Imaginary Time at the Sandwich Shop (A Lesson in Cosmology)

I’ve been re-reading A Brief History of Time by Stephen Hawking, the revised and updated edition from 1998. Oh, I know that a lot of advances have been made in theoretical cosmology, quantum mechanics, and high energy particle physics over the past ten years, but I still think that much of what Hawking wrote applies. People who know better are more than welcome to correct me on this.

One concept that caught my eye is “imaginary time”. Imaginary time, Hawking explains, can be described as a timeline perpindicular to “real time”, which is the normal experience of time that we as human beings have and that we observe as a feature of the universe we observe. I have a hard time conceptualizing this, but it seems to be useful when talking about Feynman’s notion of “sum of histories”, which is a useful concept for describing how a particle travels through the infinite number of possible paths it could take. When applied to the entire universe, we can think of the universe’s progression through real time as a sum of histories over imaginary time.

From here on, things get more confusing.

Hawking explains that in real time, the dimension of time expands and lengthens the more that the universe progresses, either expanding outwards or contracting inwards. In imaginary time, though, it’s just the opposite; in other words, imaginary time contracts in a manner that is inversely proportional to the expansion of real time. The best way I have found to think of this (and I may be completely wrong) is to think of imaginary time as a way of envisioning the number of possible paths that the universe could progress on in its expansion, by which I mean the total sum of the infinite number of possible quantum states that the universe could expand into. It’s kind of like ordering a sandwich at Togo’s. One you place your order at the desk, the number of orders that you could have placed shrinks down to one. This, in essence, means that when you walked into the sandwich shop, the value of Togo’s Imaginary Time was much wider than it is when you contract it down to the one sandwich that you order.

According to Hawking, then, while the observation of the universe in real time indicates that there was a Big Bang and there will eventually be a Big Crunch (or some other way that the universe comes to an end — I, for one, like the notion of the Big Rip, which is a notion I explored in my short story “Padma”, though you’d be hard pressed to see that in there), in imaginary time, the universe has no such grand entrance into being. It just sort of coalesces out of the infinite number of possibilities of existence that existed prior to the universe’s existence. This means that, in imaginary time, at least, the universe has no distinct beginning or end, and thus no actual boundary conditions. Thus, the universe has come into being without actually having started.

Yeah, that confuses me too, and I may have gotten it wrong.

The philosophical implications of this are that there is no need for the universe to have a “beginning” or an “end”. It just IS, without having been created ex nihilo. “What need, then, for a Creator?” he asks.

With all due respect to Hawking (and he deserves a lot), I’m not convinced that there are any philosophical or theological differences between a Big Bang model of the universe’s existence, and Hawking’s model of the universe as a closed surface (so to speak) without boundaries. Really, it goes either way. You could argue that the Big Bang model requires a Creator who created the universe from a singularity or from nothing, or you could argue that the Big Bang model does not require a Creator, that the Big Bang exploded from a singularity pre-existent in a “quantum soup” of whatever sort of particle that existed prior to that event (though, of course, any notion of “prior to the Big Bang” makes no sense from a cosmological perspective). On the other hand, you could argue that the closed surface without boundaries model negates the need for God because then the universe is just one possible state of being among an infinite series of possible states of being. On the other hand, you could just as easily argue that those infinite possible states of being had to come from nowhere, so why not God? The notion that the universe’s boundary condition is that it has no boundaries does not negate the possibility of the existence of God.

I’m not arguing for the existence of God here, of course. That’s beyond the scope of my intention right now, and I doubt I’d be able to convince anyone either way if they’ve already made their mind up. I’m just pondering a possible response to Hawking’s suggestion that the bounded without boundaries nature of the universe makes God redundant: and that is, it doesn’t. Not necessarily, at least, to someone who’s already made their mind up anyway.

Of course, if you believe the Universe came into being about four thousand years ago with all of these properties built into place to make it look like it was created fifteen billion years ago — the Omphalos hypothesis — then all bets are off, and all this speculation is pointless.

With regards to imaginary time itself… Well, people who know me already know that I have joked that I find the notion of imaginary numbers as morally reprehensible. What, the infinite set of real numbers ain’t good enough for you? You have to go so far as to make up imaginary numbers to explain things? Shame on you! So I think I will take the same moral stance on imaginary time. Shame on you, Doctor Hawking! For shame!

Thus endeth our lesson on the moral implications of modern cosmological theory. I hope you enjoyed it.

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