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Skip to 0 minutes and 15 seconds I am sure that you all associate Einstein’s theory with this equation.

Skip to 0 minutes and 24 seconds E equals mc squared. Now, in this equation, m stands for the mass, and E stands for the energy.

Skip to 0 minutes and 40 seconds And you have here the constant c, which is the velocity of light.

Skip to 0 minutes and 53 seconds The velocity of light, which plays such an important role in the special theory of relativity, the first theory of Einstein, which was conceived in 1905. Now, indeed, this equation is sort of the emblematic equation for special relativity. And it tells you that mass is a form of energy. Now, this is not the equation that summarises the general theory of relativity.

Skip to 1 minute and 27 seconds Let me first write it in words what is known as Einstein’s equation.

Skip to 1 minute and 36 seconds Einstein’s equation tells you that curvature finds its source in any form of energy. And of course, this is not an equation. Those are just words. So let me write at least once the corresponding equation in order that you can use it for your T-shirt next summer.

Skip to 2 minutes and 11 seconds Here it is.

Skip to 2 minutes and 19 seconds You see, it looks much more complex. We’re not going to use directly this equation. We are just going to use the result. But let me just tell you more or less what this is all about. On the right-hand side, you have here what corresponds to energy. On the left-hand side, this expression here has to do with the curvature of spacetime.

Skip to 2 minutes and 56 seconds And so you see that this equation is really the correspondence between, as I said, curvature and any form of energy. Now, you may notice two constants. First, the velocity of light, we call it c.

Skip to 3 minutes and 16 seconds And so that tells you that indeed, the constant which was the sort of central constant of special relativity is still present when we generalise it to the Einstein’s theory of general relativity. But even more importantly, we see another constant here, which is known as Newton’s constant because it appears in the universal law of gravity which was proposed by Newton. So that is the strength of the gravitational force. And so you see that this equation that equates curvature with energy involves Newton’s constant. And so that means this equation is an equation that describes the gravitational interaction. And that’s a generalisation of the special theory of relativity.

Skip to 4 minutes and 20 seconds Let us pause a little to see in more details what is curvature. Last time, I gave an example, which was the example of a mattress with a big stone in the middle. You have seen that the deformation due to the stone curves the trajectory of balls which I throw along the mattress. Now let me take another example, which is the example of the globe. A globe is a curved space. The curvature is measured by the radius of this globe. Now, if I consider two people walking along this globe along parallel paths starting from the equator, then you see that their parallel paths will meet at the North Pole.

Skip to 5 minutes and 7 seconds And so, in a curved space, parallels are meeting before one reaches infinity. Now, we have seen last time that a concentration of mass is curving spacetime. And so you see that, exactly like on this globe, the light rays are following the path which are the minimal path, like this line here. And so they are bent. And so, even if it started parallel, at some point they might meet because of the concentration of mass.

Skip to 5 minutes and 48 seconds Now, the natural question, then, is regarding the whole Universe. And that’s the question that Einstein is asking. Are all forms of energy present in the Universe curving the full spacetime of the Universe? Now, before answering this question, let us first try to remember what was the Universe in the days of Einstein. Of course, it was basically the same. But what was known of the Universe? Well, at the time, one only knew the stars of our own galaxy, stars which are at a fixed distance. And one imagined that there was a sort of cloud of stars, like in this drawing by the astrophysicist Herschel, which dates back to the end of the 18th century.

Skip to 6 minutes and 44 seconds But things had not changed very much in the days of Einstein. So one imagines that there was a cloud of stars in our own galaxy and a void around it. That was the Universe in the days of Einstein.

Skip to 7 minutes and 3 seconds Some people believed that there were astrophysical objects outside our own ensemble of stars, our own galaxy. For example, the philosopher Immanuel Kant was considering that the Andromeda nebula– you see a picture, one of the first photos of this Andromeda nebula, taken by the astrophysicist Isaac Roberts at the end of the 19th century. Well, Immanuel Kant was believing that this nebula was outside our galaxy, was of extragalactic nature. Now, one had to wait until 1925 and the American astrophysicist Edwin Hubble to be sure of the existence of extragalactic objects. Edwin Hubble, using the telescope of Mount Wilson, a 100-inch telescope, was studying the Cepheids, which are variable stars.

Skip to 8 minutes and 2 seconds And he identified the distance between us and these Cepheid stars as much larger then what we expected to be the size of our own galaxy. And so that means that these stars were actually of an extragalactic nature. One had discovered for the first time, and one was sure that some objects were outside our own galaxy.

Skip to 8 minutes and 33 seconds It was soon realised that these extragalactic objects are moving away from us. This was identified first by the Belgian astrophysicist Henri Lemaitre, and then by Edwin Hubble himself. And they used what is known as a Doppler effect. Let me first remind you what is the Doppler effect with sound waves. And for that, let us hear a siren and just imagine the scene. [SIREN WAILING]

Skip to 9 minutes and 13 seconds So you probably have imagined a police car or fire truck moving towards you and then moving away from you. And the reason you had this impression is because the frequency of the sound has changed. As you can see on this picture here, if you stand on the right-hand side– so if a fire truck is moving towards you– then the frequency is higher. The sound is high pitched. On the other hand, if you stand on the other side– so if a fire truck is moving away from you– then you see the frequency is smaller and this sound is low pitched.

Skip to 10 minutes and 0 seconds Now, in the case of a star and light rays, if a star is moving towards us, then the frequency of a light will be larger. If a star is moving away from us, the frequency of light will be smaller. Now, remember that the frequency of light is associated with the colour of light. You have here a prism that decomposes white light into the different colours. Well, each colour corresponds to a different frequency. And so that means that if a frequency is changed by Doppler effect, then the colour of the light will be slightly different.

Skip to 10 minutes and 45 seconds And so what Edwin Hubble observed in 1929 was that the colour, or if you want, the spectroscopic line that he was observing, of specific bodies which were burning in the star, that colour was slightly shifted. And it is shifted towards the red, it is what we call redshifted if a star is moving away from us. It is blueshifted, so shifted towards the blue, if a star is moving towards us. Now, the spectroscopic lines observed by Edwin Hubble were redshifted, which meant that these extragalactic stars, these extragalactic sources of light, were moving away from us.

Skip to 11 minutes and 42 seconds The fact that extragalactic objects are moving away from us, are receding from us, has been attributed to the expansion of the Universe. So let me illustrate the expansion of the Universe in a classic example, which is the example of an inflatable balloon.

Skip to 12 minutes and 4 seconds So I have here this balloon, which I will inflate a little.

Skip to 12 minutes and 16 seconds And I will draw on the balloon two galaxies separated by a certain distance. And of course, when I inflate more the balloon– you see that the distance has been increasing. And in fact, two galaxies separated by a larger distance than this galaxy would recede from the other one at an even larger velocity. Of course, the different galaxies are not moving on the balloon. What is changing is the texture of the balloon. So in the case of the Universe, the texture of spacetime, which makes the distance between these two crosses or the distance between two stars or two galaxies to increase. So each is static on the balloon, or in the Universe.

Skip to 13 minutes and 22 seconds It is the texture of spacetime in between that is changing. Now, the law that tells you that the velocity of recession is increasing with the distance is called the Hubble Law.

Skip to 13 minutes and 43 seconds Now, there is a question which is often asked, which has to do with the stars of our own galaxy. In the days when we were thinking that the stars of our own galaxy were the only stars in the Universe, one had not observed that recession, that motion away from us. The reason is that the stars in our galaxy are kept at a finite distance. So let me imagine stars in this galaxy. There are two different effects. There is an effect of recession, moving away. But these stars are very close by, so the velocity of recession is very small. And of course, there is the gravitational attraction between the stars of the galaxy. And so this is the important effect.

Skip to 14 minutes and 35 seconds And so that means that the stars of our own galaxy are at a finite distance. It’s only extragalactic objects, other galaxies, for example, that are moving away from us due to the expansion of the Universe.

Skip to 14 minutes and 54 seconds Let us now summarise. We have seen that Einstein’s equations of general relativity establish a connection between curvature and energy. Any form of energy. Now, in 1925 were discovered objects of extragalactic nature. And so that means that there are galaxies other than our own galaxy, which was believed until the days of Einstein. And just later, in 1929, Lemaitre and Hubble discovered that these other galaxies are moving away from us. And so, from then on, one had a model of expanding Universe, which meant that the distances between two given points, two given galaxies, these distances were growing with time.

Skip to 15 minutes and 50 seconds The stars of our own galaxy are not receding from us because of the local attraction of our own galaxy, which is stronger than the effect of the expansion.

Static or expanding universe

Einstein tries to apply his equation to the whole Universe, but what is the Universe? It turns out that it is more complex than anticipated. Henri Lemaître and Edwin Hubble find in 1929 that it is expanding (16:08 minutes: this video is somewhat longer than the others; but there is not more information, it is just that we want you to grasp properly the concept of expansion, a key concept for understanding what follows).

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