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From: Eric Gisse on 10 Sep 2006 10:13 I.Vecchi wrote: [...] > > In order to measure t a stationary observer has only to look at his > clock. Beyond the horizon however there is no way to measure r. Why do you say that? > > Cheers, > > IV
From: Tom Roberts on 10 Sep 2006 14:30 I.Vecchi wrote: > [in the external region] > A stationary (hence accelerated) observer (say, on a space-ship) can > determine her r coordinate by measuring her proper acceleration a, > weighing objects or herself with a dynamometer. Normalising everything > in sight a= -m/(r^2*sqrt(1-2m/r)) for r>2m. Yes. > The problem arises because there is no such thing as an interior > observer at fixed r. There are no stationary observers in the interior > domain, hence the above measurement of r is impossible. 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). 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. c)... surely more > I am saying that "proper measurements" of r in the interior domain are > physically impossible. Your enumeration of methods was woefully incomplete. Attempting to argue by exhaustive enumeration nearly always fails.... > 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". Daryl McCullough wrote: > I believe r can also be measured by computing the full Riemann > curvature tensor R_ijkl. The Einstein tensor G_ij is zero > everywhere, but I don't believe that the Riemann curvature > tensor is. Yes, Riemann is nonzero everywhere in the Schw. manifold. I.Vecchi wrote: > Space-time events (or points as you call them) correspond to > measurement outcomes relative to an observer. [...] Interpreted broadly that implies the moon does not exist when no observer is looking! The clear situation is that the universe exists completely independent of any observers, and so the model of it must also do so; this implies that events/points of th emanifold are completely independent of any observer. Similarly, models of physical phenomena and objects must be independent of observers and coordinates, which is why tensors are used so frequently. Tom Roberts
From: I.Vecchi on 12 Sep 2006 01:20 Daryl McCullough wrote: > I.Vecchi says... > .... > >Space-time events (or points as you call them) correspond to > >measurement outcomes relative to an observer. > > No, they really don't. > > >Any physical event (including space-time events) must be measured by > >an observer. > > No, that's not true. As I said, coordinates are just *labels*. > They don't need to correspond to anything measurable. Labels for what? What's written on them? Who writes it? 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 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. 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. Otherwise, as I said, you are talking about nothing. In the exterior domain, local procedures to determine r and t can be implemented. However, as long as a measurement protocol for r and t in the interior domain cannot be defined, the so-called interior solution is physically meaningless. Cheers, IV [1] E. Wigner "Relativistic Invariance and Quantum Phenomena" Rev. Mod. Phys. 29, 255 (1957) PS I add a further quote from [1], which I find quite relevant here, albeit indirectly: "... the measurement of position, that is, of the space coordinates, is certainly not a significant measurement if the postulates of of the general theory are adopted: the coordinates can be given any value one wants. ... Most of us have struggled with the problem of how, under these premises, the general theory of GR can make meaningful statements and predictions at all. ... This is a point that which cannot be emphasised strongly enough and it is the basis of a much deeper dilemma .... . It pervades the general theory, and to some degrees we mislead both our students and ourselves when we calculate , for instance, the mercury perihelon without explaining how our coordinate system is fixed in space, what defines it in such and such a way that it cannot be rotated, by a few seconds a year, to follow the perihelion apparent motion. ... . There must be some assumption on the nature of the coordinate system that keeps it from following the perihelion. ... . A difference in the tacit assumptions which fix the coordinate system is increasingly recognized to be at the bottom of the many conflicting results arrived at in calculations based on the general theory of relativity."
From: I.Vecchi on 12 Sep 2006 02:39 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. 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. > 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. 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. > c)... surely more Of the same kind, I suppose. > > > > I am saying that "proper measurements" of r in the interior domain are > > physically impossible. > > Your enumeration of methods was woefully incomplete. Attempting to argue > by exhaustive enumeration nearly always fails.... > > > > 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. .... > > Space-time events (or points as you call them) correspond to > > measurement outcomes relative to an observer. [...] > Interpreted broadly that implies the moon does not exist when no > observer is looking! That I never checked, but it sure implies that mathematical models are not physically relevant unless a correspondence with measurement outcomes is provided. Cheers, IV
From: Daryl McCullough on 12 Sep 2006 07:31
I.Vecchi says... >Daryl McCullough wrote: >> No, that's not true. As I said, coordinates are just *labels*. >> They don't need to correspond to anything measurable. > >Labels for what? Coordinates are labels for points on the manifold. The manifold is part of the *model* of what is going on. It's a mathematical object, and we use coordinates to describe points on that mathematical object. >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 basic. Yes, I think that you are missing the basics here. The measurements he is talking about are all *local* measurements. Those are the only things that are physically significant. Global coordinate systems are not (in general) measurable. >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. What are you saying is circular? We're *starting* with a particular solution of Einstein's field equations, and then we're exploring the features of that solution. What's circular about it? >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. No, I don't. It's a mathematical model. Mathematical models are not reducible to a finite set of observations. What's important is that you can use the model to *describe* observations. If you want to know what an observer sees at various points on the manifold, the metric above allows you to figure that out. But there is no requirement that observations allow you to figure out what manifold you are on, or that observations allow you to figure out what your coordinates are. You are coming at this from a particular philosophy of science which I certainly don't agree with, and I'm not interested in. To me, science is about building models that *predict* what observers see, not building models *from* what observers see. The latter problem is impossible---observations never uniquely pin down the model. >Otherwise, as I said, you are talking about nothing. I don't agre with your philosophy of science here, but it doesn't matter. We're talking here about a particular *model*, namely the Schwarzchild solution to Einstein's field equations. This model was developed by physicists using mathematics. It was not developed from any actual observations---nobody we know has actually fallen into a black hole, and the universe is *not* described by the Schwarzchild metric. >In the exterior domain, local procedures to determine >r and t can be implemented. You are confused. The only way that r and t are measurable in the exterior is by the *same* methods as can be used in the interior. >However, as long as a measurement protocol for r and t in the interior >domain cannot be defined, the so-called interior solution is physically >meaningless. There is no difference between the interior and the exterior as far as the measurability of t and r is concerned. >[1] E. Wigner "Relativistic Invariance and Quantum Phenomena" Rev. Mod. >Phys. 29, 255 (1957) > >PS I add a further quote from [1], which I find quite relevant here, >albeit indirectly: >"... the measurement of position, that is, of the space coordinates, is >certainly not a significant measurement if the postulates of of the >general theory are adopted: the coordinates can be given any value one >wants. ... Most of us have struggled with the problem of how, under >these premises, the general theory of GR can make meaningful statements >and predictions at all. ... He's talking here about philosophical issues that I don't think have been resolved. But they are issues about GR in *general*, not about the Schwarzchild interior versus Schwarzchild exterior solutions. -- Daryl McCullough Ithaca, NY |