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- Johannes
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Time in the Gödel universe: reality versus global consistent order
Complying with the rule that discussions on Dr. Feser's blog articles can be started only "on articles one week (7 days, inclusive) old or more", I will share an article I've just written on the subject of Dr. Feser's article "Gödel and the unreality of time", dated May 4.
Time in the Gödel universe: reality versus global consistent order and resolution of inconsistencies
1. Time: distinguishing between absolute, real, and globally consistently ordered
When speaking about time in GR, we must distinguish clearly between the absolute, real, and globally consistently ordered qualities.
Time is not absolute in any relativistic universe, neither the actual one nor any hypothetical one. This can be very easily shown in many ways, and I showed it in two ways in this forum post of last February:
Time is real in all relativistic universes, both the actual one and any hypothetical one. This can be argued for at an abstract level, as e.g. in [1], and I will show it by studying a particular hypothetical case in the Gödel universe in section 2.
I will just avoid the term "objective" because it can be understood as either "real" or "absolute".
To define clearly what I mean by time being globally consistently ordered in a universe, or equivalently by a universe being globally consistently time-ordered, I will use this convention:
tA: proper time of an observer A, i.e. the time marked by the watch on his wrist.
tB: proper time of an observer B, i.e. the time marked by the watch on his wrist.
tB.A: time of B in the frame of reference of A (so tA is also tA.A).
tA.B: time of A in the frame of reference of B (so tB is also tB.B).
A universe is globally consistently time-ordered iff
- an advancing proper time of any observer A always corresponds, in the frame of reference of any other observer B, to an advancing time of any observer in the universe, including A and B.
or, equivalently, iff
- for any time-ordered pair of events E1 and E2 in the wordline of any observer A, such that tA(E1) < tA(E2), the corresponding times in the frame of reference of any observer B are such that tx.B(E1) < tx.B(E2), where x is any observer in the universe, including A and B.
Time is globally consistently ordered in the actual universe and almost all hypothetical ones with a few exceptions, mainly the stationary cylindrically-symmetric dust models of Van Stockum (1937), later Tipler's cylinder (1974), and Gödel (1949), which allow the existence of Closed Timelike Curves (CTCs). To note, the existence of CTCs does NOT imply that time is not real in that universe, but only that time MAY not be globally consistently ordered in that universe.
Moreover, even if one of those universes actually existed, the actual use of a CTC to travel to the past would be practically unfeasible, which is why I emphasized "MAY" in the previous paragraph. In the case of the Gödel universe, this is seen in the huge ratio between the masses of a time-travelling starship at launch time and at the end of the round trip in the case of rocket propulsion [2].
2. Demonstration of the reality of time in the Gödel universe
The following study of a hypothetical case of time travel in the Gödel universe shows that time in that universe is just as real as in ours.
Let A be a scientist who lives on planet O in the Gödel universe. (I named the planet O to match the usual notation for the origen of coordinates.)
t is proper time elapsed in O, with the notation t(n) denoting succession, i.e. t(i) < t(j) if i < j.
t(0): A conceives a child.
t(1): A's child is born and soon starts to show symptoms of a rare congenital disease.
t(2): A's child dies.
At t(2), A conceives the idea of sending to herself in t(0) an unmanned starship S carrying a flash drive with the information on the disease of her child and its cure, if said cure is found during her lifetime. She starts to design and build S.
Note: S must be unmanned because the proper time elapsed on S during the roundtrip voyage is extremely long, in the order of sqrt[pi / (G · Rho_m)], where G is the universal constant of gravitation and Rho_m is the matter density in the Gödel universe. If that density is the same as that of our universe, it can be easily seen that this formula gives exactly 10 times the time elapsed since Big Bang in our universe! (Nerds trying it: assume flatness, H(to) = 1/to, and Rho_Lambda = 2.8 Rho_m(to)).
t(3): Researchers discover that daily administration since birth of a chemical compound C, already available in t(0), to children having that disease prevents it from expressing its symptoms, with an optimal outcome if the mother already takes C during pregnancy.
t(4): A launches the fully automated S carrying the flash drive on a quasi-closed timelike curve (CTC). Strictly speaking, since S will land on O at t(0) < t(4), it goes along a past-travelling timelike curve (PTC).
From t(4), the proper time tS that elapses at S decouples from the time t that elapses in O. The mapping between the always increasing tS and the corresponding value of t evolves through these stages [3]:
tS(4) = t(4): S is launched.
tS(4) < tS < tS(5): S travels within O's horizon. The corresponding t obviously increases, though not at the same pace.
tS(5): S leaves O's horizon. Let's call t(5) the corresponding t, keeping in mind that tS(5) - tS(4) ¬= t(5) - t(4).
tS(5) < tS < tS(6): S travels outside O's horizon. The corresponding t still increases, at an ever slower pace.
tS(6): The corresponding t reaches a maximum value tmax.
tS(6) < tS < tS(7): S travels farther outside O's horizon to a max. distance from O and back. The corresponding t DECREASES.
tS(7): The corresponding t reaches a minimum value tmin < t(-1) < t(0).
tS(7) < tS < tS(8): S travels outside O's horizon. The corresponding t starts to increase, at an ever faster pace.
tS(8), corresponding to t'(-1): S reenters O's horizon. Prime (') superscripts are used for times and entities in the region containing O (actually O') from this moment onwards on the assumption that the region branches at this moment (see section 3).
tS(8) < tS < tS(9): S travels within O''s horizon. The corresponding t' obviously increases.
tS(9), corresponding to t'(0): S lands on O', delivering to A' the info on the disease and its cure.
This should be enough to dispel any doubt on the reality of time in the Gödel universe: if S arrives at O' before t'(0), A's child will live a long and healthy life, whereas if S arrives at O' after t'(2), A's child will have died during babyhood. Time could not be any more real than that.
3. Options to avoid logical inconsistencies arising from time travel
I will discuss the options to avoid logical inconsistencies arising from time travel in explicitely theistic terms because, since any environment that allows time travel to the past without involving any exotic form of mater, such as the Gödel universe or Tipler's cylinder, is stationary, it could exist only as a result of it being created by God directly in that state and with the clear divine intention of allowing time travel. Thus, if God should create an environment that allows time travel, it is just logical that He would take the actions necessary to avoid logical inconsistencies if that possibility is exercised.
A logical inconsistency arises when a time-travelling starship S enters the horizon of a causally connected region R at an R-time which corresponds also to an S-previous S-time (including the S-time range S-previous to its launch), when S either was not within R or was in another place within R.
The only academic study that I am aware of on the ways to avoid logical inconsistencies arising from time travel is by John L. Bell [4], where he argued that a branching of the history of the universe is necessary for that. Developing that position, I postulate that the branching does not need to include the whole universe, but only the region affected by the inconsistency. Since that region grows as S-time advances, I call my hypothesis "growing branching". In theistic terms, the branching of a region R at a particular R-time means that God creates a numerically distinct replica of R at the state it was at R-time, now with the time travelling starship entering its horizon.
In the example above, the region including planet O must branch at t(-1) to avoid a logical inconsistency, i.e. when S reenters its horizon.
However, it is clear that, in principle, an inconsistency can also be avoided by rollback, to which an identical consideration of restriction to the (growing) affected region applies. So, in principle an avoidance of inconsistencies by way of "growing rollback" is also possible.
Now, in an universe inhabited by intelligent beings, which in a theistic framework means that each of them has a spiritual soul directly created by God at the beginning of its existence, rollback of a region R implies that those intelligent beings who were created within R during the R-time interval which is being rolled back would cease to exist, which is not a fitting course of action for God. Rollback is even more unacceptable if in that universe, as is the case with this universe, God performs a supernatural creative work (theosis) in the souls of those intelligent beings that adhere personally to Him.
Limiting ourselves then to the "growing branching" option, there are two theoretical possibilities for the fate of the original history line, as God can in principle either keep it going on or terminate it, which in turn could happen:
- as soon as the action that will change its past is taken, i.e. t(4), or
- when the time-travelling spaceship leaves its horizon, i.e. t(5), or
- at the latest, when the corresponding t to the always-increasing tS reaches a maximum value, i.e. tmax.
While the case of branching with termination of the original history line amounts at the physical level to the case of rollback, except for the numerical identity of the entities in the new branched region, there is an important difference at the spiritual level of the intelligent beings in the terminated region, as none of them would be rolled back or cease to exist, but the living would die instantaneously and go to heaven/purgatory/hell as appropriate, and the already dead would keep existing in whatever state they were. This divine course of action, while much less unfitting for God than that of rollback, seems still seriously problematic (why would God unnecessarily prevent people from living long and fruitful lives?) and is unacceptable if God has promised that the world would end with a universally seen event, such as the Second Coming of Jesus Christ in this universe, since that would not be the case in the terminated original history line.
Therefore, of the physically possible options to avoid logical inconsistencies arising from time travel, the only philosophically (and theologically) acceptable one is growing branching with continuation of the original history line, with both history lines being supernaturally merged at their ends for a shared universally seen end-of-the-world event, if God has promised such an event in that universe.
Back to our example, A did not in fact send the info on the cure to herself, but to a replica of herself, A', that God created as a result of A sending the starship. That God would do that is logical in view of what I wrote at the beginning of this section: the Gödel universe could exist only as a result of it being created by God directly in that state and with the clear divine intention of allowing time travel. Thus, if God should create an environment that allows time travel, it is just logical that He would take the actions necessary to avoid logical inconsistencies if that possibility is exercised.
References
[1] Cord Friebe, "Time Order, Time Direction, and the Presentist's View on Spacetime", Kriterion - Journal of Philosophy, 2016.
[2] José Natario, "Optimal time travel in the Gödel universe", General Relativity and Gravitation, April 2012, Volume 44, Issue 4, pp 855–874.
[3] A very helpful visualization of these stages is in fig. 11.9 b) on p. 14 of:
Frank Grave et al, "The Gödel universe: Exact geometrical optics and analytical investigations on motion", 2009.
[4] John L. Bell, "Time and Causation in Gödel's Universe", Transcendent Philosophy 3, 2002.
Avaliability for sponsorship
This article is available for sponsorship by a flash drive manufacturer, with a message such as: "BrandX Datacarrier: reliably taking your data into the future. Or the past." Any interested manufacturer should contact the forum administrator.
Last edited by Johannes (5/12/2018 4:43 pm)
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