A mass curves space time. And so a moving mass will start displacement of curvature that will be travelling throughout the universe. This is exactly what we call a gravitational wave. Now, we can take a specific example in order to get a better picture, better physical picture of a gravitational wave, which is the image of a stone that you throw in a pond. You know that when the stone reaches the surface of the water, it starts some ripples which travel through the surface of the pond. And those are deformation of a surface then we can think similarly of gravitational waves as ripples of curvature throughout the universe. We will discuss in the next sequence what are the sources of gravitational waves.
For the time being, I would like to describe the effect of the gravitational wave passing in this room, for example. I have behind me a distribution, a collection of tennis balls which represents a distribution of masses organised in a circle. Let me imagine that there is a gravitational wave passing in the room through a direction orthogonal to the plane of the circle. There are, in fact, two types of gravitational waves– we say two polarizations. In the first case, the motion of the circle of masses is either horizontal– the formation into an ellipse– or vertical. And because it’s a wave, it’s a periodic deformation. So this is the first type of polarisation.
In the second type, the deformation is now according to axes which are no longer vertical and horizontal but make an angle of 45 degrees. That’s a second polarisation of gravitational waves. And now you can imagine that if you have a material body of some size, then it will be, again, deformed in this direction and alternatively in the other direction. And so that’s through this kind of deformation of a large body that one intends to measure the presence of gravitational waves. In this example, even though we have a visual impression that the balls are moving, they are actually not moving. For example, if we had a single ball and a passing gravitational wave, the ball would just be at rest.
Now, why do we get this impression that the balls are moving? It’s because the distances between the balls are moving. For example, this distance is periodically changing. And why is that? It’s because the passing gravitational wave changes the structure of spacetime. So the texture of space and time changes. The curvature changes. And it is the relative distance between these two balls that is changing with time in a periodic way.
Before moving forward, let me address an argument which was opposed to the existence of gravitational waves. Well, the argument goes as follows. We have seen that gravitational waves are deformation of spacetime. But we are ourselves immersed in that spacetime. So is it clear that we will see any effect of a passing gravitational wave? Now, that argument was opposed by the famous physicist Richard Feynman in a conference dedicated to precisely gravitational waves. And he had the following argument. He was considering a stick of some dimension and a bead that can slide around the stick.
And the argument is called the Sticky Bead Experiment, because the bead is actually sliding along the stick, but with some friction. Now, let us imagine that a gravitational wave is passing through the room and through the setup in this direction. We have seen that the macroscopic object, which is a stick, is going to be deformed by the passing gravitational wave– so in this direction and in the horizontal direction as well. And of course, the bead itself will be set in motion relatively to the stick, because it’s much lighter. And so it would go up and down with a larger amplitude. Now, the bead is sticky, so there is friction in the motion between the stick and the bead.
And so some heat will be produced. Now, the gravitational wave has passed. It is gone. And we look at the setup. Nothing has changed. We still have the stick. We still have the bead. But there is some heat which can be measured. And because of the presence of this heat, we know that the gravitational wave has passed. So that means we can detect the passing of the gravitational waves. So it was this argument that convinced everybody that indeed, gravitational wave have some physical impact on material systems. We have seen that gravity is the weakest of all fundamental forces. So that means that the motion induced by a gravitational wave is going to be very tiny.
We’re going to quantify it in what follows. But for the time being, let me stress a complementary fact. Indeed, once a gravitational wave is produced, it will travel through the universe rather unperturbed by the mass it encounters, because it interacts very weakly with that mass. And so gravitational waves are a very good messenger carrying information from the source that produces them. A wave is a periodic phenomenon which is described by its amplitude, its frequency, and its velocity. Let us start with the amplitude of gravitational waves. And let me consider again the setup of the tennis balls organised in a circle, which I reproduced here.
What we call the amplitude of a gravitational wave is actually the relative change of distance of these balls. So that means this is a ratio between the small distance change of this ball, for example, and the size of the overall setup. And in the case of gravitational waves produced by astrophysical sources, we expect that this ratio is of the order of 10 to the minus 21 to 10 to the minus 24. So you see this is extremely small due to the weakness of the gravitational force.
Now, this means, for example that, in the case of a circle like this one with a diameter of 1 metre– that means that each individual ball will change by 10 to the minus 21 to 10 to the minus 25 metres, so basically impossible to see. If we think of a circle of much larger size, like 1 million kilometres– that’s typically 10 to the 9 metres– then the change of position, the change of distance, will be 10 to the minus 12 to 10 to the minus 15. And so 10 to the minus 12 is what one calls a picometer.
And so this tells you the kind of distance precision that we have to reach in order to be able to measure, to detect gravitational waves.
Let us turn to the frequency of gravitational waves. We’ll return to this topic, but let me just make a comparison with electromagnetic waves. In that case, we know the visible light which corresponds to a fairly narrow range of frequencies. But there is a whole wealth of frequencies actually fourteen orders of magnitude which may correspond to different astrophysical sources– x-rays, radio sources, and so on– which shows the richness of the electromagnetic spectrum. Exactly in the same way, the spectrum of gravitational waves depends very much on the source that produces those waves. And we have, again, a large wealth of frequencies which will allow us to identify different types of sources.
Finally, the velocity of gravitational waves. Well, it is predicted to be the velocity of light. And let me give you the argument. Let us return for a second to light and electromagnetic interaction. If I consider two charges, we have seen earlier that they interact between themselves by the exchange of photons. The photons are massless. They travel at the velocity of light. And the photons, because they are massless, induce a force which has an infinite range. So that means that even two extremely distant charges will interact very weakly, but they will still interact. Now, let us come to the question of interaction between two masses.
Well, the two masses will attract each other gravitationally for the same reason– because they exchange a particle, which is known as the graviton. Now, we know that the gravitational force acts at extremely large distances, even at distances of the size of the observable universe. So that means that the range of the force is basically infinite, which means that gravitons are massless. Now, since gravitons are massless, they travel at the speed of light. And in a sense, we can see gravitational waves as gravitons travelling. And so this is why we assume– we consider– that gravitational waves are travelling at the speed of light. So the velocity of light is identical to the velocity of gravitational waves.
Since I have mentioned the range of a fundamental force, let pause a second to make a very important remark.
We have seen that both the photon and the graviton are responsible for an infinite range force. So that means that even if the charges are at infinite distance or the masses are at infinite distance, they interact very weakly, but they interact. Actually, electromagnetic force and gravitational force are the only known fundamental forces with an infinite range. Now, there is a difference of behaviour which is extremely important. In the case of the electromagnetic force, we have positive charges and negative charges. So if I really put these two charges at even larger distances, there will be somewhere in the universe other charges which will screen the effect of attraction between this positive charge and this negative charge.
This is an effect called screening. And indeed, an electromagnetic force or electromagnetic charge is screened by the presence, for example, in this room of many different charges. So even though in principle the electromagnetic force has an infinite range, in practise, because of its screening of charges, it does not act at extremely large distances. Now, this is different in the case of gravitational force, because all masses are positive. So even if I put these two masses at very large distances, there are no negative masses that would screen the effect of attraction between these two masses. And so that means that both theoretically and in practise, the gravitational force acts at very long distances.
And that is very important to understand why the only force which is active at very large distances, at cosmological distances, at distances of the order of the universe– the only force which is effectively active is the gravitational force. So this is why we should not be surprised to see that the laws of the universe are due to the gravitational force at very large distances.
To sum up, gravitational waves are propagating deformations of curvature which are due to the sudden motion of large concentrations of mass. Because gravity is a weak force, these waves have an extremely weak amplitude. The good point is that they propagate without deformation in the whole universe. And we have seen that gravitational waves travel at the speed of light.