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.

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.

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.

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.

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.

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