Black hole formation watched from a distance

[bob012345]

Please read the thread. These questions have all been dealt with in the thread.

Sorry, I didn't see it through all the technical language at first pass. Now I see it. Thanks.

 

[pervect]

There is a thought experiment here that might be helpful. Consider a spaceship accelerating away from an idealized non-gravitating Earth at a constant 1g proper acceleration. The "frame" of the spaceship can be described by Rindler coordinates, the most common coordinate representation of an accelerated frame of reference. When we view things from the inertial frame of the idealized non-gravitating Earth, things are reasonably simple. Time on Earth passes as normal, the spaceship undergoes hyperbolic motion. A feature of this motion is that after approximately 1 year has elapsed on Earth (actually c / 9.8 m/s^2) signals from the Earth will no longer be able to catch up to the spaceship.

But what is the "viewpoint" from the spaceship? "The" viewpoint as a singular statement is actually misleading, but one common "viewpoint" of the spaceship can be created by the use of Rindler coordinates.

In Rindler coordinates,things are different. There is a Killing horizon approximately 1 light year behind the spaceship, and in these coordinates the coordinate time for the Earth to fall to the Killing horizon becomes infinite. By considering the Rindler metric, one form of which might be $-z^2 dt^2 + dx^2 + dy^2 + dz^2$, one might conclude that "time stops at the Rindler horizon" (z=0), similar to the way that one concludes that "time stops at the Schwarzschild horizon" in the Schwarzschild metric. And one might also conclude that the falling Earth "never reaches the Killing horizon" from the point of view of the spaceship, just as one concludes that an object falling into a black hole "never reaches the black hole", or in the case of the Openheimer-Snyder collapse, that the black hole "never forms".

But I would argue that it would be misleading to think of the Earth as "never reaching the Killing horizion". There is nothing special that happens 1 year after the spaceship takes off from the Earth as far as the Earth is concerned. The time is a bit special to the space-ship observer, though, because they never receive a signal from the Earth that is timestamped later than 1 Earth year after the departure of the spaceship, assuming the spaceship accelerates forever.

The meta-point here is that different coordinate choices are possible, and they have somewhat different descriptions of how things happen. Mathematically, one can deal with different choices via choosing a metric. Misners "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043, might be helpful in describing some of the techniques and philosophy that can be used to describe things in different coordinates.

Misner takes an interesting point of view, which is that the whole idea of an observer in General Relativity is perhaps not well defined. My view would be more along the lines that the line element of a metric defines the coordinates, a point that Misner also makes, and that thus defining the coordinates defines an associated "point of view" of an "observer". At least it does as much as that is possible.

One last point. Minkowskii and Rindler coordaintes are not the only coordinate choices here. One might also think about using radar coordinates from the spaceship, for instance, which would be an interesting exercise in "switching points of view". The first step, according to my suggestions, to look at things from this 'point of view" would be to find the associated metric. I have never done that, but it'd be the first step I'd suggest if one wanted to carry this out.

 

[PAllen]

One last point. Minkowskii and Rindler coordaintes are not the only coordinate choices here. One might also think about using radar coordinates from the spaceship, for instance, which would be an interesting exercise in "switching points of view". The first step, according to my suggestions, to look at things from this 'point of view" would be to find the associated metric. I have never done that, but it'd be the first step I'd suggest if one wanted to carry this out.

In the special case of uniform acceleration in special relativity, radar simultaneity for the rocket is identical to Rindler simultaneity. However, if distance is represented as .5 c $\tau$ (for the round trip) , then position coordinates are different from the Rindler. The result is that horizon interpretation is the same for these two coordinate choices.