Special relativity timebomb on distant planet 'paradox'?

[Matternot]

I thought of this 'paradox' which is somewhat similar to the twin paradox but can't be explained by a lack of symmetry etc. It is very similar to many paradoxes I have heard before of which the resolution is known (which is why I am mostly sure this can be resolved)

Bob is looking through his telescope and sees a distant stationary planet with a time bomb on it. In order to stop the time bomb from detonating, a clock on the planet must be destroyed within 9 years. Bob, however notes that the planet is 10 light years away and, even if he fired a laser to destroy it, it'd never reach the clock in time. Alice, meanwhile is flying past Earth at that exact moment very close to the speed of light. She is on her way to destroy the clock, regardless of what Bob thinks. She only measures the planet as being 3 light years away, but also sees the clock that will detonate the bomb tick slower and slower. Just over 3 years later she is approaching the planet. The time on the clock, she views as far less that 3 years. She sticks her sledge hammer out the window and does a drive-by smashing of the clock. Years before Bob predicted she could ever make it there. If Bob were to see her smash the clock, it would be as though she traveled far faster than the speed of light. So how does she do it?

In thinking of the resolution, it is clear that the time on the clock on the planet is mostly meaningless to Alice. It bears no resemblance of what the time is in the frame of reference in which Earth is at rest. Which, I thought would mean that the bomb would detonate regardless of what time Alice sees. This, however might introduce issues of causality. If the clock reading 9 years causes the bomb to detonate, the clock should read 9 years to all observers before the bomb detonates. The way I think I can justify it is thinking that not all observers agree on when the clock was smashed... but I can't seem to fully justify/satisfy myself with an answer.

What do you guys think? (I'm sure there are isomorphic paradoxes out there somewhere)

Thanks,

Stephen

 

[phinds]

The clock on the planet doesn't care how long Alice thinks it takes her to get there, it only cares what IT thinks and it thinks that it takes Alice probably 15 years to get there (I didn't do the math and forgive my saying a clock can "think") but of course it won't be there when she arrives because it will have blown up within 9 years.

 

[Dale]

Alice, meanwhile is flying past Earth at that exact moment very close to the speed of light

Think about what "at that exact moment" means to Alice and Bob. In particular, what is happening at the planet "at that exact moment".

 

[phyzguy]

You are wrong in thinking that Alice sees the clock on the distant planet run more slowly than her clock. Because she is approaching the planet, and because of the relativistic Doppler shift, she actually sees the clock on the distant planet run faster than hers. It is only when she takes the finite speed of light into account that she would conclude that the distant planet clock runs more slowly than hers. So she would see the distant planet clock read 9 years and the planet explode long before she arrives.

Drawing a space-time diagram of the situation with the light signals propagating from the distant planet toward the Earth will help you see this.

 

[Matternot]

The clock on the planet doesn't care how long Alice thinks it takes her to get there, it only cares what IT thinks and it thinks that it takes Alice probably 15 years to get there (I didn't do the math and forgive my saying a clock can "think") but of course it won't be there when she arrives because it will have blown up within 9 years.

It is clear that the bomb detonates from considering the simplest inertial framp. An alien on the planet, he sees Alice arrive too late. (As does the clock). The problem I start having is:

Consider the 2 events.

The clock striking 9 years

The bomb detonating.

Since the clock striking 9 years causes the detonation, Would the clock not have to strike 9 years before the bomb detonating in all reference frames (including Alice's) in the same way you can't see a bullet traveling through the air before a gun is fired?

 

[Matternot]

You are wrong in thinking that Alice sees the clock on the distant planet run more slowly than her clock. Because she is approaching the planet, and because of the relativistic Doppler shift, she actually sees the clock on the distant planet run faster than hers. It is only when she takes the finite speed of light into account that she would conclude that the distant planet clock runs more slowly than hers. So she would see the distant planet clock read 9 years and the planet explode long before she arrives.

Drawing a space-time diagram of the situation with the light signals propagating from the distant planet toward the Earth will help you see this.

This post, I believe, resolves the paradox. The ticks of the clock are, in fact more frequent to Alice. From that, she knows that the clock has already struck 9 years before she even arrives.

Thanks, Phyzguy

 

[phinds]

Since the clock striking 9 years causes the detonation, Would the clock not have to strike 9 years before the bomb detonating in all reference frames (including Alice's) in the same way you can't see a bullet traveling through the air before a gun is fired?

No, there is a reference frame in which the clock and the bomb are stationary together and thus see the same time. They do not care what time others see.

 

[Matternot]

No, there is a reference frame in which the clock and the bomb are stationary together and thus see the same time. They do not care what time others see.

Sure, they do not care what time other people see on their own clocks... but they do care what other see on the clock they are using. I now see that the paradox was wrong in construct. The line

The time on the clock, she views as far less that 3 years.

was incorrect. "the clock" is regarding the one attached to the bomb. Not the one in her space ship. This clock runs faster than the one in her space ship, and will strike 9 before she arrives. Else, causality is broken.

 

[georgir]

Check Dale's comment again, there is the key. You define a "start" event of sorts, when Alice and Bob are at the same place (Earth) even though they have different velocities. Their "surface of simultaneity (which is actually a space, and is drawn as a line on charts... don't ask me why it's called that)" or "now" at that particular event is different. Distant events that are simultaneous with the "start" for Bob are not the same as the events that are simultaneous with the "start" for Alice.
So, you are right that Alice will see the distant clock both seem closer and run slower. But still, she sees she can not make it there in time, because in her "now" at the "start", the clock already reads much closer to doom hour.

 

[Dale]

Since the clock striking 9 years causes the detonation, Would the clock not have to strike 9 years before the bomb detonating in all reference frames (including Alice's)

Yes. Special relativity preserves all causal relationships. So if A causes B then A preceeds B in all inertial frames.

 

[PeroK]

Sure, they do not care what time other people see on their own clocks... but they do care what other see on the clock they are using. I now see that the paradox was wrong in construct. The line
was incorrect. "the clock" is regarding the one attached to the bomb. Not the one in her space ship. This clock runs faster than the one in her space ship, and will strike 9 before she arrives. Else, causality is broken.

Actually, what you are missing is the relativity of simultaneity: in this case in the shape of the "leading clocks lag" rule. In Alice's frame of reference, the clock on the distant planet is several years ahead of Bob's Earth clock. It will run slower, but still reach 9 years before she gets to it.

You may be better advised to learn about the relativity of simultaneity properly, as there is no end to the supposed paradoxes you can invent without it.

 

[Matternot]

Check Dale's comment again, there is the key. You define a "start" event of sorts, when Alice and Bob are at the same place (Earth) even though they have different velocities. Their "surface of simultaneity (which is actually a space, and is drawn as a line on charts... don't ask me why it's called that)" or "now" at that particular event is different. Distant events that are simultaneous with the "start" for Bob are not the same as the events that are simultaneous with the "start" for Alice.
So, you are right that Alice will see the distant clock both seem closer and run slower. But still, she sees she can not make it there in time, because in her "now" at the "start", the clock already reads much closer to doom hour.

Of course! How foolish of me! This makes plenty of sense! I mostly used that to align co-ordinates. But when considering the time Alice reads on the planet at τ=0 at the time she thinks she passes Earth, it isn't zero. I have questioned my instinct that the clock will run slow after considering the clock's "ticks" would be blue shifted but have concluded that this would not supersede the effects of time dilation. (The clock will run slowly according to Alice after all)

Thanks,

Stephen

 

[phyzguy]

Of course! How foolish of me! This makes plenty of sense! I mostly used that to align co-ordinates. But when considering the time Alice reads on the planet at τ=0 at the time she thinks she passes Earth, it isn't zero. I have questioned my instinct that the clock will run slow after considering the clock's "ticks" would be blue shifted but have concluded that this would not supersede the effects of time dilation. (The clock will run slowly according to Alice after all)

Thanks,

Stephen

You're still getting it wrong. As Alice passes Earth, she can synchronize her clocks with Bob's so that they both agree that the moment Alice passes Earth is t=0. Since this is a single event (same time, same space point), there is no relativity of simultaneity. So this can't resolve the paradox. Alice will see the clocks on the distant planet running faster than the clock on board her ship. Draw a space-time diagram and you will see this.

 

[Matternot]

You're still getting it wrong. As Alice passes Earth, she can synchronize her clocks with Bob's so that they both agree that the moment Alice passes Earth is t=0. Since this is a single event (same time, same space point), there is no relativity of simultaneity. So this can't resolve the paradox. Alice will see the clocks on the distant planet running faster than the clock on board her ship. Draw a space-time diagram and you will see this.

But despite Bob and Alice's clock both reading 0 at the instant she passes Earth, Bob sees this as simultaneous with the clock on the planet reading 0. Alice does not. Same time, but different point, (There is relativity of simultaneity). Since the event of Alice passing Earth and the planet happens in the same position to her, she is running on proper time τ (which records the time between the 2 events faster than any other). I have drawn a diagram and will try to upload it.

 

[phyzguy]

But despite Bob and Alice's clock both reading 0 at the instant she passes Earth, Bob sees this as simultaneous with the clock on the planet reading 0. Alice does not. Same time, but different point, (There is relativity of simultaneity). Since the event of Alice passing Earth and the planet happens in the same position to her, she is running on proper time τ (which records the time between the 2 events faster than any other). I have drawn a diagram and will try to upload it.

Imagine the distant planet sending out pulses, one per year. The "t=-10" pulse arrives at Earth just as Alice passes Bob. They will both receive the "t=-10" pulse at the same time, since they are at the same location. So all three clocks are synchronized at this space-time event. If Alice and Bob each had telescopes trained on the clock on the distant planet, they would each see that clock reading -10 years at the instant that Alice passes Bob. Since Bob knows the planet is 10 light years away, and they are not in relative motion, he knows that at the instant he receives the "t=-10" pulse, that a clock on the planet would read t=0. So Alice sets her clock to t=0 at the instant she passes Bob, and all three clocks are synchronized. As Alice moves toward the distant planet at a speed close to that of light, she is intercepting the pulses and so seeing the clock on the distant planet run faster than her own. All observers will agree that the instant the clock on the distant planet reads 9, the planet explodes.

Attached is a space-time diagram. Note that Alice receives more than three pulses during one of her years, so she sees the distant clock running more than six times faster than her own (I've only drawn the eevn numbered pulses). Notice, also that she sees the planet explode (at the same time that the planet's clock reads 9) before she arrives.

(Note that I made a few edits to make the text agree with the diagram.)

 

[Matternot]

Imagine the distant planet sending out pulses, one per year. The "t=0" pulse arrives at Earth just as Alice passes Bob. They will both receive the "t=0" pulse at the same time, since they are at the same location. So all three clocks are synchronized at this space-time event. If Alice and Bob each had telescopes trained on the clock on the distant planet, they would each see that clock reading zero at the instant that Alice passes Bob. As Alice moves toward the distant planet at a speed close to that of light, she is intercepting the pulses and so seeing the clock on the distant planet run faster than her own. All observers will agree that the instant the clock on the distant planet reads 9, the planet explodes.

If you draw the spacetime diagram from the perspective of Alice, the result becomes more apparent. The frequency of photon release would be lower for her. She takes into account the distance the light had to travel to reach her in order to calculate when the photon was released. It is obvious if you think of the clock as being a light clock. It also become apparent by drawing lines of simultaneity on your diagram

 

[PeroK]

If you draw the spacetime diagram from the perspective of Alice, the result becomes more apparent. The frequency of photon release would be lower for her. She takes into account the distance the light had to travel to reach her in order to calculate when the photon was released. It is obvious if you think of the clock as being a light clock. It also become apparent by drawing lines of simultaneity on your diagram

All this talk of "what someone sees" and "photons traveling through space" may obscure the fundamental point that it is the actual measurements of time itself that are the issue. You can consider all these problems under the more realistic assumption that no one can "see" anyone else's "clock" at such distances. To do this, you need to consider the concept of time and space in different reference frames (related in this case by the Lorentz Transformation).

Considering the time light signals must travel from an event to an observer is valid, of course, but it significantly complicates the resolution of these problems and may obscure the key issue regarding the nature of time itself.

In the long run, it is also something of a dead end, as eventually you must resort to coordinate transformations between reference frames in order to analyse natural phenomena.

 

[phyzguy]

If you draw the spacetime diagram from the perspective of Alice, the result becomes more apparent. The frequency of photon release would be lower for her. She takes into account the distance the light had to travel to reach her in order to calculate when the photon was released. It is obvious if you think of the clock as being a light clock. It also become apparent by drawing lines of simultaneity on your diagram

Let me ask you this. If Alice has a telescope focused on the distant planet during her journey, do you agree that she sees the clock on the distant planet running faster than hers (which I am confident is the correct answer), or do you still think she sees the clock on the distant planet running slower than hers?

Of course, as PeroK points out, it is difficult to imagine actually seeing a clock 10 light-years away, but it is easy to imagine Alice watching the distant planet orbiting its sun and using that as a clock.

 

[Matternot]

Let me ask you this. If Alice has a telescope focused on the distant planet during her journey, do you agree that she sees the clock on the distant planet running faster than hers (which I am confident is the correct answer), or do you still think she sees the clock on the distant planet running slower than hers?

Of course, as PeroK points out, it is difficult to imagine actually seeing a clock 10 light-years away, but it is easy to imagine Alice watching the distant planet orbiting its sun and using that as a clock.

Perok's point is not that it's difficult to use movement of photons, but that by imagining the movement of photons instead on differences in the time observers measure, you can distract yourself from the fundamental nature of time.

If Alice focused on the planet she would see the clock on the planet run slower (which I am confident is the correct answer).

Here's the calculation behind it:

Δt is the time between years according to Alice
Δx is the distance between the clock ticks (of years) according to alice
Δτ is the distance between the clock ticks according to an alien on the planet
β is the speed of the planet as a fraction of c
for simplicity let c=1 so β<1

The spatial difference between the two clock ticks according to the alien is 0

consider the invariant interval (which all observers agree on)

Δs² = Δτ²=Δt²-Δx²

since

Δx= (-βt)

⇒Δτ²=Δt²-β²Δt² =(1-β²)Δt²

hence the time between years Alice sees

Δt=γΔτ>Δτ

where γ = 1/√1-β²

Think not about photons moving through space, but instead invariant quantities, else you may miss the fundamental nature of which time is measured differently

 

[phyzguy]

You're still missing the point that since Alice is approaching the planet at a speed near c, successive light pulses have less distance to travel and so arrive sooner. This means that she will see the clock on the distant planet run faster than her own. This is the essence of the relativistic Doppler shift, and what I clearly showed in my space-time diagram. However, I am getting tired of arguing about it. You might try reading this Wikipedia link about the Twin Paradox. To quote Wikipedia about what the returning twin sees as she approaches Earth at a speed near that of light,

"Then the ship turns back toward home. The clock of the staying twin shows "1 year after launch" in the screen of the ship, and during the 3 years of the trip back it increases up to "10 years after launch", so the clock in the screen seems to be advancing 3 times faster than usual."

 

[phinds]

You're still missing the point ... However, I am getting tired of arguing about it. You might try reading this Wikipedia link about the Twin Paradox.

+1 on that

If Alice focused on the planet she would see the clock on the planet run slower (which I am confident is the correct answer).

I admire your persistence but not your grasp of the situation.

 

[PeterDonis]

If Alice focused on the planet she would see the clock on the planet run slower (which I am confident is the correct answer).

Your confidence is misplaced. As phyzguy has pointed out twice now, Alice actually sees the clock on the planet running faster.

Here's the calculation behind it

Your calculation is fine as far as it goes, but it is not a calculation of what Alice actually sees through her telescope. (To calculate that you would use the relativistic Doppler formula for a source approaching the observer.) It is a calculation of the coordinate time interval in Alice's frame between the events where she is co-located with Earth and with the clock on the planet. But that calculation in itself does not tell you whether Alice reaches the clock before the bomb explodes; to know that you have to know what the clock reads at the event simultaneous, in Alice's frame, with her being co-located with Earth. To do that you need to Lorentz transform the coordinates of that event in Alice's frame (which are known) into the Earth/planet frame; the resulting time coordinate gives the clock reading on the planet's clock.

 

[georgir]

You lot... why you insist on confusing people? What Alice "actually sees" is cool and all, but the question is not about that.
She can calculate to compensate for the light travel time and Doppler shift and know the rate of the doom clock in her frame - and indeed it will be running slower, as should any process that is not at rest. That's all the OP was saying.
What she "actually sees" is a nice addition to that, as long as the two things don't get confused.

 

[Matternot]

There is plenty of doubt about the theory here. So let's ignore theory for a moment and instead think of example.

Muons form in the upper atmosphere and travel toward the Earth. They typically have a lifetime of about 30ns at rest ⇒ they should never reach the surface of the Earth... yet they do once their lifetime is lengthened by (special relativistic) time dilation. Imagine if one of these muons was used as the clock to detonate the bomb?

https://en.m.wikipedia.org/wiki/Time_dilation_of_moving_particles Look for "Experiments" sub-title

 

[Matternot]

I think I understand the disagreement. I think the disagreement occurs when thinking about measurement vs 'observation' and how she records time. I mostly think about an array of clocks in space that are synchronised by setting the time to a distance d/c from a reference clock and emitting a spherical light wave from that clock to activate the clocks. I believe using those clocks, Throughout space, Alice would measure the time on the clock moving towards her as running slowly. With regards to what she actually sees, it may be possible for her to "see" the clock running fast if she uses the photons hitting her ship at particular times, but this seems somewhat unimportant (in terms of the nature of time) compared to what she would measure on the synchronised clocks.

 

[Stephanus]

Let me ask you this. If Alice has a telescope focused on the distant planet during her journey, do you agree that she sees the clock on the distant planet running faster than hers (which I am confident is the correct answer), or do you still think she sees the clock on the distant planet running slower than hers?...

I think the telescope or whatever it is and no matter wher it focuses, the only matter is the light that COMES to your eyes or the telescope.
I think you might think that the sophisticated telescope can reach the distant planet, before the light from the distant planet reach your naked eye?
You could exchange the telescope with a very big clock on the distant planet, or you put a transmitter at the distant planet and radio EM receiver in Alice rocket. All will do the same.

 

[A.T.]

With regards to what she actually sees, it may be possible for her to "see" the clock running fast if she uses the photons hitting her ship at particular times, but this seems somewhat unimportant (in terms of the nature of time) compared to what she would measure on the synchronised clocks.

What she sees (compared to her own clock) is a physical fact. What she measures to be happening far away depends on the simultaneity convention (how the distant clock were synchronized).

 

[Vitro]

I think I understand the disagreement. I think the disagreement occurs when thinking about measurement vs 'observation' and how she records time. I mostly think about an array of clocks in space that are synchronised by setting the time to a distance d/c from a reference clock and emitting a spherical light wave from that clock to activate the clocks. I believe using those clocks, Throughout space, Alice would measure the time on the clock moving towards her as running slowly. With regards to what she actually sees, it may be possible for her to "see" the clock running fast if she uses the photons hitting her ship at particular times, but this seems somewhat unimportant (in terms of the nature of time) compared to what she would measure on the synchronised clocks.

Right, the problem comes from using the word "see" which by default would really be interpreted as what someone actually sees. Plus, the way you set up the scenario in your OP mentioning looking through a telescope that's what you actually meant. But then you said Alice sees the planet clock running slow which in that context is wrong.

Relativity is quite subtle and it's easy to confuse oneself and others by using imprecise wording or not sufficiently qualifying words which could be misinterpreted.

 

[PeterDonis]

There is plenty of doubt about the theory here.

What theory do you think there is doubt about?

 

[PeterDonis]

What Alice "actually sees" is cool and all, but the question is not about that.

I don't think that's at all clear; for example, this statement...

If Alice focused on the planet she would see the clock on the planet run slower (which I am confident is the correct answer).

...indicates that the poster is talking about what Alice actually sees through the telescope (and I took it that way in my response--and it doesn't seem like I'm the only one based on other posts).

 

[PeterDonis]

Imagine if one of these muons was used as the clock to detonate the bomb?

Sure, go ahead and imagine it. What do you come up with? I see at least one key difference in this case: the muons are moving in the Earth's frame, so they are more like Alice than the clock on the planet in your original scenario. So you'll need to clarify what scenario you are envisioning here.

 

[phyzguy]

It's very clear that the OP was in fact talking about what Alice actually sees. In the OP, the statement was:

She is on her way to destroy the clock, regardless of what Bob thinks. She only measures the planet as being 3 light years away, but also sees the clock that will detonate the bomb tick slower and slower.

Note that "the clock that will detonate the bomb" is on the distant planet. It is the incorrect statement that she sees this clock tick slowly that I have been attempting to correct.

 

[Matternot]

Whether

Sure, go ahead and imagine it. What do you come up with? I see at least one key difference in this case: the muons are moving in the Earth's frame, so they are more like Alice than the clock on the planet in your original scenario. So you'll need to clarify what scenario you are envisioning here.

The muons are traveling towards the Earth at a speed very close to the speed of light, in the same way the planet is moving towards alice at very close to the speed of light. The lifetime of the muons are extended according to people on earth. But I am certain where the confusion is and my next post should explain.

 

[Matternot]

Vitro is correct

The problem of whether she measures the clock run faster or slower depends on how she is measuring. When writing the original problem statement, the subtleties between the words see/view and measure were not immediately apparent to me. Typically when considering the time observers record events in special relativity, clocks are synchronised according to the Einstein synchronisation technique in which all observers moving at a particular velocity will agree on when an event occurs. Upon configuring these clocks and using them to measure the time between ticks, the ticks of the clock moving towards her will appear slower (τγ). There is a crucial subtlety in her looking at the clock, in particular whether or not she takes Doppler shift and the distance the light has to travel into account. If she instead measures the time between events from when the photons from the event reach her, then she will view the clock as running at a speed τ/(1+β). In setting up the problem I had assumed the Einstein synchronisation technique, and yet used words like "view the clock" due to this lack of appreciation of the subtleties of the words. I should have instead used "measured the time the clock had proceeded". This is where the crux of the disagreements appeared.

If using the Einstein synchronisation technique, the paradox is resolved as suggested by Dale, Perok, geogir and I, with Relativity of simultaneity (that the clock is far further proceeded as measured by Alice than it is bob)

If using the the time of the event as the time a photon from the event would reach you, the paradox can be resolved as suggested by Phyzguy and Phinds by the clock ticking far faster than Alice due to Doppler shift effects

Both synchronisation techniques are valid for resolving this paradox, however the Einstein synchronisation technique reveals the nature of time itself, that even if Alice is savvy and accounted for the Doppler effect. If she couldn't see the clock ticking at all and no photons passed between them, she still would not be able to make it in time, despite her being able to reach the planet in 3 years from the time she passes Earth.

The idea that muon lifetime is extended as muons move towards the Earth at very close to the speed of light is particularly interesting given you might start thinking about the lifetime being extended, yet the frequency at which you record muon decay increases if Doppler shift is ignored. In this case, because we cannot see the clock, we decide to account for the Doppler shift and calculate a lower frequency after taking the Doppler shift. Here the convention used is that of Einstein synchronisation, as opposed to the convention suggested by "seeing the events," otherwise we might even record a shorter lifetime (which makes it now harder and more confusing to think about how they can reach the Earth before they decay).

 

[PeterDonis]

the frequency at which you record muon decay increases if Doppler shift is ignored

I'm not sure what you mean by this. If you consider a series of clocks along the path of the muons, all at rest relative to the Earth and Einstein synchronized with Earth clocks, then these clocks will measure the frequency of muon decay to decrease (compared to clocks at rest relative to the muons). This has nothing to do with Doppler shift; it's just the clocks recording their times and the fraction of muons decayed as the muons pass by.

Here the convention used is that of Einstein synchronisation, as opposed to the convention suggested by "seeing the events," otherwise we might even record a shorter lifetime (which makes it now harder and more confusing to think about how they can reach the Earth before they decay).

It seems like you are thinking here of a single observer on Earth, watching the muons come towards him. This observer will in fact see (actually observe through his telescope, or muon viewer, or whatever) the muons to be decaying at a greater rate than if they were at rest relative to him--because what he sees includes the effect of the Doppler shift. But he will also see the start of the muons' flight time delayed, compared to a clock located in the upper atmosphere where the muons actually start their flight (because the light emitted by the muons when they start their flight is only traveling a little faster than the muons themselves, so it only reaches the Earth observer a little before the muons do). And this time delay more than compensates for the faster rate at which he sees the muons decay, so that there are still muons left when they reach him.

(As an instructive exercise, you might want to figure out what an observer in the upper atmosphere, at rest relative to the Earth, will see if he watches the muons flying away from him through a telescope.)