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Notions of simultaneity in strongly curved spacetime 
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#127
Nov2512, 02:51 AM

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PF Gold
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You are correct that, strictly speaking, your definition of "which clock runs faster" does not "require" a concept of "now"; you are basically using null curves as references, whereas the other definition of "which clock runs faster" uses spacelike surfaces of constant time, i.e., "now" surfaces, as references. But the difference is really immaterial: both definitions only work in static spacetime regions, so they both cover exactly the same set of cases; and one can always translate freely between them, so there is no reason other than personal preference for choosing one over the other. 


#128
Nov2512, 02:59 AM

PF Gold
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But he explains what he means with additional statements and it requires a bit of thinking over. 


#129
Nov2512, 09:17 AM

PF Gold
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To me it seems that they have different "now" and that is the main difference between them. GP is based on time of moving observers but coordinate orgin is the same as for stationary observer and radial distance too is from SC coordinates. PeterDonis: you made the same (or very similar) statement. What do you think about "now" of SC vs "now" of GP coordinates? 


#130
Nov2512, 01:38 PM

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PF Gold
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#131
Nov2512, 04:12 PM

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Offhand, I don't see any problem with your statement about the main difference between GP coordinates and SC coordinates being the assignment of the time coordinate. Perhaps problems with it will show up later, but at the moment I think it's OK. GP coordinates are sort of a hybrid coordinate system, they've got the time coordinates of the infalling observers mixed with the space coordinates of the static observers. But they're mathematically pretty convenient to use for many purposes. 


#132
Nov2512, 08:41 PM

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#133
Nov2612, 12:13 AM

PF Gold
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So where is the catch? We have two coordinate systems with different "now", object with static spatial coordinates in one coordinate system has static spatial coordinates in other coordinate system as well. 


#134
Nov2612, 12:22 AM

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In the nonstatic coordinate system (GP coordinates), the metric is not diagonal; there is a dt dr "cross term" in the line element. That means the surfaces of constant GP time are *not* orthogonal to the worldlines of objects with static spatial coordinates. And that means the definition of "now" given by GP coordinates is *different* than the definition of "now" given by the local inertial frames along the worldlines of objects with static spatial coordinates. So the sense in which the definition of "now" given by static (SC) coordinates is "unique" is that it is the only one that matches up with the definition of "now" in the local inertial frames of static observers. 


#135
Nov2612, 01:25 PM

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Now that my urgent questions concerning OppenheimerSnyder having been answered (thanks Peter), I'm returning to this thread. Atyy gave here an interesting link on which I already commented there. Retake:
I wonder if you mean that a symmetrical interpretation can be valid. That can't be correct: Eve is the one who fires the rocket engines and feels a force, in contrast to Adam. Compare https://en.wikisource.org/wiki/Relat..._of_Relativity "This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes." [..] Even though by no means all gravitational fields can be produced in this way [= from acceleration], yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. "  starting from section 20 of: https://en.wikisource.org/wiki/Relat..._of_Relativity And a modern point of view (for there is by far no unity): "A gravitational field due to matter exhibits itself as curvature in spacetime. [..] modern usage demotes the uniform "gravitational" field back to its old status as a pseudofield. "  http://math.ucr.edu/home/baez/physic...x/twin_gr.html In contrast, according to Einstein, clocks in a gravitational field go at different rates  much more different than what he should conclude according to you. 


#136
Nov2612, 02:24 PM

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#137
Nov2612, 02:35 PM

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PF Gold
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"Region of spacetime" I can see being a bit more difficult because it's not a standard term; but its physical interpretation is no more difficult than the interpretation of the term "spacetime" itself, and you don't seem to have any problem with that. Or do you? Do you think "spacetime" itself is a "mere mathematical term"? 


#138
Nov2612, 02:40 PM

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#139
Nov2612, 02:51 PM

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#140
Nov2612, 02:59 PM

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Of course, if you start saying "coordinate chart" when that's what you mean, it will become more evident that many of the things you are saying are dependent on which chart you use, meaning that they're not statements about actual physics, just about coordinate charts. 


#141
Nov2612, 11:22 PM

PF Gold
P: 1,376

I think that the catch is that in GP coordinates time coordinate is not orthogonal to space (radial) coordinate and in that sense they are not "right". 


#142
Nov2612, 11:51 PM

PF Gold
P: 1,376

But then there is an argument that "black hole" is somehow more correct extrapolation of known physical laws beyond limits of testability than "frozen star". This type of discussion is just empty. Reminds me of Feynman's comments on the field of gravity, back in the 60's in his private letter to his wife. 


#143
Nov2612, 11:55 PM

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#144
Nov2712, 12:04 AM

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PF Gold
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(IMO, even the "difference in interpretation" is somewhat strained, since the "frozen star" people agree that an object falling into the hole/frozen star/whatever it is will experience only a finite amount of proper time to the horizon. That means that if we use coordinates that are not singular at the horizon, such as GP coordinates, we can assign a *finite* time to the event of any infalling object crossing the horizon. But I don't think we'll get any further with that discussion here.) It might be possible to come up with a *different* model of a "frozen star", one which used a *different* spacetime and a *different* solution of the EFE, which actually made different physical predictions about what we would see because it was using a different model of the physics. But I've never seen one. 


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