- #211
PeterDonis
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zonde said:It's EH that is in the "infinite future" for external observer. Black hole interior is "beyond" infinite future. And because we define infinity as a limit where nothing is beyond that limit we get contradiction in terms.
Only if you switch meanings of the word "infinite" between the first and second sentences above. I have no problem with your first sentence, because I can map it easily to the actual math. Mathematically, the Schwarzschild "t" coordinate of the EH can be thought of as "plus infinity", so the region inside the horizon, which is to the future for any observer falling through the horizon, can be thought of as "beyond plus infinity". But the sense of "infinity" being used here is completely compatible, logically, with there being a region beyond infinity.
In your second sentence, you are using a different definition of the term "infinity", which explicitly rules out having a region "beyond" infinity. There are plenty of domains where that definition applies, but a black hole spacetime is not one of them. So there is no actual contradiction.
zonde said:Therefore there is no black hole interior in Schwarzschild coordinates. And it is not because of chosen coordinate system but because of chosen simultaneity.
This is correct; in *Schwarzschild coordinates* (more precisely, in Schwarzschild *exterior* coordinates--you can construct Schwarzschild coordinates for the interior region too, but that is a separate coordinate chart, disconnected from the exterior one) there is no black hole interior, because the lines of simultaneity are constructed in such a way that the chart can only cover the exterior region.
It does not follow from this, however, that the interior region doesn't exist. It only follows that the Schwarzschild exterior chart doesn't cover the interior region. These are two different statements, and the second does not require or imply the first.
zonde said:So my statement implies that simultaneity is not just convenience but rather physical fact.
Only when qualified as you do in the sentences I'm going to quote next. But the qualification is crucial, and it completely undercuts the claim you are trying to make.
zonde said:Simultaneity is defined in such a way as to get isotropic speed of light. And there is only one "correct" way how to define simultaneity for any state of motion.
The qualification, in bold, is crucial, as I said. For any given state of motion, there is a "correct" way to define simultaneity that respects the Einstein clock synchronization convention (which is what "isotropic speed of light" refers to). However, that only applies to that particular state of motion. A different state of motion can have a different "correct" simultaneity convention.
For example: if an observer is hovering at a constant radial coordinate above a black hole's horizon, there is a "correct" simultaneity convention for him, which is the one used in the Schwarzschild interior coordinates. However, if an observer is freely falling towards the hole, there is a different "correct" simultaneity convention for him, which is the one used in Painleve coordinates.
What this means is that the "lines of simultaneity" for Schwarzschild coordinates are *different lines* than the ones for Painleve coordinates. "Different lines" is an invariant, coordinate-free statement; the two sets of lines of simultaneity are different geometric objects. And it's perfectly possible for different sets of lines to cover different regions of spacetime; in this case, one set happens to reach into a region of spacetime that the other set does not. See below.
zonde said:But that's not all. It should be possible to convert consistently between reference frames that correspond to different states of motion.
Yes, certainly. But the conversion only has to be possible in a region of spacetime that is covered by both frames ("coordinate charts" would be a better term). If one chart does not cover a region (such as the black hole interior), then there is no requirement that other charts have to be able to convert to or from it in that region.
For example, anywhere in the exterior region, outside the EH, you can convert between Schwarzschild and Painleve coordinates easily. See, for example, the Wikipedia page on Painleve coordinates:
http://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates
However, inside the horizon, where the exterior Schwarzschild coordinates don't cover, you can't convert between them and Painleve coordinates. You can, however, convert between *interior* Schwarzschild coordinates and Painleve coordinates. You can also convert between Painleve coordinates and other charts that cover the interior, such as ingoing Eddington-Finkelstein or Kruskal.
zonde said:And I think there is complete chaos in GR regarding different coordinate charts and different (contradictory) simultaneity conventions they implement.
There are certainly different charts, with different simultaneity conventions. However, you have not shown that any of them are contradictory. All you have shown is that the different charts cover different regions of the spacetime, and that the "coordinate lines" on the different charts, such as the lines of simultaneity, are different geometric objects. None of this is in any way contradictory.