General Relativity is the Dynamics of Distance

kogler crazy art installation
Figure 1. This art installation by Peter Kogler at the Zagreb Museum of Contemporary Art gives a feeling of what the spacetime we live in might look like, at least in extreme cases. (Source: http://www.kogler.net/dirimart-gallery-istanbul-2011)

This is part two in a many-part series on general relativity. Last time, I described how Galileo almost discovered general relativity. In particular, I told you that gravity isn’t a force. In fact, gravity is the same as acceleration. Now, this is a completely crazy idea. After all, we’re all sitting in the gravitational field of the Earth right now, but we don’t feel like we’re moving, let alone accelerating. But let’s take this crazy idea at face value and see where it leads us.

(Of course, the Earth is spinning, which is an acceleration. And it’s orbiting the sun, which is an acceleration. And the sun is moving in the galaxy. But let’s ignore all that. It’s not important for the argument I want to make.)

But first, we need to make a brief detour  and discuss the Doppler effect.

(If you haven’t read my previous post on why gravity is acceleration, I recommend you do so now. It is here: http://www.thephysicsmill.com/2015/07/26/galileo-almost-discovered-general-relativity/)

The Doppler Effect

The Doppler effect is a bit complicated (especially for light), so I won’t go into too much depth. Instead, I’ll describe it by analogy. (I’ve given the same analogy before, in my article on the expanding universe. So if you remember, you can skip all this.)

Imagine that Paul Dirac and Leopold Kronecker are playing catch, as in figure 2. Each second, Kronecker throws a ball to Dirac, who catches it. Thus, the frequency of balls that Dirac catches is 1 Hertz (Hz)—one per second, or one inverse second.

Kronecker and Dirac Playing Catch While not Moving
Figure 2. Leopold Kronecker (left) and Paul Dirac (right) playing catch. Every second, Kronecker throws a ball to Dirac, who catches it. Thus, the frequency of balls caught is 1 Hz. (Source for Kronecker can be found here. Source for Dirac can be found here.)

But now imagine that Dirac starts backing away from Kronecker, as shown in figure 3. Kronecker continues to throw at a rate of one ball per second. However, since Dirac is moving away from the balls, each one takes longer to get to him. Thus, he catches the balls at a rate slower than one per second…say, one every 1.5 seconds.

Dirac moves away from Kronecker.
Figure 3. Dirac starts moving away from Kronecker. Because it takes the balls longer to reach Dirac, he only catches one every 1.5 seconds, even though Kronecker still throws the balls at a rate of one per second. (Source for Kronecker can be found here. Source for Dirac can be found here.)

A similar thing happens with both light and sound. (In the case of sound, we call it the acoustic Doppler effect.) Light is a wave. It has peaks and troughs which wiggle up and down in time, as shown in figure 4. The number of peaks (or troughs) per meter is called the wave number.  The speed at which it wiggles up and down in time is called the frequency. The two are related by the speed of the light wave, which is always constant, so they’re basically interchange-able.

Wave with labels
Figure 4. A light wave has peaks and troughs. The number of peaks that pass by Dirac in a given second is analogous to the frequency of the wave.

The frequency of a light wave is analogous to the frequency at which Kronecker throws balls at Dirac. Instead of counting the number of times Dirac throws the ball, we count the peaks of the wave. The frequency of a light wave also determines its color; high frequencies are blue, while low frequencies are red.

This means that if Kronecker fires a green laser at Dirac, and Dirac moves away from him, the laser light will appear more reddish to Dirac than it does to Kronecker. This is called a redshift. If Dirac were moving away from from Kronecker at an increasing rate, in other words if Dirac were accelerating, the redshift would be even more pronounced.

Gravitational Redshift

So what does all this have to do with gravity? Well remember, gravity is acceleration. So we should be able to see a Doppler-like effect just by moving from a region with strong gravity into a region with weak gravity, or vice-versa. To see what I mean, imagine that Kronecker and Dirac are up to their old tricks. But this time, imagine that Kronecker is on Earth, and Dirac is in space, as shown in figure 5.

I'd watch a movie about Dirac in space...
Figure 5. Kronecker sends a beam of green laser light from Earth (where gravity is strong) to Dirac in space (where gravity is weaker). By the time it arrives, the light is redder.

Kronecker fires a green laser up at Dirac. Now, remember: gravity is acceleration. Both Kronecker and Dirac are in a gravitational field, so they’re both accelerating. But Kronecker is in a stronger field, so he’s accelerating more. This means that, from Dirac’s perspective, Kronecker is accelerating away from him. Therefore, by the time the light reaches Dirac, he sees it redshifted because of the Doppler effect.

In the context of general relativity, we call this gravitational redshift, and it’s a real effect. We need to take it into account when we read signals sent to us from gps satellites, for example.

Redshift, Distance, Time

The peaks and troughs of light make it an extraordinarily good ruler. If you know the wave number of a wave of light, you can count the number of peaks and in the wave between two places and calculate how far away those two places are from each other. In a very real sense, distance is defined by this procedure.

How, then, do we interpret the redshifted light that Dirac sees? If light on Earth is redshifted when it goes into space, that light stretches out. The distance between adjacent peaks in the light wave grows. Does this mean that distance itself grows?

Yes. It means exactly that.

In a strong gravitational field, distances are shorter than in a weak gravitational field. Indeed, because the wave number of a wave and the frequency of a wave are interchange-able, this also means that times are longer in duration in a strong gravitational field than in a weak gravitational field.

We started with the crazy (but true!) idea that gravity is the same as acceleration. But this has lead us to an even crazier (but still true!) idea: gravity shrinks distance and stretches duration.

This is what people mean when they say that gravity is a warping of space and time (or suggestively, spacetime). The very way that we measure distance is distorted by a gravitational field.

And general relativity is the dynamics of distance.

Next time we’ll talk about how a warped spacetime creates the illusion of a gravitational force.

Further Reading

I took the gravitational redshift argument directly out of the excellent textbook Spacetime and Geometry by Sean Carrol. If you have a good background in math and you want to learn general relativity, I highly recommend it. Here are some other resources:

  • This is a nice video on the Doppler effect.
  • The PBS Spacetime Vlog has an excellent series of videos on general relativity. The first two videos cover what I’ve covered so far, but from a different perspective. You can find them here and here.

14 thoughts on “General Relativity is the Dynamics of Distance

  1. Great series of articles! Can you explain this line a little bit: “If Dirac were moving away from from Kronecker at an increasing rate, in other words if Dirac were accelerating, the redshift would be even more pronounced.” My gut instinct would be that the amount of redshift would *increase* as time passes (as one of them speeds up relative to the other), but this doesn’t necessarily imply that the redshift is more pronounced does it? I’m imaging two experiments measuring the redshift: the first with a constant relative velocity, and the other with acceleration. What if (during the period you measured acceleration) the initial and final relative velocities in the latter test were both still less than the relative velocity the former experiment (where there is no acceleration). In the 2nd experiment, would they measure an (increasing) redshift that is at all times less than the constant redshift in the 1st experiment? Or does the acceleration provide some kind of *extra* effect? If the effect of gravity can be equated with acceleration, then in the case of measuring light traveling upwards, away from the center of the earth, the acceleration must be producing some kind of redshift effect all on its own, since there is no difference in velocity between the two experimenters.

    Thanks for your insight!

    1. Hey thanks! I’m glad you liked them!

      No you’re quite right, if Dirac were accelerating, the amount of redshift would increase with time. So what I meant by it being “more pronounced” would be it would be more pronounced if you wait long enough. Thanks for the catch. It’s good to keep me honest. 🙂

  2. HI, I really enjoyed the article.
    I’m a engineering grad student, I’m not talented in Math but i have the ability to visualize concepts extremely well. Other than math based demonstration, is there a way i could learn more about relativity?

    1. Hi Ryan! I’m glad you liked it! I’ve tried my best in these posts to explain general relativity pictorially. But I’m not aware of any other resources that try to do this.

      There are some math-light textbooks, however. I might recommend the book by Bernard Schutz.

  3. Excellent article! Can I ask you one more question? I wonder if gravitational space contraction and time dilation are so related to each other to preserve speed of light constant.

    1. Thanks! I’m glad you enjoyed it! The answer to your question is a bit complicated. I will make the distinction between SPECIAL relativity and GENERAL relativity

      In SPECIAL relativity, time dilates and length contracts when you move faster. And they do so in exactly the right way so that the speed of light is constant.

      In GENERAL relativity, mass and energy are what distort space and time and these distortions are not necessarily related to each other in an obvious way. And in this case, you must be very careful what you mean by “speed.” There is a sense in general relativity in which the speed of light is not constant. Let me explain. Suppose you are very far away from a black hole and you watch a pulsing flashlight fall into the black hole. As the flashlight falls in, the light emitted by the flashlight will have a harder and harder time reaching you. In a very particular sense, the light is slowing down and the speed of light is NOT constant.

      How can special and general relativity disagree? Well, what’s happening here is that light is travelling at a constant speed through a DISTORTED spacetime. As the flashlight approaches the blackhole, distances are simply growing! By the time the flashlight passes into the blackhole, there is actually an infinite amount of distance between you and the flashlight… and light travelling at constant speed simply can’t cover all that distance to reach you. And so it looks like light is getting slower.

      1. Thank you again for response!

        How does your answer relate to the article statement: “In a strong gravitational field, distances are shorter than in a weak gravitational field.”?

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