Notions of simultaneity in strongly curved spacetime

PeterDonis

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physically reasonable case where there is vacuum everywhere--see below under the Kruskal slicing), so the slicing is no longer valid there (you can't have the same event on multiple slices).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.

harrylin

See the new thread, http://physicsforums.com/showthread.php?p=4181348

Austin0

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???

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.

PeterDonis

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.

An inertial frame (momentarily comoving or not) is not the same thing as "a moment of time".

I disagree. See above.

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.

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.

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.