Revised Twin Paradox: What's Wrong?

[dkgolfer16]

No, there are an infinite number of locations. If an observer determines that in his reference frame events A and B occurred simultaneously then all other observers at rest wrt the first also determine the events occurred simultaneously. The positions don't matter, only the fact that they are at rest wrt each other.

The reason that the positions don't matter is that each observer is intelligent and will account for the finite speed of light in determining when events A and B occured. This is why you can interchangably talk about a reference frame being a single observer accounting for finite light speed, or a system of an arbitrary number of observers all at rest wrt each other and using synchronized clocks.

I see what you are saying but aren't there only 4 locations that would experience the events happening simultaneously. No other locations could saw they say the events at the same moment in time.

 

[Fredrik]

So then whenever two events take place and are measured to have occurred at the same time (call it t_d as usual), there are always exactly 3 other locations (in the same RF that the original measurement was taken) that would conclude that these two events occurred this time, t_d?

The coordinate system assigns time coordinate t_d to those events, and that's all there is to it. Locations have nothing to do with it, unless you want to associate another coordinate system with those locations, and why would you want to do that?

I see what you are saying but aren't there only 4 locations that would experience the events happening simultaneously. No other locations could saw they say the events at the same moment in time.

I still don't understand what this means. You seem to be talking about a signal (more than one?) that's emitted at some event (which one?) and detected at 4 events (which ones and why?).

Edit: If you're talking about light signals that are emitted at both events, and are trying to figure out what events will be reached by both signals, the answer is that there's just one such event in 1+1 dimensions and infinitely many in 2+1 and 3+1 dimensions. In 2+1 dimensions the light cones are actual cones and the set of events that are reached by both signals is the curve where the two cones intersect.

 

[Dale]

I see what you are saying but aren't there only 4 locations that would experience the events happening simultaneously. No other locations could saw they say the events at the same moment in time.

I am not 100% sure what you mean by "experience the event". I assume you mean the event where an observer receives a signal emitted at a previous event. E.g. a star 20 light years away went supernova in 1970 and we "experienced the supernova" in 1990 (note that we are intelligent observers so we know that the event happened in 1970 even though we received the light in 1990).

If this is what you mean, then what you are talking about is the intersection of the light cones originating at the two events. There are also an infinite number of points in the intersection of any two light cones. It is just geometry, think of the intersection of two cones in space.

 

[dkgolfer16]

I am not 100% sure what you mean by "experience the event". I assume you mean the event where an observer receives a signal emitted at a previous event. .

Yes, this is what I meant by experience.

Sorry if my post's are becoming annoying; just trying to learn. Guess I'll try to explain my thought experiment again in a more clear manner.

I'll keep it 2-d this time. All of the following are in the same RF. Light Bulb 1 (LB 1) is placed at (2,0). Light bulb 2 (LB 2) is placed at (-2,0). Light Detector A (LD A) is placed at (0,10) and LD B is placed at (0,-10). If light from LB 1 reaches a detector first, the detector turns red. If light from LB 2 reaches a detector first, the detector turns red.

If light from LB 1 reaches a detector at the same time as light from LB 2, the detector turns green.

Lets say that LD A turns green. This must mean that LD B also turns green (remember they are in the same RF).

If you place Light Detectors anywhere else in this coordinate system (besides the y-axis), the light detectors won't turn green because they are closer to one bulb then the other (the detectors can't do math; they only respond to photons).

If they are placed on the y-axis, they will turn green, but they will experience then two flashes at a later or earlier time then LD A and LD B.

For a 2-D RF: Whenever two events are recorded as simultaneous at a location (in it's RF), there is only one other location that "experiences" these same two events at this same exact time, t_d?

It might have already been answered in another post, but I'm a slow learner so bare with me.

 

[Dale]

For a 2-D RF: Whenever two events are recorded as simultaneous at a location (in it's RF), there is only one other location that "experiences" these same two events at this same exact time, t_d?

It might have already been answered in another post, but I'm a slow learner so bare with me.

Usually you would call this 2+1 dimensions (2 space dimensions and one time dimension). Yes, in 2+1 dimensions the light cone will cause all of the detectors on the y-axis to turn green, with two detectors green at each moment in time after the detector at the origin goes green.

In 3+1 dimensions there is a whole plane of detectors that turn green with a whole circle of detectors green at each instant in time after the detector at the origin goes green.

 

[Fredrik]

I'll keep it 2-d this time. All of the following are in the same RF. Light Bulb 1 (LB 1) is placed at (2,0). Light bulb 2 (LB 2) is placed at (-2,0). Light Detector A (LD A) is placed at (0,10) and LD B is placed at (0,-10). If light from LB 1 reaches a detector first, the detector turns red. If light from LB 2 reaches a detector first, the detector turns red.

If light from LB 1 reaches a detector at the same time as light from LB 2, the detector turns green.

Lets say that LD A turns green. This must mean that LD B also turns green (remember they are in the same RF).

Just a relatively unimportant comment here: There's no information in the statement "they are in the same RF". You should say that they are both stationary in this frame (if that's what you meant).

If you place Light Detectors anywhere else in this coordinate system (besides the y-axis), the light detectors won't turn green because they are closer to one bulb then the other

Yes, only detectors on the y-axis can turn green, but there's nothing special about the positions on the y-axis you picked. Those are the positions where detectors will turn green precisely at $t=\sqrt{104}/c$ (assuming that the time coordinates of the two emission events are both 0). If you put detectors at other locations on the y axis, they will turn green at other times, but I see that you already understand that:

If they are placed on the y-axis, they will turn green, but they will experience then two flashes at a later or earlier time then LD A and LD B.

For a 2-D RF: Whenever two events are recorded as simultaneous at a location (in it's RF), there is only one other location that "experiences" these same two events at this same exact time, t_d?

First of all, a reference frame in SR is a coordinate system on spacetime, so it also assigns a time coordinate to each event. In this case we're talking about 2+1 dimensions.

And yes, it's true that if there's an event with time coordinate $\sqrt{104}/c$, where a detector detects light from simultaneous emission events, then there's at most one more event with time coordinate $\sqrt{104}/c$ where a detector can detect light from both of those emission events.

It never occurred to me that this could be what you meant. I included the dimension of time when I tried to picture what you were describing in my head, and I'm pretty sure DaleSpam did too. You didn't, and as a result we ended up with very different mental images. (Yours is a slice of ours).

 

[dkgolfer16]

Usually you would call this 2+1 dimensions (2 space dimensions and one time dimension). Yes, in 2+1 dimensions the light cone will cause all of the detectors on the y-axis to turn green, with two detectors green at each moment in time after the detector at the origin goes green.

In 3+1 dimensions there is a whole plane of detectors that turn green with a whole circle of detectors green at each instant in time after the detector at the origin goes green.

Thanks DaleSpam. That's what I had visualized. Did einstein explain this or was it just a self-explanatory part of simultaneity?

 

[dkgolfer16]

Just a relatively unimportant comment here: There's no information in the statement "they are in the same RF". You should say that they are both stationary in this frame (if that's what you meant).

Yes, only detectors on the y-axis can turn green, but there's nothing special about the positions on the y-axis you picked. Those are the positions where detectors will turn green precisely at $t=\sqrt{104}/c$ (assuming that the time coordinates of the two emission events are both 0). If you put detectors at other locations on the y axis, they will turn green at other times, but I see that you already understand that:



First of all, a reference frame in SR is a coordinate system on spacetime, so it also assigns a time coordinate to each event. In this case we're talking about 2+1 dimensions.

And yes, it's true that if there's an event with time coordinate $\sqrt{104}/c$, where a detector detects light from simultaneous emission events, then there's at most one more event with time coordinate $\sqrt{104}/c$ where a detector can detect light from both of those emission events.

It never occurred to me that this could be what you meant. I included the dimension of time when I tried to picture what you were describing in my head, and I'm pretty sure DaleSpam did too. You didn't, and as a result we ended up with very different mental images. (Yours is a slice of ours).

Yeah I was just trying to simplify it as much as possible because I was having a hard time explaining it. All in all it's an interesting part of simultaneity that I hadn't read about before. Thanks for the help. Oh and how did you calculate $\sqrt{104}/c$ part?

 

[Fredrik]

Yeah I was just trying to simplify it as much as possible because I was having a hard time explaining it. All in all it's an interesting part of simultaneity that I hadn't read about before. Thanks for the help. Oh and how did you calculate $\sqrt{104}/c$ part?

Distance divided by speed equals time. The distance is the hypothenuse of a right triangle, and you specified the length of the other sides to be 2 and 10, so the distance is $\sqrt{10^2+4^2}$.

 

[Dale]

Thanks DaleSpam. That's what I had visualized. Did einstein explain this or was it just a self-explanatory part of simultaneity?

I don't think that Einstein ever focused on your idea of "experiencing" something. In traditional SR thought experiments etc. we are always interested in what intelligent observers determine actually happened after properly accounting for the finite travel time of any light signals. Because of this we talk interchangably about an observer as being a single intelligent entity at some specific location making measurements of distant events or a system of entities at rest wrt each other making purely local measurements using synchronized clocks. Always, the concern is when something actually happened in a given reference frame, not when the signal was "experienced".