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Re: [spr] Metrics, geometries, and coordinate transformations |
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| Apr7-04, 02:13 PM | #1 |
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Re: [spr] Metrics, geometries, and coordinate transformations
<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>John Baez wrote:\n\n>In short, we\'ve got:\n\n> formulas for metrics in terms of coordinates -\n>changed by both active and passive coordinate transformations\n\n>metrics -\n>changed by active coordinate transformations but not passive ones\n\n>geometries -\n>unchanged by both active and passive coordinate transformations\n\n>I leave as a puzzle to figure out what the 4th possibility is like,\n>and whether people actually talk about this one!\n\ntopologies\nunchanged by homeomorphisms\nthey do talk about it\n\nif there is another better answer to the puzzle it\nwould be interesting\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>John Baez wrote:
>In short, we've got: > formulas for metrics in terms of coordinates - >changed by both active and passive coordinate transformations >metrics - >changed by active coordinate transformations but not passive ones >geometries - >unchanged by both active and passive coordinate transformations >I leave as a puzzle to figure out what the 4th possibility is like, >and whether people actually talk about this one! topologies unchanged by homeomorphisms they do talk about it if there is another better answer to the puzzle it would be interesting |
| Apr7-04, 02:13 PM | #2 |
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<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>baez@galaxy.ucr.edu (John Baez) wrote in message news:<c4n11l\\$8i1\\$1@glue.ucr.edu>...\n> We agree that changing coordinates doesn\'t\n> change the metric on a manifold. This is what some people call a\n> "passive" coordinate transformation. But, applying a diffeomorphism to\n> the metric *does* change it - and this is what they call an "active"\n> coordinate transformation.\n\n>\n> If you want to talk about a metric modulo diffeomorphisms, call it\n> a "geometry". Both active and passive coordinate transformations leave\n> a geometry unchanged.\n>\n> In GR, two metrics giving the same geometry give the same physics.\n>\n\nDr. Baez, in your view, is a diffeomorphism/active transformation the same\nas the transformation between two observers and is a coordinate\ntransformation/passive transformation merely a re-labelling of coordinates\nby a single observer?\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>baez@galaxy.ucr.edu (John Baez) wrote in message news:<c4n11l$8i1$1@glue.ucr.edu>...
> We agree that changing coordinates doesn't > change the metric on a manifold. This is what some people call a > "passive" coordinate transformation. But, applying a diffeomorphism to > the metric *does* change [itex]it -[/itex] and this is what they call an "active" > coordinate transformation. > > If you want to talk about a metric modulo diffeomorphisms, call it > a "geometry". Both active and passive coordinate transformations leave > a geometry unchanged. > > In GR, two metrics giving the same geometry give the same physics. > Dr. Baez, in your view, is a diffeomorphism/active transformation the same as the transformation between two observers and is a coordinate transformation/passive transformation merely a re-labelling of coordinates by a single observer? |
| Apr7-04, 02:14 PM | #3 |
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<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>John Baez wrote:\n> <ebunn@lfa221051.richmond.edu> wrote in message\n> news:c3ssmt\\$fm5\\$1@lfa222122.richmond.edu...\n>\n>\n>>It seems to me quite unwise to say that different coordinate systems\n>>give different metrics. The metric is a tensor, and a tensor\n>>is something that exists independent of what coordinates one chooses.\n>>So personally, I wouldn\'t talk about "the Rindler metric" as\n>>something distinct from the Minkowski metric; I\'d talk about\n>>Rindler coordinates.\n>>\n>>Am I alone in this?\n>\n>\n> No, this is what all mathematicians do, and many physicists - but\n> not *all* physicists.\n>\n> And even if we follow Ted Bunn\'s wise advice, we can easily get ourselves\n> confused if we\'re not careful. We agree that changing coordinates doesn\'t\n> change the metric on a manifold. This is what some people call a\n> "passive" coordinate transformation. But, applying a diffeomorphism to\n> the metric *does* change it - and this is what they call an "active"\n> coordinate transformation.\n>\n> Since there can be situations where what Herr Professor Schmidt regards\n> as a "passive" coordinate transformation is regarded by Herr Professor\n> Schultz as an "active" one, the situation is ripe for confusion. Schmidt\n> will say that the metric doesn\'t change, while Schultz will insist it does!\n\nThis reminds me of a fact I heard about in this newsgroup - that the\nequivalence of two solutions to Einstein\'s equations was in all generality an\nundecidable problem. Is my memory correct ? Can I read your portrait of\nSchmidt and Schultz as a representation of the same fact ?\n\nRegards, Boris Borcic\n--\n"The book of revelations is to sense as fiscal law is to money"\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>John Baez wrote:
> <ebunn@lfa221051.richmond.edu> wrote in message > news:c3ssmt$fm5$1@lfa222122.richmond.edu... > > >>It seems to me quite unwise to say that different coordinate systems >>give different metrics. The metric is a tensor, and a tensor >>is something that exists independent of what coordinates one chooses. >>So personally, I wouldn't talk about "the Rindler metric" as >>something distinct from the Minkowski metric; I'd talk about >>Rindler coordinates. >> >>Am I alone in this? > > > No, this is what all mathematicians do, and many physicists - but > not *all* physicists. > > And even if we follow Ted Bunn's wise advice, we can easily get ourselves > confused if we're not careful. We agree that changing coordinates doesn't > change the metric on a manifold. This is what some people call a > "passive" coordinate transformation. But, applying a diffeomorphism to > the metric *does* change [itex]it -[/itex] and this is what they call an "active" > coordinate transformation. > > Since there can be situations where what Herr Professor Schmidt regards > as a "passive" coordinate transformation is regarded by Herr Professor > Schultz as an "active" one, the situation is ripe for confusion. Schmidt > will say that the metric doesn't change, while Schultz will insist it does! This reminds me of a fact I heard about in this newsgroup - that the equivalence of two solutions to Einstein's equations was in all generality an undecidable problem. Is my memory correct ? Can I read your portrait of Schmidt and Schultz as a representation of the same fact ? Regards, Boris Borcic -- "The book of revelations is to sense as fiscal law is to money" |
| May17-04, 07:50 PM | #4 |
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Re: [spr] Metrics, geometries, and coordinate transformations
<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>In article <d8a8f2ec.0404070324.46038c0e@posting.google.com>,\nStephen Blake <stebla@ntlworld.com> wrote:\n\n>baez@galaxy.ucr.edu (John Baez) wrote in message\n>news:<c4n11l\\$8i1\\$1@glue.ucr.edu>...\n\n>> We agree that changing coordinates doesn\'t\n>> change the metric on a manifold. This is what some people call a\n>> "passive" coordinate transformation. But, applying a diffeomorphism to\n>> the metric *does* change it - and this is what they call an "active"\n>> coordinate transformation.\n\n>Dr. Baez, in your view, is a diffeomorphism/active transformation the same\n>as the transformation between two observers and is a coordinate\n>transformation/passive transformation merely a re-labelling of coordinates\n>by a single observer?\n\nI try to avoid the term "observer" when I\'m trying to think about\nthings precisely. The concept of an observer can be very helpful when\nwe\'re fumbling around trying to crack certain physics problems, but\nit\'s devilishly tricky to make precise, except in certain limited contexts.\n\nSince I guess we\'re trying to be precise here, instead of trying to\ncrack a physics problem, I\'ll avoid giving a yes-or-no answer to your\nquestion!\n\nHere\'s how I think about it. Suppose we have an n-dimensional\nspace X that admits globally defined coordinates. A "coordinate\nsystem" is then a diffeomorphism\n\nf: X -> R^n\n\nassigning to each point x in X its n-tuple of coordinates, f(x) in R^n.\n\nWe can change coordinates in two ways: actively and passively.\n\nA "passive" change of coordinates is a diffeomorphism\n\ng: R^n -> R^n\n\nIf we apply this to the coordinate system f we get a new coordinate\nsystem gf, defined by\n\ngf(x) = g(f(x))\n\nIn other words, we figure out the coordinates f(x) and then apply\nsome function to them to get some new coordinates g(f(x)).\n\nAn "active" change of coordinates is a diffeomorphism\n\ng: X -> X\n\nIf we apply this to the coordinate system f we get a new coordinate\nsystem fg, defined by\n\nfg(x) = f(g(x))\n\nIn other words, we move the point x over to the point g(x) and\nthen figure out its coordinates using f, obtaining f(g(x)).\n\nThis is sort of pretty: we see that the difference between\n"passive" and "active" changes of coordinates is just the\ndifference between something like\n\nf |-> gf\n\nand something like\n\nf |-> fg\n\nThis should make clear how active and passive coordinate changes\nact differently on things like metrics.\n\nOf course the funny thing is that since X is diffeomorphic to R^n,\nit could be that X secretly *is* R^n! In this case, coordinates\nAND active changes of coordinates AND passive changes of coordinates\nare all just diffeomorphisms of R^n - thought of in different ways!\n\nStill, I prefer to think of physical space(time) as some anonymous\nmanifold X, while our coordinates are lists of numbers, hence R^n.\n\nAlso, note that we could replace R^n by some other manifold Y throughout\nthe entire above discussion (up to but not including this paragraph).\nThis generalizes the concept of "coordinates" a little bit, in a way\nthat can be very useful sometimes. We then say:\n\na "coordinate system on X valued in Y" is a diffeomorphism f: X -> Y\n\na "passive change of coordinates" is a diffeomorphism g: Y -> Y\n\nan "active change of coordinates" is a diffeomorphism g: X -> X\n\nIf I had more time I\'d relate all this to the Schroedinger versus\nHeisenberg pictures in quantum mechanics and then maybe even tackle\nyour question about observers, but someone is yelling at me telling me\nto set the table for dinner!!!\n\nSorry to have taken so long to reply. I\'ll cc this to you but\nif you respond, please do so on sci.physics.research.\n\n\n\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>In article <d8a8f2ec.0404070324.46038c0e@posting.google.com>,
Stephen Blake <stebla@ntlworld.com> wrote: >baez@galaxy.ucr.edu (John Baez) wrote in message >news:<c4n11l$8i1$1@glue.ucr.edu>... >> We agree that changing coordinates doesn't >> change the metric on a manifold. This is what some people call a >> "passive" coordinate transformation. But, applying a diffeomorphism to >> the metric *does* change [itex]it -[/itex] and this is what they call an "active" >> coordinate transformation. >Dr. Baez, in your view, is a diffeomorphism/active transformation the same >as the transformation between two observers and is a coordinate >transformation/passive transformation merely a re-labelling of coordinates >by a single observer? I try to avoid the term "observer" when I'm trying to think about things precisely. The concept of an observer can be very helpful when we're fumbling around trying to crack certain physics problems, but it's devilishly tricky to make precise, except in certain limited contexts. Since I guess we're trying to be precise here, instead of trying to crack a physics problem, I'll avoid giving a yes-or-no answer to your question! Here's how I think about it. Suppose we have an n-dimensional space X that admits globally defined coordinates. A "coordinate system" is then a diffeomorphism f: [itex]X -> R^n[/itex] assigning to each point x in X its n-tuple of coordinates, f(x) in [itex]R^n[/itex]. We can change coordinates in two ways: actively and passively. A "passive" change of coordinates is a diffeomorphism [tex]g: R^n -> R^n[/tex] If we apply this to the coordinate system f we get a new coordinate system gf, defined by gf(x) [itex]= g(f(x))[/itex] In other words, we figure out the coordinates f(x) and then apply some function to them to get some new coordinates g(f(x)). An "active" change of coordinates is a diffeomorphism g: [itex]X -> X[/itex] If we apply this to the coordinate system f we get a new coordinate system fg, defined by fg(x) [itex]= f(g(x))[/itex] In other words, we move the point x over to the point g(x) and then figure out its coordinates using f, obtaining f(g(x)). This is sort of pretty: we see that the difference between "passive" and "active" changes of coordinates is just the difference between something like [tex]f |-> gf[/tex] and something like [tex]f |-> fg[/tex] This should make clear how active and passive coordinate changes act differently on things like metrics. Of course the funny thing is that since X is diffeomorphic to [itex]R^n,[/itex] it could be that X secretly [itex]*is* R^n![/itex] In this case, coordinates AND active changes of coordinates AND passive changes of coordinates are all just diffeomorphisms of [itex]R^n -[/itex] thought of in different ways! Still, I prefer to think of physical space(time) as some anonymous manifold X, while our coordinates are lists of numbers, hence [itex]R^n[/itex]. Also, note that we could replace [itex]R^n[/itex] by some other manifold Y throughout the entire above discussion (up to but not including this paragraph). This generalizes the concept of "coordinates" a little bit, in a way that can be very useful sometimes. We then say: a "coordinate system on X valued in Y" is a diffeomorphism f: [itex]X -> Y[/itex] a "passive change of coordinates" is a diffeomorphism g: [itex]Y -> Y[/itex] an "active change of coordinates" is a diffeomorphism g: [itex]X -> X[/itex] If I had more time I'd relate all this to the Schroedinger versus Heisenberg pictures in quantum mechanics and then maybe even tackle your question about observers, but someone is yelling at me telling me to set the table for dinner!!! Sorry to have taken so long to reply. I'll cc this to you but if you respond, please do so on sci.physics.research. |
| May19-04, 03:39 AM | #5 |
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<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>In article <mailman.1081306862.28328.spr@olympus.het.brown.edu>,\nLeonard <leonard@planck.com> wrote:\n\n>John Baez wrote:\n\n> >In short, we\'ve got:\n>\n> >formulas for metrics in terms of coordinates -\n> >changed by both active and passive coordinate transformations\n>\n> >metrics -\n> >changed by active coordinate transformations but not passive ones\n>\n> >geometries -\n> >unchanged by both active and passive coordinate transformations\n>\n> >I leave as a puzzle to figure out what the 4th possibility is like,\n> >and whether people actually talk about this one!\n\n>topologies\n>unchanged by homeomorphisms\n>they do talk about it\n>\n>if there is another better answer to the puzzle it\n>would be interesting\n\nThere has to be a better answer, since I was asking for something\nalmost *exactly* like a metric, which is changed by passive\ncoordinate transformations but not active ones!\n\nA topology is sorta vaguely like a metric, but not almost exactly.\nAlso, topologies are unchanged by both passive and active coordinate\ntransformations, where we allow our coordinate transformations to\nbe homeomorphisms rather than diffeomorphisms. So, a topology is\nmore like a watered-down "geometry" than the correct answer to my\nquestion. (Watered-down, since any geometry gives a topology, but\nthe topology is invariant under a bigger group.)\n\nIt may help to read my little discussion of active versus\npassive coordinate transformations, below.\n\nFor people who like math, this can be summarized by saying that\n"coordinate systems" form a left torsor of the group of passive\ncoordinate transformations, and a right torsor of the group of\nactive coordinate transformations. Thus, the set of coordinate\nsystems gives an example of what mathematicians call a "bitorsor".\nTorsors show up everywhere one discusses "gauge symmetries":\n\nhttp://math.ucr.edu/home/baez/torsors.html\n\nso it\'s not surprising that they show up here. BUT: you don\'t\nneed to know or care about torsors to solve my puzzle. So, don\'t\nlet this paragraph intimidate or distract you if you don\'t know\nwhat it means!\n\nI could say more, but not without giving away the answer completely.\nI\'m still hoping someone will solve the puzzle...\n\n.......................................................... .................\n\nHere\'s how I think about it. Suppose we have an n-dimensional\nspace X that admits globally defined coordinates. A "coordinate\nsystem" is then a diffeomorphism\n\nf: X -> R^n\n\nassigning to each point x in X its n-tuple of coordinates, f(x) in R^n.\n\nWe can change coordinates in two ways: actively and passively.\n\nA "passive" change of coordinates is a diffeomorphism\n\ng: R^n -> R^n\n\nIf we apply this to the coordinate system f we get a new coordinate\nsystem gf, defined by\n\ngf(x) = g(f(x))\n\nIn other words, we figure out the coordinates f(x) and then apply\nsome function to them to get some new coordinates g(f(x)).\n\nAn "active" change of coordinates is a diffeomorphism\n\ng: X -> X\n\nIf we apply this to the coordinate system f we get a new coordinate\nsystem fg, defined by\n\nfg(x) = f(g(x))\n\nIn other words, we move the point x over to the point g(x) and\nthen figure out its coordinates using f, obtaining f(g(x)).\n\nThis is sort of pretty: we see that the difference between\n"passive" and "active" changes of coordinates is just the\ndifference between something like\n\nf |-> gf\n\nand something like\n\nf |-> fg\n\nThis should make clear how active and passive coordinate changes\nact differently on things like metrics.\n\nOf course the funny thing is that since X is diffeomorphic to R^n,\nit could be that X secretly *is* R^n! In this case, coordinates\nAND active changes of coordinates AND passive changes of coordinates\nare all just diffeomorphisms of R^n - thought of in different ways!\n\nStill, I prefer to think of physical space(time) as some anonymous\nmanifold X, while our coordinates are lists of numbers, hence R^n.\n\nAlso, note that we could replace R^n by some other manifold Y throughout\nthe entire above discussion (up to but not including this paragraph).\nThis generalizes the concept of "coordinates" a little bit, in a way\nthat can be very useful sometimes. We then say:\n\na "coordinate system on X valued in Y" is a diffeomorphism f: X -> Y\n\na "passive change of coordinates" is a diffeomorphism g: Y -> Y\n\nan "active change of coordinates" is a diffeomorphism g: X -> X\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>In article <mailman.1081306862.28328.spr@olympus.het.brown.edu>,
Leonard <leonard@planck.com> wrote: >John Baez wrote: > >In short, we've got: > > >formulas for metrics in terms of coordinates - > >changed by both active and passive coordinate transformations > > >metrics - > >changed by active coordinate transformations but not passive ones > > >geometries - > >unchanged by both active and passive coordinate transformations > > >I leave as a puzzle to figure out what the 4th possibility is like, > >and whether people actually talk about this one! >topologies >unchanged by homeomorphisms >they do talk about it > >if there is another better answer to the puzzle it >would be interesting There has to be a better answer, since I was asking for something almost *exactly* like a metric, which is changed by passive coordinate transformations but not active ones! A topology is sorta vaguely like a metric, but not almost exactly. Also, topologies are unchanged by both passive and active coordinate transformations, where we allow our coordinate transformations to be homeomorphisms rather than diffeomorphisms. So, a topology is more like a watered-down "geometry" than the correct answer to my question. (Watered-down, since any geometry gives a topology, but the topology is invariant under a bigger group.) It may help to read my little discussion of active versus passive coordinate transformations, below. For people who like math, this can be summarized by saying that "coordinate systems" form a left torsor of the group of passive coordinate transformations, and a right torsor of the group of active coordinate transformations. Thus, the set of coordinate systems gives an example of what mathematicians call a "bitorsor". Torsors show up everywhere one discusses "gauge symmetries": http://math.ucr.edu/home/baez/torsors.html so it's not surprising that they show up here. BUT: you don't need to know or care about torsors to solve my puzzle. So, don't let this paragraph intimidate or distract you if you don't know what it means! I could say more, but not without giving away the answer completely. I'm still hoping someone will solve the puzzle... ....................................................................... .... Here's how I think about it. Suppose we have an n-dimensional space X that admits globally defined coordinates. A "coordinate system" is then a diffeomorphism f: [itex]X -> R^n[/itex] assigning to each point x in X its n-tuple of coordinates, f(x) in [itex]R^n[/itex]. We can change coordinates in two ways: actively and passively. A "passive" change of coordinates is a diffeomorphism [tex]g: R^n -> R^n[/tex] If we apply this to the coordinate system f we get a new coordinate system gf, defined by gf(x) [itex]= g(f(x))[/itex] In other words, we figure out the coordinates f(x) and then apply some function to them to get some new coordinates g(f(x)). An "active" change of coordinates is a diffeomorphism g: [itex]X -> X[/itex] If we apply this to the coordinate system f we get a new coordinate system fg, defined by fg(x) [itex]= f(g(x))[/itex] In other words, we move the point x over to the point g(x) and then figure out its coordinates using f, obtaining f(g(x)). This is sort of pretty: we see that the difference between "passive" and "active" changes of coordinates is just the difference between something like [tex]f |-> gf[/tex] and something like [tex]f |-> fg[/tex] This should make clear how active and passive coordinate changes act differently on things like metrics. Of course the funny thing is that since X is diffeomorphic to [itex]R^n,[/itex] it could be that X secretly [itex]*is* R^n![/itex] In this case, coordinates AND active changes of coordinates AND passive changes of coordinates are all just diffeomorphisms of [itex]R^n -[/itex] thought of in different ways! Still, I prefer to think of physical space(time) as some anonymous manifold X, while our coordinates are lists of numbers, hence [itex]R^n[/itex]. Also, note that we could replace [itex]R^n[/itex] by some other manifold Y throughout the entire above discussion (up to but not including this paragraph). This generalizes the concept of "coordinates" a little bit, in a way that can be very useful sometimes. We then say: a "coordinate system on X valued in Y" is a diffeomorphism f: [itex]X -> Y[/itex] a "passive change of coordinates" is a diffeomorphism g: [itex]Y -> Y[/itex] an "active change of coordinates" is a diffeomorphism g: [itex]X -> X[/itex] |
| May20-04, 03:46 PM | #6 |
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<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>John Baez wrote:\n\n> In article <mailman.1081306862.28328.spr@olympus.het.brown.edu>,\n> Leonard <leonard@planck.com> wrote:\n>\n>\n>>John Baez wrote:\n>\n>\n>>>In short, we\'ve got:\n>>\n>>>formulas for metrics in terms of coordinates -\n>>>changed by both active and passive coordinate transformations\n>>\n>>>metrics -\n>>>changed by active coordinate transformations but not passive ones\n>>\n>>>geometries -\n>>>unchanged by both active and passive coordinate transformations\n>>\n>>>I leave as a puzzle to figure out what the 4th possibility is like,\n>>>and whether people actually talk about this one!\n>\n>\n>>topologies\n>>unchanged by homeomorphisms\n>>they do talk about it\n>>\n>>if there is another better answer to the puzzle it\n>>would be interesting\n>\n>\n> There has to be a better answer, since I was asking for something\n> almost *exactly* like a metric, which is changed by passive\n> coordinate transformations but not active ones!\n>\n\nOK, I would like to try my hand at this puzzle.\n\nWe are looking for something that is changed under passive\ntransformations, which means it is some animal expressed in coordinates.\nBut something that is left unchanged by active transformations, which\ncan change the metric.\n\nMy guess is that it should be something topological, so that it is\nindependent of the metric, but it should be built out tensors, so that\nits form will change under coordinate changes. So how about something\nlike the Chern form, expressed in local coordinates?\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>John Baez wrote:
> In article <mailman.1081306862.28328.spr@olympus.het.brown.edu>, > Leonard <leonard@planck.com> wrote: > > >>John Baez wrote: > > >>>In short, we've got: >> >>>formulas for metrics in terms of coordinates - >>>changed by both active and passive coordinate transformations >> >>>metrics - >>>changed by active coordinate transformations but not passive ones >> >>>geometries - >>>unchanged by both active and passive coordinate transformations >> >>>I leave as a puzzle to figure out what the 4th possibility is like, >>>and whether people actually talk about this one! > > >>topologies >>unchanged by homeomorphisms >>they do talk about it >> >>if there is another better answer to the puzzle it >>would be interesting > > > There has to be a better answer, since I was asking for something > almost *exactly* like a metric, which is changed by passive > coordinate transformations but not active ones! > OK, I would like to try my hand at this puzzle. We are looking for something that is changed under passive transformations, which means it is some animal expressed in coordinates. But something that is left unchanged by active transformations, which can change the metric. My guess is that it should be something topological, so that it is independent of the metric, but it should be built out tensors, so that its form will change under coordinate changes. So how about something like the Chern form, expressed in local coordinates? |
| May24-04, 04:31 AM | #7 |
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<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\n\nbaez@galaxy.ucr.edu (John Baez) wrote in message news:<c845ea\\$rtu\\$1@glue.ucr.edu>...\n>I try to avoid the term "observer" when I\'m trying to think about\n>things precisely. The concept of an observer can be very helpful\nwhen\n>we\'re fumbling around trying to crack certain physics problems, but\n>it\'s devilishly tricky to make precise, except in certain limited\ncontexts.\n\n>Since I guess we\'re trying to be precise here, instead of trying to\n>crack a physics problem, I\'ll avoid giving a yes-or-no answer to your\n>question!\n\nI\'m puzzled by Dr. Baez\'s reluctance to identify the\ndiffeomorphisms/active transformations as the transformations that\nconnect the different viewpoints of two observers. This was something\nI thought I really understood.\n\nI\'ll argue by the analogy between a general manifold M and de Sitter\nspace. I choose dS space because it is a manifold with non-zero\ncurvature which has observer-dependent horizons, so it has the\ninteresting properties one might encounter in a general manifold, but\nthe group of congruences of dS space is SO(4,1) which is much easier\nto think about than the huge diffeomorphism group Diff(M).\n\nAn element g of SO(4,1) is a congruence that acts on the dS manifold\ng: dS -> dS. Physically, if a spacetime event appears as point p in\nthe dS manifold to observer A, then the event appears to observer B as\npoint g(p) in the dS manifold where the group element g is the\ntransformation from A\'s viewpoint to B\'s viewpoint. A similar\nsituation exists for a physical quantity. A physical quantity in dS\nspace must be a realization (or representation) of the congruence\ngroup SO(4,1). So if some physical quantity appears to observer A as\nv, the the quantity appears to observer B as T(g)v where T(g) is a rep\nof SO(4,1). The congruence group SO(4,1) gives the changes in the\nviewpoints of inertial observers in dS space, that is, observers who\nare not acted upon by forces.\n\nThe above paragraph is presumably uncontroversial, so it seems to me\nthat nothing goes wrong if I repeat the above paragraph with a few\nchanges of words to make things more general.\n\nAn element g of Diff(M) is a congruence that acts on the manifold g: M\n-> M. Physically, if a spacetime event appears as point p in the\nmanifold M to observer A, then the event appears to observer B as\npoint g(p) where the group element g is the transformation from A\'s\nviewpoint to B\'s viewpoint. A similar situation exists for a physical\nquantity. A physical quantity must be a realization (or\nrepresentation) of Diff(M). For example, a vector field on M is an\nelement of the Lie algebra of Diff(M) which is a pretty simple rep of\nDiff(M). So if some physical quantity appears to observer A as v, the\nquantity appears to observer B as T(g)v where T(g) is a rep of\nDiff(M). The congruence group Diff(M) gives the changes in the\nviewpoints of arbitrary observers in M.\n\nStephen Blake\n--\nhttp://homepage.ntlworld.com/stebla\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>baez@galaxy.ucr.edu (John Baez) wrote in message news:<c845ea$rtu$1@glue.ucr.edu>...
>I try to avoid the term "observer" when I'm trying to think about >things precisely. The concept of an observer can be very helpful when >we're fumbling around trying to crack certain physics problems, but >it's devilishly tricky to make precise, except in certain limited contexts. >Since I guess we're trying to be precise here, instead of trying to >crack a physics problem, I'll avoid giving a yes-or-no answer to your >question! I'm puzzled by Dr. Baez's reluctance to identify the diffeomorphisms/active transformations as the transformations that connect the different viewpoints of two observers. This was something I thought I really understood. I'll argue by the analogy between a general manifold M and de Sitter space. I choose dS space because it is a manifold with non-zero curvature which has observer-dependent horizons, so it has the interesting properties one might encounter in a general manifold, but the group of congruences of dS space is SO(4,1) which is much easier to think about than the huge diffeomorphism group Diff(M). An element g of SO(4,1) is a congruence that acts on the dS manifold g: [itex]dS -> dS[/itex]. Physically, if a spacetime event appears as point p in the dS manifold to observer A, then the event appears to observer B as point g(p) in the dS manifold where the group element g is the transformation from A's viewpoint to B's viewpoint. A similar situation exists for a physical quantity. A physical quantity in dS space must be a realization (or representation) of the congruence group SO(4,1). So if some physical quantity appears to observer A as v, the the quantity appears to observer B as T(g)v where T(g) is a rep of SO(4,1). The congruence group SO(4,1) gives the changes in the viewpoints of inertial observers in dS space, that is, observers who are not acted upon by forces. The above paragraph is presumably uncontroversial, so it seems to me that nothing goes wrong if I repeat the above paragraph with a few changes of words to make things more general. An element g of Diff(M) is a congruence that acts on the manifold g: M [itex]-> M[/itex]. Physically, if a spacetime event appears as point p in the manifold M to observer A, then the event appears to observer B as point g(p) where the group element g is the transformation from A's viewpoint to B's viewpoint. A similar situation exists for a physical quantity. A physical quantity must be a realization (or representation) of Diff(M). For example, a vector field on M is an element of the Lie algebra of Diff(M) which is a pretty simple rep of Diff(M). So if some physical quantity appears to observer A as v, the quantity appears to observer B as T(g)v where T(g) is a rep of Diff(M). The congruence group Diff(M) gives the changes in the viewpoints of arbitrary observers in M. Stephen Blake -- http://homepage.ntlworld.com/stebla |
| May29-04, 11:53 AM | #8 |
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<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>From baez@galaxy.ucr.edu (John Baez) wrote:\n> We agree that changing coordinates doesn\'t\n> change the metric on a manifold. This is what some people call a\n> "passive" coordinate transformation. But, applying a diffeomorphism to\n> the metric *does* change it - and this is what they call an "active"\n> coordinate transformation.\n\nThe more interesting question is: can these considerations be extended\nso as to be applicable to connections too? Let W, W\' be two\nconnections with their difference W\' - W = K. The structure equations\nyield:\nd(e^m) + W^m_n ^ e^n = T^m\nd(W^m_n) + W^m_p ^ W^p_n = R^m_n\n(summation convention used), or more concisely as:\nde + We = T; dW + WW = R,\nwith T and R being the torsion and curvature forms. A\nchange W -> W\' = W + K induces a change:\nT\' = de + (W+K)e = T + De\nR\' = d(W+K) + (W+K)(W+K) = R + dK + WK + KW + KK\nor in component form:\nT\'^m = T^m + (K^m_n ^ e^n)\nR\'^m_n = R^m_n + (DK^m_n + K^m_p ^ K^p_n)\nD = covariant exterior differential\nwhich yields the SAME theory, modulo effective spin and\nstress tensor terms:\nG\' = G + delta(G):\ndelta(G)_{mn} = delta(R)^p_{mpn}) - 1/2 g_{mn} delta(R)^{pq}_{pq}\nwith\ndelta(R)^m_{npq} = D_p K^m_{qn} - D_q K^m_{pn}\n+ K^m_{pr} K^r_{qn} - K^m_{qr} K^r_{pn}.\n\nThis is a more generalized relativity, which equates the two\ntheories by the duality:\n(W,T,R,G,S) --> (W+D,T\',R\',G+delta(G),S+delta(S))\nS = spin tensor,\nboth yielding the same observational consequences, with all the\ndifferences picked up by the tensors G and S.\n\nInterestingly, one can transform (locally) R to 0 to T to 0,\nabove, yielding either a curvature-free or torsion-free\nformulation of the same theory.\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>From baez@galaxy.ucr.edu (John Baez) wrote:
> We agree that changing coordinates doesn't > change the metric on a manifold. This is what some people call a > "passive" coordinate transformation. But, applying a diffeomorphism to > the metric *does* change [itex]it -[/itex] and this is what they call an "active" > coordinate transformation. The more interesting question is: can these considerations be extended so as to be applicable to connections too? Let W, W' be two connections with their difference [itex]W' - W = K[/itex]. The structure equations yield: [itex]d(e^m) + W^{m_n} ^ e^n = T^md(W^{m_n}) + W^{m_p} ^ W^{p_n} = R^{m_n}[/itex] (summation convention used), or more concisely as: [itex]de + We = T; dW + WW = R,[/itex] with T and R being the torsion and curvature forms. A change [itex]W -> W' = W + K[/itex] induces a change: [itex]T' = de + (W+K)e = T + DeR' = d(W+K) + (W+K)(W+K) = R + dK + WK + KW + KK[/itex] or in component form: [itex]T'^m = T^m + (K^{m_n} ^ e^n)R'^m_n = R^{m_n} + (DK^m_n + K^{m_p} ^ K^{p_n})[/itex] D = covariant exterior differential which yields the SAME theory, modulo effective spin and stress tensor terms: [itex]G' = G + \delta(G):\delta(G)_{mn} = \delta(R)^p_{mpn}) - 1/2 g_{mn} \delta(R)^{pq}_{pq}[/itex] with [itex]\delta(R)^m_{npq} = D_p K^{m_}{qn} - D_q K^{m_}{pn}+ K^{m_}{pr} K^{r_}{qn} - K^{m_}{qr} K^{r_}{pn}[/itex]. This is a more generalized relativity, which equates the two theories by the duality: (W,T,R,G,S) --> [itex](W+D,T',R',G+\delta(G),S+\delta(S))[/itex] S = spin tensor, both yielding the same observational consequences, with all the differences picked up by the tensors G and S. Interestingly, one can transform (locally) R to to T to 0, above, yielding either a curvature-free or torsion-free formulation of the same theory. |
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