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clinden
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What is the physical interpretation of Cartesian coordinates in GR? Say, e.g., a system centered at the center of a spherical mass. What are x,y, and z physically, i.e., how are they measured?
clinden said:What is the physical interpretation of Cartesian coordinates in GR? Say, e.g., a system centered at the center of a spherical mass. What are x,y, and z physically, i.e., how are they measured?
clinden said:I was looking for a direct measurement approach, rather than converting in the standard way from the spherical coordinates.
This depends on the interpretation. One can have an ether interpretation of the GR equations which contains a flat background and interprets the curved metric as distorted by the ether (the gravitational field). This "distorted" metric may be a solution of the Einstein equations but exist, in this interpretation, on some flat (undistorted) background described by some preferred coordinates.HallsofIvy said:Perhaps the reason you are asking is that "Cartesian coordinates" in particular can be used only in "flat spaces". And General Relativity is a "geometric" theory using the curvature of space to explain gravity. Space that have curvature are not flat and there do NOT EXIST global Cartesian coordinates.
Ilja said:This depends on the interpretation. One can have an ether interpretation of the GR equations which contains a flat background and interprets the curved metric as distorted by the ether (the gravitational field). This "distorted" metric may be a solution of the Einstein equations but exist, in this interpretation, on some flat (undistorted) background described by some preferred coordinates.
No, every metric is consistent with [itex]\mathbb{R}^4[/itex]. To cut an arbitrary 4D manifold in such a way that only [itex]\mathbb{R}^4[/itex] remains is trivial, what has to be thrown away is codimension 1. You need something else, like a completeness condition or so, to make the jump from a simple tensor field with (1,3) signature on [itex]\mathbb{R}^4[/itex] to a nontrivial topology.stevendaryl said:It's sort of interesting: If you think of the metric tensor as just a field on spacetime, then the "shape" of spacetime is almost independent of the metric. But not quite, because certain metrics are only consistent with certain underlying topologies.
clinden said:What is the physical interpretation of Cartesian coordinates in GR? Say, e.g., a system centered at the center of a spherical mass. What are x,y, and z physically, i.e., how are they measured?
Isn't this more than just an interpretation? Certainly, for 'almost all' observables it is the same, as all measurements are affected by 'metric field', such that they match GR. What I am not sure of is how inspiralling BH would be modeled without non-trivial topology. Unless they can, then LIGO can be argued to be an observation which distinguishes the interpretations (and falsifies the flat background). More generally, there are clear differences in prediction, it is just the most of them don't apply to any accessible part of our universe. This is different from the LET interpretation of SR which makes identical predictions in all cases, thus is truly an interpretation. What you describe is a different theory than modern classical GR. That its different predictions are hard to access does not change its nature as a different theory not just an interpretation.Ilja said:This depends on the interpretation. One can have an ether interpretation of the GR equations which contains a flat background and interprets the curved metric as distorted by the ether (the gravitational field). This "distorted" metric may be a solution of the Einstein equations but exist, in this interpretation, on some flat (undistorted) background described by some preferred coordinates.
I do not have sufficient computational power to model inspiralling BHs. But if this is done, say, in the straightforward way using harmonic coordinates (why not, this is the most common coordinate condition and essentially simplifies the equations) then this would be exactly the same mathematics as in the ether interpretation.PAllen said:What I am not sure of is how inspiralling BH would be modeled without non-trivial topology. Unless they can, then LIGO can be argued to be an observation which distinguishes the interpretations (and falsifies the flat background).
PAllen said:What I am not sure of is how inspiralling BH would be modeled without non-trivial topology.
Ilja said:Given that this interpretation excludes nontrivial topologies, and, moreover, also closed causal loops, it can be considered, formally, as a different theory. Even if the equations are identical. In this case, given that it makes additional falsifiable prediction in comparison with the spacetime interpretation, it would be preferable from the point of view of Popper's criterion of empirical power.
Right, but nobody knows how to do this yet, or whether the predictions would be identical in a complete QG theory. It cannot be done in classical GR because the topology inside a horizon is not R4. For example, inside an SC BH, it is S2 X R2.PeterDonis said:If it turns out that certain current speculations about the effects of quantum corrections are right, and there are no actual event horizons (only apparent horizons), then one actually could model the phenomena we call "black holes" using ##R^4## topology, at least in an asymptotically flat spacetime. (One still couldn't do it in asymptotically de Sitter spacetime, which is the current best candidate for our actual universe, since the topology of de Sitter spacetime is ##S^3 \times R##, not ##R^4##.)
PAllen said:It cannot be done in classical GR because the topology inside a horizon is not R4.
Again, that requires quantum corrections, and there is no well founded theory (yet) in which to do such a computation. In classical GR, there are horizons.PeterDonis said:This is only true if the horizon is an actual event horizon; it's not true if there's an apparent horizon but no event horizon. In the latter case the topology even inside the apparent horizon can be ##R^4##, as long as the spacetime is asymptotically flat, as I said.
PAllen said:I don't see this at all. For every prediction difference, you have a falsifiable prediction for both theories. Thus, I see no difference by this criterion. The only difference is which predictions are correct.
Why this? AFAIK, the observation favors a globally flat universe. (Of course, this would always allow some measurement uncertainty for the curvature, thus de Sitter would never completely excluded. And there will be always people who for philosophical reasons would prefer a finite universe. But beyond this?)PeterDonis said:... de Sitter spacetime, which is the current best candidate for our actual universe,...
Not observing such things in a situation that standard GR mandates they happen.Ilja said:No. You have solutions of GR, those with nontrivial topology or closed causal loops, which are valid GR solutions, but do not allow an ether interpretation. Observing that our universe would be described by such a solution would falsify the ether interpretation, but not the spacetime interpretation. Not observing such a solution would leave above interpretations viable. What would falsify the spactime interpretation but leave the ether interpretation valid?
It leads to accepting violation of the principle of equivalence for free fall observers approaching a supermassive BH.stevendaryl said:What I assume is the case is that if you tried to model the collapse of a star into a black hole where you treat gravity as a field theory on top of flat [itex]R^4[/itex], then you will be able to reproduce the predictions outside the event horizon, but not inside it. Since nothing inside the event horizon can affect observers outside the event horizon, that would mean (I would think) that the only way to observe a difference in the two theories would be to take a plunge into the black hole.
This brings up an issue with General Relativity that I'm not sure I understand. Mainstream physicists act as if the interior of a black hole is as real as the exterior. In terms of general coordinates, there is nothing special about the event horizon, locally, so there is no physical reason for the manifold to end at the horizon. But is that just an aesthetic choice?
That's true if you accept that the spin 2 theory is local, and is used in conjunction with a global manifold interpretation to arrive at global predictions. Then, it isn't a complete interpretation.martinbn said:I am not sure I understand the issues with the topology. Isn't the spin-2 field on flat background supposed to be only the local description? Here is an analogy that I hope will clarify my question. Consider curves (smooth) in the plane. Not all of them are graphs of functions, for example a circle is not the graph of a [itex]y=f(x)[/itex]. But locally, by the implicit function theorem, all curves can be viewed as graphs of functions, including the circle. The circle's topology is different than that of a line, but there is no problem with the statement above. So what exactly is the problem with space-times with topology different than that of [itex]\mathbb R^4[/itex]? Each event will have a neighborhood, where gravity can be described as a field on a flat background?
PAllen said:That's true if you accept that the spin 2 theory is local, and is used in conjunction with a global manifold interpretation to arrive at global predictions.
Then, it isn't a complete interpretation.
Then the ether interpretation, based on the same equations, would also mandate that they happen. So it would be falsified too.PAllen said:Not observing such things in a situation that standard GR mandates they happen.
The portion of solution you get for one R4 coordinate chart is taken to be all of reality. There is no concept of analytically continuing this manifold to a larger manifold.martinbn said:How else can one take it?
As above.martinbn said:In what way?
No it would not. It would predict geodesic incompleteness, e.g. timelike world lines that end in some finite proper time. These would be at the boundary of the R4 coordinate chart you ended up with, that is 'cut' from a full manifold solution. Two completely different predictions, either of which could be falsified.Ilja said:Then the ether interpretation, based on the same equations, would also mandate that they happen. So it would be falsified too.
How would you falsify the GR prediction in this case?PAllen said:No it would not. It would predict geodesic incompleteness, e.g. timelike world lines that end in some finite proper time. These would be at the boundary of the R4 coordinate chart you ended up with, that is 'cut' from a full manifold solution. Two completely different predictions, either of which could be falsified.
PAllen said:The portion of solution you get for one R4 coordinate chart is taken to be all of reality. There is no concept of analytically continuing this manifold to a larger manifold.
martinbn said:Isn't the spin-2 field on flat background supposed to be only the local description?
Ilja said:I don't think so. One starts on a Minkowski space, which is global. With a linear approximation of a spin 2 field. Which is global too. There is no way one can, during the iteration, obtain something different. One can imagine that the iteration fails in some parts of the Minkowski space, but then we have the result of the iteration only on some part of the Minkowski space, not on something greater.
We seem to be talking in circles. IF you treat it as local solution whose complete solution must be assembled into some larger manifold whose topology is not determined by the local analysis, you have an interpretation of standard GR.martinbn said:The local solution is not for all of R4, only for a portion of it. Just like in my analogy, locally the circle can be represented by the graph of a function, the function need not be defined on all of R.
stevendaryl said:You can do field theory on top of ##R \times S^3## (or whatever) just as well as on top of ##R^4##.
Ilja said:How would you falsify the GR prediction in this case?
You would be, once the ether prediction is not falsified, all the time on the R4 part of the solution.