PATP [Time dilation in R-W Metric?]

tessel@tum.bot

<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>On Thu, 6 Jan 2005, E. Hardy wrote:\n\n> The following question has arisen in an interdisciplinary application:\n\n<suspicious_frown>\n\nOh?\n\n</suspicious_frown>\n\n> Assuming a cosmological model with a constant mass density (motionless\n> dust):\n\nDo you mean a "Friedmann dust solution written in a comoving chart"\n(zero Lambda)? Or the "Einstein static dust" (nonzero Lambda)?\n\nIf you don\'t know, I urgently recommend this book:\n\nauthor = {Ray D\'Inverno},\ntitle = {Introducing {E}instein\'s Relativity},\npublisher = {Clarendon Press},\nyear = 1995}\n\n> Will a clock run at a different rate if the mass density parameter is\n> varied?\n\nAre we discussing gtr?\n\n(That\'s the default assumption unless you specify otherwise, in matters of\ngravity!)\n\nIf so, what do you mean by "vary the mass density parameter"?\n\nIn the Friedmann dust models, the mass density varies -in time-, but not\n"in space" (provided that we slice up the spacetime in the obvious way,\ninto spacelike hyperslices which are everywhere orthogonal to the world\nlines of the dusts; the RW charts are just one way of expresssing the fact\nthat in this case, these hyperslices all have S^3 geometry, or all have\nE^3 geometry, or all have H^3 geometry, depending on the parameters of the\nFriedmann dust model).\n\nIn the Einstein static model, the attraction of the dust is precisely\nmatched by the repulsion due to the Lambda term. This is of course an\nunstable equilibrium, and is one reason why this solution was quickly\nabandoned as a viable candidate. (It didn\'t hurt that people suddenly\nrealized that Friedmann\'s expanding models, at that time on of the few\ncompetitors to Einstein\'s model, provided a brilliant explanation of the\nrecent observations of Hubble.)\n\nThere are many exact dust solutions in gtr which can be employed as\ncosmological models, including inhomogeneous dusts. These would allow for\ndust densities varying from place to place (again slicing up the spacetime\nin the obvious way, into the hyperslices orthogonal to the world lines of\nthe dust). There are also "rotating" dust solutions, but in these, the\nworld lines of the dust do not form a hypersurface orthogonal congruence,\nso it is trickier to discuss "spaces at a time".\n\nAlso, when you say "clocks run at a different rate", of course this is\nnonsense and neither str nor gtr claim that any such thing ever happens.\nRather, they predict red shifts in certain situations (amply confirmed by\nobservations and experiments), including so called "gravitational red\nshift". But this arises from the geodesic divergence of null geodesics as\nthey climb out of a potential well, if you will, not with ideal clocks\nsomehow running at different rates. See for example\n\nauthor = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},\ntitle = {Gravitation},\npublisher = {W. H. Freeman},\nyear = 1970}\n\nwhich also discusses nonideal clocks, which -can- run at different rates,\ndue to engineering limitations, not fundamental physics.\n\n> I.e.: Suppose we have two identical Universes one with mass density A\n> and another with mass density B. Will the the clocks appear to run at\n> different rates in the two Universes?\n\nYou\'ll have to clarify your question before we can answer it.\n\n"T. Essel" (hiding somewhere in cyberspace)\n\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>On Thu, 6 Jan 2005, E. Hardy wrote:

> The following question has arisen in an interdisciplinary application:

[tex]<suspicious_frown>[/tex]

Oh?

[tex]</suspicious_frown>[/tex]

> Assuming a cosmological model with a constant mass density (motionless
> dust):

Do you mean a "Friedmann dust solution written in a comoving chart"
(zero [itex]\Lambda)[/itex]? Or the "Einstein static dust" (nonzero [itex]\Lambda)[/itex]?

If you don't know, I urgently recommend this book:

author = {Ray D'Inverno},
title = {Introducing {E}instein's Relativity},
publisher = {Clarendon Press},
year [itex]= 1995}[/itex]

> Will a clock run at a different rate if the mass density parameter is
> varied?

Are we discussing gtr?

(That's the default assumption unless you specify otherwise, in matters of
gravity!)

If so, what do you mean by "vary the mass density parameter"?

In the Friedmann dust models, the mass density varies -in time-, but not
"in space" (provided that we slice up the spacetime in the obvious way,
into spacelike hyperslices which are everywhere orthogonal to the world
lines of the dusts; the RW charts are just one way of expresssing the fact
that in this case, these hyperslices all have [itex]S^3[/itex] geometry, or all have
[itex]E^3[/itex] geometry, or all have [itex]H^3[/itex] geometry, depending on the parameters of the
Friedmann dust model).

In the Einstein static model, the attraction of the dust is precisely
matched by the repulsion due to the [itex]\Lambda[/itex] term. This is of course an
unstable equilibrium, and is one reason why this solution was quickly
abandoned as a viable candidate. (It didn't hurt that people suddenly
realized that Friedmann's expanding models, at that time on of the few
competitors to Einstein's model, provided a brilliant explanation of the
recent observations of Hubble.)

There are many exact dust solutions in gtr which can be employed as
cosmological models, including inhomogeneous dusts. These would allow for
dust densities varying from place to place (again slicing up the spacetime
in the obvious way, into the hyperslices orthogonal to the world lines of
the dust). There are also "rotating" dust solutions, but in these, the
world lines of the dust do not form a hypersurface orthogonal congruence,
so it is trickier to discuss "spaces at a time".

Also, when you say "clocks run at a different rate", of course this is
nonsense and neither str nor gtr claim that any such thing ever happens.
Rather, they predict red shifts in certain situations (amply confirmed by
observations and experiments), including so called "gravitational red
shift". But this arises from the geodesic divergence of null geodesics as
they climb out of a potential well, if you will, not with ideal clocks
somehow running at different rates. See for example

author = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},
title = {Gravitation},
publisher [itex]= {W[/itex]. H. Freeman},
year [itex]= 1970}[/itex]

which also discusses nonideal clocks, which -can- run at different rates,
due to engineering limitations, not fundamental physics.

> I.e.: Suppose we have two identical Universes one with mass density A
> and another with mass density B. Will the the clocks appear to run at
> different rates in the two Universes?

You'll have to clarify your question before we can answer it.

"T. Essel" (hiding somewhere in cyberspace)

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