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.
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-foot 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.
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 post is part of a series on cosmology. Here are the previous posts:
(On what I mean by "the universe")
(On visualizing a finite speed of light)
(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.
Footprint icon courtesy of palomaironique/Open Clip Art Library.
Spacecraft image courtesy of NASA. The artist's impression represents the planned Laser Interferometry Space Antenna, or LISA, international space mission, which would in fact be unmanned. Also, LISA is not designed to measure the angles of the triangle but temporary changes in the distances of the space probes from one another due to the passage of gravitational waves--which are themselves perturbations in the curvature of space.