General Relativity is the Curvature of Spacetime

Einstein rings are awesome!
Figure 1. The light from a distant blue galaxy is warped and distorted into a ring by the curvature of spacetime caused by a red galaxy. (Source: http://apod.nasa.gov/apod/ap111221.html)

Figure 1 shows light from a distant blue galaxy that is distorted into a so-called Einstein ring by the curvature of spacetime around a red galaxy. This is called gravitational lensing and today we’ll learn how it works.

This is part three of my many-part series on general relativity. Last time, I told you how general relativity is the dynamics of distance, which we know is a consequence of the fact that gravity is the same as acceleration. This time, I describe the consequences of the fact gravity warps distance. And in the process, we’ll learn precisely why gravity looks like a force, even though it isn’t one.

(If you haven’t read parts one and two, I recommend you do so now. You can find them here and here.)

When Distance Warps, Space Curves

First, let’s try to understand what a warping of distance means. We’re going to find that it’s the same as curvature. To understand the connection, let’s go closer to home and imagine a curved space we’re all familiar with: the surface of the Earth.

Imagine that you’re driving from your home town of City to the capital, Metropolis, and that there’s a mountain in the way, as shown on the left in figure 2. Travelling over the mountain takes more time than travelling around, both because the mountain is tall and because the vertical climb is more difficult.

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, as shown on the right in figure 2, 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.

Travell time from City to Metropolis is shorter if we go around the mountain, rather than over it.
Figure 2. 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 g green path to be the longer path, despite what we perceive to be intuitively obvious

This tells us that a curved surface (in this case, the region around City and Metropolis, which bulges out with a mountain) is the same as a surface where distance is distorted. And we can go the other way. A distortion in the way we measure distance implies curvature.

In the context of general relativity, this is what we mean when we say spacetime is curved. Distance has warped such that the straightest possible path is not what you expect.

In Curved Spacetime, Straight Paths Look Curved

Let’s get some better intuition for how curved spaces work. The curved surface we’re most familiar with is the Earth, so let’s see if we can’t get some feel for curvature by exploring how we move around on Earth.

Say you want to go from Narita, Japan to San Diego, U.S.A. What’s the shortest route? Naively, you’d look at a map and draw a straight line between the two cities. However, if you look at Japan Airline’s route map, shown in figure 3, you’ll see something quite different.

Japan Airlines Route Map
Figure 3. The naive route between Narita and San Diego is a straight line. However, aeroplanes actually fly a great deal further north than that (source).

What’s going on (other than the effects of prevailing winds)? It’ll help if we look at the Earth as a sphere instead of as a plane, as shown in figure 4. The straight line between the two cities goes through the Earth, so that’s a no-go. The naive path is just a straight line on a flat map, which in this case keeps our latitude more or less constant; this is doable, but not the best we can do. The best path is a path that goes a bit north.

Paths between Narita and San Diego
Figure 4. Because the Earth is curved, we can’t travel in a straight line between Narita and San Diego (left). A straight line on a map, the naive path, isn’t the best we can do either (center). The best path heads north (right).

What’s so special about this last path? Every path on the Earth must curve, because the sphere curves. However, there’s a portion of the curvature of the path that comes from the curvature of the Earth and there’s a portion that comes from the curvature of the path itself. The latter is called the geodesic curvature. A path that’s as straight as possible—i.e., whose only curvature is the curvature of the surface it’s on—is called a geodesic. This straightest possible path, which has no curvature of its own, will always also be the shortest possible path between two points.

The geodesics for planet Earth are the great circles. These are the circles with the same radius as the Earth; in other words, take the circle formed by the equator and rotate it to make it pass through any two points, as shown in figure 5. A great circle will always cut the Earth into two hemispheres of equal size. However, they will no longer be the Northern and Southern Hemispheres we’re used to.

Great circles are great!
Figure 5. The great circle that connects Narita and San Diego. It has the same radius as the Earth itself.

The lesson that I want you to take away from this is that, in a curved space (or spacetime!), straight doesn’t mean what you think it means. In flat space, a geodesic is a straight line. But in curved space, a geodesic is not a straight line. But it’s the closest thing to a straight line you can get. Indeed, it’s the appropriate definition of straightness.

In curved spacetime, straight lines look curved.

Gravity, Curvature, and Lensing

I told you gravity isn’t a force, but looks like one. We’re now almost ready to understand that. Let’s walk through the argument. The presence of mass, which we typically think of as gravity, distorts distance and time nearby. This, as we just learned, curves spacetime. And in a curved spacetime, straight lines don’t look straight.

Now here’s the clincher.

In the absence of an external force, objects travel along the straightest possible paths, geodesics, through spacetime.

In the absence of gravity, those paths look like the straight lines we’re all used to. But in the presence of mass, they can look very curved.

That, my friends is the gravitational “force.” And let’s be clear. It’s not a force! Particles under the influence of gravity aren’t moving, at least not in the traditional sense. (They’re moving forward in time only.) It’s just that, to us, they appear to be moving because spacetime is curved. This is why, in Galileo’s famous experiment at the leaning tower of Pisa, the feather and the bowling ball fall at the same rate: they’re not falling at all.

Gravitational Lensing

We’re now ready to discuss the gravitational lensing shown in figure 1. The red galaxy distorts the spacetime around it, very much like the “mountain” in figure 2 so that the straightest possible path light coming from the distant blue galaxy behind it is curved. The result is that light gets spread out and “lensed” to form the Einstein ring you see in the image.

Gravitational lensing is a powerful tool. We use it to search for dark matter, to measure the age and size of the universe, and even to look for planets outside the solar system. In short, it’s pretty awesome.

Spacetime Isn’t Curving Into Anything

Now, before I conclude, there’s a common misconception that I want to nip in the bud. People think that, because the universe is curved, it has to be curved into something. In other words, in the same way that the surface of the Earth is a curved two-dimensional sphere embedded in three-dimensional space, the curved four-dimensional universe must be embedded in some higher-dimensional space.

This is wrong.

The universe doesn’t need to be embedded in a bigger space to be curved. All it needs is for the way we measure distance in our own four dimensions to be distorted. We can understand and encode all the information we need about the curvature of spacetime in how distances shrink and durations stretch out. In other words, if you look at figure 2, the left picture isn’t important. Only the right picture matters.

Okay, that’s all for now, folks. Starting next time, I’m going to discuss various different cool properties of general relativity… black holes, gravitational waves, that sort of thing. Exciting!

Related Reading

If you liked this post, you may be interested in some of my older posts on gravity, curvature, and all that.

Further Reading

I took my treatment of general relativity from Sean Carroll‘s excellent text, Spacetime and Geometry. However, there are some great, less technical resources online. Currently, my favorite is this five-part series by PBS Space Time on youtube:

If you’d like some information on the history and context of general relativity and the measurements we’ve made that tell us it’s true, check out these great articles by Ethan Siegal and Brian Koberlein:

16 thoughts on “General Relativity is the Curvature of Spacetime

  1. If gravity is not a force, two questions:

    What is a graviton? Is that a real particle?

    If a body accelerates from rest under the action of the non force gravity, where does its kinetic energy come from?

    1. Hey! Thanks for reading! In order:

      1. A graviton is the particle that corresponds to gravitational waves. Whether this is a physical particle or purely a mathematical tool remains unknown. We need quantum gravity for that, which we don’t have.

      2. Great question! It comes from you, the observer, who is moving with respect to the particle. It’s the same as special relativity. If two inertial frames are moving with respect to reach other, each perceives the other as carrying kinetic energy.

  2. Galileo didn’t drop a bowling ball and a feather, those definitely don’t fall at the same rate at the surface of the Earth. Feathers have air resistance and I really doubt a bowling ball in the modern sense existed for Galileo to drop

  3. Thank you for your interesting series on GR.I think it is important to show that gravity is not a force,as this can lead to misunderstandings.

  4. You lost me in the middle where you explain how objects aren’t moving and how gravity is not a force. Do you have a different article on that particular subject?

    1. “The presence of mass, which we typically think of as gravity, distorts distance and time nearby.”. I understand that a gravitational difference distorts distance and time from the laser experiment, but the whole acceleration and mass argument confuse me.

    2. “In the absence of an external FORCE, objects travel along the straightest possible paths, geodesics, through spacetime.

    In the absence of GRAVITY, those paths look like the straight lines we’re all used to. But in the presence of mass, they can look very curved.”

    But gravity is not a force?

    3. “That, my friends is the gravitational “force.” And let’s be clear. It’s not a force! Particles under the influence of gravity aren’t moving, at least not in the traditional sense. (They’re moving forward in time only.) It’s just that, to us, they appear to be moving because spacetime is curved. This is why, in Galileo’s famous experiment at the leaning tower of Pisa, the feather and the bowling ball fall at the same rate: they’re not falling at all.”

    Huh?! 🙁

  5. Einstein NEVER said nor promoted the view “that gravity is the curvature of spacetime.” That he did is a misconception. He was emphatically against that. Einstein very much viewed gravity as a FORCE. Correct your article!

What do you think?