## Degrees of Freedom

The boundless dimensions of math and physics

# What Do You Mean, The Universe Is Flat? (Part I)

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The universe is three-dimensional.

The universe is four-dimensional—three for space, one for time.

The universe has nine, or ten or eleven dimensions.

Matter curves spacetime.

The universe is flat.

The universe is infinite.

The universe is 84 billion light-years wide.

The universe is a bubble, or an onion.

Or a hall of mirrors, shaped like soccer ball.

Or a shape out of Dante’s Divine Comedy.

Statements like these appear quite frequently in popular science magazines--including Scientific American--and they seem to be in utter contradiction with one another. But all of them are true, or at least plausible. What gives?

The subtlety is that the word “universe” has different meanings in different contexts. In colloquial English, the word is often taken to mean “everything that exists.” So this intuitive notion of universe seems like a good place to start. If we follow this line of thought, the first thing we notice is that the present tense of the verb “to exist” implicitly assumes that we are referring to “everything that exists now.”

Leaving aside the issue of whether "now" can have a universal meaning--and the even subtler ontological question of what it means to exist--it makes sense to think of the totality of space and all of its contents at the present time, and to imagine this totality as a contiguous entity.

Space or spacetime?

If we take this route, we may first notice that space appears to us to be three-dimensional. Thus, we could make the assumption that we can locate anything in the universe using three Cartesian coordinates: at this frozen moment in time that we call the present, every object occupies a certain x, y and z in our three-dimensional continuum.

So here is one natural notion of the universe: all of three-dimensional space at the present time. Call it the nowverse.

But what about all those other dimensions?

Fanciful theoretical constructs such as string theory postulate that, in fact, there is more to space than we can see, but for now those theories have no experimental evidence to support them. So, for the time being we may as well just focus on our familiar three dimensions.

In this schematic of spacetime, the disk at the top represents space at the present time; the ones below represent space at earlier times

Time, on the other hand, is indeed an additional dimension, and together with space it forms a larger, four-dimensional entity called spacetime. It is natural to think of the nowverse as a 3-D slice in this 4-D space, just like horizontal planes are 2-D slices in our 3-D world. Because most people (including yours truly) have a hard time visualizing 4-D objects, a common way of thinking of spacetime is to pretend that space had only two dimensions. Spacetime, then, would have a more manageable total of three. In this way of looking at things, the nowverse is one of many parallel planes, each of which represent the universe at a particular time of its history.

Thus, the seeming inconsistency of

The universe is three-dimensional.

The universe is four-dimensional—three for space, one for time.

The universe has nine, or ten or eleven dimensions.

is just a matter of clarifying language. For all we know, space is 3-D, and spacetime is 4-D; but if string theory is true, then space turns out to be 9-D, and spacetime 10-D.

Incidentally, when cosmologists talk about the expansion of the universe, they mean that space has been expanding, not spacetime.

Flat or Curved?

A plane and the surface of a sphere are the prototype for flat and curved space

In the last decade—you may have read this news countless times—cosmologists have found what they say is rather convincing evidence that the universe (meaning 3-D space) is flat, or at least very close to being flat.

The exact meaning of flat, versus curved, space deserves a post of its own, and that is what Part II of this series will be about [update July 31: read What Do You Mean, The Universe is Flat? Part II: In Which We Actually Answer the Question]. For the time being, it is convenient to just visualize a plane as our archetype of flat object, and the surface of the Earth as our archetype of a curved one. Both are two-dimensional, but as I will describe in the next installment, flatness and curviness make sense in any number of dimensions.

What I do want to talk about here is what it is that is supposed to be flat.

When cosmologists say that the universe is flat they are referring to space—the nowverse and its parallel siblings of time past. Spacetime is not flat. It can’t be: Einstein’s general theory of relativity says that matter and energy curve spacetime, and there are enough matter and energy lying around to provide for curvature. Besides, if spacetime were flat I wouldn’t be sitting here because there would be no gravity to keep me on the chair. To put it succintly: space can be flat even if spacetime isn't.

Moreover, when they talk about the flatness of space cosmologists are referring to the large-scale appearance of the universe. When you “zoom in” and look at something of less-than-cosmic scale, such as the solar system, space—not just spacetime—is definitely not flat. Remarkable fresh evidence for this fact was obtained recently by the longest-running experiment in NASA history, Gravity Probe B, which took a direct measurement of the curvature of space around Earth. (And the most extreme case of non-flatness of space is thought to occur inside the event horizon of a black hole, but that’s another story.)

On a cosmic scale, the curvature created in space by the countless stars, black holes, dust clouds, galaxies, and so on constitutes just a bunch of little bumps on a space that is, overall, boringly flat.

Matter curves spacetime. The universe is flat

is easily explained, too: spacetime is curved, and so is space; but on a large scale, space is overall flat.

Finite or Infinite?

If everything in the nowverse has an x, a y and a z, it would be natural to assume that we can push these coordinates to take any value, no matter how large. A spaceship flying off “along the x axis” could then go on forever. After all, what could stop her? Space would need to have some kind of boundary; most cosmologists don’t think it does.

The fact that you can go on forever however does not mean that space is infinite. Think of the two-dimensional sphere on which we live, the surface of the Earth. If you board an airplane and fly over the equator, you can just keep flying—you’ll never run into the “end of the Earth.” But after a while (assuming you have enough fuel) you would come back to the same place. Something similar could, in principle, happen in our universe: a spaceship that flew off in one direction could, after a long time, reappear from the opposite direction.

Or perhaps it wouldn’t. Cosmologists seem to believe that the universe goes on forever without coming back—and in particular, that space has infinite extension. But when pressed, most cosmologists would also admit that, in fact, they have no clue whether it's finite or infinite.

In principle, the universe could be finite and without a boundary—just like the surface of the Earth, but in three dimensions. In fact, when Einstein formulated his cosmological vision, based on his theory of gravitation, he postulated that the universe was finite. Einstein’s Weltanschauung was rooted in his deep, almost mystical sense of aesthetics; the most symmetric, aesthetically perfect three-dimensional shape is that of a three-dimensional sphere. (Some have suggested that the way Dante describes the universe in his Divine Comedy has something to do with a 3-D sphere, too: I guess that will have to wait for a future post, too.)

In more recent times, some cosmologists have taken this possibility quite seriously, and have tried to check whether space might be a 3-D sphere, or perhaps a more complicated 3-D space that is essentially a sphere wrapped around itself [see “Is Space Finite?” by Glenn D. Starkman, Jean-Pierre Luminet and Jeffrey R. Weeks; Scientific American, April 1999]. In a universe that has one of these shapes, one could observe trippy hall-0f-mirror type of effects.

The reason why we don’t know if space is finite or infinite is that we seem to have no way of observing beyond a limited horizon. The universe is 13.7 billion years old, and because nothing can travel faster than the speed of light, we don’t have any information about events that happen beyond a certain distance. (For reasons that would be too complicated to go into here, that maximum distance is actually not 13.7 billion light years.)

The observed universe

So one thing we know is what we cannot know: the universe we can observe has finite extension. Cosmologists often refer to it as the observable universe.

How large is the observable universe? That is a surprisingly difficult question, which will be the subject of yet another future post.

For now, let’s just notice that the most distant galaxies whose light we have detected emitted that light about 13.2 billion years ago. Because the universe (meaning space) has been expanding ever since, those galaxies are now at a much greater distance—some 26 billion light-years away.

Even farther away than the farthest galaxies, the most distant object we have been able to observe, the plasma that existed before the age of recombination [see Under a Blood Red Sky], existed about 13.7 billion years ago, a puny 400 millennia after the big bang. Light coming from it has taken 13.7 billion light years to reach us. The matter we “see” in that plasma has also moved farther away: that matter is now an estimated 42 billion light years away. So that’s what cosmologists talk about when they say that the observable universe has a radius of 42 billion light years. (Of course, the answer had to be 42.)

The bizarre fact about the observable universe, however, is that it is not part of the nowverse. Because light from distant galaxies took millions of years to reach us, what we see is in the past, not in the present, and the farther it is, the older it is. So if the observable universe is not part of the nowverse, how can we picture it? Where in spacetime should we place it? [to be continued]

 This post is part of a series on cosmology. Here are the other posts: What Do You Mean, the Universe Is Flat?, Part 2: In Which We Actually Answer the Question("Flat" means "not curved," but what does "curved" mean?)Being Mister Fantastic(On visualizing a finite speed of light)Under a Blood Red Sky(On the afterglow of the big bang, and why the sky used to glow red) Still to come: how do we know that the curvature of space is a fact of life; what would the world look like if space were very curved; what is the curvature (and the size) of the observable universe; and what the heck does the observable universe have to do with Dante.

Hubble Ultra Deep Field view courtesy of NASA. Sphere-and-plane image by Joe Doliner.

Many thanks to Scientific American cosmology guru George Musser.

The views expressed are those of the author and are not necessarily those of Scientific American.