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## Notions of simultaneity in strongly curved spacetime

 Quote by HomogenousCow Can't we just define a three dimensional time slice through the manifold for each coordinate at each coordinate time, then simply say that events in that slice are simultaneous?
Yes, but there are multiple ways of doing that, and some of them don't even cover the entire manifold.

In the case of Schwarzschild spacetime, for example, consider the following three slicings:

(1) The Schwarzschild slicing: slices of constant Schwarzschild coordinate time. Strictly speaking, this is *exterior* Schwarzschild coordinate time, since the "t" coordinate in the SC chart is not timelike for r <= 2m. This slicing only covers the region outside the horizon; the slices actually "converge" as you approach r = 2m, and at r = 2m they all intersect (at least, in the idealized, not physically reasonable case where there is vacuum everywhere--see below under the Kruskal slicing), so the slicing is no longer valid there (you can't have the same event on multiple slices). This is similar to the way Rindler coordinates break down at the Rindler horizon in flat Minkowski spacetime.

(2) The Painleve slicing: slices of constant Painleve coordinate time. This slicing covers the regions outside *and* inside the horizon.

(3) The Kruskal slicing: slices of constant Kruskal time. This slicing also covers the regions outside and inside the horizon; but in addition, it reveals two *other* regions (at least, it does for the idealized, not physically reasonable case where the spacetime is vacuum everywhere, i.e, there is no matter present--in any real spacetime, there would be matter present and the other two regions would not be there) that are not covered by any other slicing.

 Quote by PeterDonis Then what's the difference in Schwarzschild spacetime? That's the whole point of the Rindler horizon analogy: that if Eve does not think Adam never crosses the horizon, Eve' who is hovering above a black hole horizon should not think that Adam', who drops off her spaceship and falls into the hole, never crosses the horizon either. If you think there is a difference, what's the difference? Why can't Eve' reason the same way that Eve does, to conclude that Adam' does cross the horizon?

 Quote by harrylin 2. Peter: The term "co-moving inertial reference frame" is more precisely stated as "momentarily co-moving inertial reference frame". Evans evidently means constantly co-moving inertial reference frame, and I will explain why. According to you, Evans means that according to Eve the force she feels is due to acceleration; so that she thinks that she is one moment at rest in one inertial frame, and the next moment she is at rest in a different inertial frame. Consequently she would use the same set of inertial frames as Adam - that is standard SR. In any such reference frame there is a time for Eve when Adam passes through the horizon. It would be just an SR simultaneity disagreement. To the contrary, according to Egan there is no time for Eve when, in her co-moving inertial reference frame, Adam passes through the horizon.
 Quote by PeterDonis As you state it, this is false; you need to leave out the phrase "in her co-moving inertial reference frame" (which Egan does *not* use, and your attributing it to him is mistaken). The "time for Eve" that Egan refers to is Rindler coordinate time, which is the same as proper time along her worldline. Since she feels acceleration, i.e., feels weight, that proper time is *not* the same as the time in *any* inertial frame, even inertial frames in which she is momentarily at rest. Egan's statement simply means that there is no Rindler coordinate time at which Adam crosses the horizon; it's not referring to the time in *any* inertial frame.
 Quote by Austin0 Quote by harrylin I don't know what Egan had to say but I think you are quite mistaken regarding Rindler coordinates and the horizon. I don't think Rindler has anything to do with it. It is a coordinate artifact due to the dynamic metric in any accelerating system. This applies just as well to momentarily co-moving inertial frames. It happens because the distance to a point towards the rear shrinks due to contraction comparable to the increase in length due to system motion. SO the system asymptotically stops moving relative to points nearing the horizon as calculated . from a point within the system. So harrylin is correct that Adam never crosses the horizon in any MCRF that Eve is at rest in.
 Quote by PeterDonis No, it doesn't; at least, not in flat spacetime. In flat spacetime, any inertial frame covers the entire spacetime, including the portion of Adam's worldline at and beyond the Rindler horizon. That's a basic fact about inertial frames in flat spacetime. An MCIF is an inertial frame, so this fact applies to MCIFs in flat spacetime. Another way of saying this is that in flat spacetime, every inertial frame is global. In curved spacetime, there are *no* global inertial frames; *any* inertial frame can only cover a small patch of the spacetime. So in curved spacetime, you are correct that an MCIF at some event on an accelerated observer's worldline might not cover the horizon. But Egan's scenario is entirely set in flat spacetime, so the restrictions on inertial frames, including MCIF's, in curved spacetime doesn't apply.
This all is neither addressing my statements nor correct.
A Mpmentarily Co-moving Inertial Frame is by difinition a limited slice of spacetime. A MOMENT of constant time in the chart of that frame. To say that a MCIF is global is simply false. As far as that goes neither is a Rindler chart global so in fact there is no global chart for an accelerating system in spite of the fact it is moving through flat spacetime. Or do you disagree???

So my statement:
" So harrylin is correct that Adam never crosses the horizon in any MCRF that Eve is at rest in"
unambiguously means that at the moment Eve is at rest in any frame, the charted position of Adam according to the instantaneous metric of this frame is inside the position of the horizon.
Do you still think this is incorrect??

Can you provide an example of a case where this would not apply???

Do you understand that the relevant question is not whether the chart covers the horizon but whether Adam's instantaneous position is inside the horizon's x coordinate or not at the time of evaluation???

 Quote by PeterDonis Also, a word about "coordinate artifact". The fact that you can't assign a finite Rindler time coordinate to events at and beyond the Rindler horizon is an artifact of Rindler coordinates. But the fact that a light ray at the Rindler horizon will never intersect any of the "Rindler hyperbolas"--the curves with constant Rindler space coordinates--is not a coordinate artifact; you can express the same fact in any coordinate chart, because the curves themselves are geometric objects, not coordinate artifacts. So the existence of a "Rindler horizon" is not a coordinate artifact; there is something real and physical going on.
Your response here is not appropriate as I made no general statements about the horizon and was only talking within the limited context of the Adam and Eve example. Not related to light chasing an accelerating system. This phenomenon has nothing to do with coordinate systems (Rindler vs MCIF) per se and is just an empirical consequence of a finite light speed and a constantly accelerating system.

But as such still agrees with my statement that there is no significant effect due to Rindler coordinates as opposed to MCRFs . The effects are directly related to acceleration itself and are independent of coordinates.

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 Quote by Austin0 A Mpmentarily Co-moving Inertial Frame is by difinition a limited slice of spacetime.
In curved spacetime, yes. In flat spacetime, no. In flat spacetime, all inertial frames cover the entire spacetime. The MCIF is called "momentarily comoving" because an accelerated observer is only at rest in the MCIF for an instant; but that has nothing to do with how much of the spacetime the MCIF, or indeed any inertial frame, covers.

 Quote by Austin0 A MOMENT of constant time in the chart of that frame.
An inertial frame (momentarily comoving or not) is not the same thing as "a moment of time".

 Quote by Austin0 To say that a MCIF is global is simply false.
I disagree. See above.

 Quote by Austin0 as far as that goes neither is a Rindler chart global so in fact there is no global chart for an accelerating system in spite of the fact it is moving through flat spacetime. Or do you disagree???
It depends on what you mean by "a global chart for an accelerating system". If you mean a chart in which the accelerated object is at rest for more than an instant, then the most natural such chart, the Rindler chart, does not cover the entire spacetime. But there are other possible charts that could be used in which the accelerated object is at rest but the entire spacetime is still covered. In some recent thread or other, PAllen linked to a paper by Dolby and Gull that describes such a chart; if I can find the link I'll repost it here.

 Quote by Austin0 " So harrylin is correct that Adam never crosses the horizon in any MCRF that Eve is at rest in" unambiguously means that at the moment Eve is at rest in any frame, the charted position of Adam according to the instantaneous metric of this frame is inside the position of the horizon. Do you still think this is incorrect??
Yes, because any inertial frame, momentarily comoving with Eve or not, covers the entire spacetime, including the portion behind the horizon. The Rindler chart does not, but the Rindler chart is not an inertial frame.

 Quote by Austin0 The effects are directly related to acceleration itself and are independent of coordinates.
It depends on which "effects" you are talking about. The coordinates assigned to Adam are not "directly related to acceleration itself"; there is nothing requiring Eve to use the Rindler chart. Which light signals sent by Adam will intersect Eve's worldline *is* independent of coordinates.