaningless string of
>> words".
> 
> This is because professionals don't need to be pedantic and the difference
> between ds^2 as a tensor (which is what it really is) and its value on
> the relevant curve's tangent vector (which is a number), as well as the
> square root of this number (the "ds") is presumed understood and obvious
> and not worth elaborating. Everyone knows what's what.

You have been at least ambiguous about the correctness of the use of 
“scalar”.  You have claimed that using “scalar” instead of “tensor” would 
not be correct, and that something “represents the [spacetime] geometry” 
would be nonsense, by saying to this “incorrect” and calling it “a 
meaningless string of words”.  Neither is true.  

It would be appropriate if you admitted your mistake instead of making 
further spurious claims (with weasel words) about what “everyone knows”.
If one is explaining, one inherently starts from the assumption that _not_ 
“everyone knows”.  Being ambiguous and imprecise is the failure of the 
person explaining, not of the person something is explained to.
 
>> It is a spacetime differential, the infinitesimal (infinitely
>> small) change in spacetime position.
> 
> Well, yeah, except "infinitesimal" is an undefined notion (unless you
> are explicitly using non-standard analysis which AFAICT no physicist
> ever used or uses).

“Infinitesimal *change*” (which I said) and “infinitesimal” (which I did not 
say) are neither “undefined notion” nor “non-standard analysis”.

,-<http://mathworld.wolfram.com/Derivative.html>
| 
| The derivative of a function represents an infinitesimal change in the 
                                             ^^^^^^^^^^^^^^^^^^^^
| function with respect to one of its variables. […]
| 
| […]
| REFERENCES:
| Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical 
| Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. 
| New York: Dover, p. 11, 1972.
| 
| Amend, B. Camp FoxTrot. Kansas City, MO: Andrews McMeel, p. 19, 1998.
| 
| Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999.
| 
| calc101.com. "Step-by-Step Differentiation."
| http://www.calc101.com/webMathematica/MSP/Calc101/WalkD.
| 
| Beyer, W. H. "Derivatives." CRC Standard Mathematical Tables, 28th ed. 
| Boca Raton, FL: CRC Press, pp. 229-232, 1987.
| 
| Griewank, A. Principles and Techniques of Algorithmic Differentiation.
| Philadelphia, PA: SIAM, 2000.
| 
| Mitchell, C. W. Jr. In "Media Clips" (Ed. M. Cibes and J. Greenwood). 
| Math. Teacher 100, 339, Dec. 2006/Jan. 2007. Sloane, N. J. A. Sequence 
| A021009 in "The On-Line Encyclopedia of Integer Sequences."

,-<http://mathworld.wolfram.com/Infinitesimal.html>
|
| An infinitesimal is some quantity that is explicitly nonzero and yet 
| smaller in absolute value than any real quantity.
| 
| […]
| REFERENCES:
| Bell, J. L. A Primer of Infinitesimal Analysis. Cambridge, England: 
| Cambridge University Press, 1998.
| 
| Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 
| 2002.

,-<https://en.wikipedia.org/wiki/Differential_(infinitesimal)>
| 
| The term *differential* is used in calculus to refer to an infinitesimal 
                                                          ^^^^^^^^^^^^^^^^
| (infinitely small) change in some varying quantity. For example, if x is a 
  ^^^^^^^^^^^^^^^^^^^^^^^^^
| variable, then a change in the value of x is often denoted Δx (pronounced 
| delta x). The differential dx represents an infinitely small change in the 
| variable x. […]
| 
| Using calculus, it is possible to relate the infinitely small changes of 
| various variables to each other mathematically using derivatives. If y is 
| a function of x, then the differential dy of y is related to dx by the 
| formula
| 
|   dy = dy∕dx dx
| 
| where dy∕dx denotes the derivative of y with respect to x. This formula 
| summarizes the intuitive idea that the derivative of y with respect to x 
| is the limit of the ratio of differences Δy/Δx as Δx becomes 
| infinitesimal.

,-<https://en.wikipedia.org/wiki/Infinitesimal>
| 
| In mathematics, *infinitesimals* are things so small that there is no way 
| to measure them. The insight with exploiting infinitesimals was that 
| entities could still retain certain specific properties, such as angle or 
| slope, even though these entities were quantitatively small.[1]
| 
| […]
| [1] http://plato.stanford.edu/entries/continuity/#1

By contrast, <http://mathworld.wolfram.com/NonstandardAnalysis.html>.

> "Infinitely small" is not a number:

Yes, it most definitely is.

> for any real x, either x < 0 or x = 0 or x > 0.

Yes.

> There is no room in this scheme for "positive but smaller than all
> positive numbers" other than as a means of torturing undergraduate
> students with it.

Non sequitur.

> Finally, keep in mind Einstein's notation is a bit old-fashioned in
> places, none of the developments in global analysis due to Georges de Rham
> (Lausanne and Geneva!) and Élie Cartan seem to have made any impact on his
> writing.

AISB, you will find the *notion* in other, modern places as well.  Now you 
can even follow the references.
 
>> --
>> A neutron walks into a bar and inquires how much a drink costs.
>> The bartender replies, "For you? No charge."
>> 
>> (from: WolframAlpha)
> 
> "That's strange", said the kaon.

:)


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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#377380

On Saturday, February 27, 2016 at 1:27:22 AM UTC-8, Thomas 'PointedEars' Lahn wrote:
> JanPB wrote:
> 
> > On Friday, February 26, 2016 at 2:05:01 PM UTC-8, Thomas 'PointedEars'
> > Lahn wrote:
> >> According even to Einstein himself, ds *is* "a scalar that represents the
> >> (spacetime) geometry" indeed; that is not at all "a meaningless string of
> >> words".
> > 
> > This is because professionals don't need to be pedantic and the difference
> > between ds^2 as a tensor (which is what it really is) and its value on
> > the relevant curve's tangent vector (which is a number), as well as the
> > square root of this number (the "ds") is presumed understood and obvious
> > and not worth elaborating. Everyone knows what's what.
> 
> You have been at least ambiguous about the correctness of the use of 
> “scalar”.  You have claimed that using “scalar” instead of “tensor” would 
> not be correct, and that something “represents the [spacetime] geometry” 
> would be nonsense, by saying to this “incorrect” and calling it “a 
> meaningless string of words”.  Neither is true.  

It is. Maybe we simply differ in our tolerance to using loose terminology.
The fact is that ds^2 is a tensor, not a number. And I don't approve of statements
like "X represents the geometry" - what does it mean? If X defines the geometry,
that's how it should be phrased.

> It would be appropriate if you admitted your mistake instead of making 
> further spurious claims (with weasel words) about what “everyone knows”.

What mistake?

> If one is explaining, one inherently starts from the assumption that _not_ 
> “everyone knows”.  Being ambiguous and imprecise is the failure of the 
> person explaining, not of the person something is explained to.

It is me who is trying to be precise.

> >> It is a spacetime differential, the infinitesimal (infinitely
> >> small) change in spacetime position.
> > 
> > Well, yeah, except "infinitesimal" is an undefined notion (unless you
> > are explicitly using non-standard analysis which AFAICT no physicist
> > ever used or uses).
> 
> “Infinitesimal *change*” (which I said) and “infinitesimal” (which I did not 
> say) are neither “undefined notion” nor “non-standard analysis”.

It is an undefined object in mathematics unless you use non-standard analysis
or something similar.

> ,-<http://mathworld.wolfram.com/Derivative.html>
> | 
> | The derivative of a function represents an infinitesimal change in the 
>                                              ^^^^^^^^^^^^^^^^^^^^
> | function with respect to one of its variables. […]

It's just a pop-sci definition. It doesn't define anything.

> | […]
> | REFERENCES:
> | Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical 
> | Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. 
> | New York: Dover, p. 11, 1972.
> | 
> | Amend, B. Camp FoxTrot. Kansas City, MO: Andrews McMeel, p. 19, 1998.
> | 
> | Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999.
> | 
> | calc101.com. "Step-by-Step Differentiation."
> | http://www.calc101.com/webMathematica/MSP/Calc101/WalkD.
> | 
> | Beyer, W. H. "Derivatives." CRC Standard Mathematical Tables, 28th ed. 
> | Boca Raton, FL: CRC Press, pp. 229-232, 1987.
> | 
> | Griewank, A. Principles and Techniques of Algorithmic Differentiation.
> | Philadelphia, PA: SIAM, 2000.
> | 
> | Mitchell, C. W. Jr. In "Media Clips" (Ed. M. Cibes and J. Greenwood). 
> | Math. Teacher 100, 339, Dec. 2006/Jan. 2007. Sloane, N. J. A. Sequence 
> | A021009 in "The On-Line Encyclopedia of Integer Sequences."
> 
> ,-<http://mathworld.wolfram.com/Infinitesimal.html>
> |
> | An infinitesimal is some quantity that is explicitly nonzero and yet 
> | smaller in absolute value than any real quantity.
> | 
> | […]
> | REFERENCES:
> | Bell, J. L. A Primer of Infinitesimal Analysis. Cambridge, England: 
> | Cambridge University Press, 1998.
> | 
> | Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 
> | 2002.
> 
> ,-<https://en.wikipedia.org/wiki/Differential_(infinitesimal)>
> | 
> | The term *differential* is used in calculus to refer to an infinitesimal 
>                                                           ^^^^^^^^^^^^^^^^
> | (infinitely small) change in some varying quantity. For example, if x is a 
>   ^^^^^^^^^^^^^^^^^^^^^^^^^
> | variable, then a change in the value of x is often denoted Δx (pronounced 
> | delta x). The differential dx represents an infinitely small change in the 
> | variable x. […]

Same thing. There all hand wavings.

> | Using calculus, it is possible to relate the infinitely small changes of 
> | various variables to each other mathematically using derivatives. If y is 
> | a function of x, then the differential dy of y is related to dx by the 
> | formula
> | 
> |   dy = dy∕dx dx
> | 
> | where dy∕dx denotes the derivative of y with respect to x. This formula 
> | summarizes the intuitive idea that the derivative of y with respect to x 
> | is the limit of the ratio of differences Δy/Δx as Δx becomes 
> | infinitesimal.

This defines a limit (which is correct) but it uses incorrect terminology again: "as Δx becomes 
infinitesimal." It should read "as Δx approaches zero". (If THAT's what they mean by
"infinitesimal", then they should simply say so).

> ,-<https://en.wikipedia.org/wiki/Infinitesimal>
> | 
> | In mathematics, *infinitesimals* are things so small that there is no way 
> | to measure them.

This is ancient terminology. Today this sort of thing is used only as an intuitive
conceptual aid (perhaps).

> | The insight with exploiting infinitesimals was that 
> | entities could still retain certain specific properties, such as angle or 
> | slope, even though these entities were quantitatively small.[1]
> | 
> | […]
> | [1] http://plato.stanford.edu/entries/continuity/#1
> 
> By contrast, <http://mathworld.wolfram.com/NonstandardAnalysis.html>.
> 
> > "Infinitely small" is not a number:
> 
> Yes, it most definitely is.

"Infinitely small" is not a number.

> > for any real x, either x < 0 or x = 0 or x > 0.
> 
> Yes.

QED.

> > There is no room in this scheme for "positive but smaller than all
> > positive numbers" other than as a means of torturing undergraduate
> > students with it.
> 
> Non sequitur.

Come again? Show me the number X which is positive and smaller than all positive
real numbers:

    X = ???  <--- fill this space

--
Jan

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#377393

JanPB wrote:

> On Saturday, February 27, 2016 at 1:27:22 AM UTC-8, Thomas 'PointedEars'
> Lahn wrote:
>> JanPB wrote:
>> > On Friday, February 26, 2016 at 2:05:01 PM UTC-8, Thomas 'PointedEars'
>> > Lahn wrote:
>> >> According even to Einstein himself, ds *is* "a scalar that represents
>> >> the (spacetime) geometry" indeed; that is not at all "a meaningless
>> >> string of words".
>> > 
>> > This is because professionals don't need to be pedantic and the
>> > difference between ds^2 as a tensor (which is what it really is) and
>> > its value on the relevant curve's tangent vector (which is a number),
>> > as well as the square root of this number (the "ds") is presumed
>> > understood and obvious and not worth elaborating. Everyone knows what's
>> > what.
>> 
>> You have been at least ambiguous about the correctness of the use of
>> “scalar”.  You have claimed that using “scalar” instead of “tensor” would
>> not be correct, and that something “represents the [spacetime] geometry”
>> would be nonsense, by saying to this “incorrect” and calling it “a
>> meaningless string of words”.  Neither is true.
> 
> It is. Maybe we simply differ in our tolerance to using loose terminology.
> The fact is that ds^2 is a tensor, not a number.
                                     ^^^^^^^^^^^^
How did you get that idea?

> And I don't approve of statements like "X represents the geometry" - what 
> does it mean? If X defines the geometry, that's how it should be phrased.

But “ds” alone does _not_ define the geometry; the equation does that (most 
notably the metric tensor, which was hidden behind the signs in the equation 
given for Minkowski space(time).  We have been over this:

   ds² = c²dt² − dx² − dy² − dz²
       = c²(dx₀)² − (dx₁)² − (dx₂)² − (dx₃)²
       = g_µν dx^µ dx^ν

for the metric (tensor)

         (c²  0   0   0)
  g_µν = (0  −1   0   0)
         (0   0  −1   0)
         (0   0   0  −1)

(which – surprise! – *is* also a matrix indeed).

“ds” is the line element of the manifold only; it defines how distances in 
that manifold, Minkowski spacetime, are measured.  And in that sense, it 
represents the (spacetime) geometry.  Your claim that it does not is simply, 
provably and proven wrong.

>> It would be appropriate if you admitted your mistake instead of making
>> further spurious claims (with weasel words) about what “everyone knows”.
> 
> What mistake?

,-<news:ab39b83e-4cf4-4245-a1eb-e46f1f691c85@googlegroups.com>
| 
| > No, ds^2 is just a scalar that represents the geometry.
| 
| Incorrect (also, the phrase "scalar that represents the geometry" is
| a meaningless string of words).

*That* mistake.
 
>> If one is explaining, one inherently starts from the assumption that
>> _not_ “everyone knows”.  Being ambiguous and imprecise is the failure
>> of the person explaining, not of the person something is explained to.
> 
> It is me who is trying to be precise.

Then you are doing a pretty bad job at it.

>> “Infinitesimal *change*” (which I said) and “infinitesimal” (which I did
>> not say) are neither “undefined notion” nor “non-standard analysis”.
> 
> It is an undefined object in mathematics unless you use non-standard
> analysis or something similar.

Already shown by me to be wrong.

>> ,-<http://mathworld.wolfram.com/Derivative.html>
>> | 
>> | The derivative of a function represents an infinitesimal change in the
>>                                              ^^^^^^^^^^^^^^^^^^^^
>> | function with respect to one of its variables. […]
> 
> It's just a pop-sci definition. It doesn't define anything.

Wolfram MathWorld, with all its references to mathematics textbooks, is not 
“pop-sci”.  ISTM you just do not like being shown wrong.
 
>> | […]
>> | REFERENCES:
>> | […]
>> 
>> ,-<https://en.wikipedia.org/wiki/Differential_(infinitesimal)>
>> | 
>> | […]
> 
> Same thing. There all hand wavings.

If you summarily dismiss arguments without given any reasoning yourself, you 
do not need to quote them in full.  Your irrationality in this is less 
wastefully shown by trimming.
 
>> | Using calculus, it is possible to relate the infinitely small changes
>> | of various variables to each other mathematically using derivatives. If
>> | y is a function of x, then the differential dy of y is related to dx by
>> | the formula
>> | 
>> |   dy = dy∕dx dx
>> | 
>> | where dy∕dx denotes the derivative of y with respect to x. This formula
>> | summarizes the intuitive idea that the derivative of y with respect to
>> | x is the limit of the ratio of differences Δy/Δx as Δx becomes
>> | infinitesimal.
> 
> This defines a limit (which is correct) but it uses incorrect terminology
> again:

Utter nonsense.

> "as Δx becomes infinitesimal." It should read "as Δx approaches zero".

I beg your pardon?  That is *precisely* what it means in mathematics (see 
the *referenced* definitions that I referenced), so there is no need for a 
different wording.

  dy∕dx = d∕dx dy := lim     ∆y∕∆x
                     ∆x → 0

or

  d∕dx f(x₀) := lim   (f(x₀ + h) − f(x₀))∕h
                h → 0

where

  ∆y = f(x₀ + h) − f(x₀)
  ∆x = h

if y = f(x) is differentiable at x = x₀.

> (If THAT's what they mean by "infinitesimal", then they should
> simply say so).

Oh for crying out loud.

As I proved to you, “infinitesimal”, both as an adjective and a noun,
*is* a mathematical term with a well-defined meaning *today*.

Particularly (but as I proved to you, not exclusively), in German *today* 
even and *especially* in academia, „Infinitesimalrechnung“ refers to the 
techniques of both „Differentialrechnung“ (“calculation with differentials”) 
and „Integralrechnung“ (“calculation with integrals”).  This has to do
with the fact that the German mathematician Gottfried Wilhelm Leibniz, 
independent of the Englishman Isaac Newton, invented the basics of 
calculus – in English, the term is “calculus” now, from the historic 
“calculus of infinitesimals” (but see below).
 
>> ,-<https://en.wikipedia.org/wiki/Infinitesimal>
>> | 
>> | In mathematics, *infinitesimals* are things so small that there is no
>> | way to measure them.
> 
> This is ancient terminology. Today this sort of thing is used only as an
> intuitive conceptual aid (perhaps).

Utter nonsense.  It is a fallacy to assume that because the term “calculus 
of infinitesimals” is historic, “infinitesimal” would be as well.

>> > "Infinitely small" is not a number:
>> Yes, it most definitely is.
> 
> "Infinitely small" is not a number.

Argumentum ad nauseam.

>> > for any real x, either x < 0 or x = 0 or x > 0.
>> Yes.
> 
> QED.

You have proven nothing so far except your utter ignorance and stubbornness.
 

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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