Any coordinate system in GR?
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From: Daryl McCullough on 12 Sep 2006 09:08 I.Vecchi says... >> b) In the exterior region an observer can use any appropriate >> method; after that shows the horizon has been pased, the >> observer uses a clock. > >That's Darryl's argument, which we have been discussing. The very fact >that it relies on external observers to make sense of the interior >chart should suggest that something is amiss. No, there's nothing amiss. An internal observer in freefall can perfectly well solve the geodesic equations to compute r as a function of his proper time. However, the solution will have an arbitrary constant in it. That constant is determined by setting the maximum value of r in the interior region to be 2m. >Arguments a) and b) are circular, based on the assumption that the >metric and hence the observers' paths within the horizon are already >defined. >One needs a measurement procedure for r and t in the interior >domain in order to define the metric. There is no difference, in principle, between measuring r and t for internal observers and for external observers. >Non sequitur. In the exterior domain t is determined by clock readings. No, it isn't. A clock measures ds = square-root(|g_uv dx^u dx^v|). Clocks do not measure t except in the special case in which |g_tt| = 1. >That I never checked, but it sure implies that mathematical models are >not physically relevant unless a correspondence with measurement >outcomes is provided. The only requirement of a physical theory (or mathematical model) is that it allow us to *predict* what an observer sees. The Schwarzchild metric can perfectly well do that. Your philosophical qualms don't really have anything to do with the Schwarzchild metric. -- Daryl McCullough Ithaca, NY
From: Ahmed Ouahi, Architect on 12 Sep 2006 09:30 Space is big. You just won't believe how vastly, hugely, mind- bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space. -- Douglas Adams -- Ahmed Ouahi, Architect Best Regards! "Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote in message news:ee6bh802inb(a)drn.newsguy.com... > I.Vecchi says... > > > >> b) In the exterior region an observer can use any appropriate > >> method; after that shows the horizon has been pased, the > >> observer uses a clock. > > > >That's Darryl's argument, which we have been discussing. The very fact > >that it relies on external observers to make sense of the interior > >chart should suggest that something is amiss. > > No, there's nothing amiss. An internal observer in freefall can > perfectly well solve the geodesic equations to compute r as a > function of his proper time. However, the solution will have an > arbitrary constant in it. That constant is determined by setting > the maximum value of r in the interior region to be 2m. > > >Arguments a) and b) are circular, based on the assumption that the > >metric and hence the observers' paths within the horizon are already > >defined. > > >One needs a measurement procedure for r and t in the interior > >domain in order to define the metric. > > There is no difference, in principle, between measuring r and t > for internal observers and for external observers. > > >Non sequitur. In the exterior domain t is determined by clock readings. > > No, it isn't. A clock measures ds = square-root(|g_uv dx^u dx^v|). > Clocks do not measure t except in the special case in which > |g_tt| = 1. > > >That I never checked, but it sure implies that mathematical models are > >not physically relevant unless a correspondence with measurement > >outcomes is provided. > > The only requirement of a physical theory (or mathematical model) is > that it allow us to *predict* what an observer sees. The Schwarzchild > metric can perfectly well do that. > > Your philosophical qualms don't really have anything to do with > the Schwarzchild metric. > > -- > Daryl McCullough > Ithaca, NY >
From: Tom Roberts on 12 Sep 2006 10:23 I.Vecchi wrote: > Daryl McCullough wrote: >> coordinates are just *labels*. >> They don't need to correspond to anything measurable. > > Labels for what? What's written on them? Who writes it? Coordinates are numerical labels for the points of the manifold (sometimes called "events" to impress on the reader that time is part of the manifold in relativity). Anybody can assign them (they cannot possibly be "written" on the events themselves); that is, any analyst can use any coordinate system she chooses. > According to Wigner ([1]), "in relativity theory, the state is > described by a metric which consists of a network of points in > space-time, that is, a network of events, and the distances between > these events. If we wish to translate these general statements into > something concrete, we must decide what events are, and how we measure > the distance between events" ([1]). This is old and loosely written by modern standards. In particular, the points themselves are NOT part of the metric. The metric interrelates points in the manifold to distance between infinitesimally separated pairs of points. And the analyst does not "determine what events [points] are", those are part of the manifold which is taken as given; the analyst merely assigns coordinate values to them. The universe itself is NOT "determined" by the analyst. So our model of the universe, the manifold, is likewise not "determined" by the analyst (other than by selecting a suitable manifold to serve as the model). > This is basic. If you cannot define your space-time events in terms of > measurement outcomes you are talking about nothing and/or building > circular arguments such as yours. The modern approach is that the points of the manifold are given by the manifold, and are not "determined" by the analyst. The analyst determines the metric (by solving the field equation), using coordinate labels that she herself assigned to the points of the manifold. Assigning coordinate labels to points is merely ASSIGNMENT, not "measurement". > Specifically, when you write the Schwarzschild metric as > ds^2 = -(1-2m/r) dt^2 + 1/(1-2m/r) dr^2 + ... > you must be able to specify which measurement procedure determines (or > labels, as you prefer) r and t, i.e. who is measuring them and how. Hmmm. When you wrote that, you were implicitly thinking of such an assignment. What you measure using rulers and clocks is ds, not r,t,theta,phi. Coordinates are abstract and arbitrary. Distances between infinitesimally separated pairs of points are measurable and invariant. > "... the measurement of position, that is, of the space coordinates, Today we would say "the assignment of coordinate 4-tuples to points in the manifold". Relying on ancient texts in a modern discussion is fraught with difficulties. Tom Roberts wrote: >> > Yes, that specific method is not usable in the interior, but there are >> > other ways: >> > a) remember that a circle with a given value of r has a >> > circumference of 2*pi*r, so a collection of identical >> > observers in a circle can measure the proper distance >> > between neighbors and thus determine r anywhere, including >> > the interior (as they are all falling together). > > As far as I understand, this relies on the unwarranted (as long as the > interior metric is not defined) assumption that the parameter r is a > radial distance within a sphere of radious 2m. Not true. As I said, it relies on the fact that the CIRCUMFERENCE OF A CIRCLE is given by 2*pi*r (this comes directly from the line element), not that r is in any sense a "radial distance". Remember r is timelike in this region, and cannot possibly be anything at all like "radius". The structure of this region is not like anything you have ever seen in Euclidean geometry. You must learn to shed your preconceptions and look at what the mathematics says, which in this case is not at all familiar to you. <shrug> > I do not see how the > measurement the proper distance between infalling (hence > non-stationary) observers may work and yield a finite proper time to > r=0. You have mixed up different concepts there. The measurement of proper distance between identical infalling observers is to compute the value of r. The fact that each of them experiences a finite proper time between r=2M and r=0 is a whole different aspect of this. The former fact is due to the structure of the metric in this region; the latter fact is learned by integrating the proper time of an infalling geodesic. >> > b) In the exterior region an observer can use any appropriate >> > method; after that shows the horizon has been pased, the >> > observer uses a clock. > > That's Darryl's argument, which we have been discussing. The very fact > that it relies on external observers to make sense of the interior > chart should suggest that something is amiss. Not really. The interior region is not static, and in this case it is convenient to use the static exterior region to set up the situation such that it is easy to describe; that's all. > Arguments a) and b) are circular, based on the assumption that the > metric and hence the observers' paths within the horizon are already > defined. But they are! The metric was determined by solving the field equation, and the observers' paths are determined from that by solving the geodesic equation. This is not circular, it is just DIFFERENT from what you are used to. <shrug> > One needs a measurement procedure for r and t in the interior > domain in order to define the metric. No. One measures the distance between pairs of infinitesimally separated pairs of points to which one has already attached the coordinate labels. One then determines the metric in terms of those coordinates. Not the other way 'round. IOW: assigning coordinates is not a "measurement", it is simply a process of systematically assigning labels to points. It is more like painting than surveying. >>> > > As I wrote, since no observer can hover "at constant r" in the interior >>> > > domain, r is not a measurable quantity. >> > >> > Not true. If that were true, in the exterior region then t would be >> > likewise "not a measurable quantity". > > Non sequitur. In the exterior domain t is determined by clock readings
From: I.Vecchi on 12 Sep 2006 17:31 Daryl McCullough ha scritto: > I.Vecchi says... > > > >> b) In the exterior region an observer can use any appropriate > >> method; after that shows the horizon has been pased, the > >> observer uses a clock. > > > >That's Darryl's argument, which we have been discussing. The very fact > >that it relies on external observers to make sense of the interior > >chart should suggest that something is amiss. > > No, there's nothing amiss. An internal observer in freefall can > perfectly well solve the geodesic equations to compute r as a > function of his proper time. > However, the solution will have an > arbitrary constant in it. That constant is determined by setting > the maximum value of r in the interior region to be 2m. By setting it? And why precisely 2m? Why not 3m? What about infinity? What is the physical criterion that motivates this choice? By the way, isn't this closely related to "making some assumptions on the time it will take to reach the singularity at r=0" or "the matching at the horizon appears arbitrary and it constitutes an 'ad hoc' assumption" or "your argument assumes that to an infalling observer the inner domain appears as a sphere of radius 2m", as I wrote? > >Arguments a) and b) are circular, based on the assumption that the > >metric and hence the observers' paths within the horizon are already > >defined. > > >One needs a measurement procedure for r and t in the interior > >domain in order to define the metric. > > There is no difference, in principle, between measuring r and t > for internal observers and for external observers. > > >Non sequitur. In the exterior domain t is determined by clock readings. > > No, it isn't. A clock measures ds = square-root(|g_uv dx^u dx^v|). > Clocks do not measure t except in the special case in which > |g_tt| = 1. You measure time and space using clocks and rulers respectively. The metric encodes the results of such measurements. In the exterior domain there is a correspondence between a time interval measured by a stationary observer at r=r_0 (by his clock reading) and the corresponding interval of proper time measured by a distant observer (i.e. an observer at infinity) given by the factor ( 1 - r_S / r_0 )`^1/2, r_S being the Schwarzschild radius. Thus we can obtain t from the clock readings of a stationary observer adjusted by the above factor. > >That I never checked, but it sure implies that mathematical models are > >not physically relevant unless a correspondence with measurement > >outcomes is provided. > > The only requirement of a physical theory (or mathematical model) is > that it allow us to *predict* what an observer sees. The Schwarzchild > metric can perfectly well do that. As long as noone checks, maybe. > > Your philosophical qualms don't really have anything to do with > the Schwarzchild metric. I think they do. At least they are animating a lively and instructive discussion. Cheers, IV
From: Daryl McCullough on 12 Sep 2006 20:49
I.Vecchi says... >Daryl McCullough ha scritto: >> No, there's nothing amiss. An internal observer in freefall can >> perfectly well solve the geodesic equations to compute r as a >> function of his proper time. > >> However, the solution will have an >> arbitrary constant in it. That constant is determined by setting >> the maximum value of r in the interior region to be 2m. > >By setting it? The curvature inside the event horizon increases as r approaches 0, so r is actually measurable by computing the full Riemann curvature tensor R_ijkl. You can use that calibrate the initial value of r. >> No, it isn't. A clock measures ds = square-root(|g_uv dx^u dx^v|). >> Clocks do not measure t except in the special case in which >> |g_tt| = 1. > > >You measure time and space using clocks and rulers respectively. As I said, clocks do *not* measure the coordinate t, they measure elapsed proper time. Similarly, rulers don't measure the coordinate r, they measure proper distances. So neither r nor t can be measured directly by clocks and rulers. >The metric encodes the results of such measurements. >In the exterior domain there is a correspondence between a >time interval measured by a stationary observer at r=r_0 >(by his clock reading) and the corresponding interval of >proper time measured by a distant observer (i.e. an observer >at infinity) given by the factor ( 1 - r_S / r_0 >)`^1/2, r_S being the Schwarzschild radius. And inside the event horizon, r can be measured by computing the Riemann curvature tensor. Or it can be computed (up to a constant) by integrating the geodesic equations. I don't understand what point you are making. There is no more problem in measuring r internally than there is in measuring it externally. >> Your philosophical qualms don't really have anything to do with >> the Schwarzchild metric. > >I think they do. No, they don't. They are general qualms about the meaning of coordinates in General Relativity. -- Daryl McCullough Ithaca, NY |