For There We Are Captured—The Geometry of Spacetime

All about me there are angles—
strange angles that have no counterparts on the earth.
I am desperately afraid.
~Frank Belknap Long, The Hounds of Tindalos

Whoever…proves his point
and demonstrates the prime truth geometrically
should be believed by all the world,
for there we are captured.
~Albrecht Durer

The Hounds of Tindalos!
The hounds of Tindalos are creatures descended from angles, an important part of today’s discussion (source).

I was recently asked:

What does it mean when we say spacetime is “curved” or “flat?”

The answer lies in the interface between differential geometry and physics.

This is the latest in many articles I’ve written on Einstein’s relativity, so you might want to check out my series on faster-than-light travel. Part 1 explains why the speed of light is constant (as well as what this even means);  part 2 tells us why the constant speed of light imposes a universal speed limit; and part 3 briefly describes general relativity and uses it to predict ways we could attain superluminal speeds. I also wrote a post on Minkowski Space, which simplifies special relativity by treating time as a special kind of fourth dimension.

 Wibbley-Wobbley Timey-Wimey… Stuff

As we’ve discussed previously, the universe is made up of three spatial dimensions and one time dimension. Distance in the time direction behaves strangely; while the square of a spacelike distance is positive, the square of a distance in the time direction is negative. It is this property that distinguishes time from space. (A finicky technical detail: The time dimension itself can wobble and point different directions. Past the event horizon of a black hole, for instance, distance from the black hole behaves like time. You’re inexorably pulled towards the black hole, but not forward in time.)

Curved spacetime deflects light.
A massive object bends spacetime, which deflects a light ray (red) towards the object (source).

Mass can bend spacetime into different shapes, which is what causes gravity. But what shapes (also called geometries) are possible? And what do they mean? An infinite number of shapes are possible, most of which aren’t known—or even possible to write down. So what does it even mean to say spacetime is curved? Let’s try and answer these questions!

What is Curvature?

flat vs curved surfaces
The plane (left) is a flat surface, while the paraboloid (right) is curved.

Simply put, any path that isn’t a straight line is curved. When we think of a curved surface, we imagine something that deviates from the flat plane, like a sphere, a paraboloid, or a helicoid.

some more curved surfaces
Some more examples of curved surfaces: the sphere (left) and the helicoid (right).

However, we can also define curvature by thinking about the straightness of lines. (I’m about to summarize something I wrote in FTL part 3. If you’ve read that article, you can skip to the next section.) Imagine that you’re driving from your hometown of City to the capital, Metropolis, but there’s a mountain in the way. Travelling over the mountain takes more time than travelling around, both because the mountain is tall (adding vertical distance to the journey) and because climbing is more difficult than strolling along on a flat surface.

A three-dimensional picture of what’s going on would show that the ground is curved upward into the shape of a mountain, forcing you to go around. However, it’s possible to encode the same information in two dimensions. If we draw the two paths on a map, the path over the mountain looks straight and the path around it looks curved. However, we define the straight path to be longer than the curved one, even though our Euclidean eyes tell us otherwise.

Incidentally, if the path around the mountain is the shortest possible path between two points (in our case, City and Metropolis), that path is called a geodesic. All objects in free-fall, including light rays, travel along geodesics.

Travell time from City to Metropolis is shorter if we go around the mountain, rather than over it.
In three dimensions (left), we see that, because a mountain is in the way, the red path is shorter than the green path. However, we can encode the same information in two dimensions (right) by defining the green path as longer.

This is what I mean when I say that spacetime is curved. We can measure (using light as a meter-stick) that distance itself changes depending on our position in spacetime. We call this change in distance the curvature of spacetime. It is a curvature in spacetime, not just space, because we can measure the “temporal distance” of that change (a form of time dilation) as well as the spatial distance. Note that the curvature of spacetime affects spatial and temporal distance in ways not predicted by special relativity. There’s more going on here than the constant speed of light.

I should note that there are many types of curvature. The type I want to talk about is Riemann curvature or Gauss curvature—the two ideas are closely related, so I won’t distinguish between them. There’s another type of curvature, called mean curvature, which we won’t get into today.

Types of Riemann/Gauss Curvature

Now we’ve defined “curvature,” but we don’t have any real intuition for it. What does something being curved really mean? In reality, spacetime is a four-dimensional… thing (the technical term is manifold) floating in who knows what. (Actually, there’s no reason it has to be floating in anything. The mathematical definition of curvature doesn’t require a bigger space for our universe to exist inside!) However, since it’s nearly impossible to visualize a curved four-dimensional shape, I’ll describe curvature in two dimensions. The four-dimensional case works similarly, though with some technicalities that I’ll describe later.

Excluding flat objects, there are two types of curvature: positive and negative. I’ll describe positively curved objects first. All positively curved objects look sort of sphere-like, so let’s talk about spheres. Imagine sticking a rod into the surface of the sphere so that it is both perpendicular to and pointing away from the surface, as shown below. If you move the rod along the sphere while keeping it perpendicular to the surface, the rod will always point away from its previous position.

Loosely, this is how we define positive curvature. What’s important is that the rod consistently points in a particular direction relative to its original position. Even if we put the rod on the inside of the sphere so that it always pointed towards its original position, the sphere itself would still be positively curved because the rod’s orientation would always behave in a certain way. Differential geometry that studies objects with positive curvature is called elliptic geometry.

positive curvature image
If we take an arrow pointing away from the sphere (red) and move it along any path (blue or green), the resulting arrow points away from the original arrow. Thus, this shape has positive curvature.

Now let’s compare the sphere to the saddle surface. We stick our rod into the saddle, just like we did with the sphere…but something strange happens. If we move the rod in any one of several particular directions, it points towards its previous position. However, if go in some other direction, the rod points away from its previous position. This inconsistency of orientation is how we define negative curvature. Differential geometry on objects with negative curvature is called hyperbolic geometry.

Negative curvature on the saddle
If we move an arrow pointing away from the saddle (red) along the green path, it will point towards its original position. However, if we move the arrow along the blue path, it will point away from the arrow’s original position. We call this inconsistency negative curvature.

If the surface is flat, then the rod doesn’t change orientation at all when we move it around on the surface. For a surface to be considered flat (provided that its Riemann/Gauss curvature is not negative), there only needs to be one direction in which the rod doesn’t change orientation when we move it. So, surprisingly, a cylinder is considered “flat!”

We consider the cylinder flat!
Even though our arrow points away from its original position if we follow the blue path, we consider the cylinder to be an overall flat surface because the orientation of the arrow doesn’t change if we follow the green path.

Thus, a surface can have positive curvature and be sphere-like, have negative curvature and be saddle-like, or have zero curvature and be flat. Curvature can also be defined locally, at particular points on the surface; for example, there are surfaces that have positive curvature in some places, negative curvature in other places, and zero curvature in still other places!

(Surfaces also have a trait called normal curvature, which describes the way that the rod’s orientation changes on a local scale as you move around a particular surface. It is defined at a particular point on the surface and in a particular direction. To put it another way: Normal curvature measures what the rod does while moving in a single direction along the surface, while Riemann/Gauss curvature measures the consistency and magnitude of the normal curvature as you change direction.)

A comparison of curvatures
A surface can have positive curvature (left) and be sphere-like, have negative curvature and be saddle-like, or have no curvature and be plane-like.

Four-dimensional spacetime works similarly. The space directions work pretty much the same, but more complicated because of the extra dimensions. The time direction is different, though. Because the square of distances in the time direction are negative, positive and negative curvature do some weird things. But the two-dimensional case should give you enough sense of what’s going on for our purposes.

(An important technicality: because spacetime isn’t necessarily in a higher dimensional space, we can’t use the “rod” method to calculate curvature. We use another technique called parallel transport.)

Positively and Negatively Curved Spacetimes

The simplest positively curved spacetime is called de Sitter space, named after Willem de Sitter. de Sitter space is what happens when one direction on a sphere has negative square distance. Rather than folding in on itself, the sphere expands infinitely in the time direction and looks like a Euclidean sphere in all the space directions. This is the simplest model of the expanding universe. (For experts: It’s also a special case of the minisuperspace approximation.)

The simplest negatively curved surface is called anti de Sitter space. I wish I could say that it was named after Willem de Sitter’s evil twin, but it’s just named after de Sitter again. If we take the saddle, and change the time direction to have negative magnitude, it folds in on itself and becomes a sideways hyperboloid, sucking observers towards each other. Although anti de Sitter space looks like time repeats, we can unroll it so that this doesn’t happen. Unfortunately, however, it’s impossible to visualize in three dimensions.

(Fun fact: Lettuce leaves have the maximum negative curvature that fits into three dimensions! Because negative curvature increases surface area, more negative curvature means more leaf surface for absorbing sunlight and carbon dioxide.)

For experts: de Sitter and anti de Sitter space are examples of homogeneous spaces, which are—in some special sense—the same everywhere.

de Sitter space (left) is an expanding circle (sphere in the appropriate number of dimensions). Anti de Sitter space (right) is a saddle point that folds in on itself in time.

 Measuring Curvature

Now that we know what curvature is and what shapes our universe can take, how do we figure out which universe we actually live in? The simplest way is by measuring angles. Imagine drawing an enormous triangle on the Earth, as shown below. Start at the equator and travel along some line of longitude until you reach the North Pole. Next, go down another line of longitude that’s ninety degrees east of your original path until you reach the equator again. Finally, walk along the equator to where you started. The lines you’ve drawn are the straightest lines it is possible to draw on the Earth; the shape you’ve drawn is as close to a perfect triangle as you can get. Incidentally, this is why airplanes going from New York to London fly over the North Pole; their path travels along these straightest possible lines.

But the angles in our Earth-triangle are too large! Each angle is ninety degrees, which means that the interior angles of our triangle add up to 270 degrees! What’s going on? We’re seeing the effects of non-Euclidean geometry. The interior angles of a triangle only add up to the normal 180 degrees if the triangle is drawn on a flat surface. If the surface has positive curvature, the interior angles will always add up to more than 180 degrees. If the surface has negative curvature, the interior angles will always add up to less than 180 degrees.

The angles! The angles!
Triangles on positively curved surfaces always have interior angles that sum to more than 180 degrees.


So… What IS the Curvature of Spacetime?

Simply put, we still don’t know. Experiments tell us that spacetime near Earth is pretty much flat. However, that doesn’t mean the entire universe is flat. Just like the Earth appears flat for those of us who live on it, the universe may appear flat because we can’t see far enough away. If the universe is big enough, relatively small areas of it would appear flat even if the whole thing were quite curved. If the theory of inflation is correct and the universe is even more unimaginably huge than we already thought, we will never know the overall shape of the universe, because it will always appear locally flat to us. (On top of that, because the universe is expanding–and this expansion is accelerating—it will look more and more like de Sitter space as it gets older.)

There are a lot of mysteries in cosmology right now: dark matter, dark energy, the shape of the universe, and more. It’s an exciting time to be looking up at the stars!

Further Reading

Where to start? Differential geometry is a pretty technical topic and I’m not sure there any informal resources on it. I’m sure there are some nice popular-science books scattered around, but I haven’t run across any, so I can’t recommend a particular one. Here’s what I could dig up.

If anyone can find any better popular resources, I’d appreciate a heads up!

Questions? Comments? Hatemail?

As always, if you have any questions, comments, corrections, or insults, please don’t hesitate to let me know in the comments!