
#127
Dec612, 04:16 PM

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Btw, you are in dangerous territory. If patchwork is something mentioned in those lecture notes then it is part of mainstream physics and your claim that it isn't good physics would therefore be quite speculative. Furthermore, all of the comments on manifolds and coordinates in that section apply in simple spacetimes too. Like flat or constant curvature. 



#128
Dec612, 04:20 PM

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#129
Dec612, 04:28 PM

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The opposing view that I came up with is that Eve and Adam usually will be able to distinguish the two cases; the physical criterion that experimentally distinguishes the two cases is clear (tidal effect). I noted that in case that one or both are unable to do so (for example Adam only has a simple accelerometer and no windows), that could make them like bees that fly against a window. Surely you'll agree that nature can't care less if they did not predict the window, and the window is not "unphysical" if the bee didn't notice it before hitting it. The criterion of that opposing view is just as "mainstream" as the one you presented: real gravitation can be distinguished from acceleration in flat spacetime and is physically different from it. This weaker version of Einstein's equivalence principle remains, in the form cited in the first post of http://www.physicsforums.com/showthread.php?t=656240. Also, special Relativity is the theory of flat spacetime, without equivalence principle. That enables the use of universal ("global") descriptions such as Minkowski spacetime for negligible effects of gravitation on "clocks and "rulers" (that's extremely handy for solving Langevin's original "twin" example!) and similarly universal descriptions such as Schwarzschild spacetime for negligible effects of velocity; the two systems can be combined to globally account for both effects. That is de facto how the ECI frame is constructed. From that point of view, the use of Rindler/EEP coordinates in flat spacetime is the use of pseudo coordinates like those of Zeno. Branding such fictitious gravitational fields as pseudo gravitational fields allows for a consistent physical interpretation, globally (as far as I can see). That's enough philosophy. It will be most interesting to elaborate with numbers how a consistent physical interpretation is possible with what you call "the standard view". 



#130
Dec612, 04:32 PM

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#131
Dec612, 04:34 PM

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#132
Dec612, 05:07 PM

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With the word "combination" you may be thinking of the fact that the ECI is also a sort of "local inertial frame" for the Earth in its orbit about the Sun. This is true (with some technicalities), but note the word "local"; it is certainly not any kind of "combination" of a global Minkowski frame with a global Schwarzschild frame. If we look at the Solar System as a whole, the global frame is a Schwarzschild frame centered on the Sun. 



#133
Dec612, 05:07 PM

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#134
Dec612, 05:30 PM

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#135
Dec612, 05:33 PM

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#136
Dec612, 05:35 PM

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Let's compare what happens issue by issue. In Zeno's paradox, we have two times. Let's let the "real" time be represented by tau, and zeno time by Z. In the infalling black hole case, we have proper time tau, and Schwarzschild time t The mapping from t to tau that we worked out previously for the Schwarzschild case in great detail is: [tex]t = \tau4\,\left(3 \, \tau \right)^{\frac{1}{3}}+4\,\ln \left[ \left(3 \, \tau \right)^{\frac{1}{3}}+2 \right] 4\,\ln \left[\left(3 \, \tau \right)^{\frac{1}{3}}2 \right] [/tex] (You can probably find this in a textbook if you want to check my math). The characteristic features of this mapping is that t increases monotonically with tau, and that infinite range of tau only covers a finite range of t. This is due specifically to the term [tex] 4 \ln \left[\left(3 \, \tau \right)^{\frac{1}{3}}2 \right] [/tex] This is rather complicated, the argument will be clearest if we assume this term, which is the one that approaches infinity, is the dominant term near the event horizon, in which case we can solve for [itex]\tau[/itex] assming that this is the only term that matters. [tex]\tau \approx \frac{1}{3} \, \left[\exp^{t/4}  2\right]^3[/tex] and we see that [itex] \tau[/itex] approaches 8/3 as t> infinity (which is when the horizon is reached). In the Zeno paradox, the mapping is something like [tex] \tau = a(1\exp^{z/b}) [/tex] where a and b are some constants where [itex]\tau[/itex] approaches some constant a as Z approaches infinity And we see the issue in both cases, even though t (in one case), Z (in the other case) cover infnite ranges, [itex]\tau[/itex] does not. So, essentially Zeno never assigns a label, Z to some events. And he concludes from this that these events don't exist. And you assume the same thing  because you never assign a coordinate "t" to some events, you assume they don't happen. And this conclusion is just as unjustified when you do it, as when Zeno does it. So the notverycomplicated moral of the story is that because you can choose ANY coordinates you want, you need to be careful in your interpretation of the results. Specifically, it's possible to choose coordinates like Zeno did, that exclude important regions from analysis, because the coordinates don't label physically significant events. However, not giving something a label doesn't make it not exist, any more than closing your eyes does. At least not for most defitnitions of "existence". 



#137
Dec612, 06:21 PM

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Pervect, that's a showstopping reply !
Harrylin, pay heed. 



#138
Dec612, 07:47 PM

P: 1,657

It is easy to demonstrate that it is possible to choose coordinates that leave out part of the manifold. What reason do you have for thinking that's NOT the case with Schwarzschild coordinates? (It PROVABLY is the case, so what I'm really asking you is why you seem to believe something that is provably false.) So in Rindler coordinates, someone sees a dropped object asymptotically approach the location X=0 as time T → ∞. The correct interpretation of this situation isn't: "Rindler coordinates are wrong. Cartesian coordinates are right." The correct interpretation is "The event of the object crossing the 'event horizon' at X=0 is not an event covered by the Rindler coordinates". Rindler coordinates are perfectly fine for describing any events taking place within its chart, but it can't possibly describe events outside that chart. The same thing is true of an object crossing the event horizon in Schwarzschild coordinates. That event is not covered by Schwarzschild coordinates. Schwarzschild coordinates are perfectly good for describing events within its chart, but can't be used to describe events outside its chart. It's not a question of whether the "hovering observer" is correct and the "infalling observer" is wrong, or viceverse. The only issue is whether the event of crossing the horizon is in fact covered by this coordinate system or that coordinate system. [itex]\dfrac{4 R_s^3}{r} e^{r/R_s}(dV^2  dU^2)[/itex] where [itex]R_s[/itex] is the Schwarzschild radius, and [itex]r[/itex] is the Schwarzschild radial coordinate. This metric is defined everywhere, except at the singularity [itex]r=0[/itex]. The "time" coordinate is [itex]V[/itex]. The event horizon in these coordinates consists of all points with [itex]V^2 = U^2[/itex]. So an object can certainly cross the event horizon at a finite value for the time coordinate [itex]V[/itex]. Now, to see that this is describing the SAME situation as the Schwarzschild black hole, you note that the "patch" with [itex]U > 0[/itex] and [itex]U < V < +U[/itex] describes exactly the same region of spacetime as the Schwarzschild patch [itex]r > R_s[/itex] and [itex]\infty < t < +\infty[/itex], and the "patch" with [itex]1 > V > 0[/itex] and [itex]V < U < +V[/itex] describes exactly the Schwarzschild patch with [itex]0 < r < R_s[/itex] and [itex]\infty < t < \infty[/itex]. But the KS coordinates also covers the boundary between these two regions, the event horizon. 



#139
Dec612, 08:28 PM

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[tex]\dfrac{32 M^3}{r} e^{r/2M}(dV^2  dU^2)[/tex] 



#140
Dec712, 04:45 AM

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#141
Dec712, 05:10 AM

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It seems rather strange to me to ignore the readings of actual, physical clocks (proper time) in favor of some abstract coordinate time, but it seems alltoocommon. My speculation is that this is based on a desire for the "absolute time" of Newtonian physics. [add] Static observers do have _some_ physical significance where they exist , which is outside the event horizon. This significance is derived mostly form the Killing vector field of their timelike worldlines. The Killing vector still exists at the event horizon, but it's null, so it doesn't represent any sort of "observer". The coordinate system of static observers, where they exist, has about the same relevance to an infalling observer as the coordinate system of some "stationary" frame to somoene rapidly moving. Which in my opinion is "not very much". But I suppose opinions could vary on this point, it's not terribly critical. The biggest difference here, and another significant underlying issue, is that static observers cease to exist at the event horizon. This makes their coordinates there problematic, as you're trying to defie a coordinate system for an observer that doesn't exist anymore. This isn't any sort of breakdown in physics  it's a breakdown of the concept of static observers. For any actual physical observer, the horizon will always be approaching them at "c"  because any physical observer will have a timelike worldline, and the horizon is a null surface. This isn't really very compatabile with the event horizon as a "place". This is why spacetime diagrams that represent the event horizon as a null surface (such as the Kruskal or penrose diagram) are a good aid to understanding the physics there, and why Schwarzschild coordinates are not. Another subissue (of many) is the absolute refusal of certain posters to even consider any other coordinate systems other than Schwarzschild as having any relevance to the physics. Which gives rise to severe problems, as Schwarzschild coordinates are illbehaved at the event horizon, for the reasons I've previously aluded to (the nonexistence of static observers upon which the coordinate system is based). This ill behavior is hardly any secret  pretty much ANY textbook is going to tell you the same thing. 



#142
Dec712, 05:59 AM

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We can always do the same thing with Zeno time by judicious choice of our reference clock and our simultaneity convention. For instance, we can use a Rindlerlike simultaneity convention. As long as our reference clock asymptotically approaches the worldline of the light pulse from the arrow reaching the target then that event will be at infinite coordinate time. By varying the acceleration of the reference clock we can adjust the spacing of the time coordinate between the other points on the arrows path. 



#143
Dec712, 06:47 AM

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#144
Dec712, 06:53 AM

P: 1,657

[itex]dt = d\tau/\sqrt{1R_s/r}[/itex] I don't immediately see any simple physical interpretation for [itex]dt[/itex] at finite values of [itex]r[/itex]. 


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