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Groups > sci.physics.relativity > #424266 > unrolled thread
| Started by | "Edgar L. Owen" <Edgar@edgarlowen.com> |
|---|---|
| First post | 2017-07-08 05:05 -0700 |
| Last post | 2017-07-29 20:43 +0000 |
| Articles | 5 on this page of 125 — 24 participants |
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Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-08 05:05 -0700
Re: Why there must be an absolute frame - New Book Poutnik <poutnik4nntp@gmail.com> - 2017-07-08 14:09 +0200
Re: Why there must be an absolute frame - New Book The Starmaker <starmaker@ix.netcom.com> - 2017-07-08 12:04 -0700
Re: Why there must be an absolute frame - New Book Ursler Zilberschlag <ursler@zilbersch.org> - 2017-07-09 09:57 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-12 20:26 -0500
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-13 03:48 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-13 11:47 -0500
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-14 03:07 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-14 03:44 -0700
Re: Why there must be an absolute frame - New Book "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2017-07-14 06:46 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-14 21:12 -0500
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-14 19:49 -0700
Re: Why there must be an absolute frame - New Book "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2017-07-14 20:02 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-15 04:22 -0700
Re: Why there must be an absolute frame - New Book "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2017-07-15 06:21 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-15 08:10 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-15 08:13 -0700
Re: Why there must be an absolute frame - New Book "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2017-07-15 08:22 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-16 13:08 -0500
Re: Why there must be an absolute frame - New Book Winfred Plumer <lewru@pulwr.cn> - 2017-07-16 18:15 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-17 10:57 -0500
Re: Why there must be an absolute frame - New Book Winfred Plumer <lewru@pulwr.cn> - 2017-07-17 16:12 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-17 12:05 -0500
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-17 10:42 -0700
Re: Why there must be an absolute frame - New Book Ustin Veterre <utve@nauoe.ct> - 2017-07-17 21:45 +0000
Re: Why there must be an absolute frame - New Book Nicolaas Vroom <nicolaas.vroom@pandora.be> - 2017-07-19 05:27 -0700
Re: Why there must be an absolute frame - New Book Paparios <mrios@ing.puc.cl> - 2017-07-19 08:23 -0700
Re: Why there must be an absolute frame - New Book Nicolaas Vroom <nicolaas.vroom@pandora.be> - 2017-07-26 01:25 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-26 05:05 -0700
Re: Why there must be an absolute frame - New Book Ernard Hevalier <adrdo@tarlaab.lr> - 2017-07-26 20:49 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-27 10:59 -0500
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 09:10 -0700
Re: Why there must be an absolute frame - New Book Paparios <mrios@ing.puc.cl> - 2017-07-27 09:34 -0700
Re: Why there must be an absolute frame - New Book Odd Bodkin <bodkinodd@gmail.com> - 2017-07-27 11:59 -0500
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 12:01 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-27 12:18 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 12:49 -0700
Re: Why there must be an absolute frame - New Book Odd Bodkin <bodkinodd@gmail.com> - 2017-07-27 19:35 +0000
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 11:56 -0700
Re: Why there must be an absolute frame - New Book Ernard Hevalier <adrdo@tarlaab.lr> - 2017-07-27 19:05 +0000
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-27 12:17 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 12:47 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-28 11:01 -0700
Re: Why there must be an absolute frame - New Book Paparios <mrios@ing.puc.cl> - 2017-07-27 13:04 -0700
Re: Why there must be an absolute frame - New Book Prokaryotic Caspase Homolog <prokaryotic.caspase.homolog@gmail.com> - 2017-07-27 09:30 -0700
Re: Why there must be an absolute frame - New Book Prokaryotic Caspase Homolog <prokaryotic.caspase.homolog@gmail.com> - 2017-07-27 09:45 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-27 12:52 -0500
Re: Why there must be an absolute frame - New Book Ernard Hevalier <adrdo@tarlaab.lr> - 2017-07-27 18:06 +0000
Re: Why there must be an absolute frame - New Book Odd Bodkin <bodkinodd@gmail.com> - 2017-07-27 13:11 -0500
Re: Why there must be an absolute frame - New Book Ernard Hevalier <adrdo@tarlaab.lr> - 2017-07-27 18:04 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-26 20:11 -0500
Re: Why there must be an absolute frame - New Book kenseto <setoken@att.net> - 2017-07-19 05:59 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-15 04:07 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-16 13:22 -0500
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-17 05:26 -0700
Re: Why there must be an absolute frame - New Book shuba <thailao2558@gmail.com> - 2017-07-17 08:03 -0700
Re: Why there must be an absolute frame - New Book Winfred Plumer <lewru@pulwr.cn> - 2017-07-17 15:59 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-17 11:37 -0500
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-18 04:30 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-18 12:55 -0500
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-18 11:41 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-18 22:59 -0500
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-19 05:13 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-25 10:00 -0500
Re: Why there must be an absolute frame - New Book Jonathan Doolin <good4usoul@gmail.com> - 2017-07-25 10:23 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-25 22:04 -0500
Re: Why there must be an absolute frame - New Book Jonathan Doolin <good4usoul@gmail.com> - 2017-07-26 07:16 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-26 20:21 -0500
Re: Why there must be an absolute frame - New Book Jonathan Doolin <good4usoul@gmail.com> - 2017-07-27 07:15 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-25 13:54 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-25 21:51 -0500
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-18 11:56 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-18 12:14 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-18 13:55 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-18 23:30 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-19 01:05 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-19 04:23 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-19 11:37 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-19 11:42 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-19 13:25 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-25 10:40 -0500
Re: Why there must be an absolute frame - New Book Ernard Hevalier <adrdo@tarlaab.lr> - 2017-07-25 17:57 +0000
Re: Why there must be an absolute frame - New Book Jonathan Doolin <good4usoul@gmail.com> - 2017-07-25 11:38 -0700
Re: Why there must be an absolute frame - New Book Ernard Hevalier <adrdo@tarlaab.lr> - 2017-07-25 19:12 +0000
Re: Why there must be an absolute frame - New Book Jonathan Doolin <good4usoul@gmail.com> - 2017-07-25 14:21 -0700
Re: Why there must be an absolute frame - New Book Ernard Hevalier <adrdo@tarlaab.lr> - 2017-07-28 20:44 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-25 22:05 -0500
Re: Why there must be an absolute frame - New Book RichD <r_delaney2001@yahoo.com> - 2017-07-19 12:56 -0700
Re: Why there must be an absolute frame - New Book Dirk Van de moortel <dirkvandemoortel@hotspam.not> - 2017-07-19 22:13 +0200
Re: Why there must be an absolute frame - New Book RichD <r_delaney2001@yahoo.com> - 2017-07-19 14:49 -0700
Re: Why there must be an absolute frame - New Book "Edgar L. Owen" <Edgar@edgarlowen.com> - 2017-07-19 13:44 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-19 13:52 -0700
Re: Why there must be an absolute frame - New Book Ustin Veterre <utve@nauoe.ct> - 2017-07-20 16:58 +0000
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-19 22:04 -0500
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-14 21:05 -0500
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-18 12:08 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-18 13:22 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-26 20:45 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 00:04 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-27 09:09 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 11:55 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-27 12:16 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 12:47 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-27 13:59 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-27 23:03 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-28 10:57 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-29 00:56 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-28 11:01 -0700
Re: Why there must be an absolute frame - New Book Paparios <mrios@ing.puc.cl> - 2017-07-28 12:23 -0700
Re: Why there must be an absolute frame - New Book RichD <r_delaney2001@yahoo.com> - 2017-07-19 12:38 -0700
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-17 15:55 -0700
Re: Why there must be an absolute frame - New Book mlwozniak@wp.pl - 2017-07-17 23:25 -0700
Re: Why there must be an absolute frame - New Book Kurt Larrabee <arb@twraa.ab> - 2017-07-22 12:29 +0000
Re: Why there must be an absolute frame - New Book Sylvia Else <sylvia@not.at.this.address> - 2017-07-13 11:39 +1000
Re: Why there must be an absolute frame - New Book pcardinale@volcanomail.com - 2017-07-13 12:49 -0700
Re: Why there must be an absolute frame - New Book Jonathan Doolin <good4usoul@gmail.com> - 2017-07-25 11:21 -0700
Re: Why there must be an absolute frame - New Book "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2017-07-25 11:28 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-25 22:12 -0500
Re: Why there must be an absolute frame - New Book Jonathan Doolin <good4usoul@gmail.com> - 2017-07-26 08:22 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-26 20:35 -0500
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-27 12:13 -0700
Re: Why there must be an absolute frame - New Book Tom Roberts <tjroberts137@sbcglobal.net> - 2017-07-29 11:50 -0500
Re: Why there must be an absolute frame - New Book JanPB <filmart@gmail.com> - 2017-07-29 13:30 -0700
Re: Why there must be an absolute frame - New Book Jeèným Ludnými <hovae@ieuuio.org> - 2017-07-29 20:45 +0000
Re: Why there must be an absolute frame - New Book Jeèným Ludnými <hovae@ieuuio.org> - 2017-07-29 20:43 +0000
Page 7 of 7 — ← Prev page 1 2 3 4 5 6 [7]
| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2017-07-27 12:13 -0700 |
| Message-ID | <905e5a6a-de19-43c6-af4f-6d40138bfca8@googlegroups.com> |
| In reply to | #426231 |
On Wednesday, July 26, 2017 at 6:35:34 PM UTC-7, tjrob137 wrote:
> On 7/26/17 7/26/17 10:22 AM, Jonathan Doolin wrote:
> > On Tuesday, July 25, 2017 at 10:13:01 PM UTC-5, tjrob137 wrote:
> >> On 7/25/17 7/25/17 1:21 PM, Jonathan Doolin wrote:
> >>> If the universe has a finite mass, then there would be a center-of-mass.
> >>
> >> Why? -- I see no such necessity. You are making assumptions that are so hidden
> >> that you don't realize you are making them. In particular you assume that
> >> integration over all space is well defined (because that is how center-of-mass
> >> is defined). But on a curved manifold, as in GR, such integration is in general
> >> NOT well defined.
> >>
> >> You can integrate certain geometrical objects known as
> >> differential forms. But for center-of-mass the integrand is
> >> not such a form.
> >>
> >> Tom Roberts
> > [...]
>
> The problem goes MUCH deeper than you realize. In general you cannot perform the
> integral required [#]. This can be because it does not converge, or because it
> is not well defined and you get different values depending on which method you
> use, or because you cannot find valid boundaries to the integration region, or....
>
> [#] Though in some specific cases you can.
>
> Simple example: consider Schwarzschild spacetime with the usual coordinates, and
> the 2-sphere r=10M,t=0. This is well outside the horizon, but the volume
> enclosed within that boundary is not well defined.
>
> [I am not expert here, but I know that in general integration
> over a curved manifold is not well defined.]
Integral over an n-dim. smooth manifold (not necessarily Riemannian) is
well-defined in several different contexts, two of which are most common:
1. the manifold is oriented,
2. the manifold is not orientable.
In case 1 the integrand is an n-form. The resulting integral may still
diverge for all the usual reasons (the n-form growing too fast, the region
of integration being "too large", etc.)
In case 2 the integrand is a pseudo-n-form. Same comment on divergence
applies. Pseudoforms are a bit strange as they don't have pullbacks but
Stokes' theorem applies to them with suitable modifications.
There are many theorems guaranteeing the convergence, e.g. integrals
over compact manifolds are always convergent. (All of the above BTW
assumes smooth manifolds and smooth n-forms).
If the manifold is also Riemannian, then it has a canonical n-form
on it called "the volume form" (which is a pseudoform if the manifold
isn't orientable). The integral of the volume form is equal to the
volume of the manifold (hence the name).
In this case one can talk about integrating _functions_ over manifolds
in the sense that one is really integrating the product of that function
with the volume form. So using this language, the integral of the
function identically equal to 1 is equal to the manifold volume.
The volume inside a sphere surrounding the Schwarzschild horizon depends
obviously on the choice of the spacelike 3D slice spanning that sphere.
It's like a circle of radius R which can span a disc (area pi R^2) or
a hemisphere (area 2 pi R^2), etc. etc.
In the standard Schwarzschild coordinates the slice t=const cuts across
both spacelike and timelike domains due to those coordinates being
lopsided: they inject the region inside the horizon "lying on its side",
with the horizon itself excluded, so the slice is not even contiguous
(let alone spacelike throughout).
But other coordinate systems have good spacelike t=const. spanning slices,
for example:
1. Gullstrand–Painlevé coordinates:
ds^2 = -(1-2M/r) dt^2 + 2 sqrt(2M/r) dt dr + dr^2 +
+ r^2 dtheta^2 + r^2 sin^2(theta) dphi^2
t = const. yields:
ds^2 = dr^2 + r^2 dtheta^2 + r^2 sin^2(theta) dphi^2
...so the 3D volume form is:
dvol = r^2 sin(theta) dr /\ dtheta /\ dphi
...which means the volume is standard: 4/3 pi R^3 (taking the sphere
at r = R). The physical meaning of this number is tenuous.
2. Eddington-Finkelstein coordinates:
ds^2 = -(1-2M/r) dt^2 + 4m/r dt dr + (1+2M/r) dr^2 +
+ r^2 dtheta^2 + r^2 sin^2(theta) dphi^2
t = const. yields:
ds^2 = (1+2M/r) dr^2 + r^2 dtheta^2 + r^2 sin^2(theta) dphi^2
...so the 3D volume form is:
dvol = r^2 sin(theta) sqrt(1+2M/r)
The integral of this is nasty, the volume inside is equal to 4 pi times
the antiderivative of r^2 sqrt(1+2M/r), you can see it here (push the
"Go!" button, the "x" means our "r":
http://www.integral-calculator.com/#expr=x%5E2%2Asqrt%281%2B2m%2Fx%29
3. Kruskal-Szekeres coordinates:
This can be only done numerically as the metric is given in terms
of a function r(u,v) defined implicitly by:
(r/2M - 1) exp(-r/2M) = u^2 - v^2
...so good luck with that :-)
All these volumes are mostly of theoretical interest, although
the little factoid that Gullstrand–Painlevé "respects" both the
volume and the radius (its radial coordinate equals the arc length)
it cute.
--
Jan
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| From | Tom Roberts <tjroberts137@sbcglobal.net> |
|---|---|
| Date | 2017-07-29 11:50 -0500 |
| Message-ID | <6ICdna4oovZzIeHEnZ2dnUU7_83NnZ2d@giganews.com> |
| In reply to | #426344 |
On 7/27/17 7/27/17 - 2:13 PM, JanPB wrote: > On Wednesday, July 26, 2017 at 6:35:34 PM UTC-7, tjrob137 wrote: >> The problem goes MUCH deeper than you realize. In general you cannot perform the >> integral required [#]. > > Integral over an n-dim. smooth manifold (not necessarily Riemannian) is > well-defined in several different contexts, two of which are most common: > 1. the manifold is oriented, > 2. the manifold is not orientable. > In case 1 the integrand is an n-form. Yes, WHEN the integrand is an n-form the integral is well defined. But the case in point is to compute center-of-mass, and that integral is NOT well defined. Simple demonstration: the integral is over all SPACE, not all spacetime. That means you must choose a spacelike 3-d slice for the integral, and you have an infinite number of them to choose from; in general each will give a different answer. Tom Roberts
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2017-07-29 13:30 -0700 |
| Message-ID | <d1c84083-dd25-49e6-a3a4-43577a6ef0c6@googlegroups.com> |
| In reply to | #426547 |
On Saturday, July 29, 2017 at 9:51:01 AM UTC-7, tjrob137 wrote: > On 7/27/17 7/27/17 - 2:13 PM, JanPB wrote: > > On Wednesday, July 26, 2017 at 6:35:34 PM UTC-7, tjrob137 wrote: > >> The problem goes MUCH deeper than you realize. In general you cannot perform the > >> integral required [#]. > > > > Integral over an n-dim. smooth manifold (not necessarily Riemannian) is > > well-defined in several different contexts, two of which are most common: > > 1. the manifold is oriented, > > 2. the manifold is not orientable. > > In case 1 the integrand is an n-form. > > Yes, WHEN the integrand is an n-form the integral is well defined. But the case > in point is to compute center-of-mass, and that integral is NOT well defined. Yes. And another case of ill-defining things is integrating over spacetime (so OK) an integrand that has no addition defined for its values at different points (so not OK), the standard example being a "resultant 4-momentum" of an extended configuration: there is no such thing as vectors (even 4-vectors) at different spacetime points cannot be added together in any canonical fashion. -- Jan
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| From | Jeèným Ludnými <hovae@ieuuio.org> |
|---|---|
| Date | 2017-07-29 20:45 +0000 |
| Message-ID | <olis4l$3bf$2@gioia.aioe.org> |
| In reply to | #426559 |
JanPB wrote: >> Yes, WHEN the integrand is an n-form the integral is well defined. But >> the case in point is to compute center-of-mass, and that integral is >> NOT well defined. > > Yes. And another case of ill-defining things is integrating over > spacetime (so OK) an integrand that has no addition defined for its > values at different points (so not OK), the standard example being a > "resultant 4-momentum" of an extended configuration: there is no such > thing as vectors (even 4-vectors) at different spacetime points cannot > be added together in any canonical fashion. We don't understand.
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| From | Jeèným Ludnými <hovae@ieuuio.org> |
|---|---|
| Date | 2017-07-29 20:43 +0000 |
| Message-ID | <olis10$3bf$1@gioia.aioe.org> |
| In reply to | #426547 |
Tom Roberts wrote: > Simple demonstration: the integral is over all SPACE, not all > spacetime. That means you must choose a spacelike 3-d slice for the > integral, and you have an infinite number of them to choose from; in > general each will give a different answer. So you have in 2d, but once you know the envelope, no problemo.
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