Starship Asterisk*

Hubble Ultra Deep Field (Credit: NASA)

• 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.

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

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.

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?

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.

A plane and the surface of a sphere are the prototype
for flat and curved space (Credit: Joe Doliner)

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. 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.

Thus the seeming contradiction:

• 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-of-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]

In Which We Actually Answer the Question

Courtesy of palomaironique/Open Clip Art Library

• Stand up.
  Walk 10 feet straight ahead.
  Turn left by 90 degrees.
  Walk another 10 feet.
  Again, turn left by 90 degrees.
  Do it for a third time: walk. then turn left.

Now the next time you walk 10 feet ahead, you’ll trace the fourth and last side of a square, and you’ll end up where you started. If you turn by 90 degrees for a fourth time, you’ll face in the original direction, too.

This seems intuitively obvious, even tautological—if you trace a square on the ground, well, you trace a square on the ground—but it is actually an empirical fact. And it’s important, so I’m gonna say it out loud:

• There is no a priori reason why walking four equal sides and turning four right angles should take you exactly back to the same place. It is purely an empirical thing of our everyday experience.

As a matter of fact, it is not exactly true empirically, either. The failure to come back to the exact same spot—to precisely close a square—is not just true; it is one of the most important phenomena ever observed in the history of science. It is at the heart of everything. It is the way that gravity works the way that Einstein understood it. It tells us how black holes form and why they trap light. And it tells us if and how the universe should expand.

Our intuition tells us that every square should close. The world is far stranger than our intuition would have us believe.

In the previous part of this series, Part I, I promised that Part II would explain what it means for the universe to be flat. In this second part, I will talk about the concept—no, the phenomenon—of curved space, which is essentially when square paths fail to close, and about why flat space is where all square paths do close up.

Euclid Tried

So far I have intentionally emphasized the physical nature of this phenomenon called curvature of space. Most authors when they write about it follow a very different approach: they start with history.

You see, mathematicians came up with the idea of curvature—as a logically consistent but abstract concept—long time before anyone proved that it was relevant to reality. And measuring the curvature of space is actually very hard to do in practice, so it’s possible that no one would have tried if mathematicians had not told them that it was at least a possibility worth considering.

The mathematics required to fully make sense of curvature was invented in the mid-1800s by Georg Bernhard Riemann, and it is rather intricate. But curved space is a fact of life. In principle, you could discover it by walking around your room, without the need for mathematicians or physicists or philosophers to come up with abstract concepts first.

Euclid, the great geometer of Hellenistic Alexandria, was well aware of the fact that the closing of square paths is not a priori true. Euclid might have said it this way: the inner angles of a square (or of a rectangle or, for that matter, of a parallelogram) add up to 360 degrees. Going around a square means making four 90-degree turns.

Another way that Euclid might have put it is by stating a related fact: that the inner angles of a triangle always add up to 180 degrees. Cut any rectangle into two triangles along its diagonal, and you’ll see why: your four right angles get divided into 6 angles, but the sum is still the same.

But geometry does not have to work that way. When it does, it is called Euclidean. But in the vast majority of cases when it does not, it is called non-Euclidean geometry.

Oftentimes, the way that authors introduce the idea of non-Euclidean geometry is by giving examples of what happens when instead of tracing triangles on a plane you trace them on a curved surface—say, on the surface of the Earth.

So start at any point on the equator and head for the North Pole​. Once you get there, you’ve covered one-fourth of the circumference of the globe, or about 10,000 kilometers. Now turn left by 90 degrees and start walking south. After 10,000 kilometers, you’ll reach the equator again. But you won’t be at the place where you started. Instead, you’ll be at a place 10,000 kilometers to the west of the starting point. Now turn left by 90 degrees so that you’re facing East, and walk another 10,000 kilometers: you’ll be back where you started.

You have traced a triangle on the surface of the Earth—and the inner angles are all right angles, so they add up to 270 degrees, not 180.

Notice that you have only done three legs of your trip. If you were to follow the instructions at the beginning of this post, you would still have another 90-degree turn and another full side to walk. In this case, the failure to close the square would be rather spectacular: instead of coming back to the original point on the equator, you would have ended up at the North Pole.

Tracing squares with sides that are 10,000 kilometers long is kind of extreme, of course. If you were to try a similar experiment with sides of, say, 1,000 kilometers instead, the error would be a lot smaller, but still conspicuous. And if you tried moving in 10 feet legs, you would notice nothing amiss: the world would look perfectly Euclidean to you. You could be forgiven for thinking that the Earth is flat.

In any event, the sphere a totally legit example of a non-Euclidean geometry, but can also be confusing. “Ok, the Earth is curved,” you say, “but what does that tell me about the curvature of space?”

• “What if I had dug tunnels straight across the Earth, joining the two points on the equator and those two points with the North Pole? Together, the three tunnels would form an equilateral triangle. I could then imagine pointing lasers down the tunnels to join the three points with one another into a triangle of laser light. That triangle would surely have angles that add up to 180 degrees.”

Perhaps. But perhaps not.

Space in Outer Space

So here we come to the basic fact of life that I was referring to at the beginning of this post. The curvature of space itself.

To avoid any confusion caused by the Earth, take a trip to outer space. You could think of a spacecraft tracing a triangle or a square by traveling in space. That would not be ideal, though, because it raises all sort of thorny issues about what exactly it means for a spacecraft to fly straight ahead or to turn by 90 degrees to the left.

Instead, you and two buddies each have a spaceship, and each of the three travels to some place in the near universe. Once you’re there, you point lasers at one another and form a triangle of beams.

Laser Interferometry Space Antenna (LISA)
Artist's impression - Courtesy: NASA

Now each of you can measure the angle between the two beams that go in or out of the respective spaceship.

Fact: Those three angles won’t always add up to 180 degrees.

You could do the appropriate calculations and realize that this fact is a consequence of Einstein’s general theory of relativity. Or you could distrust math and physics and just go out to space to see for yourself.

Regardless, this is what it means for space to be curved. Whenever you can find three points in space, and join them with laser beams, and find that the triangle doesn’t have the expected 180 degrees, that means that space is curved.

And when no matter where the spacecraft are the angles add up to 180 degrees–that is what it means for space to be flat.

The mathematical machinery of Riemannian geometry goes much further and actually gives you a way to define and calculate numerical measures of curvature—not to just say if there is some or none.

There are two important special types of curved space. If in a certain region of space, no matter where you place your three spaceships the three angles they form always add up to more than 180 degrees, then the curvature is positive throughout the region. When they always add up to less than 180 degrees, that means the region has negative curvature. In the flat case, it’s precisely zero.

This of course was Part II of god-knows-how many. 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.

Voyager3 wrote:To neufer:

Posting a Victor Borge routine after a discertation about the flatness of the universe (sorry, "nowverse...?)
strangely makes absolute sense to me.

neufer wrote: I really think that you need to get some professional help, Voyager3