tessel@tum.bot
Dec14-04, 10:18 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>\nOn 17 Nov 2004 there appeared in s.a.r/s.p.r. a post by myself in the\nthread titled "Re: How important is GR in order to calc the precession of\nMercury", which began:\n\n> From tessel@tum.bot Mon Nov 29 18:25:37 2004\n> Date: Wed, 17 Nov 2004 17:03:11 +0000 (UTC)\n> From: tessel@tum.bot\n> Newsgroups: sci.physics.research, sci.astro.research\n> Subject: Re: How important is GR in order to calc the precession of Mercury\n>\n> On Wed, 10 Nov 2004, Nicolaas Vroom asked (in s.a.r.)\n>\n> > In the newsgroup sci.physics.relativity I started a posting with the\n> > subject title. The purpose\n>\n> question?\n>\n> > was how do you simulate the movement of the planets, specific the\n> > movement of Mercury.\n> >\n> > Not many people responded to my messages and as such I try in this\n> > newsgroup, maybe with a better result.\n> >\n> > The approach I take is slightly different as maybe expected and that\n> > maybe explains the low responds.\n>\n> You have asked very similar questions before in various forums including\n> s.a.r. and s.p.r., and on several previous occasions, I have gone to\n> great lengths to help you understand what gtr says about the\n> extraNewtonian precession of Mercury (and why gtr is such a satisfactory\n> theory for purposes of explaining this and a multitude of other\n> observational/experimental evidence). Unfortunately, results have been\n> unsatisfactory. But for the benefit of lurkers who may have similar\n> questions, I\'ll just restate a few general and oft-repeated\n> observations.\n\nThe post continued by summarizing various points I made on several\nprevious occasions when the precession of Mercury has been discussed in\nthese groups. Unfortunately, it seems to have led to something of a\nbrouhaha, and I\'d like to try to clear the air.\n\n(I no longer check in here very often, which explains my tardy response.)\n\nFirst of all:\n\nNicolaas, I am sorry if you got a different impression, but in fact I was\n-not- trying to address anything you said in your post. This is why:\n\n1. I couldn\'t tell whether you were asking seven questions, or claiming to\nhave -answered- them.\n\n2. If the former, I did not understand your questions.\n\n3. On the basis of past experience, I believe that our\nbackgrounds/inclinations are so different that I doubt that I -can-\nunderstand your questions (if indeed they -were- questions), even after\nyou and I have expended much effort in trying to reshape them in a form\nwhich makes more sense to me.\n\nSimilar problems have consistently bedeviled our previous attempts at\ncommunication.\n\nTo illustrate what I mean by (2), let me renounce my vow and attempt to\nlist some things which I found confusing in your seven questions/claims:\n\n> 1) Does it make sense to transform human based observations into grid\n> based positions ?\n\nI have no idea what you mean by "human based observations" or "grid based\npositions". (I realize that this might refer to some previous post which\nI missed.)\n\n> 2) Does it make sense to remove light bending as part of those\n> transformations ?\n\nI have no idea what kind of "transformation" you could possibly mean, so I\ncan\'t possibly guess what this question might mean.\n\n(When someone says "transformation" in the context of math/physics, I\nstart associating to things like "Moebius transformation", "conformal\ntransformation", "coordinate transformation", but you seem to be talking\nabout something much more murky.)\n\nThe phrase "remove light bending" also seems weird to me. I tried to guess\nwhat you might mean by this, but unfortunately the guesses I came up\nwith--- while obvious to me, because of my background/experience--- appear\nunlikely to be things you might think of:\n\nMaybe you are thinking of trying to straighten out the appearance of\ncertain null geodesics by adopting a new coordinate chart? As in the\nwell-known Eddington or Kruskal-Szekeres charts for the Schwarzschild\nvacuum? If so, be aware that those only "straighten" -radial- geodesics,\nand -only apparently so-. No coordinate transformation can -remove- light\nbending from a Lorentzian manifold in which it occurs, because this is a\ngeometric phenomenon, which has a clear geometric meaning -irrespective-\nof coordinate chart.\n\nOr maybe you are thinking of some alternative theories in which we are\nrestricted to conformally flat spacetimes, in which there really is no\nlight bending? But to make a long story short, that is inconsistent with\nthe evidence.\n\n> 3) If those transformations make sense i.e. have an advantage above\n> other methods then:\n\nI have no idea what "other methods" you might mean, or what these "other\nmethods" accomplish, if they indeed exist, if indeed you even -believe-\nthat they exist, or are in sufficiently common use that you can expect\nreaders to know them.\n\n> 4) What is the function of c within this grid or frame ?\n\nWhen I use the word "frame" in a geometrical context, I almost always\nrefer to a widely used technical notion (frame as in "frame bundle"), but\nI suspect that if you tried to explain this, after a dozen exchanges it\nmight appear that you were trying to asking about "light cones", which\nexist at a different level of structure from "frames".\n\nAs for "the function of c", I would tend to read this in the manner of\n"current as a function of resistance and voltage drop", but I guess you\nmight be asking something like this: "how does the appearance of the light\ncones in our coordinate chart vary as a function of position in our\nspacetime?"\n\n> 5) What is the function of cg within this frame ?\n\nIs "cg" supposed to be "speed of gravitational waves" or something like\nthat? If so, same comment, but with an additional comment: there are no\nadjustable parameters in gtr, and gravitational and EM radiation (light)\ntravel at precisely the same speed, in vacuum, according to gtr.\n\n> 6) Do I have to consider SR within this frame ?\n\nIs "frame" supposed to be something like "Lorentz frame"? If so, most\npeople in this context use "frame" in the sense of "frame bundle", i.e.\na collection of "local Lorentz frames" defined at each event, and varying\nsmoothly over our Lorentzian manifold.\n\n> 7) Do I need the full complexity of GR to describe the movement\n> of the stars (and planets) ?\n\n-This- question does make sense in this context, at least to me, if you\nrephrase it like this: "Do I need the full EFE to describe the movement of\nthe stars (and planets)?".\n\nThe answer is: "No, in solar system dynamics you only need the LEFE,\napplied to the far field of an isolated stationary object representing the\nSun--- which is easy--- plus a whole lot of Newtonian astrophysics---\nwhich can get quite hard, if you want enough accuracy to be able to verify\nrelativistic effects".\n\nUnfortunately, it appears likely that you actually intended to ask\nsomething quite different (something which might not make sense to me at\nall), because you added the comment:\n\n> IMO the answer on that question is NO because there are no\n> moving clocks involved\n\nThis doesn\'t make sense to me because in deriving Einstein\'s precession\nformula, you have a test particle orbiting an isolated object in a nearly\nelliptical orbit, and of course you should think of any test particle as\ncarrying an idealized clock. This is relevant to understanding the\nphysical meaning of what one means by the radial coordinate, which is one\nof the critical conceptual issues which I discussed extensively in the\nposts I cited; this issue arises because I was using a harmonic chart for\na solution to the weak-field EFE, not the Schwarzschild chart for the\nSchwarzschild vacuum, which is a solution to the full EFE.\n\n> I\'m not aware of those discussions with you but anyway thanks for\n> all the detailed information regarding perturbation theory.\n\nWell, don\'t search under "T. Essel", because I was not using that handle\nback then. Try searching under keywords (with group=s.p.r.).\n\nBTW, of course I can see that it would have been best had I simply located\nand reposted the detailed "past posts" I mentioned, but for reasons too\ntiresome to enter into, that would have been inconvenient for me--- and\nfor everyone else, there is Google!\n\n> In order to get some idea about about perturbation theory and astronomy\n> I studied the following document:\n> " Large-Scale Structure of the Universe and\n> Cosmological Perturbation Theory"\n> http://xxx.lanl.gov/abs/astro-ph/?0112551\n\nOne thing to be aware of in reading papers on metric perturbations is that\nthere are many possible gauge conditions and the LEFE (see below) can look\nquite different depending on which gauge you choose. Many papers on\ncosmological perturbations choose a different "gauge" (a different\nrestriction on the type of coordinate chart used) from the Lorentz gauge\nwhich I used in my posts. Depending on context, you might also see the\nperturbations decomposed into scalar, vector, and tensor modes; in gtr,\nonly two modes survive. Beginners might find all this variety confusing,\nbut it probably -is- good practice to sort through a few different\napproaches until you can see that they really are all talking about the\nsame thing.\n\n> My previous experience with perturbation theory was related to process\n> control.\n\nPerhaps you were using a perturbation expansion to find an approximate\nsolution to some nasty pde? If so, you can probably see where AE uses\nsimilar ideas to find an approximate solution to an ODE, the\nEinstein-Binet equation, which arises in slightly different forms\ndepending on whether you start from a weak-field solution, as did AE, or\nfrom the exact Schwarzschild solution, as do many modern textbooks.\n\n> Maybe perturbation theory is the final tool that I need in order to\n> solve the equations that describe the movements of the stars and planets\n> (in a very acurate way ?)\n\nOh dear--- I was constructing an interpretation of what you might be\ntrying to say, which momentarily appeared plausible, but then you you said\nsomething which caused me to suddenly realize I must not have even been\nclose!\n\nWhen you say "solve the equations that describe the movements of the stars\nand planets", presumably you have in mind some equations which have\nalready been written down and are thus ripe for solution. But I don\'t\nknow if you are thinking of solving the equation of motion in some\nspecific solution to the EFE, such as the Schwarzschild vacuum (i.e., if\nyou are trying to study the motion of test particles modeling planets in a\nsimple spacetime model), or if you are thinking of solving the EFE itself\n(i.e. trying to find a spacetime model in gtr), or if indeed you are\nthinking of working with gtr at all.\n\nWhen you say "in a very accurate way", it is not clear to me if you plan to\nsearch for -exact- solutions or very accurate -approximate- solutions.\nPerturbation theory is of course ideally suited to the latter task, under\nsome circumstances, but not the former.\n\nIt seems you might just possibly believe that there is no known\nmathematical model (I suppose you could say "equations", but "mathematical\nmodel" would be a far better term) which accounts for "the movements of\nthe planets in a very accurate way". Of course, there -are- such models.\nIndeed, depending upon what you mean by "very accurate", such models\nexisted by the middle of the nineteenth century (of course, not accurate\nenough to account for the extranewtonian precessions of Mercury, Venus,\nEarth,..., but very accurate nonetheless).\n\nAnd when you say "final tool", I am dismayed for yet another reason: as I\nhave mentioned in the past, "perturbation theory" as a body of\nmathematical ideas/techniques can be said to have begun with Lagrange\'s\npioneering studies of solar system dynamics, and it has played a key role\nin astrodynamics from that time right down to the present day. So, it is\nnot a "final tool", but one you need right from the start.\n\nSo, did anything I said anywhere above help at all?\n\nIf not, you probably see what I meant when I said that I have found\ncommunicating with you to be very difficult!\n\nFWIW, I can suggest some reading on pre-Einsteinian astrodynamics. I think\nthe following textbook is a terrific modern introduction to the Newtonian\ntreatment of solar system dynamics, which should convince any fair minded\nreader that good old Newtonian gravitation remains of great interest, and\ncan still provide challenges as intriguing as anything in modern\nmathematical physics:\n\nauthor = {Carl D. Murray and Stanley F. Dermott},\ntitle = {Solar System Dynamics},\npublisher = {Cambridge University Press},\nyear = 1999}\n\nThose with a less than adequate mathematical background should read this\nwith a good book on mathematical methods, such as:\n\nauthor = {Derek Richards},\ntitle = {Advanced Mathematical Methods with Maple},\npublisher = {Cambridge University Press},\nyear = 2002}\n\nI like this book because it emphasizes stuff you can -use-, which nowadays\nmostly means "stuff you can do on a computer"-- for those primitive\nbeasts, that is, who still support any AFK functionality at all :-/\n\nOK, now for the "brouhaha":\n\n> From mingstb@marssim-ss.com Mon Nov 29 19:09:02 2004\n> Date: Wed, 17 Nov 2004 22:32:57 GMT\n> From: greywolf42 <mingstb@marssim-ss.com>\n> Newsgroups: sci.astro, sci.physics\n> Subject: Re: How important is GR inorder to calc the precession of Mercury\n> (banned reply)\n>\n> The following post was banned from the sci.astro.research newsgroup ...\n> without notice, and in violation of the newsgroup charter (as is usual\n> for s.a.r).\n>\n> Not only are substantive responses blocked, but the moderators (T.\n> Essel) dual-posted a triple reply to s.p.r. and s.a.r, that was simply\n> boilerplate cheerleading for GR. Without responding to the question\n> posted by Nicholaas, of course.\n\nBarry, I am not sure what you meant to imply, but for the record, I am\n-not- a moderator for either sci.physics.research or for\nsci.astro.research. The names of the moderators are listed in the FAQs for\nthese groups.\n\n> From mingstb@marssim-ss.com Mon Nov 29 19:10:05 2004\n> Date: Thu, 18 Nov 2004 17:41:39 GMT\n> From: greywolf42 <mingstb@marssim-ss.com>\n> Newsgroups: sci.astro, sci.physics\n> Subject: Re: How important is GR inorder to calc the precession of Mercury\n> (banned reply)\n>\n> "Eric Gisse" <jowr.pi@gmail.com> wrote in message\n> news:929ed0f8.0411180145.402348b9@posting.google.c om...\n> >\n> > Their newsgroup, their rules.\n>\n> I have no problem with their rules. I just enjoy jerking the chain of\n> people who claim to have rules, then break them whenever it suits their\n> personal prejudices.\n>\n> > Its too bad (not really) that you don\'t like it, but I fail to see why\n> > you would think a) any of us side for you regarding your plight\n>\n> I don\'t have a "plight."\n>\n> > and b) that we would do anything about it\n>\n> You don\'t have to do anything.\n>\n> > and c) that posting in here about your plight will do anything.\n>\n> Posting here serves my purpose. It allows the response to exist for the\n> person who asked the question. And it shows the hypocritical actions of\n> the moderators.\n\nAs for the alleged "triple-posting", I have not been submitting multiple\nposts or anything like that. I don\'t know what you saw, or think you saw,\nbut I am sure there was no malicious intent on anyone\'s part. Possible\nexplanations include:\n\n1. I submitted a candidate post for crossposting to s.a.r and s.p.r., so\nof course this post, which was approved, should have appeared (once) in\n-each- of these two newsgroups,\n\n2. Sometimes moderation glitches might result in a given post appearing\ntwo or more times in one of the groups. I can think of several ways that\nmight happen.\n\n3. Sometimes, due to bugs arising from the way a given individual accesses\nthese newsgroups, it might appear to someone that a given post has\nappeared twice, when that is not in fact the case. If someone you know,\nwho is using a different site or ISP, says they are not seeing anything\nweird, this should indicate that the problem was not due to anything the\nmoderators did or did not do.\n\nYou also claimed that your reply had been banned (rejected?) "without\nnotice". If you didn\'t receive a rejection notice from the moderators,\nthis need not imply any "hypocritical" behavior on their part. Possible\nexplanations include:\n\n1. The moderators simply might not have gotten to your submission yet.\nSometimes the moderators of s.p.r and/or s.a.r. announce "moderation\ndelays", meaning they will be busy with something else and won\'t be able\nto tend to the newsgroup for some time. AFAIK, in such cases, by the luck\nof the draw, some recently submitted posts might be "approved" [sic]\nbefore an older submission has been considered.\n\n2. The post -was- rejected, but if your submission did not include a valid\nemail address, the moderators wouldn\'t have been able to email you to\nexplain their reasons. Think of your submissions like sending off an MS\nby snail mail to the magazine page editor of your local rag. If there\'s\nno SA on the SSAE, you might never see your MS again.\n\n3. You put your submission in the mail, but the moderators never received\nit. Mail -is- sometimes lost by spoolers. That would normally only\nhappen in circumstances all too apparent to local users (e.g. a system\ncrash), but not all the moderators are using their local system when they\ncarry out their volunteer duties here. Also, for various reasons, mail\ncan be delayed in transit, sometimes for days.\n\nAs for not responding to Nicholaas, I did say what I -was- trying to do:\n\n"But for the benefit of lurkers who may have similar\nquestions, I\'ll just restate a few general and oft-repeated observations."\n\nbut to be perfectly clear, I would have listed some things I was -not-\ntrying to do--- had the possibility of misinterpretation occurred to me!\nTo wit:\n\n1. I was not trying to respond to the questions (?) listed by Nicholaas.\n\n2. I was not trying to discuss the history of science.\n\n3. I was not trying to discuss the respective merits of gtr and its\ncompetitors.\n\nTo repeat (sigh...) several things which I\'ve explained many times in the\npast:\n\nBarry, it seems you are wont to interpret my posts as "boilerplate\ncheerleading for GR". Assuming that this implies some kind of alleged\nfailure to bring a open-minded and critical attitude to the intellectual\ntable, this must be taken as a sad indication of how badly my pedagogical\nactivity here over the years has failed, at least in your case.\n\nThe truth is that my principle pedagogical goal has always been to\n-empower- interested parties to -achieve- their own goals, which might\nvery well include something like "overthrowing" gtr. (Not such an\noutlandish goal, when you consider the fact that this would be a side\neffect of constructing a viable quantum theory of gravity, which is of\ncourse one of the principle goals of contemporary physics!)\n\nAs part of that goal, I want to help math/physics students--- at a\nsuitable level of sophistication--- to formulate their own answers to\nquestions having the general form: "just how does this thing work?". In\nparticular: "how does gtr work?"\n\nThis is not because I think that gtr is the only gravitation theory anyone\nwill ever need ---although I do think there\'ll always be a place for\nEinstein, as indeed there is for Newton--- or because I am somehow\nsatisfied with the status quo. Rather, I know that to read the research\nliterature, you need to know something about our current gold-standard\ntheory of gravitation (gtr), and its predecessor (Newtonian gravitation)--\nand I know that "something" turns out to cover a lot of territory.\n\nMaybe it will help you to think of things like this: before you even\nembark on a quest to find a gravitation theory which is even better than\ngtr, I want you to know that there are seadragons and still stranger\nhazards lurking out there upon the stormy deep. And, I believe, explorers\nshould be well qualified mariners riding a seaworthy vessel. Why? Well,\njust imagine the ignominy of driving your ship hard onto a rock just\noutside the harbour mouth, with the loss of all hands! In the long run,\ncommon sense and historical experience both suggest that proper training\nand methodical preparation will greatly increase the chances of ultimately\ndiscovering some Grand New Continent. Blind luck will no doubt also play\nan essential role, but one should avoid demanding absurd quantities of\nthat most mysterious element.\n\nOr maybe a dimensionally reduced ASCII "picture" will help:\n\n\no o The Good Ship\nWell o Endeavour ==>\nSafe Charted o (math method) Some\nHarbor Coastal ___/\\___ Brave New\n(Nwtn) Waters ~~~~~~\\______/~~~~~~~~~~~~~~~~~~~~ World,\n(gtr) scurvy, storms, shoals, Yet\n____ _ scyllae... Unknown\n| | |-|\n|____| |-|\n_||_ |_|\nVolunteer Shipwright\nand Coastal Pilot\n(I, bot)\n\nFloxian fables aside, when I ask "how does gtr work?"--- or rather, "how\nis it -supposed- to work?"--- I have in mind a three-step process:\n\n1. "How you can use gtr to construct a mathematical model suitable for\nstudying a given physical situation?",\n\n2. "How can you compute physically meaningful quantities describing\nrelevant properties of that model?",\n\n3. "How can you interpret the results?".\n\nAlas, I think many people get stuck on the easy part (computation), even\nthough there are widely available freeware tools which can compute\ncurvature tensors in a heartbeat. I find the first and third parts of the\nprocess (modeling and interpretation) most interesting, but you\ndefinitely need a strong background in applied math and geometry to\nappreciate the conceptual issues lurking behind my frequent technical\ncomments. It is certainly unfortunate that without this background, you\nare unlikely to be able to appreciate my explanations of why the real\nproblems with gtr (which are considerable) are completely different from\nwhat the cranks imagine.\n\nGtr is a mathematically sophisticated theory which cannot easily be\nunderstood without prior mastery of a rather long list of ideas/techniques\nsuch as manifolds, perturbation analysis, etc., etc., ideas and techniques\nwhich in turn require some native mathematical ability (as well as time\nand effort) to acquire. While I can easily understand why this fact\nfrustrates those who have the interest but lack the background, it -is- a\nfact. You can certainly attempt to work around it, but you can\'t ignore\nit.\n\nDon\'t forget, as a self-taught amateur myself, I have been in the position\nof studying MTW with only a high school math background (yes, -that\'s-\nchanged!), so I have a special appreciation of considerable virtues of a\nformal, curriculum-driven education.\n\nAnd BTW, while gtr is clearly an interesting and important theory, which\nunfortunately requires considerable background which requires extensive\ntime/energy to acquire/use, in the past I have often tried to popularize\ntheories/problems which are no less fascinating or timely, but which are\n-much more accessible-.\n\n>From the charters of these newsgroups, or even from the names\n"sci.*.research", I think that one might fairly might expect that these\nnewsgroups are maintained by and for the principal benefit of\nphysics/astronomy researchers and their apprentices (graduate students in\nphysics/astronomy), and that while nonphysicists are welcome, they should\nbe expected to attempt to "play by the rules" of communication in\nphysics/astronomy. E.g. by attempting to speak our language, as far as\npossible, and by trying to steer close to topics we would recognize as\nphysics-related.\n\n(As you know, I am myself a nonphysicist, but hereabouts I take care to\nplay by the rules, not just because that\'s only polite, but because it is\nvirtually -essential- for communication in such a highly developed and\ntechnical field.)\n\nUntil a few years ago, I think this model did approximately hold true in\ns.p.r., most of the time (s.a.r. is a much newer newsgroup). Nowadays it\nseems that most of the gravitation-related traffic in both groups arises\nfrom posters who are not members of the research community and who appear\nto have woefully little understanding of what members of that community do\nand why. In the last few years, I have frequently seen language\nsuggesting that some posters do not understand basic concepts of\nmathematical modeling, or even the scientific method itself, much less\nunderstand the elements of what has already been achieved (scientifically\nspeaking) in the area of gravitation physics, much less what we should try\nto achieve next, or where we can expect to go in the more distant future\nfuture.\n\nThis circumstance, while clearly tending to generate much wasted bandwidth\ndue to gross miscommunication, would be more acceptable if regular but\n"physically gauche" posters at least paid close attention to occasional\nattempts by myself or others to explain some of "the rules", for the\nbenefit of those who lack the strong background in math/physics shared by\ncard-carrying members of the research community (and certain literate\namateurs, like myself). My little fable above falls into this category.\nAlas, this is by no means the first time I have found myself trying to\nexplain to you in particular stuff which I don\'t think I should really\nneed to explain, at least not more than once, at least not to a single\nindividual, at least not in these groups.\n\nMaybe you feel you are just having fun here by "jerking my chain" (if\nthat\'s really what you meant to say in your sci.astro post), but this is\nnot fair to other participants in s.p.r./s.a.r. (some of whom still come\nhere for the purpose intended, scientific discussion of research issues of\ncurrent interest in physics and astronomy). Indeed, it is not fair to\n-me-, because you persist in attributing to me motives very different from\nthose which actually impel my occasional participation in these groups,\ndespite many attempts to disabuse you!\n\nNow, despite your complaints in sci.astro, I see that at least one post\nwhich you (Barry) wrote criticizing what I said -did- appear right here:\n\n> From mingstb@marssim-ss.com Mon Nov 29 18:25:45 2004\n> Date: Fri, 19 Nov 2004 19:30:45 +0000 (UTC)\n> From: greywolf42 <mingstb@marssim-ss.com>\n> Newsgroups: sci.physics.research, sci.astro.research\n> Subject: Re: How important is GR in order to calc the precession of Mercury\n>\n> <tessel@tum.bot> wrote in message\n> news:cncfni\\$t38\\$1@lfa222122.richmond.edu...\n> > On Wed, 10 Nov 2004, Nicolaas Vroom asked (in s.a.r.)\n> >\n\n[..]\n\n> > on several previous occasions, I have gone to great\n> > lengths to help you understand what gtr says about the extraNewtonian\n> > precession of Mercury (and why gtr is such a satisfactory theory for\n> > purposes of explaining this and a multitude of other\n> > observational/experimental evidence).\n>\n> It would be polite to provide a link to said statements.\n\nYes. In past years, I always took the trouble to provide extensive\nannotated references to both on-line and printed material of use to\nserious students, so I\'ll take this as very belated thanks for my previous\nefforts :-/\n\n> A google search of\n> your posts shows no matches against "Nicolaas Vroom". In fact, there is\n> only one post against your name (to Bill Kavanah), that contains the word\n> "mercury" or the phrase "perturbation theory":\n> http://www.google.com/groups?selm=cmq3is%24a2m%241%40lfa222122.richmond. edu\n> And in this post, you again make unreferenced statements that you\'ve "posted\n> on this (perturbation theory) very extensively before."\n\nSee my advice to Nicolaas above.\n\n> > This is needed in the problem at hand\n> > because the theoretical problem confronting Einstein in 1916 was not to\n> > explain the precession of Mercury in its orbit around the Sun, but rather\n> > to explain a small residual remaining after a perturbation theory analysis\n> > of a model in Newtonian gravity had explained all but a small part of the\n> > observed motion.\n>\n> On the contrary. The explicit, stated purpose of Einstein was to obtain\n> Newcomb\'s published value (43" per century*) for the NNPA of Mercury.\n\nSo what is the problem? The notion of a "residual"?\n\nYou said that Grossmann "remeasured" the residual, but I think you should\nhave said "recalculated". The principle issue was (and remains) not how\nto -measure- the motion (astronomers had already gotten very good at that\nbefore AE came along), but how to -explain- it theoretically. Saying that\nwe have an extranewtonian residual of 43 seconds per century amounts to\nsaying that -our best Newtonian model- leaves that much unaccounted for.\n\nYou had some other comments, which I think are mostly moot, since you\ncompletely misunderstood the purpose of my post. But I must correct one\nserious misstatement you made:\n\n> > In gtr,\n> > we have the additional complication that the full field equation (the EFE)\n> > is nonlinear, but this plays no role here because we can get away with\n> > studying solutions to a linearized version of the EFE.\n>\n> Which is simply the Newtonian equation, with an added speed-of-gravity\n> parameter (equal to the speed of light).\n\nAnyone who has studied any modern gtr textbook will know that there is\nonly one thing I could possibly mean by "a linearized version of the\nEFE"--- and I doubt that anyone who has seen it could possibly describe it\nas "simply the Newtonian equation, with an added speed-of-gravity\nparameter (equal to the speed of light)"!\n\nFor example (you did want citations?):\n\n1. See (20.28) in section 20.1 ("The linearized field equations") of the\ntextbook\n\nauthor = {Ray D\'Inverno},\ntitle = {Introducing {E}instein\'s Relativity},\npublisher = {Clarendon Press},\nyear = 1995}\n\n2. See (8.42) in section 8.3 of the textbook\n\nauthor = {Bernard F. Schutz},\ntitle = {A First Course in General Relativity},\npublisher = {Cambridge University Press},\nyear = 1985}\n\n3. See (13.14) in section 13.2 ("The fundamental equations of the\nlinearized theory") of the textbook\n\nauthor = {Hans Stephani},\ntitle = {General Relativity:\nAn Introduction of the Theory of the Gravitational Field},\npublisher = {Cambridge University Press},\nedition = {Second},\nnote = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},\nyear = 1990}\n\n4. See (7.125) of the textbook\n\nauthor = {Sean Carroll},\ntitle = {Spacetime and geometry: an introduction to general relativity},\npublisher = {Addison-Wesley},\nyear = 2004}\n\n5. See (18.8b) of section 18.1 ("The linearized theory of gravity") of\nthe textbook\n\nauthor = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},\ntitle = {Gravitation},\npublisher = {W. H. Freeman},\nyear = 1970}\n\n6. See (4.4.12) in section 4.4 ("Linearized Gravity") of the textbook\n\nauthor = {Robert M. Wald},\ntitle = {General Relativity},\npublisher = {University of Chicago Press},\nyear = 1984}\n\n7. See (53) in section 3.2 ("The linear field equations for gravitation")\nof the textbook\n\nauthor = {Hans C. Ohanian and Remo Ruffini},\ntitle = {Gravitation and Spacetime},\nedition = {Second},\npublisher = {W. W. Norton},\nyear = 1994}\n\n8. See (10.1.10) in section 10.1 of the textbook\n\nauthor = {Weinberg, Steven},\ntitle = {Gravitation and Cosmology:\nPrinciples and Applications of the General Theory of Relativity},\npublisher = {Wiley},\nyear = 1972}\n\n9. See (9) in section 4.1.2 of the textbook\n\nauthor = {C. Clarke},\ntitle = {Elementary General Relativity},\npublisher = {Wiley},\nyear = 1979}\n\nThe LEFE (linearized Einstein field equation) is a -fundamental- and\n-standard- topic which is treated at length in almost every gtr book!\nThe name LEFE (or something very similar) is also standard, as you can see\nfrom the section titles I quoted.\n\n(Are you convinced yet, or should I give even more references?)\n\nNote that the LEFE takes different forms in different gauges; most books\ndiscuss at least the "Lorenz gauge", and I have listed above the form\nwhich the LEFE takes (for trace-reversed perturbations) in that gauge.\n\nBarry, in your sci.astro post, I see that you say\n\n> MTW is not a good way to learn GR. It\'s a decent way to expand your\n> knowledge in specific areas, once you have the basics down.\n\nI think your opinion might be different once you get the basics down.\n\nMTW is one of the great scientific books of all time, one which rewards\nrepeated reading at -all- stages of your education. It is true that\nunderstanding what you read gets exponentially easier as you acquire more\nbackground and experience, but looking into this book at an early stage\ncan give you an idea of what it is you want to understand, and checking\nback periodically (and working more problems!) can reassure you that you\nare in fact making progress.\n\nBe all this as it may, the LEFE is great fun, so I urge -everyone- who\ndoesn\'t know about it to read up forthwith!\n\nDon\'t be confused by the fact that in many of the textbooks above,\ndiscussion of the LEFE occurs in the context of gravitational radiation,\nsometimes long after a discussion of the precession of nearly Keplerian\norbits. The underlying field equation is the same regardless of whether\none is discussing the "far field" of an isolated stationary object (e.g.\nfor studying precession of nearly Keplerian orbits around a star), or the\ngeneration of gravitational radiation (anything but stationary!). Of the\ntextbooks listed above, this underlying unity is probably brought out best\nby Stephani or MTW, but it should be clear enough from any of these books\nif you read with sufficient care and insight.\n\nFor the benefit of interested readers, I append a brief overview in "Baez\nemulation mode" ;-/ For the details, I hope all interested readers will\nrush right out, obtain a copy of the one of the above mentioned textbooks,\nand study them!\n\nT. Essel (spelunking somewhere in cyberspace)\n\n=================================== ==============================\n\nJust what -is- this "LEFE" business?\n\nWell, we should begin at the beginning, so let\'s begin with the EFE (the\nfull, nonlinear, Einstein field equation).\n\nIntuitively, gtr aims to directly relate geometry and matter. You\nprobably already have some intuition for what matter is, and how we can\nmeasure how much there is of it in a given place. But what about\ngeometry? To directly relate geometry to matter, we evidently need to\nsomehow quantify -geometry itself-! How can we do that? Well, according\nto Riemann, geometry always comes down to angles and distances. And to\nquantify angles and distances, we can employ the "metric tensor" of a\nLorentzian manifold.\n\nThis Lorentzian manifold, or "spacetime model", will play a dual role: it\nwill serve as our mathematical model of "gravitational phenomena as\ngeometry", in a specific physical context (e.g. modeling a planet orbiting\nthe Sun), but it will also constitute the -geometrical setting- for all\n-nongravitational- physical phenomena under study, if any. The rules for\ntransferring all our nongravitational physical theory (e.g. Maxwell\'s\ntheory of EM, hydrodynamics, the theory of elasticity) to this curved\nspacetime setting are sometimes summarized by the slogans "minimal\ncoupling" and "comma goes to semicolon". These rules are not entirely\nstraightforward, but they are not hard to use in most cases.\n\nThe full EFE reads\n\nG_(ab) = 8 pi T_(ab) (*)\n\nProbably everyone here knows that, but not everyone really knows what\nthose funky symbols actually -mean-! Fortunately, I can describe the\nbasic "ingredients" of this equation, and even try to give a hint of how\nthey fit together.\n\nOn the left, we have something mathematically representing the character\nof the geometry of our spacetime. It is G_(ab), the Einstein tensor, a\nsecond rank tensor, which is a kind of "average curvature" tensor\nconstructed from the fourth rank Riemann curvature tensor R_(abcd). The\nRiemann tensor completely describes the local geometry near each event in\nour Lorentzian manifold; it can be readily computed from the metric tensor\nby a straightforward process of double differentiation.\n\n(Going the other way is harder, but it can be done. This raises all kinds\nof fascinating mathematical issues which unfortunately I haven\'t the\nspace-time-energy to discuss, at least not here.)\n\nOn the right, we have something mathematically representing the amount and\nmotion of matter. It is T_(ab), the energy-momentum-stress tensor.\n(Multiplied by a factor of 8 pi, but I probably don\'t need to say what 8\nand pi are, huh? So I won\'t.) Mathematically speaking, T_(ab) is the\n"source term" in the EFE, modeling the stuff (anything with energy or\nmass) which generates a gravitational field. Roughly speaking, at each\nevent in our spacetime model, as measured by an appropriate "ideal\nobserver", the T_(00) component tells how much -mass- there is, the T_(0j)\ncomponents tell how much -momentum- there is, and the T_(jk) components\ntell how much stress there is.\n\nFor example, consider an isolated rigidly rotating thin disk. Here,\nT_(ab) would be nonzero only inside the disk, and if we compute the\ncomponents in an appropriate frame, T_(00) (mass density), T_(0j)\n(momentum density), and T_(ij) (stress density) will all be nonzero. The\nstress arises because in order to maintain its shape, the disk must resist\nthe tendency to fly apart due to centrifugal "forces" [sic] arising from\nits rotation, and this mechanical resistance shows up as stress.\n\nSo, the EFE does indeed say that a basic geometric phenomenon (the\naveraged curvature) is directly related to a basic physical phenomenon\n(the amount and motion of matter and energy of nongravitational fields).\n\nWe can say more: the presence of matter (nonzero source term on the RHS)\ndirectly causes a local distortion of geometry (a change in our "average\ncurvature"), which typically changes the behavior of geodesics. To see why\nthis is profoundly important, you need to know that one of the great\nbeauties of gtr is the elegant way in which it realizes the -universal-\ncharacter of gravitation (the fact that gravity acts on all forms of\nmass-energy). Namely, according to gtr, if no physical forces act on a bit\nof matter, then its world line (kinematic history as represented in our\nLorentzian manifold or spacetime model) will be a geodesic. (In a curved\nmanifold, geodesics are the analog of "straight lines": they are curves\nwhich are "as straight as possible"). So in gtr, when curvature is\npresent, this typically will affect the relative motion of free-falling\nbits of matter.\n\nNeedless to say, gtr can also handle any -nongravitational- forces acting\non matter, such as pressure acting on a fluid element. Namely, the\nmagnitude and direction of net force on some bit of matter determines the\nmagnitude and direction of the -path curvature- of its world line. If\nforces act, the world line has nonzero path curvature; if not, it doesn\'t.\nThis is just what we would do in str (which doesn\'t attempt to treat\ngravitation), except that now our spacetime is curved. Note that this\nmeans that "gravitational forces" never appear in gtr, or rather, the\nforce which a standing human applies at his feet to resist falling through\nthe floor can be called that, but really it is a -nongravitational- force\nwhich is accelerating him with a certain magnitude and direction (away\nfrom the center of the Earth), which means that his world line has nonzero\npath curvature; it is -not- a geodesic. But he steps off the balcony, he\nwill subsequently be in free fall--- until he strikes the ground--- and\nalong the free-fall portion of his kinematic history, his world line will\nhave vanishing path curvature.\n\nJohn Archibald Wheeler (the W in MTW) famously summed things up something\nlike this: "Matter (plus possible nongravitational fields) tells spacetime\nhow to curve; spacetime curvature (plus possible nongravitational forces)\ntells matter how to move".\n\nLet\'s look at how this works in a bit more detail. Mathematically, we can\ndecompose the Riemann curvature tensor like this:\n\nR_(abcd) = C_(abcd) + a kind of fourth rank "expansion" of\nRiemann Weyl curvature R_(ab) = R^m_(amb)\ncurvature Ricci curvature\n\nIn gtr, "Ricci curvature" (or equivalently, "Einstein curvature"), meaning\neither the second rank tensor or its fourth rank "expansion", is closely\nassociated with matter: the -immediate presence- of matter in some region\ncauses nonzero Ricci curvature in that region, and conversely, Ricci\ncurvature cannot exist where no matter is present.\n\n"Weyl curvature", AKA "conformal curvature", has a beautiful and\ncompelling geometric meaning, related to "light bending", but independent\nof any physical interpretation. But to save time, here I will just say\nthis: in gtr, Weyl curvature is the type of curvature which can exist\n-independently of matter-; in particular, it is the type of curvature\nwhich can propagate through a vacuum region, as a "gravitational wave".\nSuch waves are terribly important because they are ultimately the\n"physical mechanism" by which a region of spacetime (empty or otherwise)\ncan "learn" of the presence of mass/energy somewhere else, possibly far\noff, and "respond" by changing its curvature to maintain the validity of\nthe field equation.\n\nFor example, if a bomb goes off in some isolated region in space, that\nchanges the distribution and motion of matter in this region, which can\ncreate "ripples" of Weyl curvature which then propagate outwards, at the\nspeed of light, through the surrounding vacuum. Such bombs actually\nexist--- they are called "supernovae"! With LIGO, we hope to detect their\ngravitational wave signature.\n\nYou might be asking: since the EFE only mentions Einstein curvature, (or\nequivalently, Ricci curvature), how can the motion of matter possibly\ncreate propagating ripples of Weyl curvature? The answer is that the\n"differential Bianchi identity" (a mathematical identity which is true\nindependently of the EFE, and indeed, independently of any physical\ninterpretation of curvature) enforces a "coupling" between matter and Weyl\ncurvature via a first order PDE. Essentially, this relation says that the\nimmediate presence of matter in one region can cause Weyl curvature in\nthat region, and this can in turn cause more Weyl curvature in adjacent\nregions, even nearby vacuum regions. But what matters is not so much the\nmere -presence- of matter as the existence of a suitable -gradient-. And\nindeed, the Friedmann cosmological models, which are homogeneous and\nisotropic spacetimes, have nonzero Ricci curvature (since they are filled\nwith fluid or dust) but no Weyl curvature (and thus no light bending).\n\nOTH, in the famous Schwarzschild vacuum, which is an exact solution of the\nEFE representing the field outside a nonrotating spherically symmetric\nmassive object, -all- the curvature is Weyl curvature. Of course, this is\nthe same as saying we have a vacuum solution, since "no matter" is\nequivalent to "no Ricci"! But how, you might ask, did this Weyl curvature\nget there? Well, in imagination, it happened like this: when we\nconcentrated matter to form the Sun, by the conservation of mass, we had\nto move it in from somewhere, and when we did that, according to the EFE,\nthe "matter gradient" created Weyl curvature, which then propagated into\nnearby vacuum regions (at the speed of light) and began to curve them up,\nand so on. Thus, the gravitational field of the Sun is a kind of\n"historical legacy" which was created--- indirectly and gradually---\nduring the slow formation of the Sun. Any gravitational radiation\ncurrently emanating from the Sun merely represents teeny tiny\nperturbations carrying information about movements of matter inside the\nSun, information which will eventually be used by regions far away to make\n-extremely- teeny tiny "local adjustments in curvature" in order to uphold\nthe validity of the EFE. Or at least, that\'s the way gtr tells the story!\n:-/\n\nNewbies often ask "if nothing can get out of a black hole, how can\ngravitation get out?" The answer is that (in gtr) -it doesn\'t have to-.\nThe collapse of a star to make the hole probably created some strong\ngravitational radiation, but this can\'t instantly undo the Weyl curvature\nwhich was gradually created during the formation of the star.\n\nOr again, sometimes people ask: "if a black hole was created by the\ncomplete collapse of a star, and if the matter which once made up the star\nhas been crushed out of existence at a spacetime curvature singularity,\nthen surely there is nothing left to gravitate?" One answer is that it is\nprecisely because information cannot propagate from under the horizon that\nthe "bad news" about the fate of the matter which originally created all\nthat Weyl curvature (indirectly!) cannot reach the exterior region, so we\ndo not in fact need to worry about this, unless we fall into the interior\nregion, in which case we\'ve got more pressing--- and tugging--- problems\n:-/\n\nBut enough about the EFE, you say-- what about the LEFE?!\n\nOK, now I must confess that everything I just said wasn\'t the beginning at\nall, it was just stuff I had to say before I could even begin! :-/ -This-\nis the beginning:\n\nThe starting point of "linearized gtr" is a very simple idea: finding\nexact solutions of the EFE can be a daunting task, so to make things\neasier, imagine that we have, essentially, flat spacetime, except that in\nsome places we have small amounts of matter (or an EM field, or some other\nnongravitational field) which induces a "mild distortion" of our otherwise\nflat spacetime. Or ---to use the lingo--- assume that our Lorentzian\nspacetime is a "metric perturbation" of Minkowski spacetime (in fact, a\n"linear perturbation", meaning that we only carry out computations to\nfirst order in our "perturbation parameter").\n\nSo, let\'s write the metric in the form of the sum of the Minkowski space\nmetric tensor (representing the flat background) plus a perturbation\ntensor (representing the "mild distortion" due to the presence of some\nnot-too-massive object, a passing gravitational wave, whatever):\n\ng_(ab) = eta_(ab) + h_(ab)\nbackground perturbation\n\nIn weak-field gtr, the background is unobservable, so physically speaking,\nit doesn\'t exist. But mathematically speaking, we can pretend, and there\nis a rule of thumb that says that whenever you can get away with\npretending something is much simpler than is really the case, you should\nbe cynical and just grab the opportunity! So, let\'s treat the metric\nperturbation tensor h_(ab) as just another tensor field on -flat-\nspacetime. It is a second rank symmetric tensor field on a four\ndimensional manifold, which implies that it has ten algebraically\nindependent components at each event.\n\nNow, if we compute the Einstein tensor of our mildly distorted spacetime,\nusing the true metric g_(ab), but expressing our result in terms of the\nperturbation tensor h_(ab), treated as just another tensor field in flat\nspacetime, we get a big mess. But we are free to add conditions on h_(ab)\nto try to reduce the clutter, so let\'s add some! This has the effect of\nruling out the use of some coordinate charts, but it is always possible to\nfind plenty of coordinate charts which meet our "gauge conditions". There\nare many possible choices of gauge conditions, but the LEFE takes a\nparticularly simple form if we choose the "Lorenz" (or "harmonic") gauge,\nso let\'s make this choice.\n\nNext, toss in a minor trick: rewrite everything in terms of the\n"trace-reverse" of h_(ab), written hbar_(ab). -Viola!-, you get the LEFE:\n\nBox hbar_(ab) = -16 pi T_(ab) (**)\n\nIt\'s just the good old flat spacetime wave equation, times ten!\n\nOn the left, we have something acted upon by the flat spacetime wave\noperator Box(.), something which mathematically represents the mild\ndistortion of the flat spacetime geometry. It is hbar_(ab), the\ntrace-reversed metric perturbation tensor, which, as I said, we treat as a\nsecond rank symmetric tensor field on flat spacetime. So it has ten\nalgebraically independent components--- each of which satisfies the scalar\nwave equation familiar to every undergraduate sci/math student. Secretly,\nof course, the left hand side also has a beautiful and profound geometric\ninterpretation: it is the Einstein tensor, computed for our mildly\ndistorted spacetime using the -true- (physical) metric tensor g_(ab)!\n\nOn the right, we have something mathematically representing the amount and\nmotion of matter. It is T_(ab), the energy-momentum-stress tensor, treated\nas a tensor field on flat spacetime. Mathematically speaking, it serves as\nthe "source term" for our flat spacetime tensorial wave equation.\n\nIn a vacuum region (e.g. outside the disk), where the right hand side of\nthe LEFE vanishes, we are dealing with the "source-free" (tensorial) wave\nequation\n\nBox hbar_(ab) = 0 (***)\n\nIn such a region, we might have changes in the geometric distortion\npropagating as a wave (a vacuum gravitational wave).\n\nQualitatively, the LEFE says the same thing as the full EFE: "Matter\n(plus possible nongravitational fields) tells spacetime how to curve;\nspacetime curvature (plus possible nongravitational forces) tells matter\nhow to move". But mathematically, it takes on a new form which offers some\nvery suggestive analogies.\n\nLet\'s compare the LEFE with Maxwell\'s "nontrivial" field equation:\n\nBox A^b = 4 pi J^b\n\nHere, on the left, we have something on which the wave operator Box acts,\nsomething which mathematically represents "the EM field". It is A^b, the\nfour-vector potential (from which we can recover the EM field itself by\n"exterior differentiation", F = dA). On the right, we have something\nmathematically representing the amount and motion of any electrically\ncharged matter. It is J^b, the current-density four-vector. In a\ncharge-free region, we have the "source-free wave equation", and in such a\nregion, we can have an EM field propagating as a wave ---we can have\n"light".\n\nThere\'s much more to this analogy with Maxwell\'s theory of EM. For\nexample, the EM field tensor F_(ab) has two scalar Lorentz invariants\n\nF_(ab) F^(ab), F_(ab) *F^(ab)\n\nfrom which we can recognize when we have a "purely electrostatic" field,\n(in which case we can find observers who measure no magnetic field), and\nmore besides. And from the Riemann curvature tensor (the gravitational\nfield tensor) we can construct analogous scalar Lorentz invariants\n\nR_(abcd) R^(abcd), R_(abcd) *R^(abcd)\n\nand then we can recognize situations where we can find observers who\nmeasure no "magnetogravitic field", and more besides. (BTW, this scalar\ninvariant stuff actually works fine for fully-fledged gtr, not just its\nlinearized weak-field approximation.)\n\nWhy is this interesting? Well, for one thing, it points up a very\ndramatic difference between linearized gtr and Newtonian gravitation.\nConsider a rigidly rotating uniform density thin disk. In Newtonian\ngravitation, by measuring the gravitational field we cannot distinguish\nour disk from an otherwise identical but nonrotating disk. But in the\nLEFE, the source term T_(ab) takes account not only of the amount of\n-mass- in each region, but also the amount of -momentum-. This turns out\nto mean that in weak-field gtr, we -can- distinguish the rotating disk\nfrom its nonrotating twin, just by measuring its gravitational field! If\nits magnetogravitic field is nonzero (as must be case if the scalar\ninvariant R_(abcd) *R^(abcd) is nonzero), then it is rotating. If not, it\nisn\'t! Unfortunately, these are usually tiny effects, but there are\nexperiments underway or planned to test this feature of gtr using earth\norbiting satellites (Gravity Probe B and its kin).\n\n(Maybe I should remark that, while what I just said about scalar\ninvariants is true for isolated stationary objects, it turns out that\n-all- the scalar curvature invariants of vacuum gravitational plane waves\nvanish identically, but these have magnetogravitic fields comparable to\ntheir electrogravitic or tidal fields, as measured by any observer. But\neven here we have a valid analogy, since in Maxwell\'s theory of EM, the\ntwo Lorentz invariants of the field tensor F_(ab) vanish identically for\nan EM plane wave, and these have magnetic fields comparable to their\nelectric fields, as measured by any observer.)\n\nPredicting and possibly confirming new physical phenomena is always very\nexciting, but there is another reason why this observation is of\nfundamental interest. To set up our rotating disk model, we must write\ndown a boundary value problem for the LEFE. That is,\n\n1. We write down boundary conditions stipulating the size, shape,\n(uniform) mass, and (steady-state) rotation of the disk,\n\n2. We attempt to find the required vacuum solution of the LEFE (a purely\nmathematical problem),\n\n3. We attempt to interpret the resulting Lorentzian spacetime (defined\nonly in the region outside the disk) as a mathematical model of\n"gravitation as geometry".\n\nBut there is nothing in the LEFE to prevent us from removing the -mass-\nand leaving the -momentum- in our boundary conditions. As an exercise, it\nis a good idea to do just this, and to solve the new boundary value\nproblem. What you get an analogue of a "purely magnetostatic field", and\nthis turns out to be pretty darn weird: all the spacetime curvature is in\nthe t = constant family of spacelike hyperslices, so static observers\nexperience no gravitational tidal forces, and they need not fire their\nrocket engines to remain static. However, observers riding a\ngyrostabilized platform in free fall eventually find that they are\n-spinning- with respect to the distant stars!\n\nThis illustrates a very important feature of gtr: boundary conditions are\nessential, and should arise from nongravitational physics, but "garbage\nin, garbage out": if you fail to posit "physically reasonable" boundary\nconditions, your spacetime model is unlikely to be physically reasonable.\n\nOn the bright side, gtr plus a little common sense provides us with a\nstrong hint that our new boundary conditions must unreasonable, even\nbefore we discovered the weird properties of a "purely magnetogravitic\nspacetime". We need only ask, how might we "prepare" a thought experiment\ninvolving a rotating disk? Well, we can imagine a region of spacetime far\nremoved from any massive objects, and we can imagine sending in some\nmatter to form a rotating object, which will serve as the source of the\ngravitational field in our isolated region. But of course, this should\nlead to something which has positive mass and angular momentum (and not\ntoo much of the latter in comparison to the former).\n\nSpeaking of electrostatic fields--- if you know how the EM field reduces\nto an electric field in some circumstances (e.g. a static configuration of\ncharged matter, perhaps on the surface of a charged metal sphere), you\nwill recall that we can kill off all but the "time component" of the EM\npotential and the current-density four-vector. Then, our wave equation\nreduces to the Poisson equation\n\nLap phi = 4 pi sigma\n\nwhere phi is the electrostatic potential and sigma is the charge density.\nIn the case of a spherically symmetric source-free field, by introducing\nthe Lorentz force law we can recover the inverse square law of Coulomb,\nwhich is formally exactly the same as Newton\'s inverse square law. And if\nwe look at static weak gravitational fields, if we use a suitable observer\nwe will find that hbar_(ab) has only one nonvanishing component, which\ncorresponds to the Newtonian potential. Then, our wave equation again\nreduces to the Poisson equation\n\nLap phi = 4 pi rho\n\nwhere now phi is the gravitational potential and rho is the density of\nmatter. The -vacuum- field equation is the Laplace equation (again), from\nwhich we can recover Newton\'s inverse square law. But this time we don\'t\nneed to -introduce- any force law--- because in gtr, there -is- no\n"gravitational force"! This is another of those unique features which\nmakes gtr so remarkable from a mathematical point of view.\n\nNow, any solution of the three dimensional Laplace equation (such things\nare called "harmonic functions") can serve as a Newtonian potential (or as\nan electrostatic potential). From this, you might guess that you can cook\nup a storm with Green\'s functions for trace-reversed metric\nperturbations--- and you\'d be right! If you know that you can expand any\nasymptotically vanishing harmonic function as an infinite sum of\nparticularly simple ones (the "harmonics") you might guess that you can do\nsomething similar for trace-reversed metric perturbations--- and you\'d be\nright about that too! This is why, when you see the term "quadrupole\nmoment" when you are reading a paper about gravitational radiation, and\nthen see it again when you are reading another paper about the shape of\nthe Sun, you can be confident that both authors really were talking about\nthe same thing.\n\nAt this point, you might ask "what puts the L in LEFE?" Well, the wave\nequation is -linear-, which means that if we have two solutions, we can\njust add them to obtain a new solution. And the LEFE is just a tensorial\nwave equation for the trace reversed metric perturbation. This means that\nif we know two "legal" trace reversed metric perturbations (tensors which\nsolve the LEFE), we can just add them and get a new "legal" metric\nperturbation:\n\nnewsol = sol1 + sol2\n\nBut if we try the same thing with two solutions to the full EFE, we might\n(probably after a lot of work) be able to write something like this:\n\nnewsol = sol1 + sol2 + "nonlinear interaction terms"\n\nThat\'s nonlinearity in action!\n\nPhysically, we can say that the "nonlinear interaction terms" tend to\nimply that we will see stronger gravitation than we would expect in a\nlinear theory. One way to think about this is to imagine two stars which\napproach each other in some isolated region of deep space. As the total\ngravitational field gets stronger in the region between them, its energy\nincreases, but this energy itself gravitates, so the gravitational field\nincreases in strength faster than Newtonian theory would lead you to\nexpect.\n\nIf that sounds complicated, it is, and this goes a long way toward\nexplaining why the LEFE is so much easier to solve than the EFE.\n\nWhen can we use the LEFE rather than the EFE? Whenever and wherever our\nspacetime is "almost flat"! Or in other words, whenever and wherever the\ngravitational field is "weak". One place that happens is far from an\nisolated massive object like the Sun. So if hbar1_(ab) models the Sun, or\nat least the "far field" of the Sun, where gravitation is weak, and if\nhbar2_(ab) represents a gravitational wave propagating in flat spacetime,\nthen hbar1_(ab) + hbar2_(ab) represents a gravitational wave passing near\nthe Sun-- but not too near!\n\nSo, you say, if we can do all this cool stuff with the LEFE, who needs the\nEFE? Well, sometime or somewhere spacetime might not be -anywhere near\nflat-; for example, soon after the Big Bang, or near the horizon of a\nsmallish black hole!\n\nThere is also some very persuasive theoretical reasons for regarding the\nEFE as the real deal, and the LEFE as a mere approximation which is useful\nin some circumstances. One of the most intriguing is this: it turns out\nthat the LEFE is not entirely self-consistent. There is a more or less\nunique way for modifying the LEFE until eventually (after passing through\ninfinitely many stages), you have a self-consistent (but nonlinear) field\nequation. And this field equation is:\n\nooo dramatic drumroll ooo\n\nThe EFE!!\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On 17 Nov 2004 there appeared in s.a.r/s.p.r. a post by myself in the
thread titled "Re: How important is GR in order to calc the precession of
Mercury", which began:
> From tessel@tum.bot Mon Nov 29 18:25:37 2004
> Date: Wed, 17 Nov 2004 17:03:11 +0000 (UTC)
> From: tessel@tum.bot
> Newsgroups: sci.physics.research, sci.astro.research
> Subject: Re: How important is GR in order to calc the precession of Mercury
>
> On Wed, 10 Nov 2004, Nicolaas Vroom asked (in s.a.r.)
>
> > In the newsgroup sci.physics.relativity I started a posting with the
> > subject title. The purpose
>
> question?
>
> > was how do you simulate the movement of the planets, specific the
> > movement of Mercury.
> >
> > Not many people responded to my messages and as such I try in this
> > newsgroup, maybe with a better result.
> >
> > The approach I take is slightly different as maybe expected and that
> > maybe explains the low responds.
>
> You have asked very similar questions before in various forums including
> s.a.r. and s.p.r., and on several previous occasions, I have gone to
> great lengths to help you understand what gtr says about the
> extraNewtonian precession of Mercury (and why gtr is such a satisfactory
> theory for purposes of explaining this and a multitude of other
> observational/experimental evidence). Unfortunately, results have been
> unsatisfactory. But for the benefit of lurkers who may have similar
> questions, I'll just restate a few general and oft-repeated
> observations.
The post continued by summarizing various points I made on several
previous occasions when the precession of Mercury has been discussed in
these groups. Unfortunately, it seems to have led to something of a
brouhaha, and I'd like to try to clear the air.
(I no longer check in here very often, which explains my tardy response.)
First of all:
Nicolaas, I am sorry if you got a different impression, but in fact I was
-not- trying to address anything you said in your post. This is why:
1. I couldn't tell whether you were asking seven questions, or claiming to
have -answered- them.
2. If the former, I did not understand your questions.
3. On the basis of past experience, I believe that our
backgrounds/inclinations are so different that I doubt that I -can-
understand your questions (if indeed they -were- questions), even after
you and I have expended much effort in trying to reshape them in a form
which makes more sense to me.
Similar problems have consistently bedeviled our previous attempts at
communication.
To illustrate what I mean by (2), let me renounce my vow and attempt to
list some things which I found confusing in your seven questions/claims:
> 1) Does it make sense to transform human based observations into grid
> based positions ?
I have no idea what you mean by "human based observations" or "grid based
positions". (I realize that this might refer to some previous post which
I missed.)
> 2) Does it make sense to remove light bending as part of those
> transformations ?
I have no idea what kind of "transformation" you could possibly mean, so I
can't possibly guess what this question might mean.
(When someone says "transformation" in the context of math/physics, I
start associating to things like "Moebius transformation", "conformal
transformation", "coordinate transformation", but you seem to be talking
about something much more murky.)
The phrase "remove light bending" also seems weird to me. I tried to guess
what you might mean by this, but unfortunately the guesses I came up
with--- while obvious to me, because of my background/experience--- appear
unlikely to be things you might think of:
Maybe you are thinking of trying to straighten out the appearance of
certain null geodesics by adopting a new coordinate chart? As in the
well-known Eddington or Kruskal-Szekeres charts for the Schwarzschild
vacuum? If so, be aware that those only "straighten" -radial- geodesics,
and -only apparently so-. No coordinate transformation can -remove- light
bending from a Lorentzian manifold in which it occurs, because this is a
geometric phenomenon, which has a clear geometric meaning -irrespective-
of coordinate chart.
Or maybe you are thinking of some alternative theories in which we are
restricted to conformally flat spacetimes, in which there really is no
light bending? But to make a long story short, that is inconsistent with
the evidence.
> 3) If those transformations make sense i.e. have an advantage above
> other methods then:
I have no idea what "other methods" you might mean, or what these "other
methods" accomplish, if they indeed exist, if indeed you even -believe-
that they exist, or are in sufficiently common use that you can expect
readers to know them.
> 4) What is the function of c within this grid or frame ?
When I use the word "frame" in a geometrical context, I almost always
refer to a widely used technical notion (frame as in "frame bundle"), but
I suspect that if you tried to explain this, after a dozen exchanges it
might appear that you were trying to asking about "light cones", which
exist at a different level of structure from "frames".
As for "the function of c", I would tend to read this in the manner of
"current as a function of resistance and voltage drop", but I guess you
might be asking something like this: "how does the appearance of the light
cones in our coordinate chart vary as a function of position in our
spacetime?"
> 5) What is the function of cg within this frame ?
Is "cg" supposed to be "speed of gravitational waves" or something like
that? If so, same comment, but with an additional comment: there are no
adjustable parameters in gtr, and gravitational and EM radiation (light)
travel at precisely the same speed, in vacuum, according to gtr.
> 6) Do I have to consider SR within this frame ?
Is "frame" supposed to be something like "Lorentz frame"? If so, most
people in this context use "frame" in the sense of "frame bundle", i.e.
a collection of "local Lorentz frames" defined at each event, and varying
smoothly over our Lorentzian manifold.
> 7) Do I need the full complexity of GR to describe the movement
> of the stars (and planets) ?
-This- question does make sense in this context, at least to me, if you
rephrase it like this: "Do I need the full EFE to describe the movement of
the stars (and planets)?".
The answer is: "No, in solar system dynamics you only need the LEFE,
applied to the far field of an isolated stationary object representing the
Sun--- which is easy--- plus a whole lot of Newtonian astrophysics---
which can get quite hard, if you want enough accuracy to be able to verify
relativistic effects".
Unfortunately, it appears likely that you actually intended to ask
something quite different (something which might not make sense to me at
all), because you added the comment:
> IMO the answer on that question is NO because there are no
> moving clocks involved
This doesn't make sense to me because in deriving Einstein's precession
formula, you have a test particle orbiting an isolated object in a nearly
elliptical orbit, and of course you should think of any test particle as
carrying an idealized clock. This is relevant to understanding the
physical meaning of what one means by the radial coordinate, which is one
of the critical conceptual issues which I discussed extensively in the
posts I cited; this issue arises because I was using a harmonic chart for
a solution to the weak-field EFE, not the Schwarzschild chart for the
Schwarzschild vacuum, which is a solution to the full EFE.
> I'm not aware of those discussions with you but anyway thanks for
> all the detailed information regarding perturbation theory.
Well, don't search under "T. Essel", because I was not using that handle
back then. Try searching under keywords (with group=s.p.r.).
BTW, of course I can see that it would have been best had I simply located
and reposted the detailed "past posts" I mentioned, but for reasons too
tiresome to enter into, that would have been inconvenient for me--- and
for everyone else, there is Google!
> In order to get some idea about about perturbation theory and astronomy
> I studied the following document:
> " Large-Scale Structure of the Universe and
> Cosmological Perturbation Theory"
> http://xxx.lanl.gov/abs/astro-ph/?0112551
One thing to be aware of in reading papers on metric perturbations is that
there are many possible gauge conditions and the LEFE (see below) can look
quite different depending on which gauge you choose. Many papers on
cosmological perturbations choose a different "gauge" (a different
restriction on the type of coordinate chart used) from the Lorentz gauge
which I used in my posts. Depending on context, you might also see the
perturbations decomposed into scalar, vector, and tensor modes; in gtr,
only two modes survive. Beginners might find all this variety confusing,
but it probably -is- good practice to sort through a few different
approaches until you can see that they really are all talking about the
same thing.
> My previous experience with perturbation theory was related to process
> control.
Perhaps you were using a perturbation expansion to find an approximate
solution to some nasty pde? If so, you can probably see where AE uses
similar ideas to find an approximate solution to an ODE, the
Einstein-Binet equation, which arises in slightly different forms
depending on whether you start from a weak-field solution, as did AE, or
from the exact Schwarzschild solution, as do many modern textbooks.
> Maybe perturbation theory is the final tool that I need in order to
> solve the equations that describe the movements of the stars and planets
> (in a very acurate way ?)
Oh dear--- I was constructing an interpretation of what you might be
trying to say, which momentarily appeared plausible, but then you you said
something which caused me to suddenly realize I must not have even been
close!
When you say "solve the equations that describe the movements of the stars
and planets", presumably you have in mind some equations which have
already been written down and are thus ripe for solution. But I don't
know if you are thinking of solving the equation of motion in some
specific solution to the EFE, such as the Schwarzschild vacuum (i.e., if
you are trying to study the motion of test particles modeling planets in a
simple spacetime model), or if you are thinking of solving the EFE itself
(i.e. trying to find a spacetime model in gtr), or if indeed you are
thinking of working with gtr at all.
When you say "in a very accurate way", it is not clear to me if you plan to
search for -exact- solutions or very accurate -approximate- solutions.
Perturbation theory is of course ideally suited to the latter task, under
some circumstances, but not the former.
It seems you might just possibly believe that there is no known
mathematical model (I suppose you could say "equations", but "mathematical
model" would be a far better term) which accounts for "the movements of
the planets in a very accurate way". Of course, there -are- such models.
Indeed, depending upon what you mean by "very accurate", such models
existed by the middle of the nineteenth century (of course, not accurate
enough to account for the extranewtonian precessions of Mercury, Venus,
Earth,..., but very accurate nonetheless).
And when you say "final tool", I am dismayed for yet another reason: as I
have mentioned in the past, "perturbation theory" as a body of
mathematical ideas/techniques can be said to have begun with Lagrange's
pioneering studies of solar system dynamics, and it has played a key role
in astrodynamics from that time right down to the present day. So, it is
not a "final tool", but one you need right from the start.
So, did anything I said anywhere above help at all?
If not, you probably see what I meant when I said that I have found
communicating with you to be very difficult!
FWIW, I can suggest some reading on pre-Einsteinian astrodynamics. I think
the following textbook is a terrific modern introduction to the Newtonian
treatment of solar system dynamics, which should convince any fair minded
reader that good old Newtonian gravitation remains of great interest, and
can still provide challenges as intriguing as anything in modern
mathematical physics:
author = {Carl D. Murray and Stanley F. Dermott},
title = {Solar System Dynamics},
publisher = {Cambridge University Press},
year = 1999}
Those with a less than adequate mathematical background should read this
with a good book on mathematical methods, such as:
author = {Derek Richards},
title = {Advanced Mathematical Methods with Maple},
publisher = {Cambridge University Press},
year = 2002}
I like this book because it emphasizes stuff you can -use-, which nowadays
mostly means "stuff you can do on a computer"-- for those primitive
beasts, that is, who still support any AFK functionality at all :-/
OK, now for the "brouhaha":
> From mingstb@marssim-ss.com Mon Nov 29 19:09:02 2004
> Date: Wed, 17 Nov 2004 22:32:57 GMT
> From: greywolf42 <mingstb@marssim-ss.com>
> Newsgroups: sci.astro, sci.physics
> Subject: Re: How important is GR inorder to calc the precession of Mercury
> (banned reply)
>
> The following post was banned from the sci.astro.research newsgroup ...
> without notice, and in violation of the newsgroup charter (as is usual
> for s.a.r).
>
> Not only are substantive responses blocked, but the moderators (T.
> Essel) dual-posted a triple reply to s.p.r. and s.a.r, that was simply
> boilerplate cheerleading for GR. Without responding to the question
> posted by Nicholaas, of course.
Barry, I am not sure what you meant to imply, but for the record, I am
-not- a moderator for either sci.physics.research or for
sci.astro.research. The names of the moderators are listed in the FAQs for
these groups.
> From mingstb@marssim-ss.com Mon Nov 29 19:10:05 2004
> Date: Thu, 18 Nov 2004 17:41:39 GMT
> From: greywolf42 <mingstb@marssim-ss.com>
> Newsgroups: sci.astro, sci.physics
> Subject: Re: How important is GR inorder to calc the precession of Mercury
> (banned reply)
>
> "Eric Gisse" <jowr.\pi@gmail.com> wrote in message
> news:929ed0f8.0411180145.402348b9@posting.google.c om...
> >
> > Their newsgroup, their rules.
>
> I have no problem with their rules. I just enjoy jerking the chain of
> people who claim to have rules, then break them whenever it suits their
> personal prejudices.
>
> > Its too bad (not really) that you don't like it, but I fail to see why
> > you would think a) any of us side for you regarding your plight
>
> I don't have a "plight."
>
> > and b) that we would do anything about it
>
> You don't have to do anything.
>
> > and c) that posting in here about your plight will do anything.
>
> Posting here serves my purpose. It allows the response to exist for the
> person who asked the question. And it shows the hypocritical actions of
> the moderators.
As for the alleged "triple-posting", I have not been submitting multiple
posts or anything like that. I don't know what you saw, or think you saw,
but I am sure there was no malicious intent on anyone's part. Possible
explanations include:
1. I submitted a candidate post for crossposting to s.a.r and s.p.r., so
of course this post, which was approved, should have appeared (once) in
-each- of these two newsgroups,
2. Sometimes moderation glitches might result in a given post appearing
two or more times in one of the groups. I can think of several ways that
might happen.
3. Sometimes, due to bugs arising from the way a given individual accesses
these newsgroups, it might appear to someone that a given post has
appeared twice, when that is not in fact the case. If someone you know,
who is using a different site or ISP, says they are not seeing anything
weird, this should indicate that the problem was not due to anything the
moderators did or did not do.
You also claimed that your reply had been banned (rejected?) "without
notice". If you didn't receive a rejection notice from the moderators,
this need not imply any "hypocritical" behavior on their part. Possible
explanations include:
1. The moderators simply might not have gotten to your submission yet.
Sometimes the moderators of s.p.r and/or s.a.r. announce "moderation
delays", meaning they will be busy with something else and won't be able
to tend to the newsgroup for some time. AFAIK, in such cases, by the luck
of the draw, some recently submitted posts might be "approved" [sic]
before an older submission has been considered.
2. The post -was- rejected, but if your submission did not include a valid
email address, the moderators wouldn't have been able to email you to
explain their reasons. Think of your submissions like sending off an MS
by snail mail to the magazine page editor of your local rag. If there's
no SA on the SSAE, you might never see your MS again.
3. You put your submission in the mail, but the moderators never received
it. Mail -is- sometimes lost by spoolers. That would normally only
happen in circumstances all too apparent to local users (e.g. a system
crash), but not all the moderators are using their local system when they
carry out their volunteer duties here. Also, for various reasons, mail
can be delayed in transit, sometimes for days.
As for not responding to Nicholaas, I did say what I -was- trying to do:
"But for the benefit of lurkers who may have similar
questions, I'll just restate a few general and oft-repeated observations."
but to be perfectly clear, I would have listed some things I was -not-
trying to do--- had the possibility of misinterpretation occurred to me!
To wit:
1. I was not trying to respond to the questions (?) listed by Nicholaas.
2. I was not trying to discuss the history of science.
3. I was not trying to discuss the respective merits of gtr and its
competitors.
To repeat (sigh...) several things which I've explained many times in the
past:
Barry, it seems you are wont to interpret my posts as "boilerplate
cheerleading for GR". Assuming that this implies some kind of alleged
failure to bring a open-minded and critical attitude to the intellectual
table, this must be taken as a sad indication of how badly my pedagogical
activity here over the years has failed, at least in your case.
The truth is that my principle pedagogical goal has always been to
-empower- interested parties to -achieve- their own goals, which might
very well include something like "overthrowing" gtr. (Not such an
outlandish goal, when you consider the fact that this would be a side
effect of constructing a viable quantum theory of gravity, which is of
course one of the principle goals of contemporary physics!)
As part of that goal, I want to help math/physics students--- at a
suitable level of sophistication--- to formulate their own answers to
questions having the general form: "just how does this thing work?". In
particular: "how does gtr work?"
This is not because I think that gtr is the only gravitation theory anyone
will ever need ---although I do think there'll always be a place for
Einstein, as indeed there is for Newton--- or because I am somehow
satisfied with the status quo. Rather, I know that to read the research
literature, you need to know something about our current gold-standard
theory of gravitation (gtr), and its predecessor (Newtonian gravitation)--
and I know that "something" turns out to cover a lot of territory.
Maybe it will help you to think of things like this: before you even
embark on a quest to find a gravitation theory which is even better than
gtr, I want you to know that there are seadragons and still stranger
hazards lurking out there upon the stormy deep. And, I believe, explorers
should be well qualified mariners riding a seaworthy vessel. Why? Well,
just imagine the ignominy of driving your ship hard onto a rock just
outside the harbour mouth, with the loss of all hands! In the long run,
common sense and historical experience both suggest that proper training
and methodical preparation will greatly increase the chances of ultimately
discovering some Grand New Continent. Blind luck will no doubt also play
an essential role, but one should avoid demanding absurd quantities of
that most mysterious element.
Or maybe a dimensionally reduced ASCII "picture" will help:
o o The Good Ship
Well o Endeavour ==>
Safe Charted o (math method) Some
Harbor Coastal ___/\___ Brave New
(Nwtn) Waters ~~~~~~\__{____}/~~~~~~~~~~~~~~~~~~~~ World,
(gtr) scurvy, storms, shoals, Yet
__{__} _ scyllae... Unknown
| | |-|
|__{__}| |-|
_||_ |_|
Volunteer Shipwright
and Coastal Pilot
(I, bot)
Floxian fables aside, when I ask "how does gtr work?"--- or rather, "how
is it -supposed- to work?"--- I have in mind a three-step process:
1. "How you can use gtr to construct a mathematical model suitable for
studying a given physical situation?",
2. "How can you compute physically meaningful quantities describing
relevant properties of that model?",
3. "How can you interpret the results?".
Alas, I think many people get stuck on the easy part (computation), even
though there are widely available freeware tools which can compute
curvature tensors in a heartbeat. I find the first and third parts of the
process (modeling and interpretation) most interesting, but you
definitely need a strong background in applied math and geometry to
appreciate the conceptual issues lurking behind my frequent technical
comments. It is certainly unfortunate that without this background, you
are unlikely to be able to appreciate my explanations of why the real
problems with gtr (which are considerable) are completely different from
what the cranks imagine.
Gtr is a mathematically sophisticated theory which cannot easily be
understood without prior mastery of a rather long list of ideas/techniques
such as manifolds, perturbation analysis, etc., etc., ideas and techniques
which in turn require some native mathematical ability (as well as time
and effort) to acquire. While I can easily understand why this fact
frustrates those who have the interest but lack the background, it -is- a
fact. You can certainly attempt to work around it, but you can't ignore
it.
Don't forget, as a self-taught amateur myself, I have been in the position
of studying MTW with only a high school math background (yes, -that's-
changed!), so I have a special appreciation of considerable virtues of a
formal, curriculum-driven education.
And BTW, while gtr is clearly an interesting and important theory, which
unfortunately requires considerable background which requires extensive
time/energy to acquire/use, in the past I have often tried to popularize
theories/problems which are no less fascinating or timely, but which are
-much more accessible-.
>From the charters of these newsgroups, or even from the names
"sci.*.research", I think that one might fairly might expect that these
newsgroups are maintained by and for the principal benefit of
physics/astronomy researchers and their apprentices (graduate students in
physics/astronomy), and that while nonphysicists are welcome, they should
be expected to attempt to "play by the rules" of communication in
physics/astronomy. E.g. by attempting to speak our language, as far as
possible, and by trying to steer close to topics we would recognize as
physics-related.
(As you know, I am myself a nonphysicist, but hereabouts I take care to
play by the rules, not just because that's only polite, but because it is
virtually -essential- for communication in such a highly developed and
technical field.)
Until a few years ago, I think this model did approximately hold true in
s.p.r., most of the time (s.a.r. is a much newer newsgroup). Nowadays it
seems that most of the gravitation-related traffic in both groups arises
from posters who are not members of the research community and who appear
to have woefully little understanding of what members of that community do
and why. In the last few years, I have frequently seen language
suggesting that some posters do not understand basic concepts of
mathematical modeling, or even the scientific method itself, much less
understand the elements of what has already been achieved (scientifically
speaking) in the area of gravitation physics, much less what we should try
to achieve next, or where we can expect to go in the more distant future
future.
This circumstance, while clearly tending to generate much wasted bandwidth
due to gross miscommunication, would be more acceptable if regular but
"physically gauche" posters at least paid close attention to occasional
attempts by myself or others to explain some of "the rules", for the
benefit of those who lack the strong background in math/physics shared by
card-carrying members of the research community (and certain literate
amateurs, like myself). My little fable above falls into this category.
Alas, this is by no means the first time I have found myself trying to
explain to you in particular stuff which I don't think I should really
need to explain, at least not more than once, at least not to a single
individual, at least not in these groups.
Maybe you feel you are just having fun here by "jerking my chain" (if
that's really what you meant to say in your sci.astro post), but this is
not fair to other participants in s.p.r./s.a.r. (some of whom still come
here for the purpose intended, scientific discussion of research issues of
current interest in physics and astronomy). Indeed, it is not fair to
-me-, because you persist in attributing to me motives very different from
those which actually impel my occasional participation in these groups,
despite many attempts to disabuse you!
Now, despite your complaints in sci.astro, I see that at least one post
which you (Barry) wrote criticizing what I said -did- appear right here:
> From mingstb@marssim-ss.com Mon Nov 29 18:25:45 2004
> Date: Fri, 19 Nov 2004 19:30:45 +0000 (UTC)
> From: greywolf42 <mingstb@marssim-ss.com>
> Newsgroups: sci.physics.research, sci.astro.research
> Subject: Re: How important is GR in order to calc the precession of Mercury
>
> <tessel@tum.bot> wrote in message
> news:cncfni$t38$1@lfa222122.richmond.edu...
> > On Wed, 10 Nov 2004, Nicolaas Vroom asked (in s.a.r.)
> >
[..]
> > on several previous occasions, I have gone to great
> > lengths to help you understand what gtr says about the extraNewtonian
> > precession of Mercury (and why gtr is such a satisfactory theory for
> > purposes of explaining this and a multitude of other
> > observational/experimental evidence).
>
> It would be polite to provide a link to said statements.
Yes. In past years, I always took the trouble to provide extensive
annotated references to both on-line and printed material of use to
serious students, so I'll take this as very belated thanks for my previous
efforts :-/
> A google search of
> your posts shows no matches against "Nicolaas Vroom". In fact, there is
> only one post against your name (to Bill Kavanah), that contains the word
> "mercury" or the phrase "perturbation theory":
> http://www.google.com/groups?selm=cmq3is%24a2m%241%40lfa222122.richmond. edu
> And in this post, you again make unreferenced statements that you've "posted
> on this (perturbation theory) very extensively before."
See my advice to Nicolaas above.
> > This is needed in the problem at hand
> > because the theoretical problem confronting Einstein in 1916 was not to
> > explain the precession of Mercury in its orbit around the Sun, but rather
> > to explain a small residual remaining after a perturbation theory analysis
> > of a model in Newtonian gravity had explained all but a small part of the
> > observed motion.
>
> On the contrary. The explicit, stated purpose of Einstein was to obtain
> Newcomb's published value (43" per century*) for the NNPA of Mercury.
So what is the problem? The notion of a "residual"?
You said that Grossmann "remeasured" the residual, but I think you should
have said "recalculated". The principle issue was (and remains) not how
to -measure- the motion (astronomers had already gotten very good at that
before AE came along), but how to -explain- it theoretically. Saying that
we have an extranewtonian residual of 43 seconds per century amounts to
saying that -our best Newtonian model- leaves that much unaccounted for.
You had some other comments, which I think are mostly moot, since you
completely misunderstood the purpose of my post. But I must correct one
serious misstatement you made:
> > In gtr,
> > we have the additional complication that the full field equation (the EFE)
> > is nonlinear, but this plays no role here because we can get away with
> > studying solutions to a linearized version of the EFE.
>
> Which is simply the Newtonian equation, with an added speed-of-gravity
> parameter (equal to the speed of light).
Anyone who has studied any modern gtr textbook will know that there is
only one thing I could possibly mean by "a linearized version of the
EFE"--- and I doubt that anyone who has seen it could possibly describe it
as "simply the Newtonian equation, with an added speed-of-gravity
parameter (equal to the speed of light)"!
For example (you did want citations?):
1. See (20.28) in section 20.1 ("The linearized field equations") of the
textbook
author = {Ray D'Inverno},
title = {Introducing {E}instein's Relativity},
publisher = {Clarendon Press},
year = 1995}
2. See (8.42) in section 8.3 of the textbook
author = {Bernard F. Schutz},
title = {A First Course in General Relativity},
publisher = {Cambridge University Press},
year = 1985}
3. See (13.14) in section 13.2 ("The fundamental equations of the
linearized theory") of the textbook
author = {Hans Stephani},
title = {General Relativity:
An Introduction of the Theory of the Gravitational Field},
publisher = {Cambridge University Press},
edition = {Second},
note = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},
year = 1990}
4. See (7.125) of the textbook
author = {Sean Carroll},
title = {Spacetime and geometry: an introduction to general relativity},
publisher = {Addison-Wesley},
year = 2004}
5. See (18.8b) of section 18.1 ("The linearized theory of gravity") of
the textbook
author = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},
title = {Gravitation},
publisher = {W. H. Freeman},
year = 1970}
6. See (4.4.12) in section 4.4 ("Linearized Gravity") of the textbook
author = {Robert M. Wald},
title = {General Relativity},
publisher = {University of Chicago Press},
year = 1984}
7. See (53) in section 3.2 ("The linear field equations for gravitation")
of the textbook
author = {Hans C. Ohanian and Remo Ruffini},
title = {Gravitation and Spacetime},
edition = {Second},
publisher = {W. W. Norton},
year = 1994}
8. See (10.1.10) in section 10.1 of the textbook
author = {Weinberg, Steven},
title = {Gravitation and Cosmology:
Principles and Applications of the General Theory of Relativity},
publisher = {Wiley},
year = 1972}
9. See (9) in section 4.1.2 of the textbook
author = {C. Clarke},
title = {Elementary General Relativity},
publisher = {Wiley},
year = 1979}
The LEFE (linearized Einstein field equation) is a -fundamental- and
-standard- topic which is treated at length in almost every gtr book!
The name LEFE (or something very similar) is also standard, as you can see
from the section titles I quoted.
(Are you convinced yet, or should I give even more references?)
Note that the LEFE takes different forms in different gauges; most books
discuss at least the "Lorenz gauge", and I have listed above the form
which the LEFE takes (for trace-reversed perturbations) in that gauge.
Barry, in your sci.astro post, I see that you say
> MTW is not a good way to learn GR. It's a decent way to expand your
> knowledge in specific areas, once you have the basics down.
I think your opinion might be different once you get the basics down.
MTW is one of the great scientific books of all time, one which rewards
repeated reading at -all- stages of your education. It is true that
understanding what you read gets exponentially easier as you acquire more
background and experience, but looking into this book at an early stage
can give you an idea of what it is you want to understand, and checking
back periodically (and working more problems!) can reassure you that you
are in fact making progress.
Be all this as it may, the LEFE is great fun, so I urge -everyone- who
doesn't know about it to read up forthwith!
Don't be confused by the fact that in many of the textbooks above,
discussion of the LEFE occurs in the context of gravitational radiation,
sometimes long after a discussion of the precession of nearly Keplerian
orbits. The underlying field equation is the same regardless of whether
one is discussing the "far field" of an isolated stationary object (e.g.
for studying precession of nearly Keplerian orbits around a star), or the
generation of gravitational radiation (anything but stationary!). Of the
textbooks listed above, this underlying unity is probably brought out best
by Stephani or MTW, but it should be clear enough from any of these books
if you read with sufficient care and insight.
For the benefit of interested readers, I append a brief overview in "Baez
emulation mode" ;-/ For the details, I hope all interested readers will
rush right out, obtain a copy of the one of the above mentioned textbooks,
and study them!
T. Essel (spelunking somewhere in cyberspace)
================================================== ===============
Just what -is- this "LEFE" business?
Well, we should begin at the beginning, so let's begin with the EFE (the
full, nonlinear, Einstein field equation).
Intuitively, gtr aims to directly relate geometry and matter. You
probably already have some intuition for what matter is, and how we can
measure how much there is of it in a given place. But what about
geometry? To directly relate geometry to matter, we evidently need to
somehow quantify -geometry itself-! How can we do that? Well, according
to Riemann, geometry always comes down to angles and distances. And to
quantify angles and distances, we can employ the "metric tensor" of a
Lorentzian manifold.
This Lorentzian manifold, or "spacetime model", will play a dual role: it
will serve as our mathematical model of "gravitational phenomena as
geometry", in a specific physical context (e.g. modeling a planet orbiting
the Sun), but it will also constitute the -geometrical setting- for all
-nongravitational- physical phenomena under study, if any. The rules for
transferring all our nongravitational physical theory (e.g. Maxwell's
theory of EM, hydrodynamics, the theory of elasticity) to this curved
spacetime setting are sometimes summarized by the slogans "minimal
coupling" and "comma goes to semicolon". These rules are not entirely
straightforward, but they are not hard to use in most cases.
The full EFE reads
G_(ab) = 8 \pi T_(ab) (*)
Probably everyone here knows that, but not everyone really knows what
those funky symbols actually -mean-! Fortunately, I can describe the
basic "ingredients" of this equation, and even try to give a hint of how
they fit together.
On the left, we have something mathematically representing the character
of the geometry of our spacetime. It is G_(ab), the Einstein tensor, a
second rank tensor, which is a kind of "average curvature" tensor
constructed from the fourth rank Riemann curvature tensor R_(abcd). The
Riemann tensor completely describes the local geometry near each event in
our Lorentzian manifold; it can be readily computed from the metric tensor
by a straightforward process of double differentiation.
(Going the other way is harder, but it can be done. This raises all kinds
of fascinating mathematical issues which unfortunately I haven't the
space-time-energy to discuss, at least not here.)
On the right, we have something mathematically representing the amount and
motion of matter. It is T_(ab), the energy-momentum-stress tensor.
(Multiplied by a factor of 8 \pi, but I probably don't need to say what 8
and \pi are, huh? So I won't.) Mathematically speaking, T_(ab) is the
"source term" in the EFE, modeling the stuff (anything with energy or
mass) which generates a gravitational field. Roughly speaking, at each
event in our spacetime model, as measured by an appropriate "ideal
observer", the T_(00) component tells how much -mass- there is, the T_(0j)
components tell how much -momentum- there is, and the T_(jk) components
tell how much stress there is.
For example, consider an isolated rigidly rotating thin disk. Here,
T_(ab) would be nonzero only inside the disk, and if we compute the
components in an appropriate frame, T_(00) (mass density), T_(0j)
(momentum density), and T_(ij) (stress density) will all be nonzero. The
stress arises because in order to maintain its shape, the disk must resist
the tendency to fly apart due to centrifugal "forces" [sic] arising from
its rotation, and this mechanical resistance shows up as stress.
So, the EFE does indeed say that a basic geometric phenomenon (the
averaged curvature) is directly related to a basic physical phenomenon
(the amount and motion of matter and energy of nongravitational fields).
We can say more: the presence of matter (nonzero source term on the RHS)
directly causes a local distortion of geometry (a change in our "average
curvature"), which typically changes the behavior of geodesics. To see why
this is profoundly important, you need to know that one of the great
beauties of gtr is the elegant way in which it realizes the -universal-
character of gravitation (the fact that gravity acts on all forms of
mass-energy). Namely, according to gtr, if no physical forces act on a bit
of matter, then its world line (kinematic history as represented in our
Lorentzian manifold or spacetime model) will be a geodesic. (In a curved
manifold, geodesics are the analog of "straight lines": they are curves
which are "as straight as possible"). So in gtr, when curvature is
present, this typically will affect the relative motion of free-falling
bits of matter.
Needless to say, gtr can also handle any -nongravitational- forces acting
on matter, such as pressure acting on a fluid element. Namely, the
magnitude and direction of net force on some bit of matter determines the
magnitude and direction of the -path curvature- of its world line. If
forces act, the world line has nonzero path curvature; if not, it doesn't.
This is just what we would do in str (which doesn't attempt to treat
gravitation), except that now our spacetime is curved. Note that this
means that "gravitational forces" never appear in gtr, or rather, the
force which a standing human applies at his feet to resist falling through
the floor can be called that, but really it is a -nongravitational- force
which is accelerating him with a certain magnitude and direction (away
from the center of the Earth), which means that his world line has nonzero
path curvature; it is -not- a geodesic. But he steps off the balcony, he
will subsequently be in free fall--- until he strikes the ground--- and
along the free-fall portion of his kinematic history, his world line will
have vanishing path curvature.
John Archibald Wheeler (the W in MTW) famously summed things up something
like this: "Matter (plus possible nongravitational fields) tells spacetime
how to curve; spacetime curvature (plus possible nongravitational forces)
tells matter how to move".
Let's look at how this works in a bit more detail. Mathematically, we can
decompose the Riemann curvature tensor like this:
R_(abcd) = C_(abcd) +[/itex] a kind of fourth rank "expansion" of
Riemann Weyl curvature R_(ab) = R^{m_}(amb)
curvature Ricci curvature
In gtr, "Ricci curvature" (or equivalently, "Einstein curvature"), meaning
either the second rank tensor or its fourth rank "expansion", is closely
associated with matter: the -immediate presence- of matter in some region
causes nonzero Ricci curvature in that region, and conversely, Ricci
curvature cannot exist where no matter is present.
"Weyl curvature", AKA "conformal curvature", has a beautiful and
compelling geometric meaning, related to "light bending", but independent
of any physical interpretation. But to save time, here I will just say
this: in gtr, Weyl curvature is the type of curvature which can exist
-independently of matter-; in particular, it is the type of curvature
which can propagate through a vacuum region, as a "gravitational wave".
Such waves are terribly important because they are ultimately the
"physical mechanism" by which a region of spacetime (empty or otherwise)
can "learn" of the presence of mass/energy somewhere else, possibly far
off, and "respond" by changing its curvature to maintain the validity of
the field equation.
For example, if a bomb goes off in some isolated region in space, that
changes the distribution and motion of matter in this region, which can
create "ripples" of Weyl curvature which then propagate outwards, at the
speed of light, through the surrounding vacuum. Such bombs actually
exist--- they are called "supernovae"! With LIGO, we hope to detect their
gravitational wave signature.
You might be asking: since the EFE only mentions Einstein curvature, (or
equivalently, Ricci curvature), how can the motion of matter possibly
create propagating ripples of Weyl curvature? The answer is that the
"differential Bianchi identity" (a mathematical identity which is true
independently of the EFE, and indeed, independently of any physical
interpretation of curvature) enforces a "coupling" between matter and Weyl
curvature via a first order PDE. Essentially, this relation says that the
immediate presence of matter in one region can cause Weyl curvature in
that region, and this can in turn cause more Weyl curvature in adjacent
regions, even nearby vacuum regions. But what matters is not so much the
mere -presence- of matter as the existence of a suitable -gradient-. And
indeed, the Friedmann cosmological models, which are homogeneous and
isotropic spacetimes, have nonzero Ricci curvature (since they are filled
with fluid or dust) but no Weyl curvature (and thus no light bending).
OTH, in the famous Schwarzschild vacuum, which is an exact solution of the
EFE representing the field outside a nonrotating spherically symmetric
massive object, -all- the curvature is Weyl curvature. Of course, this is
the same as saying we have a vacuum solution, since "no matter" is
equivalent to "no Ricci"! But how, you might ask, did this Weyl curvature
get there? Well, in imagination, it happened like this: when we
concentrated matter to form the Sun, by the conservation of mass, we had
to move it in from somewhere, and when we did that, according to the EFE,
the "matter gradient" created Weyl curvature, which then propagated into
nearby vacuum regions (at the speed of light) and began to curve them up,
and so on. Thus, the gravitational field of the Sun is a kind of
"historical legacy" which was created--- indirectly and gradually---
during the slow formation of the Sun. Any gravitational radiation
currently emanating from the Sun merely represents teeny tiny
perturbations carrying information about movements of matter inside the
Sun, information which will eventually be used by regions far away to make
-extremely- teeny tiny "local adjustments in curvature" in order to uphold
the validity of the EFE. Or at least, that's the way gtr tells the story!
:-/
Newbies often ask "if nothing can get out of a black hole, how can
gravitation get out?" The answer is that (in gtr) -it doesn't have to-.
The collapse of a star to make the hole probably created some strong
gravitational radiation, but this can't instantly undo the Weyl curvature
which was gradually created during the formation of the star.
Or again, sometimes people ask: "if a black hole was created by the
complete collapse of a star, and if the matter which once made up the star
has been crushed out of existence at a spacetime curvature singularity,
then surely there is nothing left to gravitate?" One answer is that it is
precisely because information cannot propagate from under the horizon that
the "bad news" about the fate of the matter which originally created all
that Weyl curvature (indirectly!) cannot reach the exterior region, so we
do not in fact need to worry about this, unless we fall into the interior
region, in which case we've got more pressing--- and tugging--- problems
:-/
But enough about the EFE, you say-- what about the LEFE?!
OK, now I must confess that everything I just said wasn't the beginning at
all, it was just stuff I had to say before I could even begin! :-/ -This-
is the beginning:
The starting point of "linearized gtr" is a very simple idea: finding
exact solutions of the EFE can be a daunting task, so to make things
easier, imagine that we have, essentially, flat spacetime, except that in
some places we have small amounts of matter (or an EM field, or some other
nongravitational field) which induces a "mild distortion" of our otherwise
flat spacetime. Or ---to use the lingo--- assume that our Lorentzian
spacetime is a "metric perturbation" of Minkowski spacetime (in fact, a
"linear perturbation", meaning that we only carry out computations to
first order in our "perturbation parameter").
So, let's write the metric in the form of the sum of the Minkowski space
metric tensor (representing the flat background) plus a perturbation
tensor (representing the "mild distortion" due to the presence of some
not-too-massive object, a passing gravitational wave, whatever):
g_(ab) = \eta_(ab) + h_(ab)
background perturbation
In weak-field gtr, the background is unobservable, so physically speaking,
it doesn't exist. But mathematically speaking, we can pretend, and there
is a rule of thumb that says that whenever you can get away with
pretending something is much simpler than is really the case, you should
be cynical and just grab the opportunity! So, let's treat the metric
perturbation tensor h_(ab) as just another tensor field on -flat-
spacetime. It is a second rank symmetric tensor field on a four
dimensional manifold, which implies that it has ten algebraically
independent components at each event.
Now, if we compute the Einstein tensor of our mildly distorted spacetime,
using the true metric g_(ab), but expressing our result in terms of the
perturbation tensor h_(ab), treated as just another tensor field in flat
spacetime, we get a big mess. But we are free to add conditions on h_(ab)
to try to reduce the clutter, so let's add some! This has the effect of
ruling out the use of some coordinate charts, but it is always possible to
find plenty of coordinate charts which meet our "gauge conditions". There
are many possible choices of gauge conditions, but the LEFE takes a
particularly simple form if we choose the "Lorenz" (or "harmonic") gauge,
so let's make this choice.
Next, toss in a minor trick: rewrite everything in terms of the
"trace-reverse" of h_(ab), written \hbar_(ab). -Viola!-, you get the LEFE:
Box [itex]\hbar_(ab) = -16 \pi T_(ab) (**)
It's just the good old flat spacetime wave equation, times ten!
On the left, we have something acted upon by the flat spacetime wave
operator Box(.), something which mathematically represents the mild
distortion of the flat spacetime geometry. It is \hbar_(ab), the
trace-reversed metric perturbation tensor, which, as I said, we treat as a
second rank symmetric tensor field on flat spacetime. So it has ten
algebraically independent components--- each of which satisfies the scalar
wave equation familiar to every undergraduate sci/math student. Secretly,
of course, the left hand side also has a beautiful and profound geometric
interpretation: it is the Einstein tensor, computed for our mildly
distorted spacetime using the -true- (physical) metric tensor g_(ab)!
On the right, we have something mathematically representing the amount and
motion of matter. It is T_(ab), the energy-momentum-stress tensor, treated
as a tensor field on flat spacetime. Mathematically speaking, it serves as
the "source term" for our flat spacetime tensorial wave equation.
In a vacuum region (e.g. outside the disk), where the right hand side of
the LEFE vanishes, we are dealing with the "source-free" (tensorial) wave
equation
Box \hbar_(ab) =(***)
In such a region, we might have changes in the geometric distortion
propagating as a wave (a vacuum gravitational wave).
Qualitatively, the LEFE says the same thing as the full EFE: "Matter
(plus possible nongravitational fields) tells spacetime how to curve;
spacetime curvature (plus possible nongravitational forces) tells matter
how to move". But mathematically, it takes on a new form which offers some
very suggestive analogies.
Let's compare the LEFE with Maxwell's "nontrivial" field equation:
Box A^b = 4 \pi J^b
Here, on the left, we have something on which the wave operator Box acts,
something which mathematically represents "the EM field". It is A^b, the
four-vector potential (from which we can recover the EM field itself by
"exterior differentiation", F = dA). On the right, we have something
mathematically representing the amount and motion of any electrically
charged matter. It is J^b, the current-density four-vector. In a
charge-free region, we have the "source-free wave equation", and in such a
region, we can have an EM field propagating as a wave ---we can have
"light".
There's much more to this analogy with Maxwell's theory of EM. For
example, the EM field tensor F_(ab) has two scalar Lorentz invariants
F_(ab) F^(ab), F_(ab) *F^(ab)
from which we can recognize when we have a "purely electrostatic" field,
(in which case we can find observers who measure no magnetic field), and
more besides. And from the Riemann curvature tensor (the gravitational
field tensor) we can construct analogous scalar Lorentz invariants
R_(abcd) R^(abcd), R_(abcd) *R^(abcd)
and then we can recognize situations where we can find observers who
measure no "magnetogravitic field", and more besides. (BTW, this scalar
invariant stuff actually works fine for fully-fledged gtr, not just its
linearized weak-field approximation.)
Why is this interesting? Well, for one thing, it points up a very
dramatic difference between linearized gtr and Newtonian gravitation.
Consider a rigidly rotating uniform density thin disk. In Newtonian
gravitation, by measuring the gravitational field we cannot distinguish
our disk from an otherwise identical but nonrotating disk. But in the
LEFE, the source term T_(ab) takes account not only of the amount of
-mass- in each region, but also the amount of -momentum-. This turns out
to mean that in weak-field gtr, we -can- distinguish the rotating disk
from its nonrotating twin, just by measuring its gravitational field! If
its magnetogravitic field is nonzero (as must be case if the scalar
invariant R_(abcd) *R^(abcd) is nonzero), then it is rotating. If not, it
isn't! Unfortunately, these are usually tiny effects, but there are
experiments underway or planned to test this feature of gtr using earth
orbiting satellites (Gravity Probe B and its kin).
(Maybe I should remark that, while what I just said about scalar
invariants is true for isolated stationary objects, it turns out that
-all- the scalar curvature invariants of vacuum gravitational plane waves
vanish identically, but these have magnetogravitic fields comparable to
their electrogravitic or tidal fields, as measured by any observer. But
even here we have a valid analogy, since in Maxwell's theory of EM, the
two Lorentz invariants of the field tensor F_(ab) vanish identically for
an EM plane wave, and these have magnetic fields comparable to their
electric fields, as measured by any observer.)
Predicting and possibly confirming new physical phenomena is always very
exciting, but there is another reason why this observation is of
fundamental interest. To set up our rotating disk model, we must write
down a boundary value problem for the LEFE. That is,
1. We write down boundary conditions stipulating the size, shape,
(uniform) mass, and (steady-state) rotation of the disk,
2. We attempt to find the required vacuum solution of the LEFE (a purely
mathematical problem),
3. We attempt to interpret the resulting Lorentzian spacetime (defined
only in the region outside the disk) as a mathematical model of
"gravitation as geometry".
But there is nothing in the LEFE to prevent us from removing the -mass-
and leaving the -momentum- in our boundary conditions. As an exercise, it
is a good idea to do just this, and to solve the new boundary value
problem. What you get an analogue of a "purely magnetostatic field", and
this turns out to be pretty darn weird: all the spacetime curvature is in
the t = constant family of spacelike hyperslices, so static observers
experience no gravitational tidal forces, and they need not fire their
rocket engines to remain static. However, observers riding a
gyrostabilized platform in free fall eventually find that they are
-spinning- with respect to the distant stars!
This illustrates a very important feature of gtr: boundary conditions are
essential, and should arise from nongravitational physics, but "garbage
in, garbage out": if you fail to posit "physically reasonable" boundary
conditions, your spacetime model is unlikely to be physically reasonable.
On the bright side, gtr plus a little common sense provides us with a
strong hint that our new boundary conditions must unreasonable, even
before we discovered the weird properties of a "purely magnetogravitic
spacetime". We need only ask, how might we "prepare" a thought experiment
involving a rotating disk? Well, we can imagine a region of spacetime far
removed from any massive objects, and we can imagine sending in some
matter to form a rotating object, which will serve as the source of the
gravitational field in our isolated region. But of course, this should
lead to something which has positive mass and angular momentum (and not
too much of the latter in comparison to the former).
Speaking of electrostatic fields--- if you know how the EM field reduces
to an electric field in some circumstances (e.g. a static configuration of
charged matter, perhaps on the surface of a charged metal sphere), you
will recall that we can kill off all but the "time component" of the EM
potential and the current-density four-vector. Then, our wave equation
reduces to the Poisson equation
Lap \phi = 4 \pi \sigma
where \phi is the electrostatic potential and \sigma is the charge density.
In the case of a spherically symmetric source-free field, by introducing
the Lorentz force law we can recover the inverse square law of Coulomb,
which is formally exactly the same as Newton's inverse square law. And if
we look at static weak gravitational fields, if we use a suitable observer
we will find that \hbar_(ab) has only one nonvanishing component, which
corresponds to the Newtonian potential. Then, our wave equation again
reduces to the Poisson equation
Lap \phi = 4 \pi \rho
where now \phi is the gravitational potential and \rho is the density of
matter. The -vacuum- field equation is the Laplace equation (again), from
which we can recover Newton's inverse square law. But this time we don't
need to -introduce- any force law--- because in gtr, there -is- no
"gravitational force"! This is another of those unique features which
makes gtr so remarkable from a mathematical point of view.
Now, any solution of the three dimensional Laplace equation (such things
are called "harmonic functions") can serve as a Newtonian potential (or as
an electrostatic potential). From this, you might guess that you can cook
up a storm with Green's functions for trace-reversed metric
perturbations--- and you'd be right! If you know that you can expand any
asymptotically vanishing harmonic function as an infinite sum of
particularly simple ones (the "harmonics") you might guess that you can do
something similar for trace-reversed metric perturbations--- and you'd be
right about that too! This is why, when you see the term "quadrupole
moment" when you are reading a paper about gravitational radiation, and
then see it again when you are reading another paper about the shape of
the Sun, you can be confident that both authors really were talking about
the same thing.
At this point, you might ask "what puts the L in LEFE?" Well, the wave
equation is -linear-, which means that if we have two solutions, we can
just add them to obtain a new solution. And the LEFE is just a tensorial
wave equation for the trace reversed metric perturbation. This means that
if we know two "legal" trace reversed metric perturbations (tensors which
solve the LEFE), we can just add them and get a new "legal" metric
perturbation:
newsol = sol1 + sol2
But if we try the same thing with two solutions to the full EFE, we might
(probably after a lot of work) be able to write something like this:
newsol = sol1 + sol2 + "nonlinear interaction terms"
That's nonlinearity in action!
Physically, we can say that the "nonlinear interaction terms" tend to
imply that we will see stronger gravitation than we would expect in a
linear theory. One way to think about this is to imagine two stars which
approach each other in some isolated region of deep space. As the total
gravitational field gets stronger in the region between them, its energy
increases, but this energy itself gravitates, so the gravitational field
increases in strength faster than Newtonian theory would lead you to
expect.
If that sounds complicated, it is, and this goes a long way toward
explaining why the LEFE is so much easier to solve than the EFE.
When can we use the LEFE rather than the EFE? Whenever and wherever our
spacetime is "almost flat"! Or in other words, whenever and wherever the
gravitational field is "weak". One place that happens is far from an
isolated massive object like the Sun. So if hbar1_(ab) models the Sun, or
at least the "far field" of the Sun, where gravitation is weak, and if
hbar2_(ab) represents a gravitational wave propagating in flat spacetime,
then hbar1_(ab) + hbar2_(ab) represents a gravitational wave passing near
the Sun-- but not too near!
So, you say, if we can do all this cool stuff with the LEFE, who needs the
EFE? Well, sometime or somewhere spacetime might not be -anywhere near
flat-; for example, soon after the Big Bang, or near the horizon of a
smallish black hole!
There is also some very persuasive theoretical reasons for regarding the
EFE as the real deal, and the LEFE as a mere approximation which is useful
in some circumstances. One of the most intriguing is this: it turns out
that the LEFE is not entirely self-consistent. There is a more or less
unique way for modifying the LEFE until eventually (after passing through
infinitely many stages), you have a self-consistent (but nonlinear) field
equation. And this field equation is:
ooo dramatic drumroll ooo
The EFE!!
thread titled "Re: How important is GR in order to calc the precession of
Mercury", which began:
> From tessel@tum.bot Mon Nov 29 18:25:37 2004
> Date: Wed, 17 Nov 2004 17:03:11 +0000 (UTC)
> From: tessel@tum.bot
> Newsgroups: sci.physics.research, sci.astro.research
> Subject: Re: How important is GR in order to calc the precession of Mercury
>
> On Wed, 10 Nov 2004, Nicolaas Vroom asked (in s.a.r.)
>
> > In the newsgroup sci.physics.relativity I started a posting with the
> > subject title. The purpose
>
> question?
>
> > was how do you simulate the movement of the planets, specific the
> > movement of Mercury.
> >
> > Not many people responded to my messages and as such I try in this
> > newsgroup, maybe with a better result.
> >
> > The approach I take is slightly different as maybe expected and that
> > maybe explains the low responds.
>
> You have asked very similar questions before in various forums including
> s.a.r. and s.p.r., and on several previous occasions, I have gone to
> great lengths to help you understand what gtr says about the
> extraNewtonian precession of Mercury (and why gtr is such a satisfactory
> theory for purposes of explaining this and a multitude of other
> observational/experimental evidence). Unfortunately, results have been
> unsatisfactory. But for the benefit of lurkers who may have similar
> questions, I'll just restate a few general and oft-repeated
> observations.
The post continued by summarizing various points I made on several
previous occasions when the precession of Mercury has been discussed in
these groups. Unfortunately, it seems to have led to something of a
brouhaha, and I'd like to try to clear the air.
(I no longer check in here very often, which explains my tardy response.)
First of all:
Nicolaas, I am sorry if you got a different impression, but in fact I was
-not- trying to address anything you said in your post. This is why:
1. I couldn't tell whether you were asking seven questions, or claiming to
have -answered- them.
2. If the former, I did not understand your questions.
3. On the basis of past experience, I believe that our
backgrounds/inclinations are so different that I doubt that I -can-
understand your questions (if indeed they -were- questions), even after
you and I have expended much effort in trying to reshape them in a form
which makes more sense to me.
Similar problems have consistently bedeviled our previous attempts at
communication.
To illustrate what I mean by (2), let me renounce my vow and attempt to
list some things which I found confusing in your seven questions/claims:
> 1) Does it make sense to transform human based observations into grid
> based positions ?
I have no idea what you mean by "human based observations" or "grid based
positions". (I realize that this might refer to some previous post which
I missed.)
> 2) Does it make sense to remove light bending as part of those
> transformations ?
I have no idea what kind of "transformation" you could possibly mean, so I
can't possibly guess what this question might mean.
(When someone says "transformation" in the context of math/physics, I
start associating to things like "Moebius transformation", "conformal
transformation", "coordinate transformation", but you seem to be talking
about something much more murky.)
The phrase "remove light bending" also seems weird to me. I tried to guess
what you might mean by this, but unfortunately the guesses I came up
with--- while obvious to me, because of my background/experience--- appear
unlikely to be things you might think of:
Maybe you are thinking of trying to straighten out the appearance of
certain null geodesics by adopting a new coordinate chart? As in the
well-known Eddington or Kruskal-Szekeres charts for the Schwarzschild
vacuum? If so, be aware that those only "straighten" -radial- geodesics,
and -only apparently so-. No coordinate transformation can -remove- light
bending from a Lorentzian manifold in which it occurs, because this is a
geometric phenomenon, which has a clear geometric meaning -irrespective-
of coordinate chart.
Or maybe you are thinking of some alternative theories in which we are
restricted to conformally flat spacetimes, in which there really is no
light bending? But to make a long story short, that is inconsistent with
the evidence.
> 3) If those transformations make sense i.e. have an advantage above
> other methods then:
I have no idea what "other methods" you might mean, or what these "other
methods" accomplish, if they indeed exist, if indeed you even -believe-
that they exist, or are in sufficiently common use that you can expect
readers to know them.
> 4) What is the function of c within this grid or frame ?
When I use the word "frame" in a geometrical context, I almost always
refer to a widely used technical notion (frame as in "frame bundle"), but
I suspect that if you tried to explain this, after a dozen exchanges it
might appear that you were trying to asking about "light cones", which
exist at a different level of structure from "frames".
As for "the function of c", I would tend to read this in the manner of
"current as a function of resistance and voltage drop", but I guess you
might be asking something like this: "how does the appearance of the light
cones in our coordinate chart vary as a function of position in our
spacetime?"
> 5) What is the function of cg within this frame ?
Is "cg" supposed to be "speed of gravitational waves" or something like
that? If so, same comment, but with an additional comment: there are no
adjustable parameters in gtr, and gravitational and EM radiation (light)
travel at precisely the same speed, in vacuum, according to gtr.
> 6) Do I have to consider SR within this frame ?
Is "frame" supposed to be something like "Lorentz frame"? If so, most
people in this context use "frame" in the sense of "frame bundle", i.e.
a collection of "local Lorentz frames" defined at each event, and varying
smoothly over our Lorentzian manifold.
> 7) Do I need the full complexity of GR to describe the movement
> of the stars (and planets) ?
-This- question does make sense in this context, at least to me, if you
rephrase it like this: "Do I need the full EFE to describe the movement of
the stars (and planets)?".
The answer is: "No, in solar system dynamics you only need the LEFE,
applied to the far field of an isolated stationary object representing the
Sun--- which is easy--- plus a whole lot of Newtonian astrophysics---
which can get quite hard, if you want enough accuracy to be able to verify
relativistic effects".
Unfortunately, it appears likely that you actually intended to ask
something quite different (something which might not make sense to me at
all), because you added the comment:
> IMO the answer on that question is NO because there are no
> moving clocks involved
This doesn't make sense to me because in deriving Einstein's precession
formula, you have a test particle orbiting an isolated object in a nearly
elliptical orbit, and of course you should think of any test particle as
carrying an idealized clock. This is relevant to understanding the
physical meaning of what one means by the radial coordinate, which is one
of the critical conceptual issues which I discussed extensively in the
posts I cited; this issue arises because I was using a harmonic chart for
a solution to the weak-field EFE, not the Schwarzschild chart for the
Schwarzschild vacuum, which is a solution to the full EFE.
> I'm not aware of those discussions with you but anyway thanks for
> all the detailed information regarding perturbation theory.
Well, don't search under "T. Essel", because I was not using that handle
back then. Try searching under keywords (with group=s.p.r.).
BTW, of course I can see that it would have been best had I simply located
and reposted the detailed "past posts" I mentioned, but for reasons too
tiresome to enter into, that would have been inconvenient for me--- and
for everyone else, there is Google!
> In order to get some idea about about perturbation theory and astronomy
> I studied the following document:
> " Large-Scale Structure of the Universe and
> Cosmological Perturbation Theory"
> http://xxx.lanl.gov/abs/astro-ph/?0112551
One thing to be aware of in reading papers on metric perturbations is that
there are many possible gauge conditions and the LEFE (see below) can look
quite different depending on which gauge you choose. Many papers on
cosmological perturbations choose a different "gauge" (a different
restriction on the type of coordinate chart used) from the Lorentz gauge
which I used in my posts. Depending on context, you might also see the
perturbations decomposed into scalar, vector, and tensor modes; in gtr,
only two modes survive. Beginners might find all this variety confusing,
but it probably -is- good practice to sort through a few different
approaches until you can see that they really are all talking about the
same thing.
> My previous experience with perturbation theory was related to process
> control.
Perhaps you were using a perturbation expansion to find an approximate
solution to some nasty pde? If so, you can probably see where AE uses
similar ideas to find an approximate solution to an ODE, the
Einstein-Binet equation, which arises in slightly different forms
depending on whether you start from a weak-field solution, as did AE, or
from the exact Schwarzschild solution, as do many modern textbooks.
> Maybe perturbation theory is the final tool that I need in order to
> solve the equations that describe the movements of the stars and planets
> (in a very acurate way ?)
Oh dear--- I was constructing an interpretation of what you might be
trying to say, which momentarily appeared plausible, but then you you said
something which caused me to suddenly realize I must not have even been
close!
When you say "solve the equations that describe the movements of the stars
and planets", presumably you have in mind some equations which have
already been written down and are thus ripe for solution. But I don't
know if you are thinking of solving the equation of motion in some
specific solution to the EFE, such as the Schwarzschild vacuum (i.e., if
you are trying to study the motion of test particles modeling planets in a
simple spacetime model), or if you are thinking of solving the EFE itself
(i.e. trying to find a spacetime model in gtr), or if indeed you are
thinking of working with gtr at all.
When you say "in a very accurate way", it is not clear to me if you plan to
search for -exact- solutions or very accurate -approximate- solutions.
Perturbation theory is of course ideally suited to the latter task, under
some circumstances, but not the former.
It seems you might just possibly believe that there is no known
mathematical model (I suppose you could say "equations", but "mathematical
model" would be a far better term) which accounts for "the movements of
the planets in a very accurate way". Of course, there -are- such models.
Indeed, depending upon what you mean by "very accurate", such models
existed by the middle of the nineteenth century (of course, not accurate
enough to account for the extranewtonian precessions of Mercury, Venus,
Earth,..., but very accurate nonetheless).
And when you say "final tool", I am dismayed for yet another reason: as I
have mentioned in the past, "perturbation theory" as a body of
mathematical ideas/techniques can be said to have begun with Lagrange's
pioneering studies of solar system dynamics, and it has played a key role
in astrodynamics from that time right down to the present day. So, it is
not a "final tool", but one you need right from the start.
So, did anything I said anywhere above help at all?
If not, you probably see what I meant when I said that I have found
communicating with you to be very difficult!
FWIW, I can suggest some reading on pre-Einsteinian astrodynamics. I think
the following textbook is a terrific modern introduction to the Newtonian
treatment of solar system dynamics, which should convince any fair minded
reader that good old Newtonian gravitation remains of great interest, and
can still provide challenges as intriguing as anything in modern
mathematical physics:
author = {Carl D. Murray and Stanley F. Dermott},
title = {Solar System Dynamics},
publisher = {Cambridge University Press},
year = 1999}
Those with a less than adequate mathematical background should read this
with a good book on mathematical methods, such as:
author = {Derek Richards},
title = {Advanced Mathematical Methods with Maple},
publisher = {Cambridge University Press},
year = 2002}
I like this book because it emphasizes stuff you can -use-, which nowadays
mostly means "stuff you can do on a computer"-- for those primitive
beasts, that is, who still support any AFK functionality at all :-/
OK, now for the "brouhaha":
> From mingstb@marssim-ss.com Mon Nov 29 19:09:02 2004
> Date: Wed, 17 Nov 2004 22:32:57 GMT
> From: greywolf42 <mingstb@marssim-ss.com>
> Newsgroups: sci.astro, sci.physics
> Subject: Re: How important is GR inorder to calc the precession of Mercury
> (banned reply)
>
> The following post was banned from the sci.astro.research newsgroup ...
> without notice, and in violation of the newsgroup charter (as is usual
> for s.a.r).
>
> Not only are substantive responses blocked, but the moderators (T.
> Essel) dual-posted a triple reply to s.p.r. and s.a.r, that was simply
> boilerplate cheerleading for GR. Without responding to the question
> posted by Nicholaas, of course.
Barry, I am not sure what you meant to imply, but for the record, I am
-not- a moderator for either sci.physics.research or for
sci.astro.research. The names of the moderators are listed in the FAQs for
these groups.
> From mingstb@marssim-ss.com Mon Nov 29 19:10:05 2004
> Date: Thu, 18 Nov 2004 17:41:39 GMT
> From: greywolf42 <mingstb@marssim-ss.com>
> Newsgroups: sci.astro, sci.physics
> Subject: Re: How important is GR inorder to calc the precession of Mercury
> (banned reply)
>
> "Eric Gisse" <jowr.\pi@gmail.com> wrote in message
> news:929ed0f8.0411180145.402348b9@posting.google.c om...
> >
> > Their newsgroup, their rules.
>
> I have no problem with their rules. I just enjoy jerking the chain of
> people who claim to have rules, then break them whenever it suits their
> personal prejudices.
>
> > Its too bad (not really) that you don't like it, but I fail to see why
> > you would think a) any of us side for you regarding your plight
>
> I don't have a "plight."
>
> > and b) that we would do anything about it
>
> You don't have to do anything.
>
> > and c) that posting in here about your plight will do anything.
>
> Posting here serves my purpose. It allows the response to exist for the
> person who asked the question. And it shows the hypocritical actions of
> the moderators.
As for the alleged "triple-posting", I have not been submitting multiple
posts or anything like that. I don't know what you saw, or think you saw,
but I am sure there was no malicious intent on anyone's part. Possible
explanations include:
1. I submitted a candidate post for crossposting to s.a.r and s.p.r., so
of course this post, which was approved, should have appeared (once) in
-each- of these two newsgroups,
2. Sometimes moderation glitches might result in a given post appearing
two or more times in one of the groups. I can think of several ways that
might happen.
3. Sometimes, due to bugs arising from the way a given individual accesses
these newsgroups, it might appear to someone that a given post has
appeared twice, when that is not in fact the case. If someone you know,
who is using a different site or ISP, says they are not seeing anything
weird, this should indicate that the problem was not due to anything the
moderators did or did not do.
You also claimed that your reply had been banned (rejected?) "without
notice". If you didn't receive a rejection notice from the moderators,
this need not imply any "hypocritical" behavior on their part. Possible
explanations include:
1. The moderators simply might not have gotten to your submission yet.
Sometimes the moderators of s.p.r and/or s.a.r. announce "moderation
delays", meaning they will be busy with something else and won't be able
to tend to the newsgroup for some time. AFAIK, in such cases, by the luck
of the draw, some recently submitted posts might be "approved" [sic]
before an older submission has been considered.
2. The post -was- rejected, but if your submission did not include a valid
email address, the moderators wouldn't have been able to email you to
explain their reasons. Think of your submissions like sending off an MS
by snail mail to the magazine page editor of your local rag. If there's
no SA on the SSAE, you might never see your MS again.
3. You put your submission in the mail, but the moderators never received
it. Mail -is- sometimes lost by spoolers. That would normally only
happen in circumstances all too apparent to local users (e.g. a system
crash), but not all the moderators are using their local system when they
carry out their volunteer duties here. Also, for various reasons, mail
can be delayed in transit, sometimes for days.
As for not responding to Nicholaas, I did say what I -was- trying to do:
"But for the benefit of lurkers who may have similar
questions, I'll just restate a few general and oft-repeated observations."
but to be perfectly clear, I would have listed some things I was -not-
trying to do--- had the possibility of misinterpretation occurred to me!
To wit:
1. I was not trying to respond to the questions (?) listed by Nicholaas.
2. I was not trying to discuss the history of science.
3. I was not trying to discuss the respective merits of gtr and its
competitors.
To repeat (sigh...) several things which I've explained many times in the
past:
Barry, it seems you are wont to interpret my posts as "boilerplate
cheerleading for GR". Assuming that this implies some kind of alleged
failure to bring a open-minded and critical attitude to the intellectual
table, this must be taken as a sad indication of how badly my pedagogical
activity here over the years has failed, at least in your case.
The truth is that my principle pedagogical goal has always been to
-empower- interested parties to -achieve- their own goals, which might
very well include something like "overthrowing" gtr. (Not such an
outlandish goal, when you consider the fact that this would be a side
effect of constructing a viable quantum theory of gravity, which is of
course one of the principle goals of contemporary physics!)
As part of that goal, I want to help math/physics students--- at a
suitable level of sophistication--- to formulate their own answers to
questions having the general form: "just how does this thing work?". In
particular: "how does gtr work?"
This is not because I think that gtr is the only gravitation theory anyone
will ever need ---although I do think there'll always be a place for
Einstein, as indeed there is for Newton--- or because I am somehow
satisfied with the status quo. Rather, I know that to read the research
literature, you need to know something about our current gold-standard
theory of gravitation (gtr), and its predecessor (Newtonian gravitation)--
and I know that "something" turns out to cover a lot of territory.
Maybe it will help you to think of things like this: before you even
embark on a quest to find a gravitation theory which is even better than
gtr, I want you to know that there are seadragons and still stranger
hazards lurking out there upon the stormy deep. And, I believe, explorers
should be well qualified mariners riding a seaworthy vessel. Why? Well,
just imagine the ignominy of driving your ship hard onto a rock just
outside the harbour mouth, with the loss of all hands! In the long run,
common sense and historical experience both suggest that proper training
and methodical preparation will greatly increase the chances of ultimately
discovering some Grand New Continent. Blind luck will no doubt also play
an essential role, but one should avoid demanding absurd quantities of
that most mysterious element.
Or maybe a dimensionally reduced ASCII "picture" will help:
o o The Good Ship
Well o Endeavour ==>
Safe Charted o (math method) Some
Harbor Coastal ___/\___ Brave New
(Nwtn) Waters ~~~~~~\__{____}/~~~~~~~~~~~~~~~~~~~~ World,
(gtr) scurvy, storms, shoals, Yet
__{__} _ scyllae... Unknown
| | |-|
|__{__}| |-|
_||_ |_|
Volunteer Shipwright
and Coastal Pilot
(I, bot)
Floxian fables aside, when I ask "how does gtr work?"--- or rather, "how
is it -supposed- to work?"--- I have in mind a three-step process:
1. "How you can use gtr to construct a mathematical model suitable for
studying a given physical situation?",
2. "How can you compute physically meaningful quantities describing
relevant properties of that model?",
3. "How can you interpret the results?".
Alas, I think many people get stuck on the easy part (computation), even
though there are widely available freeware tools which can compute
curvature tensors in a heartbeat. I find the first and third parts of the
process (modeling and interpretation) most interesting, but you
definitely need a strong background in applied math and geometry to
appreciate the conceptual issues lurking behind my frequent technical
comments. It is certainly unfortunate that without this background, you
are unlikely to be able to appreciate my explanations of why the real
problems with gtr (which are considerable) are completely different from
what the cranks imagine.
Gtr is a mathematically sophisticated theory which cannot easily be
understood without prior mastery of a rather long list of ideas/techniques
such as manifolds, perturbation analysis, etc., etc., ideas and techniques
which in turn require some native mathematical ability (as well as time
and effort) to acquire. While I can easily understand why this fact
frustrates those who have the interest but lack the background, it -is- a
fact. You can certainly attempt to work around it, but you can't ignore
it.
Don't forget, as a self-taught amateur myself, I have been in the position
of studying MTW with only a high school math background (yes, -that's-
changed!), so I have a special appreciation of considerable virtues of a
formal, curriculum-driven education.
And BTW, while gtr is clearly an interesting and important theory, which
unfortunately requires considerable background which requires extensive
time/energy to acquire/use, in the past I have often tried to popularize
theories/problems which are no less fascinating or timely, but which are
-much more accessible-.
>From the charters of these newsgroups, or even from the names
"sci.*.research", I think that one might fairly might expect that these
newsgroups are maintained by and for the principal benefit of
physics/astronomy researchers and their apprentices (graduate students in
physics/astronomy), and that while nonphysicists are welcome, they should
be expected to attempt to "play by the rules" of communication in
physics/astronomy. E.g. by attempting to speak our language, as far as
possible, and by trying to steer close to topics we would recognize as
physics-related.
(As you know, I am myself a nonphysicist, but hereabouts I take care to
play by the rules, not just because that's only polite, but because it is
virtually -essential- for communication in such a highly developed and
technical field.)
Until a few years ago, I think this model did approximately hold true in
s.p.r., most of the time (s.a.r. is a much newer newsgroup). Nowadays it
seems that most of the gravitation-related traffic in both groups arises
from posters who are not members of the research community and who appear
to have woefully little understanding of what members of that community do
and why. In the last few years, I have frequently seen language
suggesting that some posters do not understand basic concepts of
mathematical modeling, or even the scientific method itself, much less
understand the elements of what has already been achieved (scientifically
speaking) in the area of gravitation physics, much less what we should try
to achieve next, or where we can expect to go in the more distant future
future.
This circumstance, while clearly tending to generate much wasted bandwidth
due to gross miscommunication, would be more acceptable if regular but
"physically gauche" posters at least paid close attention to occasional
attempts by myself or others to explain some of "the rules", for the
benefit of those who lack the strong background in math/physics shared by
card-carrying members of the research community (and certain literate
amateurs, like myself). My little fable above falls into this category.
Alas, this is by no means the first time I have found myself trying to
explain to you in particular stuff which I don't think I should really
need to explain, at least not more than once, at least not to a single
individual, at least not in these groups.
Maybe you feel you are just having fun here by "jerking my chain" (if
that's really what you meant to say in your sci.astro post), but this is
not fair to other participants in s.p.r./s.a.r. (some of whom still come
here for the purpose intended, scientific discussion of research issues of
current interest in physics and astronomy). Indeed, it is not fair to
-me-, because you persist in attributing to me motives very different from
those which actually impel my occasional participation in these groups,
despite many attempts to disabuse you!
Now, despite your complaints in sci.astro, I see that at least one post
which you (Barry) wrote criticizing what I said -did- appear right here:
> From mingstb@marssim-ss.com Mon Nov 29 18:25:45 2004
> Date: Fri, 19 Nov 2004 19:30:45 +0000 (UTC)
> From: greywolf42 <mingstb@marssim-ss.com>
> Newsgroups: sci.physics.research, sci.astro.research
> Subject: Re: How important is GR in order to calc the precession of Mercury
>
> <tessel@tum.bot> wrote in message
> news:cncfni$t38$1@lfa222122.richmond.edu...
> > On Wed, 10 Nov 2004, Nicolaas Vroom asked (in s.a.r.)
> >
[..]
> > on several previous occasions, I have gone to great
> > lengths to help you understand what gtr says about the extraNewtonian
> > precession of Mercury (and why gtr is such a satisfactory theory for
> > purposes of explaining this and a multitude of other
> > observational/experimental evidence).
>
> It would be polite to provide a link to said statements.
Yes. In past years, I always took the trouble to provide extensive
annotated references to both on-line and printed material of use to
serious students, so I'll take this as very belated thanks for my previous
efforts :-/
> A google search of
> your posts shows no matches against "Nicolaas Vroom". In fact, there is
> only one post against your name (to Bill Kavanah), that contains the word
> "mercury" or the phrase "perturbation theory":
> http://www.google.com/groups?selm=cmq3is%24a2m%241%40lfa222122.richmond. edu
> And in this post, you again make unreferenced statements that you've "posted
> on this (perturbation theory) very extensively before."
See my advice to Nicolaas above.
> > This is needed in the problem at hand
> > because the theoretical problem confronting Einstein in 1916 was not to
> > explain the precession of Mercury in its orbit around the Sun, but rather
> > to explain a small residual remaining after a perturbation theory analysis
> > of a model in Newtonian gravity had explained all but a small part of the
> > observed motion.
>
> On the contrary. The explicit, stated purpose of Einstein was to obtain
> Newcomb's published value (43" per century*) for the NNPA of Mercury.
So what is the problem? The notion of a "residual"?
You said that Grossmann "remeasured" the residual, but I think you should
have said "recalculated". The principle issue was (and remains) not how
to -measure- the motion (astronomers had already gotten very good at that
before AE came along), but how to -explain- it theoretically. Saying that
we have an extranewtonian residual of 43 seconds per century amounts to
saying that -our best Newtonian model- leaves that much unaccounted for.
You had some other comments, which I think are mostly moot, since you
completely misunderstood the purpose of my post. But I must correct one
serious misstatement you made:
> > In gtr,
> > we have the additional complication that the full field equation (the EFE)
> > is nonlinear, but this plays no role here because we can get away with
> > studying solutions to a linearized version of the EFE.
>
> Which is simply the Newtonian equation, with an added speed-of-gravity
> parameter (equal to the speed of light).
Anyone who has studied any modern gtr textbook will know that there is
only one thing I could possibly mean by "a linearized version of the
EFE"--- and I doubt that anyone who has seen it could possibly describe it
as "simply the Newtonian equation, with an added speed-of-gravity
parameter (equal to the speed of light)"!
For example (you did want citations?):
1. See (20.28) in section 20.1 ("The linearized field equations") of the
textbook
author = {Ray D'Inverno},
title = {Introducing {E}instein's Relativity},
publisher = {Clarendon Press},
year = 1995}
2. See (8.42) in section 8.3 of the textbook
author = {Bernard F. Schutz},
title = {A First Course in General Relativity},
publisher = {Cambridge University Press},
year = 1985}
3. See (13.14) in section 13.2 ("The fundamental equations of the
linearized theory") of the textbook
author = {Hans Stephani},
title = {General Relativity:
An Introduction of the Theory of the Gravitational Field},
publisher = {Cambridge University Press},
edition = {Second},
note = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},
year = 1990}
4. See (7.125) of the textbook
author = {Sean Carroll},
title = {Spacetime and geometry: an introduction to general relativity},
publisher = {Addison-Wesley},
year = 2004}
5. See (18.8b) of section 18.1 ("The linearized theory of gravity") of
the textbook
author = {Charles W. Misner and Kip S. Thorne and John Archibald Wheeler},
title = {Gravitation},
publisher = {W. H. Freeman},
year = 1970}
6. See (4.4.12) in section 4.4 ("Linearized Gravity") of the textbook
author = {Robert M. Wald},
title = {General Relativity},
publisher = {University of Chicago Press},
year = 1984}
7. See (53) in section 3.2 ("The linear field equations for gravitation")
of the textbook
author = {Hans C. Ohanian and Remo Ruffini},
title = {Gravitation and Spacetime},
edition = {Second},
publisher = {W. W. Norton},
year = 1994}
8. See (10.1.10) in section 10.1 of the textbook
author = {Weinberg, Steven},
title = {Gravitation and Cosmology:
Principles and Applications of the General Theory of Relativity},
publisher = {Wiley},
year = 1972}
9. See (9) in section 4.1.2 of the textbook
author = {C. Clarke},
title = {Elementary General Relativity},
publisher = {Wiley},
year = 1979}
The LEFE (linearized Einstein field equation) is a -fundamental- and
-standard- topic which is treated at length in almost every gtr book!
The name LEFE (or something very similar) is also standard, as you can see
from the section titles I quoted.
(Are you convinced yet, or should I give even more references?)
Note that the LEFE takes different forms in different gauges; most books
discuss at least the "Lorenz gauge", and I have listed above the form
which the LEFE takes (for trace-reversed perturbations) in that gauge.
Barry, in your sci.astro post, I see that you say
> MTW is not a good way to learn GR. It's a decent way to expand your
> knowledge in specific areas, once you have the basics down.
I think your opinion might be different once you get the basics down.
MTW is one of the great scientific books of all time, one which rewards
repeated reading at -all- stages of your education. It is true that
understanding what you read gets exponentially easier as you acquire more
background and experience, but looking into this book at an early stage
can give you an idea of what it is you want to understand, and checking
back periodically (and working more problems!) can reassure you that you
are in fact making progress.
Be all this as it may, the LEFE is great fun, so I urge -everyone- who
doesn't know about it to read up forthwith!
Don't be confused by the fact that in many of the textbooks above,
discussion of the LEFE occurs in the context of gravitational radiation,
sometimes long after a discussion of the precession of nearly Keplerian
orbits. The underlying field equation is the same regardless of whether
one is discussing the "far field" of an isolated stationary object (e.g.
for studying precession of nearly Keplerian orbits around a star), or the
generation of gravitational radiation (anything but stationary!). Of the
textbooks listed above, this underlying unity is probably brought out best
by Stephani or MTW, but it should be clear enough from any of these books
if you read with sufficient care and insight.
For the benefit of interested readers, I append a brief overview in "Baez
emulation mode" ;-/ For the details, I hope all interested readers will
rush right out, obtain a copy of the one of the above mentioned textbooks,
and study them!
T. Essel (spelunking somewhere in cyberspace)
================================================== ===============
Just what -is- this "LEFE" business?
Well, we should begin at the beginning, so let's begin with the EFE (the
full, nonlinear, Einstein field equation).
Intuitively, gtr aims to directly relate geometry and matter. You
probably already have some intuition for what matter is, and how we can
measure how much there is of it in a given place. But what about
geometry? To directly relate geometry to matter, we evidently need to
somehow quantify -geometry itself-! How can we do that? Well, according
to Riemann, geometry always comes down to angles and distances. And to
quantify angles and distances, we can employ the "metric tensor" of a
Lorentzian manifold.
This Lorentzian manifold, or "spacetime model", will play a dual role: it
will serve as our mathematical model of "gravitational phenomena as
geometry", in a specific physical context (e.g. modeling a planet orbiting
the Sun), but it will also constitute the -geometrical setting- for all
-nongravitational- physical phenomena under study, if any. The rules for
transferring all our nongravitational physical theory (e.g. Maxwell's
theory of EM, hydrodynamics, the theory of elasticity) to this curved
spacetime setting are sometimes summarized by the slogans "minimal
coupling" and "comma goes to semicolon". These rules are not entirely
straightforward, but they are not hard to use in most cases.
The full EFE reads
G_(ab) = 8 \pi T_(ab) (*)
Probably everyone here knows that, but not everyone really knows what
those funky symbols actually -mean-! Fortunately, I can describe the
basic "ingredients" of this equation, and even try to give a hint of how
they fit together.
On the left, we have something mathematically representing the character
of the geometry of our spacetime. It is G_(ab), the Einstein tensor, a
second rank tensor, which is a kind of "average curvature" tensor
constructed from the fourth rank Riemann curvature tensor R_(abcd). The
Riemann tensor completely describes the local geometry near each event in
our Lorentzian manifold; it can be readily computed from the metric tensor
by a straightforward process of double differentiation.
(Going the other way is harder, but it can be done. This raises all kinds
of fascinating mathematical issues which unfortunately I haven't the
space-time-energy to discuss, at least not here.)
On the right, we have something mathematically representing the amount and
motion of matter. It is T_(ab), the energy-momentum-stress tensor.
(Multiplied by a factor of 8 \pi, but I probably don't need to say what 8
and \pi are, huh? So I won't.) Mathematically speaking, T_(ab) is the
"source term" in the EFE, modeling the stuff (anything with energy or
mass) which generates a gravitational field. Roughly speaking, at each
event in our spacetime model, as measured by an appropriate "ideal
observer", the T_(00) component tells how much -mass- there is, the T_(0j)
components tell how much -momentum- there is, and the T_(jk) components
tell how much stress there is.
For example, consider an isolated rigidly rotating thin disk. Here,
T_(ab) would be nonzero only inside the disk, and if we compute the
components in an appropriate frame, T_(00) (mass density), T_(0j)
(momentum density), and T_(ij) (stress density) will all be nonzero. The
stress arises because in order to maintain its shape, the disk must resist
the tendency to fly apart due to centrifugal "forces" [sic] arising from
its rotation, and this mechanical resistance shows up as stress.
So, the EFE does indeed say that a basic geometric phenomenon (the
averaged curvature) is directly related to a basic physical phenomenon
(the amount and motion of matter and energy of nongravitational fields).
We can say more: the presence of matter (nonzero source term on the RHS)
directly causes a local distortion of geometry (a change in our "average
curvature"), which typically changes the behavior of geodesics. To see why
this is profoundly important, you need to know that one of the great
beauties of gtr is the elegant way in which it realizes the -universal-
character of gravitation (the fact that gravity acts on all forms of
mass-energy). Namely, according to gtr, if no physical forces act on a bit
of matter, then its world line (kinematic history as represented in our
Lorentzian manifold or spacetime model) will be a geodesic. (In a curved
manifold, geodesics are the analog of "straight lines": they are curves
which are "as straight as possible"). So in gtr, when curvature is
present, this typically will affect the relative motion of free-falling
bits of matter.
Needless to say, gtr can also handle any -nongravitational- forces acting
on matter, such as pressure acting on a fluid element. Namely, the
magnitude and direction of net force on some bit of matter determines the
magnitude and direction of the -path curvature- of its world line. If
forces act, the world line has nonzero path curvature; if not, it doesn't.
This is just what we would do in str (which doesn't attempt to treat
gravitation), except that now our spacetime is curved. Note that this
means that "gravitational forces" never appear in gtr, or rather, the
force which a standing human applies at his feet to resist falling through
the floor can be called that, but really it is a -nongravitational- force
which is accelerating him with a certain magnitude and direction (away
from the center of the Earth), which means that his world line has nonzero
path curvature; it is -not- a geodesic. But he steps off the balcony, he
will subsequently be in free fall--- until he strikes the ground--- and
along the free-fall portion of his kinematic history, his world line will
have vanishing path curvature.
John Archibald Wheeler (the W in MTW) famously summed things up something
like this: "Matter (plus possible nongravitational fields) tells spacetime
how to curve; spacetime curvature (plus possible nongravitational forces)
tells matter how to move".
Let's look at how this works in a bit more detail. Mathematically, we can
decompose the Riemann curvature tensor like this:
R_(abcd) = C_(abcd) +[/itex] a kind of fourth rank "expansion" of
Riemann Weyl curvature R_(ab) = R^{m_}(amb)
curvature Ricci curvature
In gtr, "Ricci curvature" (or equivalently, "Einstein curvature"), meaning
either the second rank tensor or its fourth rank "expansion", is closely
associated with matter: the -immediate presence- of matter in some region
causes nonzero Ricci curvature in that region, and conversely, Ricci
curvature cannot exist where no matter is present.
"Weyl curvature", AKA "conformal curvature", has a beautiful and
compelling geometric meaning, related to "light bending", but independent
of any physical interpretation. But to save time, here I will just say
this: in gtr, Weyl curvature is the type of curvature which can exist
-independently of matter-; in particular, it is the type of curvature
which can propagate through a vacuum region, as a "gravitational wave".
Such waves are terribly important because they are ultimately the
"physical mechanism" by which a region of spacetime (empty or otherwise)
can "learn" of the presence of mass/energy somewhere else, possibly far
off, and "respond" by changing its curvature to maintain the validity of
the field equation.
For example, if a bomb goes off in some isolated region in space, that
changes the distribution and motion of matter in this region, which can
create "ripples" of Weyl curvature which then propagate outwards, at the
speed of light, through the surrounding vacuum. Such bombs actually
exist--- they are called "supernovae"! With LIGO, we hope to detect their
gravitational wave signature.
You might be asking: since the EFE only mentions Einstein curvature, (or
equivalently, Ricci curvature), how can the motion of matter possibly
create propagating ripples of Weyl curvature? The answer is that the
"differential Bianchi identity" (a mathematical identity which is true
independently of the EFE, and indeed, independently of any physical
interpretation of curvature) enforces a "coupling" between matter and Weyl
curvature via a first order PDE. Essentially, this relation says that the
immediate presence of matter in one region can cause Weyl curvature in
that region, and this can in turn cause more Weyl curvature in adjacent
regions, even nearby vacuum regions. But what matters is not so much the
mere -presence- of matter as the existence of a suitable -gradient-. And
indeed, the Friedmann cosmological models, which are homogeneous and
isotropic spacetimes, have nonzero Ricci curvature (since they are filled
with fluid or dust) but no Weyl curvature (and thus no light bending).
OTH, in the famous Schwarzschild vacuum, which is an exact solution of the
EFE representing the field outside a nonrotating spherically symmetric
massive object, -all- the curvature is Weyl curvature. Of course, this is
the same as saying we have a vacuum solution, since "no matter" is
equivalent to "no Ricci"! But how, you might ask, did this Weyl curvature
get there? Well, in imagination, it happened like this: when we
concentrated matter to form the Sun, by the conservation of mass, we had
to move it in from somewhere, and when we did that, according to the EFE,
the "matter gradient" created Weyl curvature, which then propagated into
nearby vacuum regions (at the speed of light) and began to curve them up,
and so on. Thus, the gravitational field of the Sun is a kind of
"historical legacy" which was created--- indirectly and gradually---
during the slow formation of the Sun. Any gravitational radiation
currently emanating from the Sun merely represents teeny tiny
perturbations carrying information about movements of matter inside the
Sun, information which will eventually be used by regions far away to make
-extremely- teeny tiny "local adjustments in curvature" in order to uphold
the validity of the EFE. Or at least, that's the way gtr tells the story!
:-/
Newbies often ask "if nothing can get out of a black hole, how can
gravitation get out?" The answer is that (in gtr) -it doesn't have to-.
The collapse of a star to make the hole probably created some strong
gravitational radiation, but this can't instantly undo the Weyl curvature
which was gradually created during the formation of the star.
Or again, sometimes people ask: "if a black hole was created by the
complete collapse of a star, and if the matter which once made up the star
has been crushed out of existence at a spacetime curvature singularity,
then surely there is nothing left to gravitate?" One answer is that it is
precisely because information cannot propagate from under the horizon that
the "bad news" about the fate of the matter which originally created all
that Weyl curvature (indirectly!) cannot reach the exterior region, so we
do not in fact need to worry about this, unless we fall into the interior
region, in which case we've got more pressing--- and tugging--- problems
:-/
But enough about the EFE, you say-- what about the LEFE?!
OK, now I must confess that everything I just said wasn't the beginning at
all, it was just stuff I had to say before I could even begin! :-/ -This-
is the beginning:
The starting point of "linearized gtr" is a very simple idea: finding
exact solutions of the EFE can be a daunting task, so to make things
easier, imagine that we have, essentially, flat spacetime, except that in
some places we have small amounts of matter (or an EM field, or some other
nongravitational field) which induces a "mild distortion" of our otherwise
flat spacetime. Or ---to use the lingo--- assume that our Lorentzian
spacetime is a "metric perturbation" of Minkowski spacetime (in fact, a
"linear perturbation", meaning that we only carry out computations to
first order in our "perturbation parameter").
So, let's write the metric in the form of the sum of the Minkowski space
metric tensor (representing the flat background) plus a perturbation
tensor (representing the "mild distortion" due to the presence of some
not-too-massive object, a passing gravitational wave, whatever):
g_(ab) = \eta_(ab) + h_(ab)
background perturbation
In weak-field gtr, the background is unobservable, so physically speaking,
it doesn't exist. But mathematically speaking, we can pretend, and there
is a rule of thumb that says that whenever you can get away with
pretending something is much simpler than is really the case, you should
be cynical and just grab the opportunity! So, let's treat the metric
perturbation tensor h_(ab) as just another tensor field on -flat-
spacetime. It is a second rank symmetric tensor field on a four
dimensional manifold, which implies that it has ten algebraically
independent components at each event.
Now, if we compute the Einstein tensor of our mildly distorted spacetime,
using the true metric g_(ab), but expressing our result in terms of the
perturbation tensor h_(ab), treated as just another tensor field in flat
spacetime, we get a big mess. But we are free to add conditions on h_(ab)
to try to reduce the clutter, so let's add some! This has the effect of
ruling out the use of some coordinate charts, but it is always possible to
find plenty of coordinate charts which meet our "gauge conditions". There
are many possible choices of gauge conditions, but the LEFE takes a
particularly simple form if we choose the "Lorenz" (or "harmonic") gauge,
so let's make this choice.
Next, toss in a minor trick: rewrite everything in terms of the
"trace-reverse" of h_(ab), written \hbar_(ab). -Viola!-, you get the LEFE:
Box [itex]\hbar_(ab) = -16 \pi T_(ab) (**)
It's just the good old flat spacetime wave equation, times ten!
On the left, we have something acted upon by the flat spacetime wave
operator Box(.), something which mathematically represents the mild
distortion of the flat spacetime geometry. It is \hbar_(ab), the
trace-reversed metric perturbation tensor, which, as I said, we treat as a
second rank symmetric tensor field on flat spacetime. So it has ten
algebraically independent components--- each of which satisfies the scalar
wave equation familiar to every undergraduate sci/math student. Secretly,
of course, the left hand side also has a beautiful and profound geometric
interpretation: it is the Einstein tensor, computed for our mildly
distorted spacetime using the -true- (physical) metric tensor g_(ab)!
On the right, we have something mathematically representing the amount and
motion of matter. It is T_(ab), the energy-momentum-stress tensor, treated
as a tensor field on flat spacetime. Mathematically speaking, it serves as
the "source term" for our flat spacetime tensorial wave equation.
In a vacuum region (e.g. outside the disk), where the right hand side of
the LEFE vanishes, we are dealing with the "source-free" (tensorial) wave
equation
Box \hbar_(ab) =(***)
In such a region, we might have changes in the geometric distortion
propagating as a wave (a vacuum gravitational wave).
Qualitatively, the LEFE says the same thing as the full EFE: "Matter
(plus possible nongravitational fields) tells spacetime how to curve;
spacetime curvature (plus possible nongravitational forces) tells matter
how to move". But mathematically, it takes on a new form which offers some
very suggestive analogies.
Let's compare the LEFE with Maxwell's "nontrivial" field equation:
Box A^b = 4 \pi J^b
Here, on the left, we have something on which the wave operator Box acts,
something which mathematically represents "the EM field". It is A^b, the
four-vector potential (from which we can recover the EM field itself by
"exterior differentiation", F = dA). On the right, we have something
mathematically representing the amount and motion of any electrically
charged matter. It is J^b, the current-density four-vector. In a
charge-free region, we have the "source-free wave equation", and in such a
region, we can have an EM field propagating as a wave ---we can have
"light".
There's much more to this analogy with Maxwell's theory of EM. For
example, the EM field tensor F_(ab) has two scalar Lorentz invariants
F_(ab) F^(ab), F_(ab) *F^(ab)
from which we can recognize when we have a "purely electrostatic" field,
(in which case we can find observers who measure no magnetic field), and
more besides. And from the Riemann curvature tensor (the gravitational
field tensor) we can construct analogous scalar Lorentz invariants
R_(abcd) R^(abcd), R_(abcd) *R^(abcd)
and then we can recognize situations where we can find observers who
measure no "magnetogravitic field", and more besides. (BTW, this scalar
invariant stuff actually works fine for fully-fledged gtr, not just its
linearized weak-field approximation.)
Why is this interesting? Well, for one thing, it points up a very
dramatic difference between linearized gtr and Newtonian gravitation.
Consider a rigidly rotating uniform density thin disk. In Newtonian
gravitation, by measuring the gravitational field we cannot distinguish
our disk from an otherwise identical but nonrotating disk. But in the
LEFE, the source term T_(ab) takes account not only of the amount of
-mass- in each region, but also the amount of -momentum-. This turns out
to mean that in weak-field gtr, we -can- distinguish the rotating disk
from its nonrotating twin, just by measuring its gravitational field! If
its magnetogravitic field is nonzero (as must be case if the scalar
invariant R_(abcd) *R^(abcd) is nonzero), then it is rotating. If not, it
isn't! Unfortunately, these are usually tiny effects, but there are
experiments underway or planned to test this feature of gtr using earth
orbiting satellites (Gravity Probe B and its kin).
(Maybe I should remark that, while what I just said about scalar
invariants is true for isolated stationary objects, it turns out that
-all- the scalar curvature invariants of vacuum gravitational plane waves
vanish identically, but these have magnetogravitic fields comparable to
their electrogravitic or tidal fields, as measured by any observer. But
even here we have a valid analogy, since in Maxwell's theory of EM, the
two Lorentz invariants of the field tensor F_(ab) vanish identically for
an EM plane wave, and these have magnetic fields comparable to their
electric fields, as measured by any observer.)
Predicting and possibly confirming new physical phenomena is always very
exciting, but there is another reason why this observation is of
fundamental interest. To set up our rotating disk model, we must write
down a boundary value problem for the LEFE. That is,
1. We write down boundary conditions stipulating the size, shape,
(uniform) mass, and (steady-state) rotation of the disk,
2. We attempt to find the required vacuum solution of the LEFE (a purely
mathematical problem),
3. We attempt to interpret the resulting Lorentzian spacetime (defined
only in the region outside the disk) as a mathematical model of
"gravitation as geometry".
But there is nothing in the LEFE to prevent us from removing the -mass-
and leaving the -momentum- in our boundary conditions. As an exercise, it
is a good idea to do just this, and to solve the new boundary value
problem. What you get an analogue of a "purely magnetostatic field", and
this turns out to be pretty darn weird: all the spacetime curvature is in
the t = constant family of spacelike hyperslices, so static observers
experience no gravitational tidal forces, and they need not fire their
rocket engines to remain static. However, observers riding a
gyrostabilized platform in free fall eventually find that they are
-spinning- with respect to the distant stars!
This illustrates a very important feature of gtr: boundary conditions are
essential, and should arise from nongravitational physics, but "garbage
in, garbage out": if you fail to posit "physically reasonable" boundary
conditions, your spacetime model is unlikely to be physically reasonable.
On the bright side, gtr plus a little common sense provides us with a
strong hint that our new boundary conditions must unreasonable, even
before we discovered the weird properties of a "purely magnetogravitic
spacetime". We need only ask, how might we "prepare" a thought experiment
involving a rotating disk? Well, we can imagine a region of spacetime far
removed from any massive objects, and we can imagine sending in some
matter to form a rotating object, which will serve as the source of the
gravitational field in our isolated region. But of course, this should
lead to something which has positive mass and angular momentum (and not
too much of the latter in comparison to the former).
Speaking of electrostatic fields--- if you know how the EM field reduces
to an electric field in some circumstances (e.g. a static configuration of
charged matter, perhaps on the surface of a charged metal sphere), you
will recall that we can kill off all but the "time component" of the EM
potential and the current-density four-vector. Then, our wave equation
reduces to the Poisson equation
Lap \phi = 4 \pi \sigma
where \phi is the electrostatic potential and \sigma is the charge density.
In the case of a spherically symmetric source-free field, by introducing
the Lorentz force law we can recover the inverse square law of Coulomb,
which is formally exactly the same as Newton's inverse square law. And if
we look at static weak gravitational fields, if we use a suitable observer
we will find that \hbar_(ab) has only one nonvanishing component, which
corresponds to the Newtonian potential. Then, our wave equation again
reduces to the Poisson equation
Lap \phi = 4 \pi \rho
where now \phi is the gravitational potential and \rho is the density of
matter. The -vacuum- field equation is the Laplace equation (again), from
which we can recover Newton's inverse square law. But this time we don't
need to -introduce- any force law--- because in gtr, there -is- no
"gravitational force"! This is another of those unique features which
makes gtr so remarkable from a mathematical point of view.
Now, any solution of the three dimensional Laplace equation (such things
are called "harmonic functions") can serve as a Newtonian potential (or as
an electrostatic potential). From this, you might guess that you can cook
up a storm with Green's functions for trace-reversed metric
perturbations--- and you'd be right! If you know that you can expand any
asymptotically vanishing harmonic function as an infinite sum of
particularly simple ones (the "harmonics") you might guess that you can do
something similar for trace-reversed metric perturbations--- and you'd be
right about that too! This is why, when you see the term "quadrupole
moment" when you are reading a paper about gravitational radiation, and
then see it again when you are reading another paper about the shape of
the Sun, you can be confident that both authors really were talking about
the same thing.
At this point, you might ask "what puts the L in LEFE?" Well, the wave
equation is -linear-, which means that if we have two solutions, we can
just add them to obtain a new solution. And the LEFE is just a tensorial
wave equation for the trace reversed metric perturbation. This means that
if we know two "legal" trace reversed metric perturbations (tensors which
solve the LEFE), we can just add them and get a new "legal" metric
perturbation:
newsol = sol1 + sol2
But if we try the same thing with two solutions to the full EFE, we might
(probably after a lot of work) be able to write something like this:
newsol = sol1 + sol2 + "nonlinear interaction terms"
That's nonlinearity in action!
Physically, we can say that the "nonlinear interaction terms" tend to
imply that we will see stronger gravitation than we would expect in a
linear theory. One way to think about this is to imagine two stars which
approach each other in some isolated region of deep space. As the total
gravitational field gets stronger in the region between them, its energy
increases, but this energy itself gravitates, so the gravitational field
increases in strength faster than Newtonian theory would lead you to
expect.
If that sounds complicated, it is, and this goes a long way toward
explaining why the LEFE is so much easier to solve than the EFE.
When can we use the LEFE rather than the EFE? Whenever and wherever our
spacetime is "almost flat"! Or in other words, whenever and wherever the
gravitational field is "weak". One place that happens is far from an
isolated massive object like the Sun. So if hbar1_(ab) models the Sun, or
at least the "far field" of the Sun, where gravitation is weak, and if
hbar2_(ab) represents a gravitational wave propagating in flat spacetime,
then hbar1_(ab) + hbar2_(ab) represents a gravitational wave passing near
the Sun-- but not too near!
So, you say, if we can do all this cool stuff with the LEFE, who needs the
EFE? Well, sometime or somewhere spacetime might not be -anywhere near
flat-; for example, soon after the Big Bang, or near the horizon of a
smallish black hole!
There is also some very persuasive theoretical reasons for regarding the
EFE as the real deal, and the LEFE as a mere approximation which is useful
in some circumstances. One of the most intriguing is this: it turns out
that the LEFE is not entirely self-consistent. There is a more or less
unique way for modifying the LEFE until eventually (after passing through
infinitely many stages), you have a self-consistent (but nonlinear) field
equation. And this field equation is:
ooo dramatic drumroll ooo
The EFE!!