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Groups > sci.physics.relativity > #453478
| From | Thomas 'PointedEars' Lahn <PointedEars@web.de> |
|---|---|
| Newsgroups | sci.physics.relativity |
| Subject | Re: Units in spacetime |
| Date | 2018-03-22 05:57 +0100 |
| Organization | PointedEars Software (PES) |
| Message-ID | <760853805.0ifERbkFSE@PointedEars.de> (permalink) |
| References | <fgpik3F5ps8U1@mid.individual.net> <c9da30bf-feee-4de2-8bd9-e714c0ad3a5e@googlegroups.com> <p88nm5$1ee8$1@gioia.aioe.org> |
Odd Bodkin wrote:
> David (Kronos Prime) Fuller <fuller.david@hotmail.com> wrote:
>> ds²= -(cdt)² + dx² + dy² + dz².
>
> You should be careful here. The formulation of the interval above really
> only applies to spacelike intervals.
Nonsense.
Instead, the value of ds² indicates for a given infinitesimal spacetime
interval (dt, dx, dy, dz) whether the interval (i.e. the corresponding
events’ separation) is either
- spacelike (here: ds² > 0),
- lightlike (ds² = 0), or
- timelike (here: ds² < 0).
The metric signature is (−, +, +, +), and the metric tensor is
(−1 0 0 0)
[η]_µν = diag(−1, 1, 1, 1) = ( 0 1 0 0).
( 0 0 1 0)
( 0 0 0 1)
This is most obvious if you calculate ds² for the extremum cases dt > 0
(i.e., temporal separation), and dx = dy = dz = 0 (i.e., no spatial
separation); and for dt = 0 (no temporal separation), and at least one of
dx ≠ 0, dy ≠ 0, or dz ≠ 0 (some spatial separation).
c t ^ timelike lightlike (with e₁)
: e₂ .'
: . .'
: : ds .': e₃
: : .' : spacelike (with e₁)
: : .' :dt
: :'......: .
:.' e₁ dx e₄
'------------------> x
(I am not entirely sure about writing “ds” as the length of the hypothenuse
here as the metric is only *pseudo*-Euclidean, so it is _not_ that ds² = dt²
+ dx². CMIIW.)
> For timelike intervals, the interval
> is written as
> ds²= (cdt)² - dx² - dy² - dz².
Nonsense.
This is simply the same (Minkowski) metric, but with different signature
(+, −, −, −), so the metric tensor is
( 1 0 0 0)
[η]_µν = diag(1, −1, −1, −1) = ( 0 −1 0 0)
( 0 0 −1 0)
( 0 0 0 −1)
instead.
With this signature, instead:
- spacelike: ds² < 0
- lightlike: ds² = 0
- timelike: ds² > 0
(the arithmetic inverse of the above)
Which, again, you can easily see for the extremal cases dt > 0,
dx = dy = dz = 0; and dt = 0, and at least one of dx ≠ 0, dy ≠ 0, or dz ≠ 0.
If that helps to understand it, assume
∆s² = −(c ∆t)² + ∆x² + ∆y² + ∆z²
= −c² ∆t² + ∆x² + ∆y² + ∆z²
instead, i.e. all events are connected by straight lines in a (flat)
spacetime diagram. Then for given events (corresponding to the diagram
above)
e₁ = (1, 1, 0, 0) = (t₁, x₁, y₁, z₁)
e₂ = (2, 1, 0, 0) = (…)
e₃ = (2, 1 + c, 0, 0) = (…)
e₄ = (1, 5, 0, 0) = (…)
[In time interval ∆t light has traveled from spatial coordinate x₁ = 1
exactly c ∆t units of length, and ∆t = 1 here. So x₃ = x₁ + c ∆t
= 1 + c × 1 = 1 + c.]
we find, for example
∆s²(e₁, e₂) = −c² (t₂ − t₁)² + (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²
= −c² (2 − 1)² + 0 + 0 + 0
= −c²
< 0 → timelike ∎
∆s²(e₁, e₃) = −c² (t₃ − t₁)² + (x₃ − x₁)² + (y₃ − y₁)² + (z₃ − z₁)²
= −c² (2 − 1)² + (1 + c − 1)² + 0 + 0
= −c² + c²
= 0 → lightlike ∎
∆s²(e₁, e₄) = −c² (t₄ − t₁)² + (x₄ − x₁)² + (y₄ − y₁)² + (z₄ − z₁)²
= −c² (1 − 1)² + (5 − 1)² + 0 + 0
= 0 + 16
= 16
> 0 → spacelike ∎
which was probably intuitively obvious in the case of (e₁, e₂), and (e₁,
e₄), and not as obvious with (e₁, e₃) (without a spacetime diagram); but now
we have *mathematical* *proof* which is *a lot* better than intuition.
PointedEars
--
Q: What did the nuclear physicist order for lunch?
A: Fission chips.
(from: WolframAlpha)
Back to sci.physics.relativity | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread
Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-13 10:06 +0100
Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-13 07:15 -0700
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-13 14:36 +0000
Re: Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-13 07:51 -0700
Re: Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-13 08:34 -0700
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-13 09:20 -0700
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-13 16:27 +0000
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-13 16:28 +0000
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-13 09:45 -0700
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-13 18:39 +0100
Re: Units in spacetime Térov Balónky <iytn@yeicenlw.ik> - 2018-03-13 17:49 +0000
Re: Units in spacetime Térov Balónky <iytn@yeicenlw.ik> - 2018-03-13 17:49 +0000
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-14 13:54 +0100
Re: Units in spacetime Térov Balónky <iytn@yeicenlw.ik> - 2018-03-14 16:02 +0000
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-14 21:47 +0100
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-13 18:02 +0000
Re: Units in spacetime Térov Balónky <iytn@yeicenlw.ik> - 2018-03-13 18:12 +0000
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-13 22:00 +0100
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-13 22:55 +0000
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-14 09:19 -0700
Re: Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-14 09:32 -0700
Re: Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-14 09:52 -0700
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-14 18:07 +0000
Re: Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-14 12:03 -0700
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-14 13:18 -0700
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-14 21:38 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-14 16:27 -0700
Re: Units in spacetime Python <python@python.invalid> - 2018-03-15 01:05 +0100
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-15 03:52 +0000
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-14 23:10 -0700
Re: Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-15 00:16 -0700
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-15 11:55 +0000
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-15 14:40 +0100
Re: Units in spacetime Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2018-03-15 16:02 +0100
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-15 15:20 +0000
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-15 09:49 -0700
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-15 19:49 +0000
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-15 11:26 +0100
Re: Units in spacetime mlwozniak@wp.pl - 2018-03-15 04:03 -0700
Re: Units in spacetime Keith Stein <keithstein111@gmail.com> - 2018-03-15 15:50 +0000
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-15 21:15 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-15 10:30 -0700
Re: Units in spacetime Python <python@python.invalid> - 2018-03-15 18:32 +0100
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-15 19:51 +0000
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-15 20:56 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-15 23:20 -0700
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-16 10:11 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-18 22:25 -0700
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-19 09:17 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-19 10:36 -0700
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-19 17:57 +0000
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-19 11:16 -0700
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-19 18:43 +0000
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-19 13:02 -0700
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-19 21:24 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-20 09:32 -0700
Re: Units in spacetime Python <python@python.invalid> - 2018-03-20 17:40 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-20 09:55 -0700
Re: Units in spacetime Dirk Van de moortel <dirkvandemoortel@notmail.com> - 2018-03-20 18:25 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-20 14:21 -0700
Re: Units in spacetime Dirk Van de moortel <dirkvandemoortel@notmail.com> - 2018-03-20 22:57 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-20 16:00 -0700
Re: Units in spacetime Darryl Fletcher <lu@ewiwwe.tt> - 2018-03-20 23:38 +0000
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-22 14:01 +0100
Re: Units in spacetime Koobee Wublee <koobee.wublee@gmail.com> - 2018-03-22 13:45 -0700
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-27 20:18 +0200
Re: Units in spacetime xip1415926@gmail.com - 2018-03-27 14:40 -0700
Re: Units in spacetime xip1415926@gmail.com - 2018-03-26 07:38 -0700
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-27 19:53 +0200
Re: Units in spacetime xip1415926@gmail.com - 2018-03-27 12:21 -0700
Re: Units in spacetime Tom Roberts <tjroberts137@sbcglobal.net> - 2018-03-27 16:47 -0400
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-28 09:34 +0200
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-20 21:12 +0100
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-16 10:32 +0100
Re: Units in spacetime "Paul B. Andersen" <relativity@paulba.no> - 2018-03-16 12:33 +0100
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-17 21:29 +0100
Re: Units in spacetime Odd Bodkin <bodkinodd@gmail.com> - 2018-03-14 20:51 +0000
Re: Units in spacetime "David (Kronos Prime) Fuller" <fuller.david@hotmail.com> - 2018-03-20 15:41 -0700
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-15 14:04 +0100
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-15 20:10 +0100
Re: Units in spacetime Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2018-03-22 05:57 +0100
Re: Units in spacetime Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2018-03-27 19:57 +0200
Re: Units in spacetime numbernumber1964@gmail.com - 2018-03-20 13:49 -0700
Re: Units in spacetime Keith Stein <keithstein111@gmail.com> - 2018-03-21 07:26 +0000
Re: Units in spacetime numbernumber1964@gmail.com - 2018-03-22 15:28 -0700
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-23 09:57 +0100
Re: Units in spacetime xip1415926@gmail.com - 2018-03-25 18:54 -0700
Re: Units in spacetime numbernumber1964@gmail.com - 2018-03-23 10:48 -0700
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-25 19:03 +0200
Re: Units in spacetime xip1415926@gmail.com - 2018-03-26 06:39 -0700
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-03-26 21:43 +0200
Re: Units in spacetime Keith Stein <keithstein111@gmail.com> - 2018-03-30 11:00 +0100
Re: Units in spacetime Will Bos <sllu@wsws.sb> - 2018-03-30 10:52 +0000
Re: Units in spacetime Will Bos <sllu@wsws.sb> - 2018-03-30 11:02 +0000
Re: Units in spacetime Thomas Heger <ttt_heg@web.de> - 2018-04-03 06:45 +0200
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