So I just finished reading A Brief History of Time the other day. I realize it’s a bit dated, so I was hoping someone might be able to explain this concept to me (or explain why it’s no longer viable).
Hawking describes space-time as having four dimensions and being finite, yet having no boundaries. How can something be finite with no boundaries?
Also, if anyone’s willing to suggest another book or two on this sort of thing, I’d love to hear your recommendations.
The way he seemed to explain it is that you can’t go outside of the universe, it IS all there is. So, if you got close to the ‘edge’ (and how could you tell?), then you would never break through. in passing any imagined edge, you would be carrying the universe with you, including all the laws which govern it.
Thus, there are no boudaries, as there in not anyway to pass from it.
Of course, a negative cannot be proved to be fact, and the math he uses is well beyond me.
Now, the physics majors!
Imagine a bubble. The bubble is limited – it has a diameter, and its surface has a measurable area. That surface, however, has no boundaries – you just go around and around forever. If you are limited to the two dimensions of the surface, you never directly perceive the limits.
Similarly, space-time is a four-dimensional construct which we perceive in three dimensions. It has limits, but we cannot directly perceive them. instead, we travel around on the “surface.”
There are many very sinple things that are finite but without limits. A bubble, or sphere has been mentioned. A circle is likewise finite but contains an infinite number of points in its perimiter line. Any closed line or surface has the same properties, except possibly some pathological cases that someone is sure to mention now that I’ve said this.
A line from 0 to 1 is finite and has an inlimited number of numbers in its length. The distance of any number n down to but not including 0 has no number that you can point to and say, “That’s the end of the line.” Yet it is what most people would call a finite entity.
This is driving me nuts. I know there’s a short topological term that exactly describes the short of shape we’re talking about, but I cannot for the life of me remember it.
It’s a closed manifold. The term “closed” includes both compact (what’s being referred to here as “finite”) and without boundary. Hence the surface of a sphere is a closed manifold, but not a solid disc (the interior and boundary of a circle), which has a boundary, and not an infinite plane, which has no boundary but is not compact.
I never took a course in topology so I don’t know the technical definition of a manifold–but is that what you’re thinking of?
The sphere is an example of a finite 2d surface with no boundry. Another is the asteroids screen. The left edge of the screen is identified with the right and the top is identified with the bottom. When you go off the left edge, you emerge on the right. The sphere has positive curvature while the asteroids screen is flat. It’s topology is that of a torus.
It is, in fact, a short term. One might even call it compact.
Ha! I kill me.
In most (although not all) theoretical physics, attention is restricted to boundaries without manifold, so “compact” is sufficient. Once you introduce boundaries you have to start worrying about boundary conditions when solving your equations, and that can be a real pain in the Heine-Borel.
I’m sorry, I’m in an odd mood tonight.
Indeed, what would a boundary in our universe’s manifold look like, physically? Would it behave like a huge, frictionless, perfect mirror that reflected everything? Or would it absorb and destroy everything, like a black hole?
Or is it folly to even speculate in this direction?
Um, the open interval that was mentioned isn’t a closed manifold. All we’ve specified in this thread is a boundaryless manifold. The reason that “closed manifold” is what the physicists mean is rather subtle, and hinges on the precise definition of “finite”.
Are you a pirate with a steering wheel sticking out of your pants? 
That’s so weird. I just finished reading it for the first time a couple of weeks ago. The cover said it was an updated and expanded 10th anniversary edition of the book. Is that the one you read?
[HomerSimpsonVoice]…hmmm…donuts…[/HomerSimpsonVoice]
Don’t think so. I picked my copy up at a used bookstore.
The last option, really. Since we’ve never found a boundary in spacetime, the question can only be answered by looking at what the mathematics allows. Mathematics on manifolds with boundaries is perfectly well-defined, but you need to specify boundary conditions in addition to specifying what’s going on in the interior of the manifold. And the different kinds of boundary conditions mathematicians have thought up would (I’m pretty sure)[sup]1[/sup] look different to us poor schmoes in the interior.
[sup]1[/sup] The above hedge is made because it’s late and I need sleep.
BTW, thanks to all the responses. I get it now, or at least I understand it well enough for a layman.
GQ can really be cool sometimes!
You could get away with pretty much any sort of boundary conditions on the Universe, provided you put the boundaries far enough away that we poor schoes couldn’t see them. But then you’re putting humans in a privelidged place in the Universe, which is something that modern cosmologists are reluctant to do.
What if you picked antianthropic boundary conditions?
I know that people in general cringe at the idea of the anthropic principle, but it seems to me that a weak enough version is pretty much tautological. If we find, say, a theory that explains everything but the fine structure constant, maybe that’s a free variable. There “exist” (in that there can) universes with values of that variable nowhere near 1/137, but atoms aren’t possible there, let alone sentient life to ask. I’d love there to be a uniqueness proof for the universe, but I’ll settle for a moduli space of universes with a very small section capable of supporting sentient life.
In the current discussion, what I mean is this: what if the properties of the boundary of the universe were such that nothing could exist stably within its forward light-cone? That is, we can’t see it because if we could see it from here, sentient life would never have developed here. It doesn’t require a priviledged position, except in that some positions are closer to the boundary than others and those that are too close are too dangerous for observers (like us) to have arisen in the first place.
Pick up “The Fabric of the Cosmos” by Brian Greene. It’s a good current survey of Cosmology and touches on some of this space-time topology stuff. IIRC, Hawking’s model of the universe that shows up in his diagrams in ABHoT implies a “big crunch” in the future. The jury is still out, but these days the big crunch seems less and less likely to happen. Rather, we’re starting to worry about a “big rip”.
Of course, you can have a boundaryless, finite universe and no big crunch; any of the few hypothesised outcomes is possible within those contraints, though a big-ripping universe is approaching infinite size very quickly. The most up-to-date understanding of the shape of the universe includes the much-improved cosmic microwave background anisotropy data, as well as supernova data that indicates the rate of universal expansion is accelerating, revealing the possible existence of “dark energy”, which, as it turns out, makes up more of the universe than anything else.