Implications of Relative Simultaneity: Carl & Shirley's Observations

[Approximation]

Let's say you have a planet rigged to explode only if two bowling balls hit respective sensors at the same time. An unfortunate observer, call him Carl, releases two bowling balls at the same time from his reference frame. Meanwhile, Shirley passes by at nearly the speed of light. From Shirley's perspective, the bowling balls do not hit the sensors at the same time. However, since the sensors and the explosives share a reference frame with Carl and his simultaneous release, the planet explodes. Does Shirley observe the non-simultaneous bowling ball impacts followed by the planet (paradoxically) exploding?

Thanks!

 

[Nugatory]

Are the two sensors in the same exact place? If not, you cannot construct a mechanism that will work the way you describe because the sensors must send signals to one another and/or the detonator and these signals must have some travel time. All observers will agree about whether the explosion happens or not, but they will disagree about the travel time of the signals. Thus, Shirley will see the sensors trigger at different times and the explosion happens; Carl will see the sensors trigger at the same time and the explosion happens.

Conversely, if the sensors are located in the exact same place as the detonator, then all observers will agree about whether the bowling balls arrived at that place at the same time and there's no problem building a device that will trigger the explosion only if that happens. However, if the balls arrive at the device at the same time, Carl and Shirley will find that the balls were released at different times and traveled at different speeds for different distances to produce this result.

 

[phinds]

Let's say you have a planet rigged to explode only if two bowling balls hit respective sensors at the same time. An unfortunate observer, call him Carl, releases two bowling balls at the same time from his reference frame. Meanwhile, Shirley passes by at nearly the speed of light. From Shirley's perspective, the bowling balls do not hit the sensors at the same time. However, since the sensors and the explosives share a reference frame with Carl and his simultaneous release, the planet explodes. Does Shirley observe the non-simultaneous bowling ball impacts followed by the planet (paradoxically) exploding?

Thanks!

Yes.

The first bowling ball hitting its sensor is an event. The second bowling ball hitting its sensor is an event. The planet exploding is an event (well, it's a bit bigger than what an event is defined to be technically, but for the sake of this discussion ...).

Shirley sees all 3 events. She disagrees w/ Carl as to when they happen but she does not disagree that they happen.

EDIT: I see Nugatory beat me to it.

 

[A.T.]

Let's say you have a planet rigged to explode only if two bowling balls hit respective sensors at the same time.

At the same time in which reference frame?

 

[Approximation]

Are the two sensors in the same exact place? If not, you cannot construct a mechanism that will work the way you describe because the sensors must send signals to one another and/or the detonator and these signals must have some travel time. All observers will agree about whether the explosion happens or not, but they will disagree about the travel time of the signals. Thus, Shirley will see the sensors trigger at different times and the explosion happens; Carl will see the sensors trigger at the same time and the explosion happens.

Conversely, if the sensors are located in the exact same place as the detonator, then all observers will agree about whether the bowling balls arrived at that place at the same time and there's no problem building a device that will trigger the explosion only if that happens. However, if the balls arrive at the device at the same time, Carl and Shirley will find that the balls were released at different times and traveled at different speeds for different distances to produce this result.

The sensors would be in slightly different locations, but I didn't think that would be significant. Now I'm questioning that. Practically, there would be a signal propagation delay difference and this error tolerance must be built into the design of the system. Since this mechanism must be time-dependent, I think Shirley would see this tolerance increase such that the signal events always occur within the tolerance sufficient to cause the explosion.

It's hard to extract a rule from that which quashes the apparent paradox. Is there a relativistic reason why a detector can't exist that produces an output only if two signals exactly coincide in time with no tolerance? If such a device exists, then Shirley's velocity would be unable to dilate that tolerance time.

 

[Approximation]

At the same time in which reference frame?

Carl's. But more significantly, the reference frame of the explosives or the AND gate that decides when they go off.

 

[A.T.]

Carl's

Then there is nothing paradoxical about Shirley observing the non-simultaneous bowling ball impacts followed by the planet exploding.

 

[Approximation]

Then there is nothing paradoxical about Shirley observing the non-simultaneous bowling ball impacts followed by the planet exploding.

Wouldn't Shirley see the system fail for no apparent reason? The ball impacts were not simultaneous thus the planet should not explode.

 

[phinds]

It's hard to extract a rule from that which quashes the apparent paradox.

There IS no paradox. If the balls and sensors are in the same frame of reference and the sensors feed a detonator half-way between them and the balls hit at the same time, then the detonator is triggered and it does not matter whether someone in a different frame of reference see the balls hit at the same time. Reread post #3

 

[phinds]

Wouldn't Shirley see the system fail for no apparent reason? The ball impacts were not simultaneous thus the planet should not explode.

If she doesn't understand frames of reference and the relativity of simultaneity then she might be puzzled, but that's her problem (and for the moment, yours, until you get this figured out), not the universe's.

 

[Orodruin]

Wouldn't Shirley see the system fail for no apparent reason? The ball impacts were not simultaneous thus the planet should not explode.

No. The system is rigged to explode if the events are simultaneous in Carl's frame. Shirley will agree that Carl's clocks had the same readings for both events and therefore there is no paradox.

Shirley will not consider Carl's clocks to be synchronised.

 

[A.T.]

Wouldn't Shirley see the system fail for no apparent reason? The ball impacts were not simultaneous...

.. in her frame, which is not relevant for the trigger. Carl's is, as you said.

 

[Nugatory]

Carl's. But more significantly, the reference frame of the explosives or the AND gate that decides when they go off.

Right - both inputs of the AND gate have to be active at the same time for the explosion to happen. Carl and Shirley will both agree that either this happened or it didn't, and there is no relativity of simultaneity issue because the AND gate is at a single point in space. Relativity of simultaneity can only affect the ordering of spatially separated points, and the AND gate isn't spatially separated from itself.

There are actually two versions of the paradox, and it's important not to confuse them. In one version, the detectors and the AND gate are close enough that the signal transmission delays don't matter; they're within the tolerances of the circuity and we can treat the detectors and the detonator as a single device at a single point in space. In this version, both observers agree that the balls arrived at that point at the same time so the explosion happened. However, if the balls were released at different locations simultaneously according to one observer, the other one will find that the release events were not simultaneous.

In the second version of the paradox, the detectors are spatially separated by enough distance that the propagation times do matter. In this version, the simultaneous arrival of the signals at the AND gate implies that they did not leave the detectors simultaneously according to at least one of the observers.

 

[Ibix]

To paraphrase all of the above, you can build a device that detects whether or not two events were simultaneous in some chosen frame. Viewed from any other frame, the device is designed to detect two events offset by some time $\Delta t=v\Delta x/c^2$.

An easy to analyse example is two detectors which turn on lights, plus a sensor in the middle that only triggers if light from both sensors reaches it at the same time. Say that in frame S the two detectors are triggered at $t=0$, $x=\pm L$. Work out the time and place of the lights turning on and of the central sensor reacting. Then Lorentz transform the three events. You'll find that the sensor is in exactly the right place to react to the two lights, even though they are triggered at different times in each frame.

You can carry out an equivalent analysis for any sensor design.

 

[Approximation]

This is fascinating stuff. I had my first lecture on this idea today so it's very new to me. I am being a little stubborn because I want to make sure I have this idea conceptually correct before I go deeper. I'm trying to comprehend just how big this idea is since most of our experience of reality seems to depend on the simultaneous occurrence of events or a particular ordering. However, the idea is a little easier to stomach upon realizing that the Lorentz transformations are linear, and, for instance, Shirley would still be able to construct consistent rules to describe Carl's planet. From all of your comments, I gather that Carl's planet destroyer isn't broken, it just operates in a consistently different manner from Shirley's perspective. Is this correct?

 

[Ibix]

More precisely, they have two different descriptions of its function. It's related to the way you might say "the train went past the tree" while I, on the train, might say that "the tree went past my window". They're two descriptions of the same thing, but they differ because we disagree about the definition of "stationary". Relativity just says that when we disagree about "stationary" we also disagree about clock rates, clock synchronisation and ruler lengths.