Novikov self-consistency principle

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The Novikov self-consistency principle, also known as the Novikov self-consistency conjecture, is a principle developed by Russian physicist Igor Dmitriyevich Novikov in the mid-1980s to solve the problem of paradoxes in time travel, which is theoretically permitted in certain solutions of general relativity (solutions containing what are known as closed timelike curves). The principle asserts that if an event exists that would give rise to a paradox, or to any "change" to the past whatsoever, then the probability of that event is zero. It would thus be impossible to create time paradoxes.

History of the principle[edit]

Physicists have long been aware that there are solutions to the theory of general relativity which contain closed timelike curves, or CTCs—see for example the Gödel metric. Novikov discussed the possibility of CTCs in books written in 1975 and 1983, offering the opinion that only self-consistent trips back in time would be permitted. In a 1990 paper by Novikov and several others, Cauchy problem in spacetimes with closed timelike curves,[1] the authors state:

The only type of causality violation that the authors would find unacceptable is that embodied in the science-fiction concept of going backward in time and killing one's younger self ("changing the past"). Some years ago one of us (Novikov10) briefly considered the possibility that CTCs might exist and argued that they cannot entail this type of causality violation: Events on a CTC are already guaranteed to be self-consistent, Novikov argued; they influence each other around a closed curve in a self-adjusted, cyclical, self-consistent way. The other authors recently have arrived at the same viewpoint.
We shall embody this viewpoint in a principle of self-consistency, which states that the only solutions to the laws of physics that can occur locally in the real Universe are those which are globally self-consistent. This principle allows one to build a local solution to the equations of physics only if that local solution can be extended to a part of a (not necessarily unique) global solution, which is well defined throughout the nonsingular regions of the spacetime.

Among the coauthors of this 1990 paper were Kip Thorne, Mike Morris, and Ulvi Yurtsever, who in 1988 had stirred up renewed interest in the subject of time travel in general relativity with their paper Wormholes, Time Machines, and the Weak Energy Condition,[2] which showed that a new general relativity solution known as a traversable wormhole could lead to closed timelike curves, and unlike previous CTC-containing solutions it did not require unrealistic conditions for the universe as a whole. After discussions with another coauthor of the 1990 paper, John Friedman, they convinced themselves that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent through the wormhole.[3]:509

In response, another physicist named Joseph Polchinski sent them a letter in which he argued that one could avoid questions of free will by considering a potentially paradoxical situation involving a billiard ball sent through a wormhole which sends it back in time. In this scenario, the ball is fired into a wormhole at an angle such that, if it continues along that path, it will exit the wormhole in the past at just the right angle to collide with its earlier self, thereby knocking it off course and preventing it from entering the wormhole in the first place. Thorne deemed this problem "Polchinski's paradox".[3]:510–511

After considering the problem, two students at Caltech (where Thorne taught), Fernando Echeverria and Gunnar Klinkhammer, were able to find a solution beginning with the original billiard ball trajectory proposed by Polchinski which managed to avoid any inconsistencies. In this situation, the billiard ball emerges from the future at a different angle than the one used to generate the paradox, and delivers its younger self a glancing blow instead of knocking it completely away from the wormhole, a blow which changes its trajectory in just the right way so that it will travel back in time with the angle required to deliver its younger self this glancing blow. Echeverria and Klinkhammer actually found that there was more than one self-consistent solution, with slightly different angles for the glancing blow in each case. Later analysis by Thorne and Robert Forward showed that for certain initial trajectories of the billiard ball, there could actually be an infinite number of self-consistent solutions.[3]:511–513

Echeverria, Klinkhammer and Thorne published a paper discussing these results in 1991;[4] in addition, they reported that they had tried to see if they could find any initial conditions for the billiard ball for which there were no self-consistent extensions, but were unable to do so. Thus it is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven.[5]:184 This only applies to initial conditions which are outside of the chronology-violating region of spacetime,[5]:187 which is bounded by a Cauchy horizon.[6] This could mean that the Novikov self-consistency principle does not actually place any constraints on systems outside of the region of spacetime where time travel is possible, only inside it.

Even if self-consistent extensions can be found for arbitrary initial conditions outside the Cauchy Horizon, the finding that there can be multiple distinct self-consistent extensions for the same initial condition—indeed, Echeverria et al. found an infinite number of consistent extensions for every initial trajectory they analyzed[5]:184—can be seen as problematic, since classically there seems to be no way to decide which extension the laws of physics will choose. To get around this difficulty, Thorne and Klinkhammer analyzed the billiard ball scenario using quantum mechanics,[3]:514–515 performing a quantum-mechanical sum over histories (path integral) using only the consistent extensions, and found that this resulted in a well-defined probability for each consistent extension. The authors of Cauchy problem in spacetimes with closed timelike curves write:

The simplest way to impose the principle of self-consistency in quantum mechanics (in a classical space-time) is by a sum-over-histories formulation in which one includes all those, and only those, histories that are self-consistent. It turns out that, at least formally (modulo such issues as the convergence of the sum), for every choice of the billiard ball's initial, nonrelativistic wave function before the Cauchy horizon, such a sum over histories produces unique, self-consistent probabilities for the outcomes of all sets of subsequent measurements. ... We suspect, more generally, that for any quantum system in a classical wormhole spacetime with a stable Cauchy horizon, the sum over all self-consistent histories will give unique, self-consistent probabilities for the outcomes of all sets of measurements that one might choose to make.

Potential implications for paradoxes[edit]

The Novikov Principle is able to circumvent most commonly cited temporal paradoxes which are often alleged to exist should time travel be possible (and are often claimed to make it impossible). A common example of the principle in action is the idea of preventing disasters from happening in the past and the potential paradoxes this may cause (notably the idea that preventing the disaster would remove the motive for the traveller to go back and prevent it and so on). The Novikov self-consistency principle states that a time traveller would not be able to do so. An example is the Titanic sinking; even if there were time travelers on the Titanic, they obviously failed to stop the ship from sinking. The Novikov Principle does not allow a time traveler to change the past in any way at all, but it does allow them to affect past events in a way that produces no inconsistencies—for example, a time traveler could rescue people from a disaster, and replace them with realistic corpses if history recorded that bodies of victims had been found. Provided that the rescuees were not known to have survived prior to the date that the time traveler stepped into the time machine (perhaps because they were taken forward in time to a later date, or because their identities were hidden), the time traveler's motivation to travel back in time and save them will be preserved. In this example, it must always have been true that the people were rescued by a time traveler and replaced with realistic corpses, and there would be no "original" history where they were actually killed, since the notion of changing the past is deemed impossible by the self-consistency principle.

Assumptions of the Novikov self-consistency principle[edit]

The Novikov consistency principle assumes certain conditions about what sort of time travel is possible. Specifically, it assumes either that there is only one timeline, or that any alternative timelines (such as those postulated by the many-worlds interpretation of quantum mechanics) are not accessible.

Given these assumptions, the constraint that time travel must not lead to inconsistent outcomes could be seen merely as a tautology, a self-evident truth that cannot possibly be false, because if you make the assumption that it is false this would lead to a logical paradox. However, the Novikov self-consistency principle is intended to go beyond just the statement that history must be consistent, making the additional nontrivial assumption that the universe obeys the same local laws of physics in situations involving time travel that it does in regions of spacetime that lack closed timelike curves. This is made clear in the above-mentioned Cauchy problem in spacetimes with closed timelike curves,[1] where the authors write:

That the principle of self-consistency is not totally tautological becomes clear when one considers the following alternative: The laws of physics might permit CTC's; and when CTC's occur, they might trigger new kinds of local physics which we have not previously met. ... The principle of self-consistency is intended to rule out such behavior. It insists that local physics is governed by the same types of physical laws as we deal with in the absence of CTC's: the laws that entail self-consistent single valuedness for the fields. In essence, the principle of self-consistency is a principle of no new physics. If one is inclined from the outset to ignore or discount the possibility of new physics, then one will regard self-consistency as a trivial principle.

Time loop logic[edit]

Time loop logic, coined by the roboticist and futurist Hans Moravec,[7] is the name of a hypothetical system of computation that exploits the Novikov self-consistency principle to compute answers much faster than possible with the standard model of computational complexity using Turing machines. In this system, a computer sends a result of a computation backwards through time and relies upon the self-consistency principle to force the sent result to be correct.

A program exploiting time loop logic can be quite simple in outline. For example, to compute one prime factor of the natural number N in polynomial time (no polynomial time factorization algorithm is known in traditional complexity theory; see integer factorization):

  1. If N is 0 or 1, abort.
  2. Allocate a communication channel c.
  3. Receive one prime factor, F, of N from the future on channel c.
  4. Test that FN, that F divides N (time complexity O(log N)), and that F is prime (polynomial time; see AKS primality test).
    1. If so, send F backwards in time on channel c.
    2. If not, send F + 1 backwards in time on channel c. Note that this results in a paradox, as the number received in step 3 above is not the same as that sent in this step.

The self-consistency principle guarantees that the sequence of events generating the paradox in the nested conditional has zero probability. Note that if N is itself prime, i.e., there is no such prime FN, then some event will prevent the execution of step 3 that receives the value F from the future. Assuming the machine executing the program itself continues to function, it can detect this failure and abort.

Physicist David Deutsch showed in 1991 that this model of computation could solve NP problems in polynomial time,[8] and Scott Aaronson later extended this result to show that the model could also be used to solve PSPACE problems in polynomial time.[9][10]

See also[edit]

References[edit]

  1. ^ a b Friedman, John; Michael Morris; Igor Novikov; Fernando Echeverria; Gunnar Klinkhammer; Kip Thorne; Ulvi Yurtsever (1990). "Cauchy problem in spacetimes with closed timelike curves". Physical Review D 42 (6): 1915. Bibcode:1990PhRvD..42.1915F. doi:10.1103/PhysRevD.42.1915. 
  2. ^ Thorne, Kip; Michael Morris; Ulvi Yurtsever (1988). "Wormholes, Time Machines, and the Weak Energy Condition". Physical Review Letters 61 (13): 1446–1449. Bibcode:1988PhRvL..61.1446M. doi:10.1103/PhysRevLett.61.1446. PMID 10038800. 
  3. ^ a b c d Thorne, Kip S. (1994). Black Holes and Time Warps. W. W. Norton. ISBN 0-393-31276-3. 
  4. ^ Echeverria, Fernando; Gunnar Klinkhammer; Kip Thorne (1991). "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory". Physical Review D 44 (4): 1077. Bibcode:1991PhRvD..44.1077E. doi:10.1103/PhysRevD.44.1077. 
  5. ^ a b c Earman, John (1995). Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press. ISBN 0-19-509591-X. 
  6. ^ Nahin, Paul J. (1999). Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction. American Institute of Physics. p. 508. ISBN 0-387-98571-9. 
  7. ^ Hans Moravec (1991). "Time Travel and Computing". Retrieved 2008-07-28. 
  8. ^ Deutsch, David (1991). "Quantum mechanics near closed timelike lines". Physical Review D 44 (10): 3197–3217. Bibcode:1991PhRvD..44.3197D. doi:10.1103/PhysRevD.44.3197. 
  9. ^ "The Limits of Quantum Computers". Scientific American: 68–69. March 2008. 
  10. ^ Aaronson, Scott; John Watrous (2009). "Closed Timelike Curves Make Quantum and Classical Computing Equivalent". Proceedings of the Royal Society A 465 (2102): 631–647. arXiv:0808.2669. Bibcode:2009RSPSA.465..631A. doi:10.1098/rspa.2008.0350. 

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