First, what do we mean by the words “universe” or “cosmos?” The start of the Wikipedia entry for “universe” seems to me to sum it up nicely, especially because it says pretty much what I hoped it would say: “The Universe comprises everything perceived to exist physically, the entirety of space and time, and all forms of matter and energy.”
That second clause, “the entirety of space and time,” raises a point that I think a lot of people overlook: The universe is not in space, rather, all space is in the universe. This means the universe cannot have a spatial boundary: You can’t travel to the edge of the universe and find that the universe ends while space continues. As a result, the shape or structure of the universe must be such that if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other.
It seems to me that there are only two ways this can be possible. The first is if the universe is infinitely large. This would certainly allow for a never-ending trajectory, but it also creates serious difficulties.
First, given the equivalence between space and time, it seems necessary that an infinitely large universe also be infinitely old. Even if there were no inherent reasons for rejecting an infinite universe (and I think there are), this requirement that it be infinitely old would directly contradict the prevailing scientific cosmological model, the Big Bang theory, which holds that the universe is “only” about 13 billion years old.
Also contradicting the Big Bang theory would be any requirement that matter/energy be consistently distributed throughout an infinite space. Such consistency is required if we aren’t to allow different regions of space to have drastically different properties; in other words, if we do want to guarantee that universal laws really are universal. But if we distribute matter/energy similarly throughout an infinite universe, it’s clear that we must have an infinite amount of matter/energy to distribute.
Since matter/energy can be neither created nor destroyed, an infinite universe must have contained the same amount – i.e., an infinite amount – of matter/energy from its beginning. In other words, a less-than-infinite volume of space must have held an infinite quantity of matter/energy. This strikes me as very unlikely. In short, I don’t see any possibility that an infinite universe could have had a beginning in time (without divine agency), because it must already have been infinite at its beginning, something that contradicts the entire basis of the Big Bang theory.
But if an infinitely large universe must also be infinitely old, then one would expect certain physical phenomena to have advanced long ago to their extremes. Most obviously, the second law of thermodynamics requires that the total entropy of the universe increase over time. If it has already been increasing for an infinite time, then one might reasonably expect that entropy would long ago have reached its maximum, known as “heat death,” in which no free energy is available to cause motion or sustain order. Clearly, that is not the case.
So I’m convinced that infinity is just not on, which means we need to find another structure “such that if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other,” as I said earlier.
To explain what I think is the best alternative to infinity, I’ll start with an object that may be familiar to many readers:
Möbius strip (Adapted from Wikipedia)
This is the famous Möbius strip, discovered by German mathematician August Ferdinand Möbius. It’s just a strip of paper with the ends glued or taped together, with the important requirement that the paper be given a half-twist before gluing or taping, but it has a number of interesting properties, which many of you are undoubtedly familiar with.
For one, if you take a pen and draw a line along the length of the strip, the line will return to its starting point, having covered both “sides” of the paper, without lifting the pen off the paper at any time.
The standard explanation for this is that mathematically (topologically), despite all appearances to the contrary, the strip has only one surface. But there’s another way of explaining it:
If you took a strip of paper and stuck the ends together without the half-twist, the only way to draw a line on both sides of the paper would be to lift the pen from the paper and move it around to the other side. In other words, the pen would have to leave the two-dimensional surface of the paper and travel through three-dimensional space to the other side.
But the half-twist turns the Möbius strip as a whole into a three-dimensional object, and so the tip of your pen does in fact make that trip through 3D space as it makes its “orbit” of the entire strip, without ever leaving the (apparently) 2D plane.
If you consider the twist as spread evenly over the whole of the strip, then in effect the dimension of the surface at any one point is slightly more than 2D; it has a dimension of 2.00…n, with the magnitude of the fractional part at any point depending on how long the strip is. When you add the fractional parts over the whole length, the sum will be 1.0, which, when added to the nominal 2.0 dimension of the surface, makes the dimension of the object as a whole 3.0.
Now, suppose we take a three-dimensional object – a sphere, for example – and give it a half-twist and join its ends together. Obviously, this isn’t something we can actually do in three-dimensional space – a sphere has no “ends” in 3D space. (By the same token, a 2D being couldn’t make a Möbius strip.) But I hope that by bootstrapping up from the example of the Möbius strip, we can conceptualize the result of this procedure.
As was the case with the Möbius strip, the half-twist raises the dimension of the object as a whole by 1.0, in this case to 4.0. And as with the Möbius strip, the dimension at any point within the sphere will appear to be unchanged. But in fact, it will be 3.00…n, with the fractional part again adding up over a full orbit of the space to 1.0, making the total dimension 4.0.
And again as in the case of the Möbius strip, a straight-line trajectory in any direction will eventually return to its starting point, having traversed both “sides” of the sphere. But mathematically, just as the Möbius strip has only one surface, this “Möbius sphere” has only one volume, though it would appear to a four-dimensional observer to have separate “inside” and “outside” volumes, just as a three-dimensional observer (you or me) sees an “inside” and an “outside” surface on the Möbius strip.
A space of this kind would satisfy the requirement that “if you travel through it in a straight line, you never reach a point where the universe is on one side and something else is on the other.” In fact, in this twisted universe, no matter which direction you travel, there will always be as much of the universe in front of you as behind, and as much above as below. In other words, for an ordinary (physically constituted) traveler, such a universe is likely inescapable.
There are a number of other implications of this form as a cosmological model, some of which I find rather odd. But this is already a rather lengthy post, so I’ll have to take up the ramifications next time.
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