View Full Version : Order of indices in tensors...
Flip Tomato
Aug24-04, 04:57 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>Hello--I\'m doing some intro-GR (using Carroll\'s new book), and I am confused\nabout a subtle point:\n\nWhat is the significance of the order of indices in a tensor? I understand\nthat the convention is for upper indices to sum with lower indices and vice\nversa when the tensor acts on the appropriate object, however, what is the\nsignificance of having the upper index listed first or the lower index\nlisted first (horizontally)?\n\nThanks,\nFlip\nflipt (at) stanford\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>Hello--I'm doing some intro-GR (using Carroll's new book), and I am confused
about a subtle point:
What is the significance of the order of indices in a tensor? I understand
that the convention is for upper indices to sum with lower indices and vice
versa when the tensor acts on the appropriate object, however, what is the
significance of having the upper index listed first or the lower index
listed first (horizontally)?
Thanks,
Flip
flipt (at) stanford
David Park
Aug25-04, 02:42 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>"Flip Tomato" <flipt@stanford.edu> wrote in message\nnews:cgbmkh\\$2rv\\$1@news.Stanford.EDU.. .\n> Hello--I\'m doing some intro-GR (using Carroll\'s new book), and I am\nconfused\n> about a subtle point:\n>\n> What is the significance of the order of indices in a tensor? I understand\n> that the convention is for upper indices to sum with lower indices and\nvice\n> versa when the tensor acts on the appropriate object, however, what is the\n> significance of having the upper index listed first or the lower index\n> listed first (horizontally)?\n>\n> Thanks,\n> Flip\n> flipt (at) stanford\n>\n\nFor dummy indices it doesn\'t matter which comes first. You could premultiple\nby the up metric tensor and the down metric tensor and thus raise the down\nindex and lower the up index. But the two metric tensors are just the\nidentity - so it makes no significant change. Hence you can swap the up/down\npositions of dummy indices changing their "horizontal" order.\n\nFor differentiated dummy indices you have to be careful because raising or\nlowering an index does not generally commute with partial differentiation,\nbecause the partial derivative of the metric is not generally zero. However,\nif a dummy index is a covariant derivative index you can raise and lower it\nbecause the covariant derivative of the metric is zero.\n\nSometimes dummy indices are shown one over the other because it doesn\'t\nmatter which comes first. The same is often done with the Kronecker. For\ncomputer implementation of tensors it is much easier and clearer to have a\none index - one (up/down) slot convention. (Otherwise, if you lower, say a\nnondummy index, do you put it before or after the other down indices?)\n\nI hope to see other comment on this by better experts.\n\nDavid Park\ndjmp@earthlink.net\nhttp://home.earthlink.net/~djmp/\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>"Flip Tomato" <flipt@stanford.edu> wrote in message
news:cgbmkh$2rv$1@news.Stanford.EDU...
> Hello--I'm doing some intro-GR (using Carroll's new book), and I am
confused
> about a subtle point:
>
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and
vice
> versa when the tensor acts on the appropriate object, however, what is the
> significance of having the upper index listed first or the lower index
> listed first (horizontally)?
>
> Thanks,
> Flip
> flipt (at) stanford
>
For dummy indices it doesn't matter which comes first. You could premultiple
by the up metric tensor and the down metric tensor and thus raise the down
index and lower the up index. But the two metric tensors are just the
identity - so it makes no significant change. Hence you can swap the up/down
positions of dummy indices changing their "horizontal" order.
For differentiated dummy indices you have to be careful because raising or
lowering an index does not generally commute with partial differentiation,
because the partial derivative of the metric is not generally zero. However,
if a dummy index is a covariant derivative index you can raise and lower it
because the covariant derivative of the metric is zero.
Sometimes dummy indices are shown one over the other because it doesn't
matter which comes first. The same is often done with the Kronecker. For
computer implementation of tensors it is much easier and clearer to have a
one index - one (up/down) slot convention. (Otherwise, if you lower, say a
nondummy index, do you put it before or after the other down indices?)
I hope to see other comment on this by better experts.
David Park
djmp@earthlink.net
http://home.earthlink.net/~djmp/
Marcus Wellpoth
Aug25-04, 02:44 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>Flip Tomato wrote:\n\n> Hello--I\'m doing some intro-GR (using Carroll\'s new book), and I am\n> confused about a subtle point:\n>\n> What is the significance of the order of indices in a tensor? I understand\n> that the convention is for upper indices to sum with lower indices and\n> vice versa when the tensor acts on the appropriate object, however, what\n> is the significance of having the upper index listed first or the lower\n> index listed first (horizontally)?\n>\n> Thanks,\n> Flip\n> flipt (at) stanford\n-------------------------------------------------------------------------------------\nIf you lift or lower a spatial component (1,2,3) the sign changes, the\nlowering or lifting of a time component (0) does change nothing.\nExamples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =\nA^{00}, A_{0}^{1} = A^{01}, ....\nHope that helps\nmw\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>Flip Tomato wrote:
> Hello--I'm doing some intro-GR (using Carroll's new book), and I am
> confused about a subtle point:
>
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and
> vice versa when the tensor acts on the appropriate object, however, what
> is the significance of having the upper index listed first or the lower
> index listed first (horizontally)?
>
> Thanks,
> Flip
> flipt (at) stanford
-------------------------------------------------------------------------------------
If you lift or lower a spatial component (1,2,3) the sign changes, the
lowering or lifting of a time component (0) does change nothing.
Examples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =A^{00}, A_{0}^{1} = A^{01}, ....
Hope that helps
mw
Rufus Anton
Aug25-04, 02:46 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>Tensors can have permutational symmetries. The symmetry with respect\nto a pair of indices is even if interchanging them changes nothing and\nodd when the tensor changes sign under the exchange. Of course, there\nneeds to be no such symmetry in which case you must take care of the\norder. This is the significance.\n\nNotice that in general (in the presence of a non-trivial metric) the\npermutational symmetry must be assessed with all indices either raised\nor lowered.\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>Tensors can have permutational symmetries. The symmetry with respect
to a pair of indices is even if interchanging them changes nothing and
odd when the tensor changes sign under the exchange. Of course, there
needs to be no such symmetry in which case you must take care of the
order. This is the significance.
Notice that in general (in the presence of a non-trivial metric) the
permutational symmetry must be assessed with all indices either raised
or lowered.
Igor Khavkine
Aug25-04, 02:46 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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh\\$2rv\\$1@news.Stanford.EDU>...\n> Hello--I\'m doing some intro-GR (using Carroll\'s new book), and I am confused\n> about a subtle point:\n>\n> What is the significance of the order of indices in a tensor? I understand\n> that the convention is for upper indices to sum with lower indices and vice\n> versa when the tensor acts on the appropriate object, however, what is the\n> significance of having the upper index listed first or the lower index\n> listed first (horizontally)?\n\nThe order of upper indices matters, the order of lower indices matters,\nthe order of the lower indices relative to the upper ones (and vice versa)\ndoes not matter. However, when you raise and lower indices using the metric\ntensor, you need to care where the new upper (lower) index will be placed\nrelative to the other upper (lower) indices. Sometimes people fix an\nabsolute order for all lower and upper indices to avoid ambiguity. this\nconvention can be denoted by leaving a blank right above or below each\nindex.\n\nThe reason the order of the indices matters at all is simple. A 2 index\ntensor a_{ij} can always be written as a linear combination of terms\nof the form b_i c_j, but clearly b_i c_j != c_i b_j.\n\nHope this helps.\n\nIgor\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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh$2rv$1@news.Stanford.EDU>...
> Hello--I'm doing some intro-GR (using Carroll's new book), and I am confused
> about a subtle point:
>
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and vice
> versa when the tensor acts on the appropriate object, however, what is the
> significance of having the upper index listed first or the lower index
> listed first (horizontally)?
The order of upper indices matters, the order of lower indices matters,
the order of the lower indices relative to the upper ones (and vice versa)
does not matter. However, when you raise and lower indices using the metric
tensor, you need to care where the new upper (lower) index will be placed
relative to the other upper (lower) indices. Sometimes people fix an
absolute order for all lower and upper indices to avoid ambiguity. this
convention can be denoted by leaving a blank right above or below each
index.
The reason the order of the indices matters at all is simple. A 2 index
tensor a_{ij} can always be written as a linear combination of terms
of the form b_i c_j, but clearly b_i c_j != c_i b_j.
Hope this helps.
Igor
Rene Meyer
Aug25-04, 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>\n\n"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh\\$2rv\\$1@news.Stanford.EDU>...\n> What is the significance of the order of indices in a tensor? I understand\n> that the convention is for upper indices to sum with lower indices and vice\n> versa when the tensor acts on the appropriate object, however, what is the\n> significance of having the upper index listed first or the lower index\n> listed first (horizontally)?\n\nHi Flip,\n\nyou should distinguish between two aspects:\n\n1) Tensors can be symmetric, antisymmetric or without any symmetry\nwithin their indices. For example, the metric tensor is symmetric, the\nLevi-Civita tensor antisymmetric in all indices. Can you write a\ntensor without special symmetry? [1]\n\n2) If you are dealing with tensors of order 2, i.e. with two indices,\nyou can write them down as a matrix, i.e. the metric tensor. Then, the\norder of writing of the indices also becomes important when you are\ncontracting tensors by doing matrix multiplication.\n\nRene.\n\n[1] T^abc_def = delta^a_d delta^b_e delta^c_f\n- delta^a_e delta^b_d delta^c_f\n- delta^a_f delta^b_d delta^c_e\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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh$2rv$1@news.Stanford.EDU>...
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and vice
> versa when the tensor acts on the appropriate object, however, what is the
> significance of having the upper index listed first or the lower index
> listed first (horizontally)?
Hi Flip,
you should distinguish between two aspects:
1) Tensors can be symmetric, antisymmetric or without any symmetry
within their indices. For example, the metric tensor is symmetric, the
Levi-Civita tensor antisymmetric in all indices. Can you write a
tensor without special symmetry? [1]
2) If you are dealing with tensors of order 2, i.e. with two indices,
you can write them down as a matrix, i.e. the metric tensor. Then, the
order of writing of the indices also becomes important when you are
contracting tensors by doing matrix multiplication.
Rene.
[1] T^{abc_def} = \delta^a_d \delta^b_e \delta^c_f- \delta^a_e \delta^b_d \delta^c_f- \delta^a_f \delta^b_d \delta^c_e
Danny Ross Lunsford
Aug26-04, 04:07 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"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh\\$2rv\\$1@news.Stanford.EDU>...\n> Hello--I\'m doing some intro-GR (using Carroll\'s new book), and I am confused\n> about a subtle point:\n>\n> What is the significance of the order of indices in a tensor? I understand\n> that the convention is for upper indices to sum with lower indices and vice\n> versa when the tensor acts on the appropriate object, however, what is the\n> significance of having the upper index listed first or the lower index\n> listed first (horizontally)?\n\nIt\'s not the order but the symmetry that is important. The order can\nbe thought to represent the (arbitrary) choice of independent\ndirections in a local frame, while the symmetry properties represent\npersistent geometrical facts about the objects they represent.\nParticularly important are totally antisymmetric tensors (changes sign\non interchange of any two indices). These represent primitive "space\nelements" - line segments, surface elements, volume elements etc.\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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh$2rv$1@news.Stanford.EDU>...
> Hello--I'm doing some intro-GR (using Carroll's new book), and I am confused
> about a subtle point:
>
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and vice
> versa when the tensor acts on the appropriate object, however, what is the
> significance of having the upper index listed first or the lower index
> listed first (horizontally)?
It's not the order but the symmetry that is important. The order can
be thought to represent the (arbitrary) choice of independent
directions in a local frame, while the symmetry properties represent
persistent geometrical facts about the objects they represent.
Particularly important are totally antisymmetric tensors (changes sign
on interchange of any two indices). These represent primitive "space
elements" - line segments, surface elements, volume elements etc.
-drl
branen1
Aug26-04, 04:29 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>Hi,\n\nperhaps I am missing the main point of your question, but the order of\nthe different upper/lower indices is first of all a matter of\nconvention. In the Riemann tensor for example you have to fix your\nconventions once and that\'s it.\nAfter that the order in a certain tensor usually really matters. Only\nin the case of further symmetries you can still interchange them\nwithout affecting the outcome!\n\nRegards,\n\nbranen1\n\n"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh\\$2rv\\$1@news.Stanford.EDU>...\n> Hello--I\'m doing some intro-GR (using Carroll\'s new book), and I am confused\n> about a subtle point:\n>\n> What is the significance of the order of indices in a tensor? I understand\n> that the convention is for upper indices to sum with lower indices and vice\n> versa when the tensor acts on the appropriate object, however, what is the\n> significance of having the upper index listed first or the lower index\n> listed first (horizontally)?\n>\n> Thanks,\n> Flip\n> flipt (at) stanford\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,
perhaps I am missing the main point of your question, but the order of
the different upper/lower indices is first of all a matter of
convention. In the Riemann tensor for example you have to fix your
conventions once and that's it.
After that the order in a certain tensor usually really matters. Only
in the case of further symmetries you can still interchange them
without affecting the outcome!
Regards,
branen1
"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh$2rv$1@news.Stanford.EDU>...
> Hello--I'm doing some intro-GR (using Carroll's new book), and I am confused
> about a subtle point:
>
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and vice
> versa when the tensor acts on the appropriate object, however, what is the
> significance of having the upper index listed first or the lower index
> listed first (horizontally)?
>
> Thanks,
> Flip
> flipt (at) stanford
Frank Hellmann
Aug26-04, 04:32 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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh\\$2rv\\$1@news.Stanford.EDU>...\n> Hello--I\'m doing some intro-GR (using Carroll\'s new book), and I am confused\n> about a subtle point:\n>\n> What is the significance of the order of indices in a tensor? I understand\n> that the convention is for upper indices to sum with lower indices and vice\n> versa when the tensor acts on the appropriate object, however, what is the\n> significance of having the upper index listed first or the lower index\n> listed first (horizontally)?\n>\n> Thanks,\n> Flip\n> flipt (at) stanford\n\nAs a general rule: Order matters unless proven otherwise.\nThat could be contractions or symmetries as others have pointed out,\nor more subtle stuff, but unless proven otherwise order matters, even\nbetween "up" and "down" indices!\nA_mu^nu = g_mu_lambda A^lambda^nu\nA^nu_mu = g_mu_lambda A^nu^lambda\n\nso unless A is symmetric you need to keep track!\n\n---\nfrank\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>"Flip Tomato" <flipt@stanford.edu> wrote in message news:<cgbmkh$2rv$1@news.Stanford.EDU>...
> Hello--I'm doing some intro-GR (using Carroll's new book), and I am confused
> about a subtle point:
>
> What is the significance of the order of indices in a tensor? I understand
> that the convention is for upper indices to sum with lower indices and vice
> versa when the tensor acts on the appropriate object, however, what is the
> significance of having the upper index listed first or the lower index
> listed first (horizontally)?
>
> Thanks,
> Flip
> flipt (at) stanford
As a general rule: Order matters unless proven otherwise.
That could be contractions or symmetries as others have pointed out,
or more subtle stuff, but unless proven otherwise order matters, even
between "up" and "down" indices!
A_{mu}^\nu = g_{mu_lambda} A^\lambda^\nuA^\nu_mu = g_{mu_lambda} A^\nu^\lambda
so unless A is symmetric you need to keep track!
---
frank
Kefka G
Aug30-04, 02:25 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>Marcus Wellpoth wrote:\n\n>If you lift or lower a spatial component (1,2,3) the sign changes, the\n>lowering or lifting of a time component (0) does change nothing.\n>Examples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =\n>A^{00}, A_{0}^{1} = A^{01}, ....\n>Hope that helps\n>mw\n\nI hope it doesn\'t help, because it\'s bound to lead to trouble if you take it to\nheart! To be fair, this would be correct in special relativity if you\'re using\nthe metric convention {1, -1, -1, -1}. However, in general relativity there is\nno universal relation between the raised and lowered components of a tensor.\nThey are to be raised and lowered with the metric, which no longer has the form\ndiag{1, -1, -1, -1}, but is now some complicated function of the coordinates.\n\nAs far as index ordering, I like the response that someone else already gave\nthe best - assume that the order matters unless you\'ve explicitly shown\notherwise, i.e. that the tensor is symmetric like T_uv or something. Even\nthen, it\'s better to think about it as if you\'re using the identity T_uv = T_vu\ninstead of just neglecting the order of indices - you essentially define what\norder the indices have meaning in (example: the Riemann tensor "slots" have\ndifferent meanings depending on which way you define the tensor, and it is\n_disastrous_ to forget which convention you\'re using).\n\nAs a simple example (a little closer to the basics than Riemann) of something\nwhere order certainly matters, just look at F_uv = d_u A_v - d_v A_u (d\'s are\nsupposed to be partials) in special relativity (so that the metric is just diag\n{+1, -1, -1, -1}). This tensor has F_uv = - F_vu, so we most definitely need\nto keep the order in mind.\n\n-Eric\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>Marcus Wellpoth wrote:
>If you lift or lower a spatial component (1,2,3) the sign changes, the
>lowering or lifting of a time component (0) does change nothing.
>Examples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =>A^{00}, A_{0}^{1} = A^{01}, ....
>Hope that helps
>mw
I hope it doesn't help, because it's bound to lead to trouble if you take it to
heart! To be fair, this would be correct in special relativity if you're using
the metric convention {1, -1, -1, -1}. However, in general relativity there is
no universal relation between the raised and lowered components of a tensor.
They are to be raised and lowered with the metric, which no longer has the form
diag{1, -1, -1, -1}, but is now some complicated function of the coordinates.
As far as index ordering, I like the response that someone else already gave
the best - assume that the order matters unless you've explicitly shown
otherwise, i.e. that the tensor is symmetric like T_{uv} or something. Even
then, it's better to think about it as if you're using the identity T_{uv} = T_{vu}
instead of just neglecting the order of indices - you essentially define what
order the indices have meaning in (example: the Riemann tensor "slots" have
different meanings depending on which way you define the tensor, and it is
_disastrous_ to forget which convention you're using).
As a simple example (a little closer to the basics than Riemann) of something
where order certainly matters, just look at F_{uv} = d_u A_v - d_v A_u (d's are
supposed to be partials) in special relativity (so that the metric is just diag
{+1, -1, -1, -1}). This tensor has F_{uv} = - F_{vu}, so we most definitely need
to keep the order in mind.
-Eric
Danny Ross Lunsford
Aug31-04, 08:05 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\nkefkag@aol.com (Kefka G) wrote in message news:<20040830141919.15177.00000056@mb-m21.aol.com>...\n> Marcus Wellpoth wrote:\n>\n> >If you lift or lower a spatial component (1,2,3) the sign changes, the\n> >lowering or lifting of a time component (0) does change nothing.\n> >Examples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =\n> >A^{00}, A_{0}^{1} = A^{01}, ....\n> >Hope that helps\n> >mw\n>\n> I hope it doesn\'t help, because it\'s bound to lead to trouble if you take it to\n> heart! To be fair, this would be correct in special relativity if you\'re using\n> the metric convention {1, -1, -1, -1}. However, in general relativity there is\n> no universal relation between the raised and lowered components of a tensor.\n\nThis is not so. There is no relation between upper and lower indices\nin an *affine* tensor. Once a metric is at hand there is a determined\nrelation. It\'s practically the definition of the metric! In GR, a\nmetric is always at hand.\n\n> They are to be raised and lowered with the metric, which no longer has the form\n> diag{1, -1, -1, -1}, but is now some complicated function of the coordinates.\n\nThe point is not the components of g_mn, but the relation of the\ncovariant to the contravariant components, found by solving\n\ng_mn g^np = delta_n^p\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>kefkag@aol.com (Kefka G) wrote in message news:<20040830141919.15177.00000056@mb-m21.aol.com>...
> Marcus Wellpoth wrote:
>
> >If you lift or lower a spatial component (1,2,3) the sign changes, the
> >lowering or lifting of a time component (0) does change nothing.
> >Examples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =
> >A^{00}, A_{0}^{1} = A^{01}, ....
> >Hope that helps
> >mw
>
> I hope it doesn't help, because it's bound to lead to trouble if you take it to
> heart! To be fair, this would be correct in special relativity if you're using
> the metric convention {1, -1, -1, -1}. However, in general relativity there is
> no universal relation between the raised and lowered components of a tensor.
This is not so. There is no relation between upper and lower indices
in an *affine* tensor. Once a metric is at hand there is a determined
relation. It's practically the definition of the metric! In GR, a
metric is always at hand.
> They are to be raised and lowered with the metric, which no longer has the form
> diag{1, -1, -1, -1}, but is now some complicated function of the coordinates.
The point is not the components of g_{mn}, but the relation of the
covariant to the contravariant components, found by solving
g_{mn} g^{np} = \delta_n^p-drl
Ken S. Tucker
Sep1-04, 03:21 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>antimatter33@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.0408310457.7e128b2d@posting.google. com>...\n> kefkag@aol.com (Kefka G) wrote in message news:<20040830141919.15177.00000056@mb-m21.aol.com>...\n> > Marcus Wellpoth wrote:\n> >\n> > >If you lift or lower a spatial component (1,2,3) the sign changes, the\n> > >lowering or lifting of a time component (0) does change nothing.\n> > >Examples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =\n> > >A^{00}, A_{0}^{1} = A^{01}, ....\n> > >Hope that helps\n> > >mw\n> >\n> > I hope it doesn\'t help, because it\'s bound to lead to trouble if you take it to\n> > heart! To be fair, this would be correct in special relativity if you\'re using\n> > the metric convention {1, -1, -1, -1}. However, in general relativity there is\n> > no universal relation between the raised and lowered components of a tensor.\n>\n> This is not so. There is no relation between upper and lower indices\n> in an *affine* tensor. Once a metric is at hand there is a determined\n> relation. It\'s practically the definition of the metric! In GR, a\n> metric is always at hand.\n>\n> > They are to be raised and lowered with the metric, which no longer has the form\n> > diag{1, -1, -1, -1}, but is now some complicated function of the coordinates.\n>\n> The point is not the components of g_mn, but the relation of the\n> covariant to the contravariant components, found by solving\n>\n> g_mn g^np = delta_n^p\n>\n> -drl\n\nI think the fact controversy arose in this thread from the\nOP\'s OP suggests a notation that can at times be ambiguous.\nI find raising and lowering indice in the R_abcd requires\nattention and caution. Maybe an improved notation is\nrequired for some of the more complicated tensors, ie,\n\n.e.. eb\nR = g R\na.cd abcd\n\n*might* be an improvement in real text. That\'s the way\nSpeigel wrote his tensors in complicated association,\nin Schaum\'s Vector Analysis, see for example his prob.\n40(c) in his Tensor Analysis Chapter.\n\nThe purpose of notation convention is clarity, and at\ntimes the conveyance of the evolution of the resulting\nequation. At any rate, there should never be ambiguity.\nRegards\nKen S. Tucker\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>antimatter33@yahoo.com (Danny Ross Lunsford) wrote in message news:<2b93dd16.0408310457.7e128b2d@posting.google.com>...
> kefkag@aol.com (Kefka G) wrote in message news:<20040830141919.15177.00000056@mb-m21.aol.com>...
> > Marcus Wellpoth wrote:
> >
> > >If you lift or lower a spatial component (1,2,3) the sign changes, the
> > >lowering or lifting of a time component (0) does change nothing.
> > >Examples : A_{00} = A^{00}, A_{01} = -A^{01}, A_{11} = A^{11}, A_{0}^{0} =
> > >A^{00}, A_{0}^{1} = A^{01}, ....
> > >Hope that helps
> > >mw
> >
> > I hope it doesn't help, because it's bound to lead to trouble if you take it to
> > heart! To be fair, this would be correct in special relativity if you're using
> > the metric convention {1, -1, -1, -1}. However, in general relativity there is
> > no universal relation between the raised and lowered components of a tensor.
>
> This is not so. There is no relation between upper and lower indices
> in an *affine* tensor. Once a metric is at hand there is a determined
> relation. It's practically the definition of the metric! In GR, a
> metric is always at hand.
>
> > They are to be raised and lowered with the metric, which no longer has the form
> > diag{1, -1, -1, -1}, but is now some complicated function of the coordinates.
>
> The point is not the components of g_{mn}, but the relation of the
> covariant to the contravariant components, found by solving
>
> g_{mn} g^{np} = \delta_n^p
>
> -drl
I think the fact controversy arose in this thread from the
OP's OP suggests a notation that can at times be ambiguous.
I find raising and lowering indice in the R_{abcd} requires
attention and caution. Maybe an improved notation is
required for some of the more complicated tensors, ie,
.e.. eb
R = g R
a.cd abcd
*might* be an improvement in real text. That's the way
Speigel wrote his tensors in complicated association,
in Schaum's Vector Analysis, see for example his prob.
40(c) in his Tensor Analysis Chapter.
The purpose of notation convention is clarity, and at
times the conveyance of the evolution of the resulting
equation. At any rate, there should never be ambiguity.
Regards
Ken S. Tucker
<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>antimatter33@yahoo.com wrote:\n>kefkag@aol.com (Kefka G) wrote in message\n>news:<20040830141919.15177.00000056@mb-m21.aol.com>...\n[snip]\n>> I hope it doesn\'t help, because it\'s bound to lead to trouble if you take\n>it to\n>> heart! To be fair, this would be correct in special relativity if you\'re\n>using\n>> the metric convention {1, -1, -1, -1}. However, in general relativity\n>there is\n>> no universal relation between the raised and lowered components of a\n>tensor.\n>\n>This is not so. There is no relation between upper and lower indices\n>in an *affine* tensor. Once a metric is at hand there is a determined\n>relation. It\'s practically the definition of the metric! In GR, a\n>metric is always at hand.\n\nYes, I suppose I should have spoken more precisely. I guess by "universal\nrelation" I meant a constant one, like multiplying by 1 or -1 - certainly if we\nhave a metric we can get back and forth at will using F^u_v = F_nv g^nu. But I\nthink Marcus was under the impression that it was as simple as in special\nrelativity, just flip a sign or two, which is false - often we forget in SR\nthat in raising or lowering we\'re actually using n_uv, and we just play around\nwith the signs by hand, which can lead to misconceptions in GR.\n\nAs you\'ve correctly noted, when dealing with differential geometry without a\nmetric, it is important to remember that we can\'t just haphazardly raise and\nlower indices. Lucky for us, GR is a metric theory, so life is much easier.\n\n>\n>> They are to be raised and lowered with the metric, which no longer has the\n>form\n>> diag{1, -1, -1, -1}, but is now some complicated function of the\n>coordinates.\n>\n>The point is not the components of g_mn, but the relation of the\n>covariant to the contravariant components, found by solving\n>\n>g_mn g^np = delta_n^p\n>\n\nI\'ll agree with the identity if we switch that second n to an m so the right\nside reads "delta_m^p," but I don\'t know about "the point" - I certainly don\'t\nthink the components are irrelevant to a practical computation, even though the\nabstract tensor has meaning apart from components in a particular coordinate\nsystem. However, I do agree that it\'s vital to understand how the metric ends\nup functioning as a raising/lowering tool when we pick coordinates, which may\nbe what you were getting at.\n\n-Eric\n\n>-drl\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>antimatter33@yahoo.com wrote:
>kefkag@aol.com (Kefka G) wrote in message
>news:<20040830141919.15177.00000056@mb-m21.aol.com>...
[snip]
>> I hope it doesn't help, because it's bound to lead to trouble if you take
>it to
>> heart! To be fair, this would be correct in special relativity if you're
>using
>> the metric convention {1, -1, -1, -1}. However, in general relativity
>there is
>> no universal relation between the raised and lowered components of a
>tensor.
>
>This is not so. There is no relation between upper and lower indices
>in an *affine* tensor. Once a metric is at hand there is a determined
>relation. It's practically the definition of the metric! In GR, a
>metric is always at hand.
Yes, I suppose I should have spoken more precisely. I guess by "universal
relation" I meant a constant one, like multiplying by 1 or -1 - certainly if we
have a metric we can get back and forth at will using F^{u_v} = F_{nv} g^\nu. But I
think Marcus was under the impression that it was as simple as in special
relativity, just flip a sign or two, which is false - often we forget in SR
that in raising or lowering we're actually using n_{uv}, and we just play around
with the signs by hand, which can lead to misconceptions in GR.
As you've correctly noted, when dealing with differential geometry without a
metric, it is important to remember that we can't just haphazardly raise and
lower indices. Lucky for us, GR is a metric theory, so life is much easier.
>
>> They are to be raised and lowered with the metric, which no longer has the
>form
>> diag{1, -1, -1, -1}, but is now some complicated function of the
>coordinates.
>
>The point is not the components of g_{mn}, but the relation of the
>covariant to the contravariant components, found by solving
>
>g_{mn} g^{np} = \delta_n^p
>
I'll agree with the identity if we switch that second n to an m so the right
side reads "\delta_m^p," but I don't know about "the point" - I certainly don't
think the components are irrelevant to a practical computation, even though the
abstract tensor has meaning apart from components in a particular coordinate
system. However, I do agree that it's vital to understand how the metric ends
up functioning as a raising/lowering tool when we pick coordinates, which may
be what you were getting at.
-Eric
>-drl
>
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