Can Information Be Preserved on Closed Timelike Curves?

[JPBenowitz]

The arrow of time is globally derived from the global increase of entropy. In an information theory sense as a system evolves in time it becomes more random, the system can be in more possible configurations otherwise known as states. Likewise from the second law of thermodynamics a closed system cannot be reversed. It cannot evolve backwards in time to its initial conditions. However irreversibly of a system is ambiguous in the sense that if enough information is known and enough computational power is available then an observer can reverse the system close to its initial conditions but not precisely.

The increase of randomness in a system over time is deterministic but tends towards chaos in the sense that the error associated with calculating the initial conditions increases exponentially. So as the arrow of time marches forward it becomes exponentially difficult to calculate the initial conditions of any closed system to the point where it is fundamentally impossible due to a physical computational limit. Therefore, if this fundamental computational limit lies on a closed timelike curve there wouldn't be sufficient enough information to determine causality and thus can said to be preserved.

In an analogy consider an observer with an infinite amount of memory and records its entire journey. The CTC is so large that eventually its memory starts to decay and break down, conserving information but scrambling it. When it returns to its initial position it would have retained nothing from its journey.

Is this possible?

 

[Simon Bridge]

Therefore, if this fundamental computational limit lies on a closed timelike curve there wouldn't be sufficient enough information to determine causality and thus can said to be preserved.

Thus what can be said to be "preserved"?

if this fundamental computational limit lies on a closed timelike curve

How would a "computational limit" exist as a space-time event?

In an analogy consider an observer with an infinite amount of memory and records its entire journey. The CTC is so large that eventually its memory starts to decay and break down, conserving information but scrambling it. When it returns to its initial position it would have retained nothing from its journey.

Is this possible?

Is what possible?
These statements are too vague.

 

[JPBenowitz]

Causality can be preserved.

 

[Simon Bridge]

The total causality in a closed system is a constant?
How would one experimentally verify this?
How are you defining "causality"?

What about the other two questions?

 

[Demystifier]

This may help:
http://xxx.lanl.gov/abs/gr-qc/0403121 [Found.Phys.Lett. 19 (2006) 259-267]

 

[Simon Bridge]

That kinda implies that you can have coordinate-time travel without paradoxes?
(reads)

Anyway - I tried, and failed, to find where the phrase "conservation of causality" is used that is not pseudoscience. That may not mean anything. I still think OP is too vague and would benefit by refining the questions.

 

[JPBenowitz]

That kinda implies that you can have coordinate-time travel without paradoxes?
(reads)

Anyway - I tried, and failed, to find where the phrase "conservation of causality" is used that is not pseudoscience. That may not mean anything. I still think OP is too vague and would benefit by refining the questions.

Yes, I am trying to build a closed timelike curve with a chronology protection conjecture using information theory. I never used conservation of causality, I said causality is preserved meaning you can distinguish which event took place first.

 

[PeterDonis]

I said causality is preserved meaning you can distinguish which event took place first.

Isn't this false by hypothesis if the two causally connected events lie on a closed timelike curve?

 

[JPBenowitz]

Isn't this false by hypothesis if the two causally connected events lie on a closed timelike curve?

Not if it is undetectable by an observer.

 

[JPBenowitz]

http://www.scientificamerican.com/a...ental-physical-limits-of-computation&offset=6

Here is an interesting article pertaining to a physical computational limit.

And here is a paper: http://arxiv.org/pdf/quant-ph/9908043.pdf

 

[PeterDonis]

Not if it is undetectable by an observer.

Your definition of causality didn't say anything about observers, unless I'm misunderstanding it. I took "you can distinguish which event took place first" to mean "there is a definite time ordering to the pair of events", regardless of whether any observer actually detects that ordering. This is false for any pair of events that lie on a CTC, period; it doesn't matter whether any observer detects the events or not.

If by "you can distinguish..." you meant "there must exist some observer who distinguishes...", then you are saying that, by your definition, "causality" is observer-dependent. Is that the position you are trying to take?

 

[JPBenowitz]

Your definition of causality didn't say anything about observers, unless I'm misunderstanding it. I took "you can distinguish which event took place first" to mean "there is a definite time ordering to the pair of events", regardless of whether any observer actually detects that ordering. This is false for any pair of events that lie on a CTC, period; it doesn't matter whether any observer detects the events or not.

If by "you can distinguish..." you meant "there must exist some observer who distinguishes...", then you are saying that, by your definition, "causality" is observer-dependent. Is that the position you are trying to take?

Essentially yes it is. If two events lie on a CTC sufficiently far enough then no meaningful information can be communicated from either event.

 

[PeterDonis]

Essentially yes it is. If two events lie on a CTC sufficiently far enough then no meaningful information can be communicated from either event.

Hm. In that case I don't think this is really a GR question; it's more of a thermodynamics/quantum mechanics question. You do need GR to provide a background spacetime with CTCs in it (such as the Godel spacetime), but that's all, and I don't think it's enough to answer the "is it possible?" question you pose.

 

[JPBenowitz]

The rational is essentially the same as an entangled pair of particles. An observer cannot determine which system collapsed first and thus no information has been sent. I am speculating the reverse must also be true such that if no information is sent then causality cannot be distinguished. What I mean by "no information being sent" is that the channel's thermal noise reached a maximum insofar that the error associated with retrieving such information is physically impossible.

 

[JPBenowitz]

Hm. In that case I don't think this is really a GR question; it's more of a thermodynamics/quantum mechanics question. You do need GR to provide a background spacetime with CTCs in it (such as the Godel spacetime), but that's all, and I don't think it's enough to answer the "is it possible?" question you pose.

Right, however it is also necessary to incorporate the holographic principle to construct unique surfaces retaining all the information of the physical system.

 

[lukesfn]

Can something like the Poincaré recurrence theorem help in how a CTC might make sense?

If an observer where exist at a particular state of the system, and the state is repeated at a later time, obviously, the observer an be aware of the original occurrence, since it is apart of the system, and must have the same observation and memory in both occurrence.

But, I don't understand CTCs my self at all. I'm curious what determines the recurrence time in a CTC? Is this set by GR? Or is a CTC simply a separate piece of space time?

 

[JPBenowitz]

Can something like the Poincaré recurrence theorem help in how a CTC might make sense?

If an observer where exist at a particular state of the system, and the state is repeated at a later time, obviously, the observer an be aware of the original occurrence, since it is apart of the system, and must have the same observation and memory in both occurrence.

But, I don't understand CTCs my self at all. I'm curious what determines the recurrence time in a CTC? Is this set by GR? Or is a CTC simply a separate piece of space time?

The Poincaré recurrence theorem could theoretically be used if and only if information is lost (not destroyed) after every recurrence. There is actually a paper on the Poincaré Recurrence time of the universe which was calculated to be 10^10^10^1.08 years if I remember correctly. However, what I am trying to derive is that information cannot be retained or processed over such time scales due to an increase of entropy. This problem is of a fundamental physical limit of computation, namely how long can a memory space last before it decays? the PRT is apart of ergodic theory which statistically states that if a system evolves over a long enough time is "forgets" its initial state. So yes the PRT would fundamentally be apart of a CTC.

 

[lukesfn]

There is actually a paper on the Poincaré Recurrence time of the universe which was calculated to be 10^10^10^1.08 years if I remember correctly.

Interesting, but I guess that number must be very hypothetical given that the size of the universe is unknown. Also, can anybody tell me if it is actually known if the PTR can be applied to the universe? I am assuming it is unknown.

However, what I am trying to derive is that information cannot be retained or processed over such time scales due to an increase of entropy. This problem is of a fundamental physical limit of computation, namely how long can a memory space last before it decays? the PRT is apart of ergodic theory which statistically states that if a system evolves over a long enough time is "forgets" its initial state. So yes the PRT would fundamentally be apart of a CTC.

I guess it all depends on the nature of the memory. But even if you imagine a memory that is immune to decay, the size of the memory will still be limited. The time that the memory space can last would depend on its design. You could imagine an observer that could freeze itself for an arbitrary amount of time, only making extremely rare observations, that might cause recurrence of the entire system to be less frequent.

 

[Simon Bridge]

@JPBenowitz: <rereads> oh yes, and I noticed at the time too ... must be the 2am effect :( But notice how being more specific about your terms gets better responses?

 

[JPBenowitz]

Interesting, but I guess that number must be very hypothetical given that the size of the universe is unknown. Also, can anybody tell me if it is actually known if the PTR can be applied to the universe? I am assuming it is unknown.I guess it all depends on the nature of the memory. But even if you imagine a memory that is immune to decay, the size of the memory will still be limited. The time that the memory space can last would depend on its design. You could imagine an observer that could freeze itself for an arbitrary amount of time, only making extremely rare observations, that might cause recurrence of the entire system to be less frequent.

Even if the instrument only took measurements at discrete time intervals I couldn't imagine it would be enough to determine the initial conditions of the system.

All of it is speculation but they are indeed interesting questions. Also there are fundamental limits on memory space regardless of design http://arxiv.org/pdf/quant-ph/9908043.pdf look at page 6. I guess what I want to do is be bold and redefine the arrow of time in a quantum information theory perspective and thus prove the chronology protection conjecture (maybe prove is a strong too strong of a word). But if someone were to build a time machine and planned on killing there grandfather the second they went back in time there entire memory would be wiped clean and thus could never kill their grandfather.

 

[lukesfn]

Even if the instrument only took measurements at discrete time intervals I couldn't imagine it would be enough to determine the initial conditions of the system.

Of course not, but you mentioned the question of how long a memory can last without decay. How long a memory can last without decay depends on its accuracy.

I guess what I want to do is be bold and redefine the arrow of time in a quantum information theory perspective and thus prove the chronology protection conjecture (maybe prove is a strong too strong of a word).

I think the suggestion that the arrow of time can be derived from the global increase in entropy, although often stated, has some issues. I would suggest that it works the other way around. Time between events always moving in the same direction makes a decrease in entropy impossible. I'm not sure that entropy can be a global property of a system, because it is observer dependent. It is about information rather then anything physically real. A system can return to an earlier state, but it is impossible to observe that process from with in the system.

I think that to start talking about quantum information theory, or entropy, is to actually move further away from the problem, rather then closer to it. Time travel backwards in time on a CTC would allow an event to occur with no detectable cause putting cause an effect in trouble, which would then lead to issues with information.

When thinking about the possibly of time travel. I like to imagine building a computer that can send a signal to itself to itself in the past. The signal will contain a number. If the machine receives a number from the future, it will add 1 then store it. When choosing what number to send back to itself in the past, it will send the stored number. Such a machine would seem impossible to build under the Novikov self-consistency principle. I suppose there must be no way for information to survive in a time loop with a single time line and cause and effect. But how could the building of such a machine be prevented, where as building a machine that didn't add 1 could work?

 

[JPBenowitz]

Of course not, but you mentioned the question of how long a memory can last without decay. How long a memory can last without decay depends on its accuracy.I think the suggestion that the arrow of time can be derived from the global increase in entropy, although often stated, has some issues. I would suggest that it works the other way around. Time between events always moving in the same direction makes a decrease in entropy impossible. I'm not sure that entropy can be a global property of a system, because it is observer dependent. It is about information rather then anything physically real. A system can return to an earlier state, but it is impossible to observe that process from with in the system.

I think that to start talking about quantum information theory, or entropy, is to actually move further away from the problem, rather then closer to it. Time travel backwards in time on a CTC would allow an event to occur with no detectable cause putting cause an effect in trouble, which would then lead to issues with information.

When thinking about the possibly of time travel. I like to imagine building a computer that can send a signal to itself to itself in the past. The signal will contain a number. If the machine receives a number from the future, it will add 1 then store it. When choosing what number to send back to itself in the past, it will send the stored number. Such a machine would seem impossible to build under the Novikov self-consistency principle. I suppose there must be no way for information to survive in a time loop with a single time line and cause and effect. But how could the building of such a machine be prevented, where as building a machine that didn't add 1 could work?

Information is a physical manifestation though. It's simply the physical property of being distinguishable. I do not agree that pursing this from an information theory perspective is moving in the wrong direction but that is besides the point. However your analogy makes a very good comparison to causality and information. A time machine computer generates a 1 and sends it to itself in the past and adds it to 0, however the paradox is that it could never add the 1 to 0 because the 0 became a 1. The moment the machine is turned on you would have essentially added an infinite amount of numbers which requires an infinite amount of memory which does not physically exist. The question is what happens to this machine after you exceed its memory?

One way to reconcile information theory with the causality paradox is to simply state that because it is impossible to distinguish the initial conditions of the 0 and the 1 then no information is actually stored on the device.

 

[PeterDonis]

A time machine computer generates a 1 and sends it to itself in the past and adds it to 0, however the paradox is that it could never add the 1 to 0 because the 0 became a 1. The moment the machine is turned on you would have essentially added an infinite amount of numbers which requires an infinite amount of memory which does not physically exist. The question is what happens to this machine after you exceed its memory?

The Novikov Self-Consistency Principle would say that such a machine is not physically possible, because a closed timelike loop has to be self-consistent.

http://en.wikipedia.org/wiki/Novikov_self-consistency_principle

 

[JPBenowitz]

The Novikov Self-Consistency Principle would say that such a machine is not physically possible, because a closed timelike loop has to be self-consistent.

http://en.wikipedia.org/wiki/Novikov_self-consistency_principle

Right, but if the device were in a state which is physically impossible to distinguish whether or not the loop is self-consistent then why couldn't it be constructed? You cannot send a message back in time but theoretically you could send information. It would just appear to be truly random noise.

 

[PeterDonis]

Right, but if the device were in a state which is physically impossible to distinguish whether or not the loop is self-consistent then why couldn't it be constructed?

If it's impossible to tell whether or not the loop is self-consistent, then it's not self-consistent as far as the principle is concerned. See below.

You cannot send a message back in time but theoretically you could send information. It would just appear to be truly random noise.

At the microphysical level, "random noise" is just "information" that meets particular statistical tests for randomness.

For example, consider the simple closed loop you described earlier: a one-bit register that sends a signal through an adder that adds 1 to it, then feeds the result back into the same one-bit register, going around a closed timelike curve. The state of the one-bit register is not consistent: if it's 0, then it's 1, and if it's 1, then it's 0. So the circuit is not physically possible.

Now consider a huge number of such circuits, where we say that the initial values of the bits are randomly chosen. The whole ensemble is still not physically possible, because there is no possible consistent state for any of the bits. The fact that the ensemble as a whole looks like random noise to us, rather than a discernible pattern, makes no difference.

Similar remarks apply if we are unable to measure the bits. Say, for example, that to the best of our knowledge, each bit is in some quantum superposition of the "0" state and the "1" state. That still doesn't make any difference to the self-consistency of each bit: the unitary evolution of each bit, passing around the CTC through the adder, is still inconsistent. In quantum terms, this means that there is zero amplitude for a signal to pass through the circuit; the set of possible evolutions of a signal passing through the circuit will destructively interfere, for a net total amplitude of zero.

 

[JPBenowitz]

If it's impossible to tell whether or not the loop is self-consistent, then it's not self-consistent as far as the principle is concerned. See below.



At the microphysical level, "random noise" is just "information" that meets particular statistical tests for randomness.

For example, consider the simple closed loop you described earlier: a one-bit register that sends a signal through an adder that adds 1 to it, then feeds the result back into the same one-bit register, going around a closed timelike curve. The state of the one-bit register is not consistent: if it's 0, then it's 1, and if it's 1, then it's 0. So the circuit is not physically possible.

Now consider a huge number of such circuits, where we say that the initial values of the bits are randomly chosen. The whole ensemble is still not physically possible, because there is no possible consistent state for any of the bits. The fact that the ensemble as a whole looks like random noise to us, rather than a discernible pattern, makes no difference.

Similar remarks apply if we are unable to measure the bits. Say, for example, that to the best of our knowledge, each bit is in some quantum superposition of the "0" state and the "1" state. That still doesn't make any difference to the self-consistency of each bit: the unitary evolution of each bit, passing around the CTC through the adder, is still inconsistent. In quantum terms, this means that there is zero amplitude for a signal to pass through the circuit; the set of possible evolutions of a signal passing through the circuit will destructively interfere, for a net total amplitude of zero.

I suppose if the bits are not in a consistent state, i.e all 1's and 0's are indistinguishable then the device would retain no information. But this doesn't necessarily mean such a device isn't possible to construct. It means the information you sent to itself back in time could never be registered by the device. Why couldn't it dissipate into the environment? The Novikov Self-Consistency Principle doesn't exclude CTC's from be possible only that backwards time travel must be self-consistent. And there are plenty of self-consistent solutions, see "Polchinski's paradox" which also has self-consistent solutions in QM. The principle merely states it is impossible to change the past so the future becomes inconsistent before the time traveling began. The random noise present in the environment as a result of the device produces no observable change to the past and thus the timeline will remain self-consistent.

 

[lukesfn]

But this doesn't necessarily mean such a device isn't possible to construct. It means the information you sent to itself back in time could never be registered by the device.

If the device can't register the information it sends back to its self, doesn't this actually mean the device can't work, so was impossible to be constructed?

I guess that to be self consistent, if you send back random information, the random information would have to be consistent, as in always the same.

I'm curious about considering a machine similar to the CTC adder, except that weather or not to add 1 or not can be decided before activating the machine. It doesn't seem plausible that there should be a physical reason why the machine is possible if the adding 1 is disabled, but why the machine would not be possible if the adding 1 was disabled. I can't imagine a physical reason why a CTC, with the Novikov self-consistency principle, would allow one case but not the other, so I would imagine that both would be impossible. I would imagine this means that there would be severe limitation on the types of trajectories that could be taken through a CTC.

I could imagine a sub class of CTCs where it is possible for some information to enter the CTC, continue traveling forward in time, then travel backwards in time, but never be able to travel back in time or space to a point where it could meet its self, then leave the CTC, after it entered. There would be no inconsistencies possible, backwards time travel can take place, but it would be impossible to tell, or exploit, but I don't even know if this would be considered a CTC. I am really very ignorant on this topic.

 

[PeterDonis]

I suppose if the bits are not in a consistent state, i.e all 1's and 0's are indistinguishable then the device would retain no information.

In which case you have changed the specification of the problem. Either you have a device that can store bits, or you don't. If you don't, then trying to analyze whether the "device" can send signals to itself through a CTC is pointless; there are no "signals" to begin with, because you've ruled out the existence of anything that can detect them.

 

[JPBenowitz]

In which case you have changed the specification of the problem. Either you have a device that can store bits, or you don't. If you don't, then trying to analyze whether the "device" can send signals to itself through a CTC is pointless; there are no "signals" to begin with, because you've ruled out the existence of anything that can detect them.

Hypothetically if the signal were sent to the device in the past but the device is incapable of storing that information then the information would dissipate into the atmosphere as if it were being deleted at which point is possible to measure an increase in thermal noise around the device.

 

[yuiop]

I am curious about the definition of a CTC. Is it sufficient for wordline of a test particle to return to a given coordinate time to qualify as a CTC or must the particle return to a given coordinate time and location?

Does the proper time of an observer whose worldline is a CTC always continue to advance or can proper time be reversed as well?

Does that observer on a CTC always continue to observe that entropy is increasing locally and that for him the arrow of time still applies?

 

[PeterDonis]

I am curious about the definition of a CTC. Is it sufficient for wordline of a test particle to return to a given coordinate time to qualify as a CTC

My understanding is, no.

or must the particle return to a given coordinate time and location?

My understanding is, yes.

Does the proper time of an observer whose worldline is a CTC always continue to advance

On purely classical assumptions, yes. But this is really tied into your next questions:

Does that observer on a CTC always continue to observe that entropy is increasing locally and that for him the arrow of time still applies?

I don't think these are answerable purely in the context of GR. You would need a model of the possible microstates of the observer and the matter surrounding him.

However, I think it is possible to say that no "ordinary" model of the microstates of the observer and the matter surrounding him are compatible with a CTC, because that would entail the same sort of inconsistency in the local state at a given event as we have been discussing in this thread. That is, if the observer's entropy, for example, continues to increase with his proper time, then when he returns to the same event on the CTC as he was at "before" (i.e., at an earlier value of his proper time), there is no consistent way to assign him an entropy.

So I think that there is at least an apparent contradiction between our normal understanding of things like the second law of thermodynamics, the "arrow of time", etc., and a belief that CTCs are possible.

 

[PeterDonis]

Hypothetically if the signal were sent to the device in the past but the device is incapable of storing that information then the information would dissipate into the atmosphere as if it were being deleted at which point is possible to measure an increase in thermal noise around the device.

Increase relative to what? See my response to yuiop below. Any hypothetical scenario along these lines, it seems to me, would end up having to assign multiple incompatible values for physical quantities such as temperature, entropy, etc., at the same event.

 

[JPBenowitz]

Increase relative to what? See my response to yuiop below. Any hypothetical scenario along these lines, it seems to me, would end up having to assign multiple incompatible values for physical quantities such as temperature, entropy, etc., at the same event.

Keep in mind this hypothetical device would only send a signal back in time from the moment it is turned on. The hypothesis is that no meaningful information can be sent backwards in time because it requires you to send the information down a maximally noisy channel. The information can never reach the device because it is lost by virtue of traveling through the CTC. Whether or not the hypothesis is testable is not known but what I want to know is if it is mathematically possible.

 

[PeterDonis]

Keep in mind this hypothetical device would only send a signal back in time from the moment it is turned on.

I'm not sure how this helps. See below.

Whether or not the hypothesis is testable is not known but what I want to know is if it is mathematically possible.

I'm not sure this question is answerable as it stands. Or rather, I suspect that what you have proposed so far is *not* mathematically possible, but not because of any problems with transmitting signals around a CTC requiring a "noisy channel". I think it's because you don't yet have a consistent formulation of the scenario in question to start with.

You are talking about a device sending signals around a CTC, and asking whether having it do so can "change" the state of a detector in the past of the event of emission. Just formulating the question that way to begin with seems inconsistent to me, because the state of the detector (or the thermal environment, or any other place you want to stash the "information" in the signal) can't be two different things at the same event; it has only one state at a given event, and so the arrival of the signal can't "change" the state; the signal has to already be part of the initial conditions that determine what the state *is*.

I would suggest stepping back and carefully formulating a scenario that is entirely self-consistent, *including* the signal traveling around a CTC, and only assigns one state at any given event to the detector, or the thermal environment, or anything else. Doing that will, I think, help a lot in trying to answer the question you appear to be trying to ask.

 

[JPBenowitz]

I'm not sure how this helps. See below.



I'm not sure this question is answerable as it stands. Or rather, I suspect that what you have proposed so far is *not* mathematically possible, but not because of any problems with transmitting signals around a CTC requiring a "noisy channel". I think it's because you don't yet have a consistent formulation of the scenario in question to start with.

You are talking about a device sending signals around a CTC, and asking whether having it do so can "change" the state of a detector in the past of the event of emission. Just formulating the question that way to begin with seems inconsistent to me, because the state of the detector (or the thermal environment, or any other place you want to stash the "information" in the signal) can't be two different things at the same event; it has only one state at a given event, and so the arrival of the signal can't "change" the state; the signal has to already be part of the initial conditions that determine what the state *is*.

I would suggest stepping back and carefully formulating a scenario that is entirely self-consistent, *including* the signal traveling around a CTC, and only assigns one state at any given event to the detector, or the thermal environment, or anything else. Doing that will, I think, help a lot in trying to answer the question you appear to be trying to ask.

Your're right I definitely need to reformulate it to be entirely self consistent because as it stands it is not entirely self-consistent. But I do suspect that there is a self-consistent solution with the approach I am taking. Thanks for clearing a lot of things up though!