<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>Hi Physicists,\nThis is a correction to the previous post. Here is what I meant to\nwrite for the two different Eta n_(i,j):\n\n| 1 0 0 0 |\n| 0 1 0 0 |\nn = | 0 0 1 0 |\n| 0 0 0 -1 |\n\n[found in G.L. Naber\'s book.]\n\n| 1 0 0 0 |\n| 0 -1 0 0 |\nn = | 0 0 -1 0 |\n| 0 0 0 -1 |\n\n[found in Penrose and Pimler\'s book.]\n\nThe different Eta\'s aren\'t just -1 times each other, which is what is\ncausing me confusion.\n\nSo I\'ll repost my questions:\n1. Does that reverse the convention of which vectors are time like and\nwhich vectors are space like?\n2. Which convention is the modern one (i.e. which one is found in\ncurrent literature)?\n\nSo overall, I\'m thinking that this will do something odd like require\nyou to consider the inverse of a Minkowski space when switching from\nconvention to the other. I am probably totally off, but maybe someone\ncan clue me in to why its acceptable either way.\n\nYour buddy,\nAndy\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>Hi Physicists,
This is a correction to the previous post. Here is what I meant to
write for the two different \Eta n_(i,j):
| 1 |
| 1 |
n = | 1 |
| -1 |
[found in G.L. Naber's book.]
| 1 |
| -1 |
n = |-1 |
| -1 |
[found in Penrose and Pimler's book.]
The different \Eta's aren't just -1 times each other, which is what is
causing me confusion.
So I'll repost my questions:
1. Does that reverse the convention of which vectors are time like and
which vectors are space like?
2. Which convention is the modern one (i.e. which one is found in
current literature)?
So overall, I'm thinking that this will do something odd like require
you to consider the inverse of a Minkowski space when switching from
convention to the other. I am probably totally off, but maybe someone
can clue me in to why its acceptable either way.
Your buddy,
Andy
davidoff404
Jul2-04, 05:32 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>\nAndy Y wrote:\n\n> Hi Physicists,\n> This is a correction to the previous post. Here is what I meant to\n> write for the two different Eta n_(i,j):\n>\n> | 1 0 0 0 |\n> | 0 1 0 0 |\n> n = | 0 0 1 0 |\n> | 0 0 0 -1 |\n>\n> [found in G.L. Naber\'s book.]\n>\n> | 1 0 0 0 |\n> | 0 -1 0 0 |\n> n = | 0 0 -1 0 |\n> | 0 0 0 -1 |\n>\n> [found in Penrose and Pimler\'s book.]\n>\n> The different Eta\'s aren\'t just -1 times each other, which is what is\n> causing me confusion.\n>\n\nThere are two different conventions, depending on what branch of physics\nyou work in. Particle physicists usually take the Minkowski metric to\nhave the following form\n\n| n_00 n_01 n_02 n_03 | | 1 0 0 0 |\n| n_10 n_11 n_12 n_13 | | 0 -1 0 0 |\nn = | n_20 n_21 n_22 n_23 | = | 0 0 -1 0 |\n| n_30 n_31 n_32 n_33 | | 0 0 0 -1 |\n\nwhereas differential geometers and those working with classical\nrelativity use\n\n| -1 0 0 0 |\n| 0 1 0 0 |\nn = | 0 0 1 0 |\n| 0 0 0 1 |\n\nYou\'ll usually find this metric denoted in the literature by\nn=diag(-1,1,1,1) or n=(-+++), and is considered to be the more \'modern\'\nconvention. In older texts the convention is usually to take the\nspacetime metric to be of signature n=(+---); I believe this represents\na reluctance on the part of authors to associate a negative sign with\nthe time component of the metric -- they considered it to be more\nnatural to let the time component be positive and the spatial components\nbe negative. As far as I know, this practice began to decline with the\nintroduction of spacetime diagrams and Synge\'s book. Of course, it is\nonly a convention and doesn\'t detract from the physics in any way.\n\n> So I\'ll repost my questions:\n> 1. Does that reverse the convention of which vectors are time like and\n> which vectors are space like?\n\nYes. It may be helpful for you to consider what a metric actually *is*,\napart from taking the form of a matrix. Given a four-dimensional\nmanifold we introduce a metric *tensor field* g. A tensor is nothing\nmore than a linear mapping of vectors to the real number line, and the\n"field" bit means that this tensor is defined over the whole (or most\nof) the manifold. As a case in point, the metric tensor field is\nrequired to be a symmetric rank 2 tensor field -- what this means is\nthat you can think of the metric as a function which takes two vectors\nas arguments and spits out a real number. The "symmetric" part of the\ndefinition simply means that g(U,V = g(V,U) for any vectors U,V.\n\nIf we say that the metric has signature n=(-+++) for example, we can\nthen divide all of the vectors on a manifold into three classes\ndepending on how their scalar product behaves:\n\ng(U,U) < 0 means U is timelike,\ng(U,U) = 0 means U is null,\ng(U,U) > 0 means U is spacelike.\n\nIf, on the other hand, we started with a metric of signature n=(+---),\nwe would divide the vectors into the three classes as follows:\n\ng(U,U) < 0 means U is spacelike,\ng(U,U) = 0 means U is null,\ng(U,U) > 0 means U is timelike.\n\nSo you see that the convention is reversed depending on which definition\nof the metric you take. The important thing is that you decide on a\nparticular convention and then stick with it.\n\n> 2. Which convention is the modern one (i.e. which one is found in\n> current literature)?\n>\n\nAgain, it depends what area you\'re working in. Modern relativity and\ndifferential geometry applied to relativity almost always uses the\nconvention that n=(-+++) or, in the case of a more general spacetime\nmetric, g=(-+++).\n\nAs an aside, you may find Sean Carroll\'s notes on general relativity to\nbe a help. You can find them at the link below.\n\nhttp://pancake.uchicago.edu/~carroll/notes/\n\ndavidoff\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>Andy Y wrote:
> Hi Physicists,
> This is a correction to the previous post. Here is what I meant to
> write for the two different \Eta n_(i,j):
>
> | 1 |
> | 1 |
> n = | 1 |
> | -1 |
>
> [found in G.L. Naber's book.]
>
> | 1 |
> | -1 |
> n = |-1 |
> | -1 |
>
> [found in Penrose and Pimler's book.]
>
> The different \Eta's aren't just -1 times each other, which is what is
> causing me confusion.
>
There are two different conventions, depending on what branch of physics
you work in. Particle physicists usually take the Minkowski metric to
have the following form
whereas differential geometers and those working with classical
relativity use
| -1 |
| 1 |
n = | 1 |
| 1 |
You'll usually find this metric denoted in the literature by
n=diag(-1,1,1,1) or n=(-+++), and is considered to be the more 'modern'
convention. In older texts the convention is usually to take the
spacetime metric to be of signature n=(+---); I believe this represents
a reluctance on the part of authors to associate a negative sign with
the time component of the metric -- they considered it to be more
natural to let the time component be positive and the spatial components
be negative. As far as I know, this practice began to decline with the
introduction of spacetime diagrams and Synge's book. Of course, it is
only a convention and doesn't detract from the physics in any way.
> So I'll repost my questions:
> 1. Does that reverse the convention of which vectors are time like and
> which vectors are space like?
Yes. It may be helpful for you to consider what a metric actually *is*,
apart from taking the form of a matrix. Given a four-dimensional
manifold we introduce a metric *tensor field* g. A tensor is nothing
more than a linear mapping of vectors to the real number line, and the
"field" bit means that this tensor is defined over the whole (or most
of) the manifold. As a case in point, the metric tensor field is
required to be a symmetric rank 2 tensor field -- what this means is
that you can think of the metric as a function which takes two vectors
as arguments and spits out a real number. The "symmetric" part of the
definition simply means that g(U,V = g(V,U) for any vectors U,V.
If we say that the metric has signature n=(-+++) for example, we can
then divide all of the vectors on a manifold into three classes
depending on how their scalar product behaves:
g(U,U) < means U is timelike,
g(U,U) = means U is null,
g(U,U) > means U is spacelike.
If, on the other hand, we started with a metric of signature n=(+---),
we would divide the vectors into the three classes as follows:
g(U,U) < means U is spacelike,
g(U,U) = means U is null,
g(U,U) > means U is timelike.
So you see that the convention is reversed depending on which definition
of the metric you take. The important thing is that you decide on a
particular convention and then stick with it.
> 2. Which convention is the modern one (i.e. which one is found in
> current literature)?
>
Again, it depends what area you're working in. Modern relativity and
differential geometry applied to relativity almost always uses the
convention that n=(-+++) or, in the case of a more general spacetime
metric, g=(-+++).
As an aside, you may find Sean Carroll's notes on general relativity to
be a help. You can find them at the link below.
http://pancake.uchicago.edu/~carroll/notes/
davidoff
carlip@no-physics-spam.ucdavis.edu
Jul4-04, 08:40 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>\nAndy Y <andy314159pi@yahoo.com> wrote:\n> Hi Physicists,\n> This is a correction to the previous post. Here is what I meant to\n> write for the two different Eta n_(i,j):\n\n> | 1 0 0 0 |\n> | 0 1 0 0 |\n> n = | 0 0 1 0 |\n> | 0 0 0 -1 |\n\n> [found in G.L. Naber\'s book.]\n\n> | 1 0 0 0 |\n> | 0 -1 0 0 |\n> n = | 0 0 -1 0 |\n> | 0 0 0 -1 |\n\n> [found in Penrose and Pimler\'s book.]\n\n> The different Eta\'s aren\'t just -1 times each other, which is what is\n> causing me confusion.\n\nThey\'re almost -1 times each other. The only other change is the position\nof the time coordinate. Physicists almost universally use coordinates\nx^0, x^1, x^2, x^3, where x^0 is time. This is the convention in your\nsecond metric. Mathematicians occasionally prefer x^1, x^2, x^3, x^4,\nwith x^4 time. That\'s the convention in your first metric. (You can\ntell which entry represents time, because it comes in with the opposite\nsign from the spatial coordinates.)\n\nSo just swap x^4 and x^0, and refer back to the answers to your first\npost.\n\n(In physics, this *is* standardized -- time is almost always x^0 in the\ncurrent literature. I\'m quite surprised that Naber would use a different,\nvery old-fashioned convention.)\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>Andy Y <andy314159pi@yahoo.com> wrote:
> Hi Physicists,
> This is a correction to the previous post. Here is what I meant to
> write for the two different \Eta n_(i,j):
> | 1 |
> | 1 |
> n = | 1 |
> | -1 |
> [found in G.L. Naber's book.]
> | 1 |
> | -1 |
> n = |-1 |
> | -1 |
> [found in Penrose and Pimler's book.]
> The different \Eta's aren't just -1 times each other, which is what is
> causing me confusion.
They're almost -1 times each other. The only other change is the position
of the time coordinate. Physicists almost universally use coordinates
x^0, x^1, x^2, x^3, where x^0 is time. This is the convention in your
second metric. Mathematicians occasionally prefer x^1, x^2, x^3, x^4,
with x^4 time. That's the convention in your first metric. (You can
tell which entry represents time, because it comes in with the opposite
sign from the spatial coordinates.)
So just swap x^4 and x^0, and refer back to the answers to your first
post.
(In physics, this *is* standardized -- time is almost always x^0 in the
current literature. I'm quite surprised that Naber would use a different,
very old-fashioned convention.)