(This is Part 2 of a series on Hawking’s no-boundary proposal. Read Part 1 first.)

I thought the whole point of this program was that we couldn’t just…get to the beginning of the universe, and now, thanks to the magic of Wheeler and DeWitt, we have the precise machinery we need to solve the beginning of the universe….if only we knew one thing, just one tiny piece of information, just one measly morsel…and we could do it.

Hawking did it. Well, he had an idea, which is more than anyone else had at the time.

Two decades after Wheeler and DeWitt, Stephen Hawking comes along and connects, as he usually does, several different lines of thought, and he realizes that the problem for one thing is actually the solution to another.

Check this out: Hawking is staring at the Wheeler-DeWitt equation. It’s a puzzle that reveals the universe, but all the puzzle pieces are scattered around, AND we don’t have the picture on the front of the box. If we know how to place just ONE piece, we can put together the quantum wave function of the universe.

We just need the first piece. The boundary condition that defines the beginning state of the universe. But we can’t measure it, we can’t read any device or look through any telescope to figure it out. Nothing gives us access to the first moment of the big bang. AND we have no theory of quantum gravity around the corner to just ASK.

So that leaves us with…taking a wild guess and seeing if it sticks.

Hawking decided that his wild guess would be as grounded as possible. He argued that the best boundary condition of the universe, the best statement that you can make about how it all gets started, had to be SELF-JUSTIFYING. That means the guess about the beginning of the universe couldn’t come from anywhere else: you can’t point to God or Wheeler or ANYTHING to just HAND you the answer, because the universe is every single thing to ever exist, in totality, and you can’t reach OUTSIDE that. There’s no hidden corner of the cabinet that exists outside the universe to just give us the boundary.

And the most self-justifying statement Hawking could make about the beginning of the universe is that it had no beginning.

In other words, what if the reason you can’t find the boundary at the beginning of the universe is not because it’s hidden or inaccessible, but because it genuinely isn’t there?

Now this is a very lovely thought to have. But we’re not here for lovely thoughts, we’re here for down and dirty physics. It’s one thing to say something crazy, it’s another to turn that into a working theory of nature. Thankfully, Hawking had exactly what he needed.

One of the defining features of the Wheeler-DeWitt equation is that it doesn’t involve time. It doesn’t know or care that the universe evolves, expands, does interesting things, heads out to dinner on a Tuesday night just because it’s wild like that.

The key that can unlock the Wheeler-DeWitt equation is a solid statement ABOUT time, specifically, the most important time of all: the beginning of the universe. So we NEED to involve time SOMEHOW in all this mess if we’re going to make progress.

So Hawking…involves time. Instead of just looking at space, he stitches together these geometries back to back like frames in a film. He makes a sequence of them, representing an evolution to the history of the universe. These frames tell the story of the cosmos. Now, we don’t know what that story is (I mean, from Wheeler and DeWitt’s machine; we can observe it and so we KNOW it, but we’re trying to EXPLAIN it), so Hawking constructs all these…paths. Possible histories of the universe. Trajectories, evolutions, stories. In some stories, the universe gets really big really fast and fizzles out to nothing. In others, it never even expands. In still others, there’s nothing but matter. And others, nothing but…nothing.

Now all of these paths, all of these histories of the universe, all share one thing in common: they’re in the usual spacetime that we know and love. Cause and effect, past and future, speed of light, all that. And they all have a BEGINNING. A first frame in the movie of the universe.

Well, what if we just made time behave differently? Now I’m going to share a term with you, and when I say it it’s going to sound really wrong, like icky, deep in your gut. But I’m going to say it, then I’m going to explain it. It…will still feel icky, but at least it will have an explanation.

Here goes: imaginary time.

Yeah, imaginary time. Time, but imaginary. Listen, I don’t know how much you know about imaginary numbers. But they’re really cool and fun and DEFINITELY worth bringing up in your next workplace all-hands meeting. The core idea behind imaginary numbers is to pretend to take the square root of, I don’t know, negative 4. The square root of regular four is 2, but the square root of negative four is…uh….what? In normal grade school math this is where your teacher scoffs at you and says you can’t take the square root of a negative number.

But this isn’t grade school. We’re not going to take the square root of a negative number. Instead, we’re going to say that the square roots of negative numbers are an ENTIRELY NEW KIND OF NUMBER. A brand new category. You have whole numbers, rational numbers, negatives, and now you have imaginary numbers, which are all the square roots of the negative numbers.

Phew, I swear I’m going somewhere with this.

The trick Hawking pulled was that he took all these histories of spacetime and replaced “time” with “imaginary time”. He multiplied the passage of time by the square root of negative one. Now, we actually do this in quantum mechanics all the time (or should I say imaginary time?) as a TRICK. Sometimes when we get equations that are really, really hard to solve, we replace time with imaginary time and they become easier. Then we solve them, then we swap back. Just a little reshuffling in the backend to work through some thorny problems.

But when Hawking does this to the spacetime of the universe, he gets a bonus. It’s not a party trick anymore. It’s a statement. You see, in normal spacetime, the universe has a beginning. But when you replace time with imaginary time, the beginning…goes away. This procedure actually puts space and time on equal footing. It makes them all creatures of curvature and geometry, with no separate identity. Which makes the beginning of the universe no special time at all. It becomes like the south pole, which is really just any other point on the globe. You reach the south pole and keep walking, and it’s only ever north from there. You reach the beginning of the universe, and it’s not special or unique (maybe a little hot); all you have is the future in front of you.

No beginning. No start. No boundary. A universe that justifies itself.

By making the switch to imaginary time, Hawking could ENCODE his “the universe has no beginning” idea, AND he could crunch through the math.

Voila: a key that unlocks the Wheeler-DeWitt equation and the know-how to run the mathematical machinery.

And what do you get for all this work? Nothing less than a wave function for the universe.

In Part 3, the wave function delivers something Hawking didn’t even ask for: our universe, more or less, for free.