**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.

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.

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.

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.

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.

- Last week, I talked about how general relativity is the dynamics of distance.
- The week before that, I talked about how Galileo almost discovered general relativity.
- If you want to know more about the curvature of spacetime, you can read my article on it.
- If you want to know how general relativity is relevant for the history of the universe, you may enjoy my article on the expanding universe.
- If you want to know more about why space and time are united into a single spacetime, you might enjoy my article on Minkowski space.

## 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:

- Are Space and Time an Illusion?
- Is Gravity an Illusion?
- Can a Circle Be a Straight Line?
- Can You Trust Your Eyes in Spacetime?
- General Relativity and the Curvature of Spacetime Explained!

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:

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?

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.

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

Totally true! In fact, Galileo’s experiment was likely apocryphal. I was just using those two objects as an example. I have no idea what they really were. But check out this vacuum experiment where it is a bowling ball and a feather. 🙂

https://youtu.be/E43-CfukEgs

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.

Thanks, David! I’m glad you enjoyed it! And thanks for reading! 🙂

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?! 🙁

Hey thanks for reading! Yes I have two articles on how gravity is not a force! Here they are:

http://www.thephysicsmill.com/2015/07/26/galileo-almost-discovered-general-relativity/

http://www.thephysicsmill.com/2015/08/03/general-relativity-is-the-dynamics-of-distance/

The first link makes the acceleration and mass argument that I think you’re confused by. Hopefully it helps. 🙂