Proposed Model of the Origin of Tensed Time and Entropy

Models of time come in two basic variants. The tenseless theory, given by modern cosmological theories such as General Relativity and Quantum Field Theory, holds that time is essentially another spatial dimension (with a negative curvature parameter, as opposed to positive curvature paramaters for normal space dimensions, in the anti-DeSitter solution to Relativity, which is basically the mainstream model of how the universe we observe developed over time). Therefore, if time is "tenseless", then the cosmos taken as a whole is in a sense a timeless mathematical object. On the other hand, less a part of physics, and a more a part of philosophy, the tensed theory of time holds that time is not another spatial dimension but is "real" in some sense, and thus the models of physics that given time as a type of dimension are just that, models, but are not accounting for the true nature of time. Here I want to give a possible account of how a tensed ("real") sense of time can emerge. First, I will digress to talk about Cosmological Natural Selection, for that is what I was thinking about when I came up with this particular accounting of the origin of tensed time (and, congruently with it, also entropy).

Lee Smolin published a book, Life of the Cosmos, which I read as a teenager and have recently returned to thinking about. Essentially he accounts for the constants of nature by a type of natural selection. To be very brief, in Smolin's model, black holes create new, "baby" universes, outside of our own. They "inherit" features such as for example the ratio of the electromagnetic force to the gravitational force, or, say, Boltzmann's Constant, etc., from their "parent" universe, with some variations (call them "mutations"). Over time, universes with lots of black holes dominate the "fitness landscape" because the ones with the most black holes reproduce the most. Thus, in terms of probability, one is likely to find oneself in a universe fine-tuned to produce black holes. A prediction of this is that neutron stars must be less than 2 solar masses. This has since been confirmed in the years since the publication of the book. So the theory, while of course not proven, has withstood scrutiny and falsifiability so far. 

One issue with the theory has to do with information. If objects that fall into black holes "leave behind" their quantum information at the boundary (event horizon) of the black hole, then how is it that any "baby universe" produced by the black hole can "inherit" features from its "parent" universe? This was and remains a challenge, but I have a possible solution, which will then lead me into my main topic about an origin for a tensed model of time (and entropy while we are at it).

The nice thing about a part of General Relativity know as Weyl Curvature, which measures how objects are distorted or bent out of shape in a gravitational field (similar to how the oceans are affected by the moon's gravity that produce tides), is that Weyl Curvature is conformally invariant. This basically means that it holds on whatever scale we are talking about, and in whatever reference frame. To make a silly example: If I have a large basketball with very thin rubber, on, say, Jupiter, this basketball may be however slightly distorted in its shape due to Jupiter's gravity. Now, say I am on a NASA station on Jupiter, and I then put a second basketball into a rocket and launch it into orbit. The basketball in orbit (or, technically, free fall) around Jupiter will on the one hand be affected differently or distorted differently than the basketball at my ground station, but, if I know how the basketball at the ground station is distorted, and I know the velocity of the rocket, I can work out some math to tell me how the basketball in the rocket is distorted. This is conformal invariance. Another even more simple example is if I have a map of New Bedford that lies flat on my table, I can figure out how to get from my home to City Hall, just as well as if my map of New Bedford is on a globe. Either way I can get to City Hall. Conformal invariance means that if I change my reference frame, or how I create a map, I can still preserve some properties between these transformations. Thus, going back to black holes, a child universe produced by a black hole could "inherit" properties from a parent universe due to the conformal invariance of the Weyl tensor. So, even if much or most "information" about the parent universe is lost, a suprising amount could likely still be transfered via how space is curved or distorted in the vicinity of the black hole as measured by the Weyl Tensor. I digress to this subject to point out that I think Smolin's Cosmological Natural Selection may well hold up under scrutiny and the biggest critique against it, the issue of information transfer, may not be so big after all.

As an instructional aide, here is a smart-sounding fellow in a British accent explaining Cosmological Natural Selection:

Well now, this then raises the question of if the universe(es) or cosmos at the grandest scale is a Darwinian process of "successful" ones being the ones that "reproduce" the most (by creating black holes, specifically from stellar collapse of stars that are spinning, known as Kerr Black Holes), then does this process go on infinitely to the past, or must have it begun at some point? Technically, if as I believe, the "clock" of entropy is "reset" in each "baby universe", due to how the geometry of space looks like coming out of the black hole in the new "baby universe", then in a teneseless model of time, this process of cosmological selection can indeed extend into the past infinity. But if we have a tensed model of time, this cannot be so, for then you have the logical paradox of needing an infinite succession of causes and events to arrive at any one point in time. Smolin himself has sided with the tensed model camp so (one presumes) he would also see the need for the cosmological natural selection (CNS) process to have begun at some finite point in the past. Or if not then one must find a way to explain how anything happens at all, given that for anything to happen you need an infinite succession of causation in a tensed theory of time that does not have a starting point.

So now we arrive at the age-old conundrum. This (again) is not so much a conundrum a tenesless theory of time. You can simply posit (say) a Big Bang, and say that such a point is the t = 0 point and you cannot have points of time earlier than that anymore than you can have points south of the south pole. There really is not a problem because the spacetime metric is seen as finally a "timeless" geometrical object (for example the Hartle-Hawking No Boundary Proposal (1983) ). But if one insists upon a tensed theory of time the situation becomes murkier indeed. What I propose here I can certainly not "prove" in a Popperian falsifiability type of way, but it seems to be a logical starting point.

When we talk about "the origin of (tensed) time" we already are in an odd situation in terms of the language - the word "origin" pre-supposes time, so to speak of an "origin" of time we are in a meta-narrative or self-referential situation of asking for what is the "origin" of the notion of "origins". It is tricky, doubtless. But here goes anyway.

Let us say "in the beginning" there was an infinite expanse of a undefined (maybe infinite) number of dimensions of space anterior to the process of cosmological natural selection. The only "things" in this expanse were small, primordial pieces of curvature scattered about on this expanse, ripples, so to speak. As a thought experiment, think of it as perhaps an infinite Jackson Pollock painting, with ink blotches as these curved "ripples" or distortions (our friend the Weyl Tensor again). Some of them would be grouped together in collections or sets of ripples. So we could draw imaginary boxes around these "ink botches" of ripples and create collections or sets of ripples in the infinite Jackson Pollock painting of the cosmos anterior to the cosmological natural selection process. With me so far? Good. Now, there are two rules about making sets: Sets cannot contain themselves, and sets cannot have duplicate members. In real life, I can have a coin collection with say two Gold Buffalo Ounces. But in Set Theory this is not so. I can only have 1 Gold Buffalo Ounce - or one Jackson Pollock inblot of a given sort, per set. I also cannot contain the set as a member of the set itself. To explain that, say I got a list of books from my local library. Then I got lists of books in all the libraries in the world. Then say I made another list, call it the Master List, of all the lists of library contents in the world. You might say the Master List contains all the lists (of library books) possible, but one list the Master List does not contain, namely, the Master List does not contain itself. This is Bertrand Russell's Set Paradox.

As another instructional aide, here is a animated video further explaining Russell's Set Paradox, using the analogy of the Barber of Seville, a Barber who shaves all the men of Seville who do not shave themselves. If he shaves all the men of Seville except himself, he has left out one man in Seville who does not shave himself, namely, himself, and if he does shave himself, by shaving himself, he has violated the rule that he must only shave those men in Seville who do not shave themselves. This video of course follows the well-known Internet Rule that animation always makes math less intimidating:

OK, let us review. So we have an infinite Jackson Pollock painting where the canvas is space, the ink blots are gravitational "ripples" and we can take an imaginary pen and draw squares around these ripples to make sets of the ripples. Only rules are, we cannot make a set that has duplicate elements, and we cannot make a set that contains itself, for that would violate the rule that Sets cannot contain themselves, anymore than we can have barbers in Seville who shave themselves in violation of the rule that barbers in Seville must only shave those who do not shave themselves. Well and good.

But "time" has not entered the picture yet. We just have a static object. But hang on. The rule about no sets contain themselves? Well let's violate that. Boom. If a set, say, Set A, also contains itself (call it Set A1) as a member object, then we immediately have an infinite recursion. Set A1 then has its own self as a member, call it, Set A1.1, which in turn has - say - Set A1.1.1 ...). Let's stop that before we get a headache.

Time may very well begin so to speak as a "violation" of the first rule of Set Theory. As soon as you allow sets to contain themselves, you get an infinite sequence, which may in fact be time.

But, hang on a moment, time is not just an infinite recursion of a set of objects. Time is change. Successive moments have different configurations of particles, etc. So what gives? Here is where we violate the second rule, we allow duplicates. Going back to our friend, Set A, say A has two objects which are mirrors of itself, A1_I and A1_II. So then if we go to the "next generation" of Set A, we have two child sets, A1_I and A1_II (each containing themselves, etc.). If we go to the "second level" or "next generation" in our recursion and randomly pick sets, we are more likely to pick sets that had duplicates, for the simple reasons that sets with duplicates birthed more recursive descendents. Allow this process to continue, N generations into the recursion, sets with duplicates dominate the ensemble, but, and here is the key point, "from the outside" of a set, we may not know which ones have duplicates. Pause here. Think. The further down the rabbit hole we go, at any given scale of sets containing other sets and so on, the more likely it is that sets will have duplicates, but unless we examine each one, a set with a duplicate will "look" like a set without a duplicate in terms of how it "behaves" as a set that is a member element of yet a higher set. Duplcates only serve to replicate sets with more frequency but (since they are not really part of Set Theory anyway) they  don't change anything else. Where are we going with this? 

Recall that entropy is simply a matter of objects that have internal states that are invisible from the outside. Think of an impressionist painting. If you change a couple of lines in it you won't notice a difference. The increase of entropy means that as you go along in time the more objects you run into that have internal states whose changes are not noticable. Allowing an arbitrary or infinite number of sets containing themselves at the start of this whole thing to have duplicates means that as you go down the recursive rabbit hole, the deeper you go, when you randomly sample sets at a given level of the recursion, you will find more and more sets having duplicates, analagous to the idea of entropy of "macroscopic" objects having an increasing amount of internal states the further in time you go.

So, example:

Level 1:

A = {Foo, Bar_I, Bar_II, A1_I, A1_II,}

Level 2:

A1_I = {Foo, Bar_I, Bar_II, A1.1_I, A1.1_II}
A1_II = {Foo, Bar_I, Bar_II, A1.1_I, A1.1_II}

Level 3:

A1.1_I = {Bar_I, Bar_II, Foo, A1.2_I, A1.2_II}

A1.1_I = {Bar_I, Bar_II, Foo, A1.2_I, A1.2_II}

etc.

At each generation you can have duplicate recursive elements spawing new levels, and as you saw I sneakily did, you can change the order of elements from level to level. [Sets need not have ordered elements, but traditionally most do, so by allowing ordering to not be enforced, we give them flexibility.]

If we allowed each level of recursion to have varying numbers of duplicate elements, we can change the number of "internal states" each Set has, without changing the Set's "behavior." Since entropy works by allowing macroscopic objects to have an increasing amount of internal states, we can model entropy by levels of recursive sets having increasing numbers of duplicate elements.

Furthermore, by changing the ordering of elements could - for instance - change the distrubution of matter/energy from moment to moment.

So we have violated two rules and one tradition of set theory. We violate the rule that sets cannot contain themselves, we violate the rule that sets must have no duplicates, and we violate the tradition that ordering is preserved in sets. By doing this we can model the passage of time and also entropy.

What does all this have to do with cosmological natural selection? Well this is just a lower level, mathematical description of how it all works at the higher level. At the higher level, you start with a blank canvas of nothing but energy ripples. Time chugs along (in a tensed manner, i.e., it is "real"), black holes form, making baby universes, that make more black holes, ad infinitum. All that stuff about sets was to give a mathematical description of how this gets going in the first place, because tensed time's boundary is trickier than tenseless time's boundary. In tenseless theories, the beginning of time is like the South Pole - you cannot have times "earlier" than the the beginning of time just like you cannot have points "more south" of the South Pole, because time is treated like a mathematical object. In a tensed theory of time you must have a beginning to avoid logical paradoxes, but then you are in the uncomfortable position of time "appearing out of thin air" as it were. If time comes finally down to violations of the basic rules of Set Theory, so much the worse perhaps for Set Theory, but so much the better perhaps for (tensed) time.

To try to give a non-technical stab at what I am trying to say, here would be an approach. The beginning of the cosmos at the start of Cosmological Natural Selection is indeed like a Jackson Pollock painting with random splotches of energy scattered about. You can think of it as chaos or darkness or void. Much like the creation myth in Beresheit (meaning "Beginning" in Hebrew), the first book of Torah - "there was chaos and darkness in the void, and the wind from G-d blew over the waters". The "void" here is space and the "waters" or "wind" if you will are the ripples in the fabric of space. Then - to channel again the poetry of Beresheit - "G-d said, let there be light." You can think of it as a disruption of the rules of logic (as we known them), and this disruption of the rules of logic, so to speak, caused time itself to come into being. What caused the disruption you might ask? Here we must go to Wittgenstein: Whereof one cannot speak, Thereof one must be silent. For the sake of conversation let us simply say that the cosmological process of Creativity disrupted the nature of things, a disruption that nucleated in the same instance both time and entropy. And following on from that began the never-ending cycle of universes birthing other universes, with each birth resetting the entropy clock so that each universe history would have its own unique trajectory of evolution and stellar (and consequently, black hole) formation, a process producing an ever-increasing ensemble of universes, into time unending.

As a disclaimer, even if this particular approach of using Set Theory to model the origin of time and entropy at the start of Cosmological Natural Selection proves to be a less than optimal approach, I think the basic intuition here is correct, and that is, that the cosmos is initially a blank infinite canvas of arbitrary dimensionality with energy in the form of metric disturbances scattered randomly about, and then a "disruption" occurs which instantiates both the flow of (real, tensed) time, and also the concept of the increase of entropy through time. If as is perhaps probable better technical models of this process come forth in the future, I think this essential intuition will hold.

[As an additional disclaimer, I do note that I do not take the view of mathematics, including Set Theory, as being some sort of timeless collection of truths, but rather I take the view that mathematics is a human-made project of finding ways to model the reality that we experience, so my approach here to use Set Theory to model the origin of the flow of time and entropy is simpy a technical model, and is not meant to be taken in a literalist manner, consistent with my viewing here of time as "tensed" or "real" and hence not seeing anything, including mathematics, as being in some way "timeless" or "outside" of the experienced reality that we try to find ways to describe. Hence for instance I would reject the idea of there being a collection "out there" of different types of mathematical worlds, one of which being our own, and rather I would take the view that there is simply the ensemble of universes generated in real time by the process of Cosmological Natural Selection, wherein we reside.]

It is instructive perhaps that we can think of how time and entropy nucleated to kick-off the process of Cosmological Natural Selection as a kind of disruption of the rules as we know them. This is perhaps the ultimate nature of the cosmological process of Creativity, a sequence of disruptions of the status quo to produce that which is truly new. Whatever the technical descriptions may be, the unassailable fact appears to remain that the disruption of the old to create the new is the most fundamental feature of existence.

As the late Debbie Reynolds memorably sang in the animated film, Charlotte's Web, "how very special are we, for just a moment to be, part of life's eternal rhyme."