Stopped Clock Paradox - Analysis

The Stopped Clock Paradox establishes a scenario wherein a train with a special stop-clock is passing a station that has an identical stop-clock. The train’s stop-clock is mounted exactly in the center of a car and at the instant the two stop-clocks are aligned, flashers mounted fore and aft on that car are triggered by special sidetrack arms that touch the flashers as they pass by. The stop-clocks are of a type that will only stop when they experience simultaneous photon strikes from both sides, as from the two flashers.

The question being asked by the paradox is “which clock(s) will be stopped by the flashers?”

Visually looking at both reference frames so as to compare what each might see, we have the following;

Due to the clocks being centered at the moment the flashers are triggered, the distance from each flasher to each clock is identical and each frame of reference can know that this must be true merely because of the physical setup. And it is also understood that the fact that the flashers were moving when they flashed has no affect upon the speed of the photons leaving them. Photons travel independent of their source.

Time dilation is usually the first thought that comes to mind, but actually time dilation merely refers to what time each clock will be reading at various moments. But the paradox isn’t about what time the clocks will read when and if they stop, but rather merely IF either stops and which one if not both.

Length contraction is usually the next thought. The concern of length dilation is that the train will be shorter from the station’s perspective than it would be if the train wasn’t moving and from that of the train’s perspective. That means that the trigger arms must be set at a compensated distance “Xs-” that properly reflects that shorter length.

But it gets a little more complicated because from the train’s perspective, it is the station and distance between the trigger arms that is shorter. So the trigger arms must be special in each having two trigger fingers; one to compensate for the shorter train, and the other to compensate for the shorter station, “Xs+” that reflects that shorter station.

By such an arrangement, both frames would have to agree that the flashers would flash at the moment of alignment. And interestingly, they would argue about which trigger fingers on the trigger arms actually caused the flash. But that is a different paradox. Right now, we are only concerned to ensure that both reference frames have sufficient cause for the flashers to be simultaneously triggered.

So the length contraction issue is eliminated by having special trigger arms with multiple fingers to trigger at each length contracted distance that might be involved. The criteria is that the clocks be centered between (equal distance fore and aft) whatever triggers the flashers.

“Relativity of Simultaneity” is the third issue typically raised. And it is the issue of simultaneity, deduced by the premise that light travels at the same speed for all observers, that is actually being tested by the scenario. If a clock stops, it means that in that frame of reference, the clock perceived the flashes to trigger simultaneously. And if a clock does not stop, it means that in that frame, the flashes were perceived as not being simultaneous. And that is where the paradox comes in.

The physical arrangement is such that there is no alternative to the flashes being simultaneous and for both frames of reference.

By the defined conditions of the scenario, the theory that the speed of light must be measured to be the same speed regardless of the motion of the observer leads to a contradiction of the station insisting that the train’s clock could not be stopped and the train insisting that it is the station’s clock that could not be stopped. Yet both must insist that their own clock will stop.

But after the event, the clocks either stopped or they didn’t.

I’ll go with the obvious answer.

One clock with stop and the other will not. Which one stops depends on how you set up the triggers. You can either set the triggers so the flash is simultaneous in the train frame of reference or you can set the triggers so that the flash is simultaneous in the station frame of reference. But you can’t do both.

Basically, you are trying to place the triggers to compensate for relativity of simultaneity issues. But you can’t beat the universe.

Do I win a prize?

Yeah, you win today’s prize for not reading with comprehension.

What makes you think that those specified trigger arms cannot trigger together in BOTH frames?
Explain your assertion. The arms have a dual-finger set at BOTH possible locations for both front and rear flashers.
I thought the picture made that pretty clear.

The train has two lengths that could be used in calculation. One is the length as it appears from the station. We know that it is contracted because it moves. The second is the length as it is measured from the train, which would be the same as the contracted moving length run through a Lorenz transform.

The photons appear to arrive at the stations clock and at the trains clock from a distance determined by the trains contracted length. Seen from the station, the photons have to travel the contracted distance.

From the trains reference frame, the photons appear to arrive at both clocks from a distance determined by its uncontracted length. The distance in meters between the flashers as seen from the train differs from that distance as measured from the station. Seen from the train, the photons have to travel the uncontracted distance to hit both clocks.

Regardless of what happens to be the outcome, more time has to have elapsed on the train than on the station between the flash and the stopping of the clocks.

Wherever the flashers are mounted, the distance between them is moving and thus contracted in terms of at least one of the frames. It does not matter how fast either of the flashers moves separately, what matters is that the distance between them moves.

The speed of light is not influenced by the motion of the flashers, but the distance it has to travel is influenced by the changing location of that distance with respect to one of the frames.

If there are no triggers and the flashers happen to go off just as the midpoint of the train passes the station then we have Einstein’s ‘train and lighting strike’ example where he explains that the flashes are seen as simultaneous in one frame but not the other.

Any attempt to get around that means placing the triggers in a ‘special place’ such that simultaneity is forced in the frame where it did not previously exist. In Einstein’s example, it would like moving back from the midpoint of the train so that the lightning appears to be simultaneous.

The “calculation” doesn’t matter after the setup of the trigger arms. It is the trigger arms that guarantee that no matter what length the train happens to be, the flashes still trigger together. Both cases of potential length contraction have been accounted for.

The issue isn’t “how far did the photons have to travel”, but rather whether both photons (front and rear) had the SAME distance to travel and if they were both triggered together.

The physical arrangement requires that both flashers triggered together, no matter which length is involved. And it also requires that both photons have the same distance to travel in an “inertial frame” (being presumed by both observers as their own).

Irrelevant to the paradox issue.

Both fore and aft flashers are moving at the same speed, yielding identical length contraction. It is only any difference in those lengths that would matter.

You are focusing on what value Xs has. It doesn’t matter what Xs is as long as it is the same for both sides of both clocks within their own frame. And it is.

True, but there ARE triggers so as to remove the guess work on the part of the observers.
Read my last post on that other thread.

Yes, the Stopped Clock Paradox is NOT proposed to be the same situation as Einstein’s proposal. In Einstein’s proposal, neither observer has any means to know the truth, so they each have to presume. Then he defines the word “simultaneous” to mean what ever they presume.

He is saying that reality is only whatever you see, not what logic tells you that it must be regardless of what you see. He is raising presumption, the seed of all sin/error, to the level of God/Truth.

It is a religious act injected into Science.

But it is like Einstein’s example. You are treating simultaneity are something fixed and absolute… a fixed truth. You are trying to create a simultaneity which cannot exist. That’s why you see a paradox.

Or is it that you are treating simultaneity as only something relative to what someone wants to believe?

The paradox doesn’t care what anyone wants to believe. The clocks either stop or they don’t.

I don’t think it’s purely a question of what someone believes. It seems to be a logical consequence of SP and the parts of SP have been experimentally confirmed.

The only thing that has been “experimentally confirmed” is that considering subjective perspective is MORE accurate than presuming that all subjective perspectives are identical. Hell, I could have told you that.

Yes it is true that Newtonian presumption of constant time is false. And it is true that Lorentz relative time is MORE accurate than “constant time” presumption.

The problem is that such doesn’t make it true, merely closer.

And yes it “seems to be” a logical consequence, but only because the word “simultaneous” was redefined to mean whatever someone perceives as simultaneous without thinking (void of logic).

This paradox requires that you either set the word “simultaneous” back to its original meaning or that you deny the physical reality as presented in the scenario. And denying that physical reality not only requires that “A is A” is false, but also has consequences that makes ALL other physical experiments and their deductions null and void.

Can you find anything in that scenario that simply cannot be physically arranged? The presumption that light MUST always be perceived to be at one particular speed by all observers requires that you deny that paradox scenario as impossible to setup. But each element in the scenario seems pretty easily arranged to me. Do you see something that can’t be done?

I’m beginning to feel like I am arguing with Eugene again…
{{I really shouldn’t that that}}
… yet. [-o<

It doesn’t seem impossible to me.

The issue is also that the time in seconds required for the photon to bridge the distance is unequal in both frames.
This makes it seem difficult to decide on a simultaneous event.

Why? If the crucial point is simultaneity then we have to be thinking about progression of time, what it means for a moment in one frame to be a moment in another. A moment compared to which other points in time?

What matters for the concept of simultaneity is the difference in length as the object is seen moving and as it is from its own frame.

By length contraction and time dilation, the smallest moment of PtA change in frame A would have to appear different from the smallest moment of PtA change in B, no matter that this moment has an effect in each frame. If this is true, how can absolute time be established in either frame? How can it apply to a real situation?

Do we divide the energy of each reference frame by infinite, or do we divide an objective scale by infinite so as to build each reference frame from the ground up?

I need to see how an objective standard for infinitesimal PtA can be successfully applied to two different reference frames.

You can physically construct everything and you can place the triggers equidistant from the center of the station.

However, the configuration will only line up in the spacetime frame which contains both train and station with no relative motion.

Actually, I am somewhat baffled about how your trigger compensation could possibly work. I mean to compensate for length contraction, the triggers would have to placed less than the stopped length of the train apart. But the train in motion also sees the station as contracted and therefore the distance between the triggers (which is already reduced) appears very short.

Phyllo, your questions are too intelligent too often for me to believe that you can’t read. So what about the following portion of the OP doesn’t register with you?

The trigger arms have 4 “fingers”, two each. They are vertically separated so as to not trigger the wrong flasher. Not knowing which length is actually going to be the case for either frame, both possible train lengths (Xs+ and Xs-) have fingers ready at all times for either frame to use. No matter which length either frame experiences, there are already fingers waiting at that length. Considering the picture, where is the question?

That is the whole paradox issue itself. The physical construction guarantees that the flashers are simultaneous, yet if they really are, it is obvious that the time of flight for the photons won’t yield a coherent end result… ASSUMING the theory to be correct.

Well that makes sense to me. So let’s do that;

I take it that you agree that time is forward sequential void of reversals or random jumping around. So it seems to me that we could begin a sequential examination of each relevant instant of time passage… IN ORDER of their occurrence.

So the action starts with the triggering of the flashers, so let’s start there at “t0”;

Do you agree that picture represents (separately) each of the reference frame’s view at the instant of trigger?
The upper station’s frame has the train reduced in length and touching the darker orange, “inner” fingers. The lower train’s perspective has the station reduced in length (shown as a longer train for my drawing convenience) and touching the outer lighter orange fingers.

At that point in time, “t0”, both frames experience the fingers touching the flashers. The train would argue that the outer fingers touched and the station would argue that the inner fingers touched. We don’t really care as long as they both agree that both flashers are being triggered at that very instant of clock alignment.

See any issue with that pic?

Let’s resolve the issue of whether Relativity theory can answer this before we get into RM:AO’s answer.

Looking at your last picture where the train is aligned with the Xs+ and Xs- triggers:
You are assuming that length compression takes place relative to the center of train and station, so that midpoints line up and the triggers line up as well. Let’s say it happens that way.
In the train frame, the front of the train is touching the Xs+ trigger.
In the station frame, the front of the train is only touching the red Xs- trigger. So at a specific time, the front of the train is in two different locations in space. Yet you suggest that simultaneous triggering happens at this time. Strange.

The other way to look at length compression is that the front of the train is at the same location in both frames and the remaining length is compressed.
Let’s see what happens then:
In the train frame, the front of the train is again touching the Xs+ trigger.
In the station frame, the front of the train has also touching the orange Xs+ trigger. However, the rear of the train has already passed the rear Xs+ trigger. The front of the train has also passed both red Xs- triggers.

Each case seems to have problems.

Not exactly.
I’m stipulating that regardless of the length of the train, it is the center alignment that is to be what initiates the flashing. Without the flashing occurring due to the center alignment, the whole scenario has no meaning.

SR requires that “space bends” or distorts between two motion perspectives (length contraction). The two outer walls of the station are in different “picture frame space locations” if viewed by the train and also by the station within the same picture frame.

The picture frames that show two different perspectives are not proposed as a perspective itself (although it can be taken as a mental perspective), but rather as a comparison tool. I show both together so that we can compare each of them, not sum them as though they were a single reality.

But since you mentioned it, the proposal that things get shorter merely because they started moving leads to some interesting contradictions (and related to all of this).

The centers are aligned and there are two triggering events at that ‘time’… one at the orange Xs+ and the other at the red Xs-… One for the benefit of the station clock and one for the train clock.
You don’t think that’s some kind of fiddling?

I think that the problems of triggering actually demonstrate the relativity of simultaneity.

It is one event SEEN differently by different observers.

If I show you a picture of what a clear sighted person sees as an object and in the same picture frame include what a visually impaired person sees as that same object, are there two objects? Or merely one object seen differently by different people?

The triggering is a single event, but it appears to occur at different locations by each observer.

Read this and read about relativity of simultaneity. Notice how similar the concepts are. Phrases like ‘SEEN differently by different observers’ and ‘appears to occur at different locations’ are leading to certain logical conclusions.