Our Local Spacetime

Gravity Probe B circling Earth
Figure 1. People usually imagine the distortion of spacetime due to the earth as something like this: a dip in the fabric of space. As we’ll see, the actual distortion is quite different. (Source: Gravity Probe B)

General relativity tells us that mass (and energy) bend spacetime. And when people visualize the effect of a planet on spacetime, they usually imagine something like in figure 1, where the planet creates a “dip” in spacetime much like a “gravitational well.” But today I’m going to show you what spacetime actually looks like near a planet… and it doesn’t look anything like the common picture.

This is the fifth part in my many-part series on general relativity. Here are the first four parts:

Dropping the Ball

As we learned, general relativity tells us that gravity is really a distortion in how we measure distance and duration. In the presence of mass, spacetime distorts so that distances are longer or shorter and time flows more or less quickly. Then objects (under no forces) travel along the straightest possible path through this distorted spacetime. And this motion, which doesn’t look straight, is what we perceive as gravity.

But what does this curvature look like? It’s hard to visualize. And as a result, I often get the following question: how does all this work on Earth? If I stand at the top of a cliff and drop a bowling ball, as shown in figure 2, what causes it to accelerate towards the Earth? How does the structure of spacetime make that happen? Why doesn’t it, for example, simply fall at a constant speed? Or simply hold still in the air?

I dropped the ball.
Figure 2. Me dropping a ball (red) off of a cliff.

To understand this, we’re going to try and visualize our local spacetime.

Minkowski Space

Before we talk about curved spacetime, though, I want to remind you what spacetime looks like in the absence of gravity… i.e., when it’s flat. That’s the domain of special relativity. Flat spacetime is called Minkowski space.

In Minkowski space, we give each point (or event) a position in space and a position in time, as shown in figure 3.

Two events in Minkowski Space
Figure 3. Two events in Minkowski space. Event B happens after event A, but both happen at different places.

In Minkowski space, people and objects exist at all times (between birth and death at least), but move between places. The line representing someone or something’s path through space and time is called a worldline. If an object is stationary, the worldline is vertical. If an object is moving, the worldline is at an angle, and the slope of the line is based on the speed at which the object is moving, as shown in figure 4.

Worldlines in Minkowski Space
Figure 4. Worldlines in Minkowski space. Because Jack isn’t moving, his worldline is vertical. Jill, on the other hand, is moving away from Jack at a constant rate, so her worldline is angled at a constant slope.

When working in Minkowski space, it is customary to work in units where the speed of light is one. We do this so that we can convert between position and time, which we treat as two different types of distance. (For instance, a second is the amount of time it takes light to travel 3\times 10^8 meters.)

Using such units, the worldline of a photon is a line forty-five degrees off of each axis–i.e., a line whose slope is one. The worldlines of light traveling away from a point in every direction thus form the light cone for that point.The light cones traveling into the future are future-directed and the light cones traveling into the past are  past-directed, as shown in figure 5.

Light cones in Minkowski Space
Figure 5. Future- and past-directed light cones emanating from event A.

Because nothing can travel faster than light, the light cones determine what events in the past can affect current events and what events in the future can be affected by the present. As shown in figure 6, if event B is in the past-directed light cone of event A, it would be possible for event B to affect event A. However, since event C is outside of the light cone, it can’t possibly affect event A.

Past-directed causality
Figure 6. Because event B is in the past-directed light cone of event A, it can affect event A. However, because event C is outside the light cone, it cannot affect event A.

Visualizing Far From the Earth

Since we can’t visualize a four-dimensional spacetime, we’re going to make some simplifying assumptions. We’re going to imagine that spacetime only depends on how far we are from the Earth, and we’re going to ignore things like lattitude and longitude. This brings us from a four-dimensional spacetime to a two-dimensional one, which we can visualize by putting it into a three-dimensional volume.

However, things are still tricky because we want distances one travels on our two-dimensional spacetime to match up with the distances one travels in the real four-dimensional spacetime. And this is going to distort the image slightly from what we would intuitively expect. Because our visualization preserves distances in this way, it’s called an isometric embedding.

Far from the Earth, we can get the shape of spacetime in our visualization by taking piece of paper with the graph of Minkowski space in figure 3, putting one hand each on the top and bottom of the paper, and lifting it so that the centre sags, as shown in figure 7. Because paper isn’t stretchy and the graph paper didn’t rip, we know distances were preserved.

the arrow of time is blue
Figure 7. Spacetime in our visualization far from the Earth. The red arrow points towards the Earth and the blue arrow points towards the future. Lines parallel to the red arrow are lines of constant time while lines parallel to the blue arrow are lines of constant distance from the Earth.

But wait! I said that spacetime far from the Earth was flat! So in that case, shouldn’t it just look like figure 3 and not be bent like it is in figure 7 at all? It turns out that, in the sense that we care about, both figure 3 and figure 7 are flat. The kind of curvature we’re interested in is exactly equivalent to a distortion of how we measure distance. If the graph paper doesn’t rip, it’s flat. In this sense, any shape you can make from a sheet of paper is flat.

This type of curvature is called intrinsic curvature. A two-dimensional shape is intrinsically curved if one would need to stretch or distort or cut a piece of paper to make it. In other words, if distance changes on the surface of the shape. (There are higher-dimensional generalizations of this too.) There’s another type of curvature called extrinsic curvature, which describes how a surface looks when you put it in a volume. Figure 7 is extrinsically curved while figure 3 is not.

But why do we insist on figure 7 if both figures are flat? Well, flat spacetime certainly could look like figure 3, but if it did, we would run into trouble when we got closer to the Earth. Not all two-dimensional shapes fit in three dimensions and if we want the shape of spacetime near the Earth to fit, while at the same time preserving distances, then the bit of spacetime far from the Earth has to look like figure 7.

Our Local Spacetime

Now that we know what spacetime looks like far from the Earth, we’re ready to explore what it looks like near Earth. Our local spacetime is shown in figure 8.

spacetime!
Figure 8. The shape spacetime near Earth. The red arrow points towards the Earth, the surface of which is a solid black tube. The blue arrow points into the future.

The lines parallel to the red arrow are lines of constant time, and the lines parallel to the blue arrow are lines of constant distance from the Earth. Notice that the surface of the Earth, the big solid black line, is not a point but a line. This is the worldline of the surface of the Earth. Notice also that the lines scrunch together as you approach the surface of the Earth. This is because lengths and durations are actually shrinking near the Earth. We age slightly slower at sea level than we do on an airplane. (This is related to the gravitational redshift I discussed in an older post.)

If it looks like that scrunching together would eventually lead to the lines of constant distance lying on top of each other, you’re right! If I made the surface of the Earth a smaller and smaller radius, then the lines would eventually lie on top of each other. And that would be the event horizon of a black hole. The spacetime wouldn’t stop at the event horizon, of course. It would happily continue. But that’s a story for another time.

I should note that to make the curvature more visible, I’ve stretched out the axis along the red arrow. This means light travels at about 30 degrees off of horizontal, not 45 degrees.

Dropping the Ball Again

So what happens when I stand on a cliff and drop a ball from the top of the cliff? The ball wants to take the straightest possible path through spacetime. Since I don’t throw the ball, I just drop it, it starts in a path roughly like that of the blue arrow. This is a path of constant radius where the only motion is forward in time. It should be roughly visible in the picture that such a path is extremely bendy. The more the ball moves either towards or away from the Earth, the straighter the path.

Of course, because the ball can’t travel faster than light. So a path like that of the red arrow, which is almost a straight line, isn’t valid. The ball has to be within my light cone. Therefore, the worldline of the ball will be some path that travels both forward in time and towards the Earth. And because of the way space and time curve, this will appear as an “accelerating” path.

ballin!
Figure 9. The path of the ball after I drop it. The rainbow line is the path of the ball. Angle changed to make curvature of surface and path clearer.

I plot the geodesic for the ball in figure 9. Note that it approaches a straight line. That’s because as it accelerates it’s approaching the speed of light (we are neglecting air resistance and exaggerating the distance from the surface of the Earth to make that happen). Note also that the speed of light is a straight line that’s wider than 45 degrees. That’s because of the stretched axis. The path of the ball is curved—it curves with the surface, after all. But it’s as straight as it possibly can be. And that’s what makes it a geodesic.

It’s worth noting that a path away from the Earth would also be a valid worldline. And indeed, it would be just as straight as the path towards the Earth. If, instead of dropping my ball, I threw it upwards at escape velocity, this is indeed the worldline it would choose.

If we’d somehow included lattitude and longitude in our visualization, we could have seen worldlines where the ball orbited the Earth too.

Cool, huh? I think that’s enough for now.

Spacetime Isn’t Curved Into Anything

Our visualization exercise today may have lead you to believe that spacetime must be curved inside some higher-dimensional space. After all, to show you the curvature of spacetime near the Earth, I took a two-dimensional spacetime and put it in a three-dimensional volume. But I did this out of convenience, to help us understand what goes on near a planet. In truth, all you need for spacetime to be curved is for distances and durations to distort. And they can distort all by themselves, without depending on a higher-dimensional space.

Play With it Yourself

If you’re interested in exploring our local spacetime, good news! I wrote a Python script that generates the surface I showed you in figure 8. You can find it in the following github repository:

Your plots won’t look exactly like figure 8, because I generated that figure using Maple 16, which makes nicer 3d plots. But it should still be fun to explore.

Further Reading

I created my visualization using the excellent paper Spacetime Embedding Diagrams for Black Holes by Don Marolf. You can find a preprint of the paper here:

I used a black hole to describe the spacetime around the Earth because far from the event horizon, the spacetimes are the same.

Related Reading

This post relied on a a fair amount of special relativity. If you want to learn more about that, you may want to check out some of my older posts on special relativity:

Thanks for reading, everyone! See you next time!

8 thoughts on “Our Local Spacetime

  1. “Time passes slightly faster for us at sea level than it does on an airplane.”

    I thought it was the other way around? Time passes slightly *slower* at sea level than at a higher altitude?

    1. Ah, right. I should have been more clear here. So we age more slowly at sea level than at a higher altitude. (If you think about the pop knowledge about black holes, this is why things appear to freeze near the event horizon.)

      But if you age slower at sea level and look up at an airplane, it appears to be moving faster. That’s what I meant. Sorry for the confusion!

  2. Great post! I think someday when I teach hs physics, I should have a relativity day because I know how badly students want to learn about this. The traditional basketball on a rubber sheet picture definitely has its place for helping students to imagine flat spacetime and mass distorting it, but your investigation and 3d plots have revealed several problems with it:

    1) In that picture, space stretches out more closer to massive objects. As you point out, it should be the opposite.

    2) That picture (basketball on rubber sheet) is so effective because it seems obvious why things accelerate towards big massive objects: when they get close to massive stuff, they go downhill! But whoops, in general relativity there is no more force of gravity, and thus not really a downhill.

    3) Lastly, that picture prevents us from seeing spatialized time, which IMO is the Big Idea in relativity, at least for someone learning about it for the first time.

    I have one small quibble where you write “In Minkowski space, people and objects exist at all times, but move between places.” It sounds like you mean that they exist at all times between their creation and annihilation, but it could be read as suggesting particles have a world-line before they’re created. A philosophically oriented lay reader may be tempted to imagine particles travelling through spacetime before they’re created.

    Like I said, a quibble, not sure why that caught my attention.

    1. Thanks, Vincent! You caught on to exactly the three reasons why I wanted to do this post. 🙂

      I never liked the rubber sheet picture for exactly the reasons you’re describing. When I first saw that analogy, I think it confused me more than it helped me. It bothered me because the story would go something like “mass bends space like a ball on a rubber sheet. And then a particle takes the straightest possible path on the sheet.” But I never understood what people meant by that… mass attracts stuff right? But the straightest possible path isn’t towards the planet, it’s around it.

      (Of course, that’s exactly how gravitational lensing works, but anyway.)

      And thanks for the correction regarding Minkowski space. You’re right about what I meant and I added a clarification. 🙂

What do you think?