Relativity Paradox: Solving for Different Observers in the Cloud and Ship

[Nick J Harris]

I'm trying to solve this problem but I get a different answer depending on which observer I solve it for.

For the observer in the cloud, the parallel light has traveled 2.294cs, but this is only 0.23cs in front of the ship. 0.23 cs is only 0.086cs in the perspective of the ship due to length contraction.

But if you solve this for the observer on the ship, the light will have traveled 1cs in front of the ship and the cloud would have traveled 0.9 cs past it. A total length of 1.9cs into the cloud would have a relative length of 4.418 to the observer in the cloud. So when the vertical light strikes the ship is it 4.418cs or 2.294cs into the cloud?

 

[jbriggs444]

For the observer in the cloud, the parallel light has traveled 2.294cs, but this is only 0.23cs in front of the ship. 0.23 cs is only 0.086cs in the perspective of the ship due to length contraction.

You start out with a number 2.294 cs seemingly pulled out of thin air. Where did that number come from? The answer I get is quite different.

Of course there is room for disagreement between the two frames. The question asks about "how far into the cloud" and "when the vertical laser returns to the ship". That is a question about what is happening over there simultaneous with an event over here. Whose clocks are we using to judge simultaneity?

The idea of a "vertical" pulse of light is also not without its complexities.

 

[FactChecker]

You start out with a number 2.294 cs seemingly pulled out of thin air. Where did that number come from? The answer I get is quite different.

I also got a different answer.

The idea of a "vertical" pulse of light is also not without its complexities.

Does the angle change due to a rotation? If so, I didn't account for that in anything.

 

[jbriggs444]

I also got a different answer.Does the angle change due to a rotation? If so, I didn't account for that in anything.

I did not work the problem through completely in the cloud frame, but in that frame the "vertical" light pulse clearly does not traverse a vertical path.

 

[Janus]

I'm trying to solve this problem but I get a different answer depending on which observer I solve it for.

View attachment 239157For the observer in the cloud, the parallel light has traveled 2.294cs, but this is only 0.23cs in front of the ship. 0.23 cs is only 0.086cs in the perspective of the ship due to length contraction.

But if you solve this for the observer on the ship, the light will have traveled 1cs in front of the ship and the cloud would have traveled 0.9 cs past it. A total length of 1.9cs into the cloud would have a relative length of 4.418 to the observer in the cloud. So when the vertical light strikes the ship is it 4.418cs or 2.294cs into the cloud?

The issue I see here is that for the laser fired ahead of the ship you have to take the relativity of simultaneity into account. The reason you get different answers for the cloud frame vs the ship frame is that the two frame measure simultaneity differently. Let's call the point of the cloud where the ship is when the vertical laser returns to the ship as point A. The point in the cloud the forward laser has reached when the ship is at point A as measured by the ship is point B, and the point the laser reached when the ship is at point A as measured by the cloud as point C (different than point B).
Thus the ship reaching A and the laser reaching B are simultaneous events according to the ship, but according to the cloud frame, it is the Ship reaching point A and the laser reaching point C that are the simultaneous events.

 

[Nick J Harris]

The issue I see here is that for the laser fired ahead of the ship you have to take the relativity of simultaneity into account. The reason you get different answers for the cloud frame vs the ship frame is that the two frame measure simultaneity differently. Let's call the point of the cloud where the ship is when the vertical laser returns to the ship as point A. The point in the cloud the forward laser has reached when the ship is at point A as measured by the ship is point B, and the point the laser reached when the ship is at point A as measured by the cloud as point C (different than point B).
Thus the ship reaching A and the laser reaching B are simultaneous events according to the ship, but according to the cloud frame, it is the Ship reaching point A and the laser reaching point C that are the simultaneous events.

Thanks for the reply. Are you sure relativity of simultaneity applies here? Ignoring the horizontal laser, the vertical laser would hit the ship on the same point in the cloud, at a length of .9cs for the ship's observer and 2.0646 cs for the observer in the cloud. Due to length contraction the light hits the ship on the same point in the cloud in both frames. Would adding the horizontal laser in the initial frame be considered a separate event even if it's fired from the same time and place as the first?

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[jbriggs444]

The horizontal displacement of the vertical laser is clearly 0 cs for the ship observer. Not 0.9 cs.

If you are going to adopt a reference frame, adopt the reference frame.

 

[Janus]

Thanks for the reply. Are you sure relativity of simultaneity applies here? Ignoring the horizontal laser, the vertical laser would hit the ship on the same point in the cloud, at a length of .9cs for the ship's observer and 2.0646 cs for the observer in the cloud. Due to length contraction the light hits the ship on the same point in the cloud in both frames. Would adding the horizontal laser in the initial frame be considered a separate event even if it's fired from the same time and place as the first?

It's probably easier to see what happening with an an animation.
First from the frame of the ship. The small dot represents the laser path. it starts from the ship when the the laser reaches the edge of the cloud and the point of the cloud the laser is at when it returns is marked off by the change between green and yellow in the cloud.

In this frame the cloud is length contracted.

Now from the cloud frame:

Here the ship is length contracted and the cloud isn't. The the timing for the round trip of the laser trip is time dilated.
The laser pule returns to the ship when the lasers is at the dividing line between green and yellow. And even though the length of the green section is longer as measured by the ship, so is the length of the cloud by the same factor. This puts the dividing line at the same relative position of the cloud in both frames.

You will note however that in the ship frame the edge of the cloud is just about even with the back of the ship when the the laser pulse returns, while in the cloud frame, the back of the ship has gone well past the edge of the cloud by the time the laser returns. This is where relativity of simultaneity comes in. In the ship frame, laser returning and the rear of the ship aligning with the edge of the cloud are simultaneous events, while in the cloud frame they are not.
If you were to put clocks at the edge and dividing line, which are synchronized in the cloud frame, then both frames would agree as to what reading was on the edge clock when the laser is aligned with it, and both frame would agree as to what time the dividing line clock reads when the laser reaches it. However, the ship frame would not agree that the two clocks were in sync, but would say that the clock at the dividing line was always reading some time ahead of the edge clock.
When you add the horizontal laser, you run into this same relativity of simultaneity issue.

 

[Nick J Harris]

It's probably easier to see what happening with an an animation.
First from the frame of the ship. The small dot represents the laser path. it starts from the ship when the the laser reaches the edge of the cloud and the point of the cloud the laser is at when it returns is marked off by the change between green and yellow in the cloud.
View attachment 239235

In this frame the cloud is length contracted.

Now from the cloud frame:
View attachment 239236
Here the ship is length contracted and the cloud isn't. The the timing for the round trip of the laser trip is time dilated.
The laser pule returns to the ship when the lasers is at the dividing line between green and yellow. And even though the length of the green section is longer as measured by the ship, so is the length of the cloud by the same factor. This puts the dividing line at the same relative position of the cloud in both frames.

You will note however that in the ship frame the edge of the cloud is just about even with the back of the ship when the the laser pulse returns, while in the cloud frame, the back of the ship has gone well past the edge of the cloud by the time the laser returns. This is where relativity of simultaneity comes in. In the ship frame, laser returning and the rear of the ship aligning with the edge of the cloud are simultaneous events, while in the cloud frame they are not.
If you were to put clocks at the edge and dividing line, which are synchronized in the cloud frame, then both frames would agree as to what reading was on the edge clock when the laser is aligned with it, and both frame would agree as to what time the dividing line clock reads when the laser reaches it. However, the ship frame would not agree that the two clocks were in sync, but would say that the clock at the dividing line was always reading some time ahead of the edge clock.
When you add the horizontal laser, you run into this same relativity of simultaneity issue.

Oh I see. Great explanation, thank you!