Re: What is the Physical Significance of the Metric Tensor of the Text

Arnold Neumaier

Jul28-04, 11:11 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\nFranz Heymann wrote:\n>\n> Time really *is* different from space. That difference is formalised\n> in the negative sign associated with the time component of the metric.\n> If it had not been there, the situation would have been so symmetrical\n> that it would have been impossible to say which of the four components\n> referred to time.\n\nEven so, it is impossible to say what is time, since SR is Lorentz\ninvariant. Time and space cannot be separated naturally without choosing\nfirst a preferred timelike vector to indicate the time direction.\nOnly the future/past relation,\nx later than y iff d=x-y satisfies d_0>sqrt(d_1^2+d_2^2+d_3^2),\nhas invariant physical significance, but it is a partial order\n(on spacetime points) only.\n\n\nArnold Neumaier\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>Franz Heymann wrote:
>
> Time really *is* different from space. That difference is formalised
> in the negative sign associated with the time component of the metric.
> If it had not been there, the situation would have been so symmetrical
> that it would have been impossible to say which of the four components
> referred to time.

Even so, it is impossible to say what is time, since SR is Lorentz
invariant. Time and space cannot be separated naturally without choosing
first a preferred timelike vector to indicate the time direction.
Only the future/past relation,
x later than y iff d=x-y satisfies d_0>\sqrt(d_1^2+d_2^2+d_3^2),
has invariant physical significance, but it is a partial order
(on spacetime points) only.

Arnold Neumaier

Danny Ross Lunsford

Jul29-04, 02:04 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>\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4107AB26.6070100@univie.ac.at>...\n> Franz Heymann wrote:\n> >\n> > Time really *is* different from space. That difference is formalised\n> > in the negative sign associated with the time component of the metric.\n> > If it had not been there, the situation would have been so symmetrical\n> > that it would have been impossible to say which of the four components\n> > referred to time.\n>\n> Even so, it is impossible to say what is time, since SR is Lorentz\n> invariant. Time and space cannot be separated naturally without choosing\n> first a preferred timelike vector to indicate the time direction.\n\nBut this is why we have the parameter C! That is, the *overall* lost\nisotropy due to mixed + and - dimensions instantly kicks out a fixed\nreal parameter relating the measurement scale on each type of axis.\nSo, it\'s not that some spacetime directions are "preferred", it\'s just\nthat there are two classes of them.\n\nThere must be a projective argument lying around in which Euclidean\nspace is also characterized by a constant, and it\'s probably the one\nappearing in the Caley distance formula\n\nD = k log (cross ratio)\n\nIn Euclidean geometry the referent quadric is the "plane at infinity"\nand k is imaginary.\n\n-drl\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4107AB26.6070100@univie.ac.at>...
> Franz Heymann wrote:
> >
> > Time really *is* different from space. That difference is formalised
> > in the negative sign associated with the time component of the metric.
> > If it had not been there, the situation would have been so symmetrical
> > that it would have been impossible to say which of the four components
> > referred to time.
>
> Even so, it is impossible to say what is time, since SR is Lorentz
> invariant. Time and space cannot be separated naturally without choosing
> first a preferred timelike vector to indicate the time direction.

But this is why we have the parameter C! That is, the *overall* lost
isotropy due to mixed + and - dimensions instantly kicks out a fixed
real parameter relating the measurement scale on each type of axis.
So, it's not that some spacetime directions are "preferred", it's just
that there are two classes of them.

There must be a projective argument lying around in which Euclidean
space is also characterized by a constant, and it's probably the one
appearing in the Caley distance formula

D = k log (cross ratio)

In Euclidean geometry the referent quadric is the "plane at infinity"
and k is imaginary.

-drl

Arnold Neumaier

Jul29-04, 03:27 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>\nDanny Ross Lunsford wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4107AB26.6070100@univie.ac.at>...\n>\n>>Franz Heymann wrote:\n>>\n>>>Time really *is* different from space. That difference is formalised\n>>>in the negative sign associated with the time component of the metric.\n>>>If it had not been there, the situation would have been so symmetrical\n>>>that it would have been impossible to say which of the four components\n>>>referred to time.\n>>\n>>Even so, it is impossible to say what is time, since SR is Lorentz\n>>invariant. Time and space cannot be separated naturally without choosing\n>>first a preferred timelike vector to indicate the time direction.\n>\n> But this is why we have the parameter C! That is, the *overall* lost\n> isotropy due to mixed + and - dimensions instantly kicks out a fixed\n> real parameter relating the measurement scale on each type of axis.\n> So, it\'s not that some spacetime directions are "preferred", it\'s just\n> that there are two classes of them.\n\nNot quite. In fact, c should be any timelike 4-vector, and position q\n(in the 3-space orthogonal to c) and time t of a spacetime point are\ncomputed from\nt = c dot x/c^2, q = x - tc\n(for the +--- metric). For the preferred vector c = (c_0 0 0 0)^T,\nwhere c_0 is the speed of light, we get the standard form.\n\n\nArnold Neumaier\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>Danny Ross Lunsford wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4107AB26.6070100@univie.ac.at>...
>
>>Franz Heymann wrote:
>>
>>>Time really *is* different from space. That difference is formalised
>>>in the negative sign associated with the time component of the metric.
>>>If it had not been there, the situation would have been so symmetrical
>>>that it would have been impossible to say which of the four components
>>>referred to time.
>>
>>Even so, it is impossible to say what is time, since SR is Lorentz
>>invariant. Time and space cannot be separated naturally without choosing
>>first a preferred timelike vector to indicate the time direction.
>
> But this is why we have the parameter C! That is, the *overall* lost
> isotropy due to mixed + and - dimensions instantly kicks out a fixed
> real parameter relating the measurement scale on each type of axis.
> So, it's not that some spacetime directions are "preferred", it's just
> that there are two classes of them.

Not quite. In fact, c should be any timelike 4-vector, and position q
(in the 3-space orthogonal to c) and time t of a spacetime point are
computed from
t = c dot x/c^2, q = x - tc
(for the +--- metric). For the preferred vector c = (c_00)^T,
where c_0 is the speed of light, we get the standard form.

Arnold Neumaier

Franz Heymann

Jul31-04, 10:15 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"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> wrote in message\nnews:41094EA8.2090805@univie.ac.at...\n>\n> Danny Ross Lunsford wrote:\n> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message\nnews:<4107AB26.6070100@univie.ac.at>...\n> >\n> >>Franz Heymann wrote:\n> >>\n> >>>Time really *is* different from space. That difference is\nformalised\n> >>>in the negative sign associated with the time component of the\nmetric.\n> >>>If it had not been there, the situation would have been so\nsymmetrical\n> >>>that it would have been impossible to say which of the four\ncomponents\n> >>>referred to time.\n> >>\n> >>Even so, it is impossible to say what is time, since SR is Lorentz\n> >>invariant. Time and space cannot be separated naturally without\nchoosing\n> >>first a preferred timelike vector to indicate the time direction.\n> >\n> > But this is why we have the parameter C! That is, the *overall*\nlost\n> > isotropy due to mixed + and - dimensions instantly kicks out a\nfixed\n> > real parameter relating the measurement scale on each type of\naxis.\n> > So, it\'s not that some spacetime directions are "preferred", it\'s\njust\n> > that there are two classes of them.\n>\n> Not quite. In fact, c should be any timelike 4-vector, and position\nq\n> (in the 3-space orthogonal to c) and time t of a spacetime point are\n> computed from\n> t = c dot x/c^2, q = x - tc\n> (for the +--- metric). For the preferred vector c = (c_0 0 0 0)^T,\n> where c_0 is the speed of light, we get the standard form.\n\nYou seem to be making heavy weather of something quite\nstraightforward:\nThe quantity c is simply an invariant which is the conversion\ncoefficient between the units of space and the units of time. It is\nnot a vector at all.\n\nFranz\n\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>"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> wrote in message
news:41094EA8.2090805@univie.ac.at...
>
> Danny Ross Lunsford wrote:
> > Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message
news:<4107AB26.6070100@univie.ac.at>...
> >
> >>Franz Heymann wrote:
> >>
> >>>Time really *is* different from space. That difference is
formalised
> >>>in the negative sign associated with the time component of the
metric.
> >>>If it had not been there, the situation would have been so
symmetrical
> >>>that it would have been impossible to say which of the four
components
> >>>referred to time.
> >>
> >>Even so, it is impossible to say what is time, since SR is Lorentz
> >>invariant. Time and space cannot be separated naturally without
choosing
> >>first a preferred timelike vector to indicate the time direction.
> >
> > But this is why we have the parameter C! That is, the *overall*
lost
> > isotropy due to mixed + and - dimensions instantly kicks out a
fixed
> > real parameter relating the measurement scale on each type of
axis.
> > So, it's not that some spacetime directions are "preferred", it's
just
> > that there are two classes of them.
>
> Not quite. In fact, c should be any timelike 4-vector, and position
q
> (in the 3-space orthogonal to c) and time t of a spacetime point are
> computed from
> t = c dot x/c^2, q = x - tc
> (for the +--- metric). For the preferred vector c = (c_00)^T,
> where c_0 is the speed of light, we get the standard form.

You seem to be making heavy weather of something quite
straightforward:
The quantity c is simply an invariant which is the conversion
coefficient between the units of space and the units of time. It is
not a vector at all.

Franz