Communication with Rindler Observer: Is It Possible?

[kent davidge]

Is it possible that some light signals will never reach the Rindler observer?

Just to be clear, by Rindler observer I mean an observer undergoing constant acceleration (as measured in his own rest frame).

 

[haushofer]

Wel, yes, that's the idea of a Rindler horizon, right? :)

 

[Dale]

Yes. There is a "Rindler horizon" and any light signals emitted beyond the horizon will never reach the Rindler observer.

Edit: @haushofer for the win!

 

[kent davidge]

Wel, yes, that's the idea of a Rindler horizon, right? :)

Yes. There is a "Rindler horizon" and any light signals emitted beyond the horizon will never reach the Rindler observer.

so this Rindler observer is a great example of an "object" traveling slower than light and that is still "disconnected" from part of the universe?

 

[haushofer]

so this Rindler observer is a great example of an "object" traveling slower than light and that is still "disconnected" from part of the universe?

Yes. You don't need a black hole to obtain event horizons. And given the fact that locally an acceleration is indistinguishable from gravity, this shouldn't surprise you too much. But a Rindler observer can always decelerate to a constant speed such that his horizon disappears. The horizon of a black hole is truly a geometric property of spacetime.

 

[haushofer]

Yes. There is a "Rindler horizon" and any light signals emitted beyond the horizon will never reach the Rindler observer.

Edit: @haushofer for the win!

Sorry, was annoying other people in the GR-forum about questions concerning Lie derivatives, so was already here :P

 

[Dale]

Sorry, was annoying other people in the GR-forum about questions concerning Lie derivatives, so was already here :P

No need to apologize. Hearing the same answer from multiple sources adds confidence in the answer

 

[Pencilvester]

But a Rindler observer can always decelerate to a constant speed such that his horizon disappears.

Tiny nitpick: the Rindler observer wouldn’t have to decelerate—she would just have to cut her engines to become an inertial observer with no horizon.

 

[PeterDonis]

You don't need a black hole to obtain event horizons.

Yes, you do. The Rindler horizon is not an event horizon. Light signals emitted from behind the Rindler horizon can still reach infinity (more precisely, future null infinity); they just will never reach the Rindler observer.

 

[kent davidge]

Light signals emitted from behind the Rindler horizon can still reach infinity (more precisely, future null infinity); they just will never reach the Rindler observer.

Now that's confusing. If they do reach infinity, they should reach the Rindler observer, since the observer must be somewhere between a given point and "the infinity".

 

[PeterDonis]

If they do reach infinity, they should reach the Rindler observer

No, they don't. Both statements (that the light signals do reach infinity, and that they don't reach the Rindler observer) should be obvious from looking at a spacetime diagram, such as the one here:

https://en.wikipedia.org/wiki/Rindler_coordinates#Characteristics_of_the_Rindler_frame
The Rindler observer always stays within the right "wedge" shown in the diagram. A light signal emitted from behind the horizon is emitted from above and to the left of the 45 degree line that is the upper boundary of the wedge, and travels on a 45 degree line up and to the right (i.e., parallel to the boundary line). Obviously, such a line cannot reach the Rindler observer (since it never crosses the boundary, being parallel to it), and obviously it does reach infinity, since it keeps on going up and to the right indefinitely.

The misconception you probably have here is that "infinity" is a single point. It isn't.

 

[Dale]

Now that's confusing. If they do reach infinity, they should reach the Rindler observer, since the observer must be somewhere between a given point and "the infinity".

No. The Rindler observer also reaches infinity and gets there first.

 

[PeterDonis]

The Rindler observer also reaches infinity and gets there first.

This assumes that "infinity" is a single point. It isn't.

A more technically precise way of stating it would be that the Rindler observer ends up at a point on future null infinity that is earlier (i.e., further away from future timelike infinity) than the point on future null infinity where the light ray emitted from behind the Rindler horizon ends up. (The point on future null infinity where the Rindler observer ends up is the same as the point where the Rindler horizon ends up.)

 

[Dale]

This assumes that "infinity" is a single point. It isn't.

It doesn’t assume that. The phrase “reach infinity” is yours. I wouldn’t have chosen it precisely for the confusion it engendered in the OP and now apparently in yourself. In the sense that, as you said, the light reaches infinity, the observer reaches it first. I.e. in the limit the observer still always remains ahead of the light.

 

[PeterDonis]

The phrase “reach infinity” is yours.

As I originally used it, I immediately qualified it with:

(more precisely, future null infinity)

That was intended by me to make clear that "infinity" as I was using the term was not a single point. But I probably should have accompanied that with an explanation of what "future null infinity" actually is, or given a reference like the following:

https://en.wikipedia.org/wiki/Penrose_diagram
This article calls it "lightlike infinity" but it's the same thing.

In the sense that, as you said, the light reaches infinity, the observer reaches it first.

But the places they "reach" are not the same point. They are different points along the future null infinity line. Note that I also qualified "earlier" when I used it:

earlier (i.e., further away from future timelike infinity)

 

[haushofer]

Yes, you do. The Rindler horizon is not an event horizon. Light signals emitted from behind the Rindler horizon can still reach infinity (more precisely, future null infinity); they just will never reach the Rindler observer.

Ok, maybe I'm sloppy with terms.

 

[Dale]

As I originally used it, I immediately qualified it with:

That was intended by me to make clear that "infinity" as I was using the term was not a single point.

The problem with "reach infinity" is not the "infinity" part but the "reach" part. The fact that it is not a single point is not an issue. Delaware is also not a single point, but I can "reach" Delaware in a way that I cannot "reach" infinity.

In any case, I am kind of irritated for you to correct me on sloppy terminology that was introduced by you in the first place.

 

[PeterDonis]

The problem with "reach infinity" is not the "infinity" part but the "reach" part.

In the sense that "infinity" is not actually part of spacetime and nothing can ever reach it, sure, I agree. However, please bear in mind that the original reason I introduced the term "infinity" was to correct @haushofer's statement about the Rindler horizon being an event horizon. Without a definition of future null infinity and what it means for light signals to reach it, we cannot define what an event horizon is.

I am kind of irritated for you to correct me on sloppy terminology that was introduced by you in the first place.

I apologize for that. I didn't realize when I corrected @haushofer that what I said would naturally seem confusing to the OP and would cause further problems. I should have introduced the more precise terminology earlier.