Mathematically Allowable Instantaneousness [Archive]

Perfectly Innocent

Jul6-04, 01:47 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nAre there pseudo-Riemannian spacetimes having the property that no\nlogical contradiction exists in the mere conjecture of an\ninstantaneous comparison of time rates for all observers? It seems\nto me that the Einstein Universe has this property.\n\nhttp://www.everythingimportant.org/viewtopic.php?t=605\n\nAre there any simple, airtight proofs for or against this possibility\nthat can be understood by students of physics and mathematics at the\nundergraduate level?\n\nEugene Shubert\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Are there pseudo-Riemannian spacetimes having the property that no
logical contradiction exists in the mere conjecture of an
instantaneous comparison of time rates for all observers? It seems
to me that the Einstein Universe has this property.

http://www.everythingimportant.org/viewtopic.php?t=605

Are there any simple, airtight proofs for or against this possibility
that can be understood by students of physics and mathematics at the
undergraduate level?

Eugene Shubert

carlip@no-physics-spam.ucdavis.edu

Jul13-04, 12:00 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nPerfectly Innocent <perfectlyInnocent@as-if.com> wrote:\n\n> Are there pseudo-Riemannian spacetimes having the property that\n> no logical contradiction exists in the mere conjecture of an\n> instantaneous comparison of time rates for all observers?\n\nI\'m not sure I understand the question. Are you asking whether\nthere are spacetimes that admit a well-defined time slicing,\nthat is, a collection of ``hypersurfaces of simultaneity\'\' that\ncan be unambiguously defined?\n\nIf so, the answer is yes -- but there are many such slicings,\nwhich differ from each other as to what events are considered\nsimultaneous, and there seems to be no particular physical reason\nto prefer any one over another. For example:\n\n-- For spacetimes with topology RxS, where S is either compact or\nasymptotically flat, one can often use York\'s ``extrinsic time,\'\'\nthat is, a slicing by spatial hypersurfaces of constant mean\nextrinsic curvature TrK = t. This is often possible for S compact\n(see Marsden and Tipler, Phys. Rep. 66 (1980), or Anderson, Comm.\nMath. Phys. 222 (2001) 533, for example), though there are some\ncases known for which it is not (Isenberg and Rendall, Class.\nQuant. Grav. 15 (1998) 3679).\n\n-- For spacetimes with topology RxS with S noncompact, one can often\nchoose ``maximal slicing,\'\' that is, selecting time slices with\nvanishing mean extrinsic curvature. Depending on the geometry and\ntopology, this slicing can be unique, can require additional data\nat infinity, or can be impossible.\n\n-- For globally hyperbolic spacetimes with an initial singularity, it\nis often possible to define a ``cosmological time function\'\' as,\nessentially, the longest proper time to the singularity (see Andersson\net al., Class. Quant. Grav. 15 (1998) 309). The level sets of this\nfunction -- that is, the surfaces of constant time -- are differentiable\nalmost everywhere (in the technical sense. i.e., except on a set\nof measure zero), but typically have ``bends\'\' or ``pleats\'\'; see\nBenedetti and Guadagnini, Nucl. Phys. B613 (2001) 330 for a fairly\nexplicit (though fairly complicated) description for hyperbolic\nmanifolds in 2+1 dimensions.\n\n-- One can choose the Brady-Creighton-Thorne slicing, in which the\ntime-time and time-space components of the metric obey a certain\ndifferential equation that, in some sense, minimizes the ``strain\'\'\nas one goes from one time slice to the next. This often gives a unique\nslicing, although added boundary data may be needed -- see Garfinkle\net al., Class. Quant. Grav. 17 (2000) 3899.\n\n-- For noncompact spatial hypersurfaces, it is sometimes possible to\ndefine a slicing by demanding that the hypersurfaces of constant time\nare asymptotically hyperbolic; see Isenberg and Park, Class. Quant.\nGrav. 14 (1997) A189.\n\n-- For many choices of three-manifold topology, any metric is conformal\nto a unique or almost unique metric with scalar curvature 1, -1, or 0\n(the "Yamabe metric"). It is sometimes possible to define a slicing\nof a spacetime by demanding that the conformal factor be equal to t.\n\nThere are a great many other possibilities -- the question is of some\nimportance in numerical relativity, in which a ``good\'\' choice of slicing\ncan grealy improve the approximation. You might look at section 3.1.2\nof Lehner\'s recent review, Class. Quant. Grav. 18 (2001) R25, for a\ndescription of some of the other choices that have been considered.\n\nIn short: the problem isn\'t that one can\'t define simultaneity. It\'s not\neven that one can\'t find a ``nice\'\' definition. The problem is that for\na typical pseudo-Riemannian manifold, there are lots of ``nice\'\' definitions,\neach with its own use, but no general reason to prefer one over another.\n\nSteve Carlip\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Perfectly Innocent <perfectlyInnocent@as-if.com> wrote:

> Are there pseudo-Riemannian spacetimes having the property that
> no logical contradiction exists in the mere conjecture of an
> instantaneous comparison of time rates for all observers?

I'm not sure I understand the question. Are you asking whether
there are spacetimes that admit a well-defined time slicing,
that is, a collection of ``hypersurfaces of simultaneity'' that
can be unambiguously defined?

If so, the answer is yes -- but there are many such slicings,
which differ from each other as to what events are considered
simultaneous, and there seems to be no particular physical reason
to prefer any one over another. For example:

-- For spacetimes with topology RxS, where S is either compact or
asymptotically flat, one can often use York's ``extrinsic time,''
that is, a slicing by spatial hypersurfaces of constant mean
extrinsic curvature TrK = t. This is often possible for S compact
(see Marsden and Tipler, Phys. Rep. 66 (1980), or Anderson, Comm.
Math. Phys. 222 (2001) 533, for example), though there are some
cases known for which it is not (Isenberg and Rendall, Class.
Quant. Grav. 15 (1998) 3679).

-- For spacetimes with topology RxS with S noncompact, one can often
choose ``maximal slicing,'' that is, selecting time slices with
vanishing mean extrinsic curvature. Depending on the geometry and
topology, this slicing can be unique, can require additional data
at infinity, or can be impossible.

-- For globally hyperbolic spacetimes with an initial singularity, it
is often possible to define a ``cosmological time function'' as,
essentially, the longest proper time to the singularity (see Andersson
et al., Class. Quant. Grav. 15 (1998) 309). The level sets of this
function -- that is, the surfaces of constant time -- are differentiable
almost everywhere (in the technical sense. i.e., except on a set
of measure zero), but typically have ``bends'' or ``pleats''; see
Benedetti and Guadagnini, Nucl. Phys. B613 (2001) 330 for a fairly
explicit (though fairly complicated) description for hyperbolic
manifolds in 2+1 dimensions.

-- One can choose the Brady-Creighton-Thorne slicing, in which the
time-time and time-space components of the metric obey a certain
differential equation that, in some sense, minimizes the ``strain''
as one goes from one time slice to the next. This often gives a unique
slicing, although added boundary data may be needed -- see Garfinkle
et al., Class. Quant. Grav. 17 (2000) 3899.

-- For noncompact spatial hypersurfaces, it is sometimes possible to
define a slicing by demanding that the hypersurfaces of constant time
are asymptotically hyperbolic; see Isenberg and Park, Class. Quant.
Grav. 14 (1997) A189.

-- For many choices of three-manifold topology, any metric is conformal
to a unique or almost unique metric with scalar curvature 1, -1, or
(the "Yamabe metric"). It is sometimes possible to define a slicing
of a spacetime by demanding that the conformal factor be equal to t.

There are a great many other possibilities -- the question is of some
importance in numerical relativity, in which a ``good'' choice of slicing
can grealy improve the approximation. You might look at section 3.1.2
of Lehner's recent review, Class. Quant. Grav. 18 (2001) R25, for a
description of some of the other choices that have been considered.

In short: the problem isn't that one can't define simultaneity. It's not
even that one can't find a ``nice'' definition. The problem is that for
a typical pseudo-Riemannian manifold, there are lots of ``nice'' definitions,
each with its own use, but no general reason to prefer one over another.

Steve Carlip

Perfectly Innocent

Jul16-04, 08:19 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ncarlip@no-physics-spam.ucdavis.edu wrote in message news:<cd1433\\$pv8\\$2@woodrow.ucdavis.edu>...\n> Perfectly Innocent <perfectlyInnocent@as-if.com> wrote:\n>\n> > Are there pseudo-Riemannian spacetimes having the property that\n> > no logical contradiction exists in the mere conjecture of an\n> > instantaneous comparison of time rates for all observers?\n>\n> I\'m not sure I understand the question. Are you asking whether\n> there are spacetimes that admit a well-defined time slicing,\n> that is, a collection of ``hypersurfaces of simultaneity\'\' that\n> can be unambiguously defined?\n\n> If so, the answer is yes -- but there are many such slicings,\n> which differ from each other as to what events are considered\n> simultaneous, and there seems to be no particular physical reason\n> to prefer any one over another. For example:\n>\n> [snip] [snip] [snip] ...\n>\n> In short: the problem isn\'t that one can\'t define simultaneity. It\'s not\n> even that one can\'t find a ``nice\'\' definition. The problem is that for\n> a typical pseudo-Riemannian manifold, there are lots of ``nice\'\' definitions,\n> each with its own use, but no general reason to prefer one over another.\n\nMy question is one level of complexity above defining simultaneity\nglobally; I don\'t have a geometric definition for you. I\'m relying on\nintuition that is based on the simplest model of pseudo-Riemannian\nspacetime known to exist. That\'s the spacetime cylinder SxR where\nspace S is just a circle.\n\nWhen special relativity is derived on a circle, and consequently on\na sphere or hypersphere also, because of this constraint, spacetime\ndisplays physically distinguished frames of reference and a restricted\nprinciple of relativity. A natural, unavoidable, physically\ndistinguished definition of universal clock synchronization appears.\nAs a consequence, in all but one "inertial frame of reference," the\nspeed of light is physically, intrinsically and objectively\nanisotropic. The end result is "mathematically allowable\ninstantaneousness," i.e., the observation that, in a very physically\nmeaningful way, events that are simultaneous for one observer are\nsimultaneous for all.\n\nI suppose that my first question should be as easy as possible.\nIs there any other way of defining simultaneity for all frames of\nreference in SxR to arrive at some other fundamentally different\ndefinition of mathematically allowable instantaneousness?\n\nhttp://www.everythingimportant.org/viewtopic.php?t=79\nhttp://www.everythingimportant.org/viewtopic.php?t=605\nhttp://cornell.mirror.aps.org/abstract/PRD/v8/i6/p1662_1\nhttp://arxiv.org/PS_cache/gr-qc/pdf/0101/0101014.pdf\nhttp://qcd.th.u-psud.fr/page_perso/Uzan/fileps/art_2002_ullp_ejp23.pdf\n\nEugene Shubert\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>carlip@no-physics-spam.ucdavis.edu wrote in message news:<cd1433$pv8$2@woodrow.ucdavis.edu>...
> Perfectly Innocent <perfectlyInnocent@as-if.com> wrote:
>
> > Are there pseudo-Riemannian spacetimes having the property that
> > no logical contradiction exists in the mere conjecture of an
> > instantaneous comparison of time rates for all observers?
>
> I'm not sure I understand the question. Are you asking whether
> there are spacetimes that admit a well-defined time slicing,
> that is, a collection of ``hypersurfaces of simultaneity'' that
> can be unambiguously defined?

> If so, the answer is yes -- but there are many such slicings,
> which differ from each other as to what events are considered
> simultaneous, and there seems to be no particular physical reason
> to prefer any one over another. For example:
>
> [snip] [snip] [snip] ...
>
> In short: the problem isn't that one can't define simultaneity. It's not
> even that one can't find a ``nice'' definition. The problem is that for
> a typical pseudo-Riemannian manifold, there are lots of ``nice'' definitions,
> each with its own use, but no general reason to prefer one over another.

My question is one level of complexity above defining simultaneity
globally; I don't have a geometric definition for you. I'm relying on
intuition that is based on the simplest model of pseudo-Riemannian
spacetime known to exist. That's the spacetime cylinder SxR where
space S is just a circle.

When special relativity is derived on a circle, and consequently on
a sphere or hypersphere also, because of this constraint, spacetime
displays physically distinguished frames of reference and a restricted
principle of relativity. A natural, unavoidable, physically
distinguished definition of universal clock synchronization appears.
As a consequence, in all but one "inertial frame of reference," the
speed of light is physically, intrinsically and objectively
anisotropic. The end result is "mathematically allowable
instantaneousness," i.e., the observation that, in a very physically
meaningful way, events that are simultaneous for one observer are
simultaneous for all.

I suppose that my first question should be as easy as possible.
Is there any other way of defining simultaneity for all frames of
reference in SxR to arrive at some other fundamentally different
definition of mathematically allowable instantaneousness?

http://www.everythingimportant.org/viewtopic.php?t=79
http://www.everythingimportant.org/viewtopic.php?t=605
http://cornell.mirror.aps.org/abstract/PRD/v8/i6/p1662_1
http://arxiv.org/PS_cache/gr-qc/pdf/0101/0101014.pdf
http://qcd.th.u-psud.fr/page_perso/Uzan/fileps/art_2002_ullp_ejp23.pdf

Eugene Shubert

carlip@no-physics-spam.ucdavis.edu

Aug12-04, 08:29 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nPerfectly Innocent <perfectlyInnocent@as-if.com> wrote:\n\n> My question is one level of complexity above defining simultaneity\n> globally; I don\'t have a geometric definition for you. I\'m relying on\n> intuition that is based on the simplest model of pseudo-Riemannian\n> spacetime known to exist. That\'s the spacetime cylinder SxR where\n> space S is just a circle.\n\nAssuming you mean a *flat* SxR, that\'s too simple a model -- its\nsymmetries are much too strong to allow a generalization to less\nsymmetric situations. Try thinking about SxR with curvature, or\neven flat T^2xR (T^2 is the two-torus), and you\'ll see that there\nis no unique or obviously preferred global time-slicing.\n\n> When special relativity is derived on a circle, and consequently on\n> a sphere or hypersphere also,\n\nI don\'t understand your ``consequently\'\' here. S^2xR is not flat,\nand special relativity shouldn\'t hold at all.\n\n> because of this constraint, spacetime displays physically distinguished\n> frames of reference and a restricted principle of relativity.\n> A natural, unavoidable, physically distinguished definition of\n> universal clock synchronization appears.\n\nFor flat SxR this is true (except for the word "unavoidable" --\nthere is nothing in principle to prevent an "unnatural" choice).\nBut as soon as you go to higher dimensions or more complicated\ngeometries, this claim is no longer correct.\n\n[...]\n> I suppose that my first question should be as easy as possible.\n> Is there any other way of defining simultaneity for all frames of\n> reference in SxR to arrive at some other fundamentally different\n> definition of mathematically allowable instantaneousness?\n\nSure, though the alternatives are ugly. Pick some arbitrary everywhere\nspacelike closed curve around S^1, and define all points on that curve\nas being simultaneous. Translate the curve parallel to itself up\nand down the R axis -- this is well-defined, because the spacetime\nis flat -- and you will have a collection of closed spacelike curves\nthat foliate SxR. These give you a consistent mathematically allowable\ndefinition of instantaneousness.\n\nSteve Carlip\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Perfectly Innocent <perfectlyInnocent@as-if.com> wrote:

> My question is one level of complexity above defining simultaneity
> globally; I don't have a geometric definition for you. I'm relying on
> intuition that is based on the simplest model of pseudo-Riemannian
> spacetime known to exist. That's the spacetime cylinder SxR where
> space S is just a circle.

Assuming you mean a *flat* SxR, that's too simple a model -- its
symmetries are much too strong to allow a generalization to less
symmetric situations. Try thinking about SxR with curvature, or
even flat T^{2xR} (T^2 is the two-torus), and you'll see that there
is no unique or obviously preferred global time-slicing.

> When special relativity is derived on a circle, and consequently on
> a sphere or hypersphere also,

I don't understand your ``consequently'' here. S^{2xR} is not flat,
and special relativity shouldn't hold at all.

> because of this constraint, spacetime displays physically distinguished
> frames of reference and a restricted principle of relativity.
> A natural, unavoidable, physically distinguished definition of
> universal clock synchronization appears.

For flat SxR this is true (except for the word "unavoidable" --
there is nothing in principle to prevent an "unnatural" choice).
But as soon as you go to higher dimensions or more complicated
geometries, this claim is no longer correct.

[...]
> I suppose that my first question should be as easy as possible.
> Is there any other way of defining simultaneity for all frames of
> reference in SxR to arrive at some other fundamentally different
> definition of mathematically allowable instantaneousness?

Sure, though the alternatives are ugly. Pick some arbitrary everywhere
spacelike closed curve around S^1, and define all points on that curve
as being simultaneous. Translate the curve parallel to itself up
and down the R axis -- this is well-defined, because the spacetime
is flat -- and you will have a collection of closed spacelike curves
that foliate SxR. These give you a consistent mathematically allowable
definition of instantaneousness.

Steve Carlip