The Lay of the Landscape

Let’s say you have a theory, and, let’s say it happens to have a whole lot of solutions. Maybe it’s the theory you’re thinking about, but it doesn’t have to be. Nothing of what I’m going to say depends on any details besides this surfeit of solutions.

I should begin by saying what it means for a theory to have a lot of solutions. In fact, given any set of equations, one will almost always have many solutions. A polynomial of degree n will generically have n roots, for example. In classical physics, instead of a finite number of solutions, you will almost always have an infinite number. If you look at the equations that describe an object moving in a gravitational field, you don’t know the orbit until you fix the initial position and velocity. Once you fix enough of these “initial conditions”, the future solution is complete determined, however, and we can make predictions.

When I played mini-golf a long time ago at Malibu Fun and Games, there was a hole which had three consecutive hills of roughly the same height right after the tee. The goal was to hit the ball so that it ended up in the middle valley. This is, of course, an example of what I’m talking about. We know the initial position, and by varying the initial velocity, we can control where the ball will end up. Here, the solutions have the nice property that, after some time, you always end up in one particular valley.

While on this particular mini-golf course, the valleys all looked the same, there’s no reason they have to. We can have a landscape (there’s that word….) of peaks and valleys, and a vast array of different shapes for the valleys. If we know how hard we hit the ball and in which direction, we can still predict where the ball ends up, but we’re not always so lucky.

So by now everyone knows where I’m going. Let’s say our theory of peaks and valleys is in fact some sort of theory of everything. Each of the valleys is a possible universe, and the shape of the valley describes the (low-energy) physics we’d observe if we were to live in that universe. The problem with theories of everything, however, is that they aren’t necessarily theories of everything. Besides a theory that governs how things change, we also need a theory of initial conditions. Without that, we can’t predict which valley we end up in. Not good. And even then, with quantum mechanics sneaking up on us, maybe we could be in all of the valleys at once (a “superposition of states”).

But even without quantum mechanics, the golf ball analogy has some problems. A slightly better analogy is that every point in our universe is a golf ball. Since golf balls tend towards the bottom of valleys (ie, things roll towards the center of the valley), the universe will divide up into regions labelled by which valley that part of the universe is in. Instead of considering a single universe sitting at the bottom of a valley, we have a multiverse of regions in different valleys with walls between them.

In general relativity, more can happen: the universe itself can expand. In certain theories, small regions of the universe called “bubbles” can spontaneously head into a different valley, and then, often, begin to rapidly expand surrounded by a wall. Thus, one might say that universes can reproduce. (Some people have postulated that black holes may bubble off new universes, too, but I know of no scientific theory which describes such a process.)

The upshot of all of this is that if it happens to be true, we’ve got a bit of a problem. If we have no way of telling ab initio which valley our part of the universe is in, how can we get anything out of our theory? Well, if you’re a conservative guy like me, there’s an obvious answer: we measure the shape of our valley. That’s what we’ve been doing for years, after all, and we’ve been pretty successful at it. What we need to understand, then, is what all the valleys look like. Even if we can’t nail down precisely what valley we’re in, if we can narrow it down enough, we can find properties of all the valleys consistent with what we know and obtain predictions. We can also look at facets of our theory that go beyond the simple shape of the valley. Even if we can’t predict beforehand that such a thing has to happen, if we were to see it happen, it would give us evidence that we’re on the right track.

This is all old-school science, but it’s hard work with the tools we have available at the moment. There can be a lot of valleys. Really a lot. And finding their shape isn’t so easy. Thus, one might wonder if there might not be other ways to make predictions. Maybe one can use statistics to understand our place in the multiverse. This is a radical idea that seems to be a significant break from how we’ve always done physics. It is, as they say, somewhat controversial. In the next post, I will give some of the arguments supporting this point of view. While I personally disagree with this approach as a technique for doing science, I hope to explain that maybe it’s not as easy to dismiss as one might have expected or hoped.

The posts in this series are:
The Multiverse: An Apology
The Lay of the Landscape
Twisty Little Universes, All Alike
Alone in the Multiverse

Comments

  1. #1 Jonathan Vos Post
    August 20, 2007

    Or maybe I should have posted this here, and not in the thread “The Multiverse: An Apology”:

    # 3 | Jonathan Vos Post | August 20, 2007 09:04 AM

    First, how many solutions are there currently thought to be in the Landscape?

    I’ve seen these 3 values a lot in the literature:

    (a) 10^500

    (b) 10^1000

    (c) Infinity.

    Secondly: Why are the many science fiction stories and novels about the Multiverse different from what String Theory tells us?

    Thirdly: how well are actual anlyses of data (WMAP etc) doing at defining WHICH universe we happen to inhabit?

    Fourth: The Axis of Evil. Real, or artefact?

  2. #2 Uncle Al
    August 20, 2007

    Is BRST invariance true? Are massive body and accelerating geometry effects indistinguishable without empirical exception? Affine, teleparallel, and noncommutative theories wholly contain subset GR. They also describe Equivalence Principle violation, ending string theory if demonstrated. The Einstein-Cartan Lagrangian includes terms

    1) epsilon_{abcd} e^a ^ e^b ^ R^{cd} — for the curvature scalar
    2) epsilon_{abcd} e^a ^ e^b ^ e^c ^ e^d — for the Cosmological constant, if any.
    3) epsilon^{mnrs} R_{mnrs} or e^a ^ e^b ^ R_{ab} — for parity breaking. The metric-affine geometry from which Einstein-Cartan arises embraces such generalization. The result echoes parity-violating teleparallel gravitation.

    String theory can be falsified by founding postulate empirical counterexample. The math could be rigorous, self-consistent, and physically irrelevant. Somebody should look.

  3. #3 Thomas Larsson
    August 21, 2007

    The wave equation has infinitely many solutions. If you pluck a violin string, it could vibrate with the fundamental frequency, with the first overtone, or with the 4711:th overtone. In fact, it is most likely that the frequency will be even higher, since there are infinitely many overtones above the 4711:th but only finitely many below it. Hence the violin frequency will be a GHz or higher.

    This is bad news for orchestras.

  4. #4 cet
    March 26, 2009

    thanks, By Brooklyn

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