Any coordinate system in GR?
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From: hilbertsnutsack on 22 Aug 2006 09:09 I replied to this post last night through mathforum.org, and for some reason my reply still hasn't made it to google groups. Here it is again, apologies if it now appears twice. ******************************************* I'm not entirely sure I understand your question, but I will have a go at answering it; let me know if I am barking up the wrong tree. > The offhand claim is sometimes made that GR, which I take it means the >formal machinery of GR, treats all coordinate systems equally. Is this >true? For some definition of "all coordinate systems", yes. More precisely, spacetime is assumed to be equipped with a differentiable structure, i.e. an equivalence class of local coordinate charts. What this means is that, given some coordinate chart x that maps part of spacetime to a subset, say U, of R^n, defining a new coordinate system y by y=f(x), where f is a smooth bijection with smooth inverse from U to some other subset V of R^n, y will be a valid coordinate system. The equations of GR are constructed in such a way that they will look the same in both the x and y coordinate systems. >Consider a 1-dimensional manifold, a real coordinate x, and the metric >ds = |dx| OK. For simplicity let's take the manifold to be R^1. >Next, stretch this manifold by a postive factor a(x), also >stretching the coordinates (so that points on the manifold are still >labeled by the same numbers) I am not sure that I understand what you mean by "stretch this manifold"... it seems like it should mean "change the metric on the manifold by ds |-> a(x)ds, is this what you had in mind? >Now, instead of stretching the manifold, compress the coordinates by >this same function a( ), so that dx = a(u)du. The most natural >transformation law for the metric would again be >ds = a(u)|du| Yes. It seems that what you have shown, if I have understood what you are doing, is that the "stretched" manifold is isometrically equivalent to the original manifold, i.e. there is a diffeomorphism between the two which preserves the metric. >So, if we allow "any coordinate system", we would apparently be >unable to distinguish the (physical) case of the stretched manifold >from the (unphysical) case of the compressed coordinates. >From a purely mathematical standpoint, there is no difference, since the spaces are equivalent considered as manifolds with metrics. From a physics standpoint, we can tell the difference if there is anything in this spacetime: suppose there are two planets in the original manifold at x=0 and x=1, then if the space is stretched by some supernatural being changing the metric but leaving the planets fixed at x=0 and x=1, the inhabitants of this universe would be able to tell the difference, since, for example, the metric determines how the laws of electromagnetism look in a given coordinate system, and if the metric gets bigger, then the difference in x-values of two adjacent atoms in ruler gets smaller. Therefore if the inhabitants use a ruler to measure the distance between the planets before and after the stretching, they will get different answers. On the other hand if you simply change to using u-coordinates then the ruler and planets will both appear different, but the length between the planets, as measured by the ruler, will stay the same. Does that help at all?
From: Igor on 22 Aug 2006 13:15 Edward Green wrote: > The offhand claim is sometimes made that GR, which I take it means the > formal machinery of GR, treats all coordinate systems equally. Is this > true? Aside from questions about smoothness, it seems to me there is > at least some other implicit condition on permissible coordinate > systems. > > Consider a 1-dimensional manifold, a real coordinate x, and the metric > ds = |dx|. Next, stretch this manifold by a postive factor a(x), also > stretching the coordinates (so that points on the manifold are still > labeled by the same numbers). Calling the new, stretched, coordinates > u, the most natural transformation law for the metric would be > > ds = a(u)|du| > > Now, instead of stretching the manifold, compress the coordinates by > this same function a( ), so that dx = a(u)du. The most natural > transformation law for the metric would again be > > ds = a(u)|du| > > So, if we allow "any coordinate system", we would apparently be > unable to distinguish the (physical) case of the stretched manifold > from the (unphysical) case of the compressed coordinates. > > It may be objected that by compressing the coordinates, we have > implicitly changed physical units, and this is not allowed. But that's > a new rule, isn't it? What's a "unit"? We need additional structure > to specify what we mean by this; we must equip our manifold with some > physics -- a local property establishing the natural scale of the > coordinates. This will insure that we don't arbitrarily distort the > coordinates, and any distortion can be attributed to the manifold. > > Am I right or wrong? Stretching or compressing the coordinate system just redefines it mathematically. There's nothing physical there. You can define a coordinate transformation any way you want to provided that it is invertible. But that doesn't result in any new physics. It may result in so-called fictictious forces arising in the new coordinate system that weren't there previously, but those are never really physical either. And changing units can indeed be allowed under coordinate transformations. Think of going from rectangular to polar or vice versa. Rectangular has both coordinates with length units, and polar has one with length and the other with angular units. The metric will always compensate to make up for the difference.
From: Shmuel (Seymour J.) Metz on 22 Aug 2006 20:30 In <1156252169.062521.249040(a)74g2000cwt.googlegroups.com>, on 08/22/2006 at 06:09 AM, hilbertsnutsack(a)hotmail.co.uk said: >I replied to this post last night through mathforum.org, and for some >reason my reply still hasn't made it to google groups. Google is irrelevant; this is Usenet. >>From a purely mathematical standpoint, there is no difference, since >the spaces are equivalent considered as manifolds with metrics. That is only true for a spacetime without matter. Plug in, e.g., an E-M field and there is a Mathematical difference between a co?rdinate change and stretching the metric. -- Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel> Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap(a)library.lspace.org
From: Bill Hobba on 22 Aug 2006 21:59 "Edward Green" <spamspamspam3(a)netzero.com> wrote in message news:1156222614.472337.73970(a)p79g2000cwp.googlegroups.com... > Bill Hobba wrote: > >> "Edward Green" <spamspamspam3(a)netzero.com> wrote ... > > <stretching the manifold vs. compressing the coordinate system> > >> First why is one unphysical and the other not? > > Well, that depends on whether it is "physical" to stretch the manifold. Since it is not a material then obviously it is a 'word salad's with no meaning like saying stew smells like isomorphism. > In a simple minded analogy, the one dimensional manifold might be > represented by a rubber band: stretching the rubber band is a > physically distinct situation from compressing the coordinates -- one > might be able to determine the state of strain by local measurements, > or at least determine the relative strain for two locations. In GR the > "stretch" might correspond to a gravitational field vs. flat spacetime. > Not _everything_ is an artifact of the coordinate system. You were on the right track above - stretching is not a property of the space-time manifold. Thanks bill > >> Secondly since all rulers >> are compressed or stretched by the same amount nothing will change. > > This might be the case, but I think you assume too much. It may be the > case that "nothing will change" locally -- meaning we cannot tell the > difference by local measurements alone -- but that we can very > definitely tell that something is changing over paths. This is the > situation in GR, I believe. I made my "stretching" dependent on > position BTW -- not the very simplest situation one could imagine -- to > try to avoid the impression that this was solely about an arbitrary > choice of units. > > As a silly illustration of principle about a possible physical > difference between distortion of a manifold and distortion of > coordinates, imagine an elastic manifold which turns blue beyond a > certain strain state. Now,eitehr stretching the manifold in strata, > or, on the other, inversely compressing the coordinates, we either see > blue striations or we don't -- physically distinct situations! > >> But >> that is not really the claim of GR - it does not claim like SR does for >> inertial frames that if we take the same experiment and shift it to an >> other >> frame then exactly the same result will occur - shift it to an >> accelerated >> frame and it will be different. Here lies the crucial difference - SR by >> being restricted to inertial frames has that property - GR does not. In >> both >> cases the laws of physics are still the same. >> >> GR obeys the principle of general invariance (not to be confused with the >> principle of covariance). As pointed out by Kretchmann (and eventually >> agreed by Einstein) any law can be put into covariant from so the >> principle >> of general covariance contains no actual physics. Its physics lies in >> the >> fact that when in such a form all absolute terms (ie things like the >> speed >> of light, planks constant etc) remain unchanged and any other terms must >> be >> dynamical - technically this is called general invariance. But you will >> find a lot of articles confuse one with the other - but as long as you >> understand what is happening no problems will arise - at least I have >> never >> found any. This immediately implies that the metric tensor for example >> is >> not an absolute term so must be dynamical - which is a cornerstone of >> GR - >> in fact all by itself it pretty much implies the EFE's. > > Well, you have confused me, though I claim I am alive to the kinds of > issues you raise, and you have failed to persuade me, if that was your > purpose, that my question is somehow misguided. However, I thank you > very cordially for your serious answer, and especially your detailed > reference, available free through the miracle of pdf, and I certainly > must be compelled to study it. > > That post was rejected by the moderator of s.p.r., BTW, where I only > repaired because that's where the experts are, and the cranks aren't. > The reason given was "mainly mathematical, and containing elementary > errors". I've been led to believe that s.p.r. was waiting with open > arms for the disillusioned amateur to escape the trials and > tribulations of trolls and orcs, provided he only asked respectful > questions. I hardly went ranting about how GR is wrong, only mentioned > a common aside which I suspected was wrong, and if not, could the error > in my ointment please be strained out? I may be wrong -- my usual > refrain -- but I'm not _that_ obviously wrong. Your thoughtful answer > shames this miscreant, but unfortunately does not remove him from his > position. >
From: Edward Green on 24 Aug 2006 19:00
> Edward Green wrote: > > The offhand claim is sometimes made that GR, which I take it means the > > formal machinery of GR, treats all coordinate systems equally. Is this > > true? Aside from questions about smoothness, it seems to me there is > > at least some other implicit condition on permissible coordinate > > systems. > > > > Consider a 1-dimensional manifold, a real coordinate x, and the metric > > ds = |dx|. Next, stretch this manifold by a postive factor a(x), also > > stretching the coordinates (so that points on the manifold are still > > labeled by the same numbers). Calling the new, stretched, coordinates > > u, the most natural transformation law for the metric would be > > > > ds = a(u)|du| > > > > Now, instead of stretching the manifold, compress the coordinates by > > this same function a( ), so that dx = a(u)du. The most natural > > transformation law for the metric would again be > > > > ds = a(u)|du| > > > > So, if we allow "any coordinate system", we would apparently be > > unable to distinguish the (physical) case of the stretched manifold > > from the (unphysical) case of the compressed coordinates. > > > > It may be objected that by compressing the coordinates, we have > > implicitly changed physical units, and this is not allowed. But that's > > a new rule, isn't it? What's a "unit"? We need additional structure > > to specify what we mean by this; we must equip our manifold with some > > physics -- a local property establishing the natural scale of the > > coordinates. This will insure that we don't arbitrarily distort the > > coordinates, and any distortion can be attributed to the manifold. Igor wrote: > Stretching or compressing the coordinate system just redefines it > mathematically. There's nothing physical there. Of course. > You can define a > coordinate transformation any way you want to provided that it is > invertible. But that doesn't result in any new physics. Obviously, my dear man. > It may result > in so-called fictictious forces arising in the new coordinate system > that weren't there previously, but those are never really physical > either. We are straying from the question. > And changing units can indeed be allowed under coordinate > transformations. Think of going from rectangular to polar or vice > versa. Rectangular has both coordinates with length units, and polar > has one with length and the other with angular units. Ok. Good point. > The metric will always compensate to make up for the difference. So you claim. I believe I have found the answer to my own question (as usual), and while I did not state the requirement 100% correctly, I was on the right track. I happened upon the correct formulation in Schwarzschild's 1916 paper on the field of a mass point: "The field equations ... have the fundamental property that they preserve their form under the substitution of other arbitrary variables in lieu of x1,x2, x3, x4, as long as the determinant of the substitution is equal to 1." http://arxiv.org/PS_cache/physics/pdf/9905/9905030.pdf If the old coordinates were orthogonal and the new coordinates remained orthogonal (which requirement I did not state) and we kept the same units, mine would be a sufficient condition to meet the above requirement on the named determinate. In general we have more latitude than that, _but_ we are not free to chose the new coordinates arbitrarily, even if they are invertible in a neighborhood. The unit of _volume_ at least must be preserved. In the case of a one-dimensional manifold this reduces to the requirement to keep the same units. Are we in agreement now? |