The Geodetic Effect: Measuring the Curvature of Spacetime

gravity probe B in curved spacetime
Figure 1. A promotional image created by P. Eekels for Gravity Probe B (GP-B). It shows the GP-B satellite orbiting Earth in a spacetime distorted by Earth’s rotating mass.

A couple of weeks ago, I described the so-called “classical tests of general relativity,” which were tests of early predictions of the theory. This week, I want to tell you about a much more modern, difficult, and convincing test: A direct measurement of the curvature of spacetime. It’s called the geodetic effect. This is the eighth post in my howgrworks series. Let’s get to it.

We know from general relativity that gravity is a distortion of how we measure distance and duration. And that we can interpret this distortion as the curvature of a unified spacetime. When particles travel through this curved environment, they attempt to travel in straight lines. However, the straightest possible path is curved, and this gives the illusion of an acceleration, which we perceive as the force of gravity.

Triangles

General relativity provides us with a very geometric interpretation of gravity, so it is perhaps not surprising that there are measurable geometric consequences of the theory. For example, the presence of curvature changes the way angles work. To see how, let’s consider the humble triangle, like the one shown in figure 2.

a humble triangle
Figure 2. A humble triangle. The sum of interior angles (angles a, b, and c) of every triangle is always 180 degrees.

In flat space (i.e., what we’re used to), the interior angles of a triangle (that’s angles a, b, and c in figure 2) always add up to 180 degrees, no matter the triangle. It’s a theorem, a mathematical fact. (Although it’s not that hard to prove.)

But what about if the space is curved? To answer this question, we first have to define what a triangle is in a curved space. Intuitively, a triangle is a collection of three straight lines. However, as we discussed, in a curved space, there may not be any straight lines! But we must make do with what we have. So in curved space, we build triangles out of lines that are as close to straight lines as possible. These are called geodesics.

As a brief example, let’s try and construct a triangle on the surface of the Earth, which is definitely curved. On the surface of the Earth, the geodesics are segments of the great circles, which are the largest possible circles you can draw on the Earth. The equator is one great circle, as is every line of longitude. The lines of latitude are not. One possible triangle is shown in figure 3.

triangle on the earth
Figure 3. One possible triangle on the surface of the Earth. The edges are segments of the great circles.

But there’s a funny feature of our triangle in figure 3. Can you spot it? Each of the angles a, b, and c are ninety degrees! So the sum of the interior angles is 270 degrees! In flat space, the total sum of interior angles of a triangle is always 180 degrees. But in curved space, this isn’t the case. And indeed, the curvature controls the sum of the interior angles in a triangle. This amazing fact is summarized in figure 4. The difference between the sum of the interior angles of a triangle and 180 degrees is called the deficit angle.

a comparison of triangles
Figure 4. In flat space, the sum of interior angles of a triangle adds up to 180 degrees. Not so for curved space.

What this means is that we can use deficit angles to measure curvature.

Fingers.

In the case of the Earth, we can measure the interior angles of a triangle by simply walking around it with a protractor (or gigantic version thereof).

So imagine that you are a cartoon character on a cartoon world painted with lines and letters as in the illustration. You are at angle a. You are so small that the world looks flat and the lines look straight (and you’ve been told they are straight). You have an arrow. Your assignment: walk along the lines from a to b to c to a, as shown in figure 6. while keeping the arrow pointing in the same direction (it’s pointing toward b now, as shown in figure 5). You set off along ab, carefully keeping the arrow pointing b-ward. You get to b and note that your arrow points at right angles to bc. So you set off along bc, carefully keeping the arrow at right angles to your path. You get to c and note that your arrow is pointing backward along ac. You set off along ac carefully keeping the arrow pointing to your rear. Arriving at a you note that the arrow is now at right angles to ab. (For experts, we are using your arrow as  a stand-in for a tangent vector.) Moving the arrow in this way is called parallel transport.

the beginning
Figure 5. To measure the deficit angle of the triangle, we start by aiming an arrow from angle c to angle b.
gyroscopes man, how do they work?
Figure 6. We carry our arrow around the edges of the triangle. We find that, quite naturally, the arrow rotates.

Notice something? The arrow rotates ninety degrees, as shown in figure 7! It’s no coincidence that the amount that the arrow rotates is the same as the size of the deficit angle. It turns out that parallel transport is one way to define curvature. (For experts, I am exploiting a special case of the Gauss-Bonnet theorem.)

ninety degrees!
Figure 7. If we parallel transport our arrow along all of the edges of the triangle, we find that it’s rotated by ninety degrees.

Spacetime and the Geodetic Effect

So, we we know that we can use parallel transport on the surface of the earth to extract the curvature of the Earth. Let’s go back to general relativity. Can we use the lessons we learned on the surface of the Earth to calculate the curvature of spacetime due to the mass of the Earth? The answer is a resounding yes!

It turns out that in a curved three-dimensional space, a gryoscope works exactly like your arrow stuck on a curved surface. In other words, a real-world gyroscope is a good stand-in for a three-dimensional tangent vector.

If we take our gyroscope and put it in orbit around the Earth, the direction it points will rotate due to the curvature of spacetime sure to the mass of the Earth. I show an extremely exaggerated version of this in figure 8. In reality, the rotation of the gyroscope is not detectable by eye. This rotation is called the geoedetic effect.

Figure 8. An exaggerated representation of the geodetic effect. A gyroscope placed in orbit about the earth precesses due to the curvature of space around the Earth.

Frame Dragging

The geodetic effect as I’ve described is actually one of two effects that cause our gyroscope to precess. The other effect is called frame dragging and it comes from the rotation of the Earth. Intuitively, as the Earth rotates, it drags spacetime with it, causing additional curvature as space and time mix into each other.

A real explanation of frame dragging is a bit too technical for me to get into now, although it’s something I’d like to cover in the future. So for now, I’ll just mention that this is a related effect that we have to take into account. As figure 9 shows, the geodetic effect rotates the gyroscope in the polar direction, whereas frame dragging rotates it in the azimuthal direction.

Figure 9. The geodetic effect and the frame dragging effect rotate the gyroscope in different directions. Image due to the Gravity Probe B collaboration.
Figure 9. The geodetic effect and the frame dragging effect rotate the gyroscope in different directions. Image due to the Gravity Probe B collaboration.

Gravity Probe B

So, this geodetic deviation stuff is all well and good. It’s a nice idea. But is it really measurable? Can we really do this? The answer is a resounding yes! The most direct measurement was made by Gravity Probe B, shown in figure 10.

Gravity Probe B before launch
Figure 10. The Gravity Probe B satellite before launch.

Gravity Probe B is a real experiment almost exactly like the theoretical one I just described. They made ultra-precise gyroscopes and put them on a satellite, which orbited the Earth. And then they watched the gyroscopes precess. The gyroscopes on Gravity Probe B, are real marvels of engineering, by the way. At the time they were created, the quartz spheres used in the gyroscopes, shown in figure 11, were the most perfect spheres ever created by humans. They deviate from perfect spheres by no more than 40 atoms in thickness.

EINSTEIIIIIN
Figure 11. A close-up of part of the gyroscope in Gravity Probe B. It’s being used as a lens for the picture of Einstein in the background. Credit: Gravity Probe B collaboration.

Gravity Probe B measured the geoedetic effect with an accuracy better than 0.03% and the frame dragging effect with an accuracy of better than 1%. Both measurements agreed perfectly with the predictions of general relativity. As amazing as it sounds, we can directly measure the curvature of spacetime.

Lunar Ranging

Gravity Probe B was not the first experiment to measure the geodetic effect. Although it may not be as aesthetically perfect, any spinning object in space can serve as a gyroscope, so long as we can keep track of the axis about which it rotates. For example… what about the moon?

During of the Apollo missions (11, 14, and 15), astronauts planted reflectors (basically fancy mirrors) on the lunar surface. The one from Apollo 11 is shown in figure 12. This allows us to shoot lasers at the moon (yes you read that right—a ground station is shown in figure 13) and have them be reflected back at us. And this in turn, let’s us measure all sorts of things: the distance between the Earth and the moon, the rotation of the moon, the precession of the moon’s axis of rotation, and more.

moon mirror!
Figure 12. The lunar reflector planted during the Apollo 11 mission. Image due to the NASA Apollo archive.

These hugely important measurements are called lunar ranging measurements, and we can learn an awful lot from them. For example, they told us that the moon is moving away from the Earth at a rate of 3.8 centimeters per year. They also told us a lot about the makeup of the moon and allowed us to test the strong equivalence principle, which is a foundational idea behind general relativity.

MOON LASER!
Figure 13. Lunar ranging in action, as viewed from the ground facility at NASA Goddard Space Center. Source: NASA Picture of the Day.

You may also recognize some of the properties I listed as exactly what we need to measure the geodetic effect. And indeed, the lunar ranging experiments did it long before Gravity Probe B, and to similar accuracy. (Gravity Probe B was designed to be much more accurate, but it had some problems with the gyroscopes discovered after launch.)

LAGEOS

We can apply the same principle to human-made satellites as well. And in fact, we custom-built some satellites precisely for this purpose, the LAGEOS satellites, one of which is shown in figure 14.

reflecting sphere
Figure 14. The LAGEOS 1 satellite. Image courtesy of NASA.

The LAGEOS satellites are basically just reflecting spheres sent into space that we can bounce lasers off of. The LAGEOS satellites have actually been used not only to measure geoedetic deviation, but also frame dragging. The original claim is that they only measured frame dragging to an accuracy of about 10%. However, many people are still trying to use the satellites to extract a more accurate measurement, perhaps one that can even rival Gravity Probe B.

Concluding Thoughts

Through general relativity, Einstein provides us with a purely geometric interpretation of gravity. Measurements of gravitational redshift like the Pound-Rebka experiment, which directly measure the distortion of distance due to gravity, are one direct measurement of the geometry of spacetime. But parallel transport and geodetic deviation provide us with another direct measurement, one that makes the curvature of space and time manifest. And that’s very satisfying.

I should note that these experiments have only been performed in situations where gravity is weak. The Earth’s gravity holds us on the surface, but it is far from the most extreme situations…. situations like black holes and neutron stars. Even though these experiments agree with Einstein, we shouldn’t use them to rule out the possibility that general relativity fails for extremely massive objects. We need different experiments for that. And I plan to tell you about one of those in the near future.

Related Reading

This is the seventh part of my series on general relativity. Here are the first parts:

Further Reading

Here is the relevant peer-reviewed and popular material for frame dragging, the geodetic effect and measurements thereof.

Parallel Transport on the Surface of the Earth

I introduced the geodetic effect by describing parallel transport on the surface of the Earth. I learned this material from Differential Geometry and its Applications, by Oprea.

The Geodetic Effect and Frame Dragging

The original calculations of the geodetic effect are in german. However, a translation and modern analysis of the work is available here. Unfortunately, it’s behind a paywall.

Lunar Ranging

There is quite a lot of literature on tests of general relativity using lunar ranging. It’s almost a field into itself. Therefore, I figured the best thing to share with you would be these two review articles, which are free to read and both quite good.

LAGEOS

  • Here‘s the original paper measuring frame dragging using the LAGEOS satellites.
  • Here‘s a Nature News article on the discovery.
  • Here‘s a review of measurements of frame dragging using satellites. (This is a preprint, but it was published in Space Science Reviews.
  • Here‘s a proposal to measure frame dragging with an accuracy of 1% using laser ranging.

Gravity Probe B

  • Because lunar ranging beat it to the geodetic effect and LAGEOS beat it to frame dragging, there’s some controversy about the merit of Gravity Probe B. It’s summarized in this Nature News article.
  • The press release for the results of Gravity Probe B is available on youtube. You can find it here.
  • The Gravity Probe B results paper is available here, but it’s behind a paywall.
  • The journal Classical and Quantum Gravity has released a focus issue on Gravity Probe B, with many free-ro-read papers on the subject. You can find it here.
  • Of particular interest is the summary paper, found here.

Acknowledgements

Thanks to Reddit user John_Hassler for corrections.

 

9 thoughts on “The Geodetic Effect: Measuring the Curvature of Spacetime

  1. At the beginning of your article, you said that it was the 8th article about GR. At the end of the article, you said it was the 7th. Which is it?

    1. Oh oops. It’s both I suppose. When I wrote the article on “our local spacetime.” I had a follow-up article too. “Geodesics in our local spacetime.” But I modified the first article to include all the info that was in the second.

      1. Alright, makes sense. Was a tad bit confused about the numbering, and where the missing article had ended up. Really enjoying your series on GR, and the suggested reading materials are excellent!

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