In his Special theory of Relativity, which we were considering in the previous chapter, Einstein resolved the physical conflict which regarded the speed of light and our intuitive understanding of motion. He showed that our perceptions of motion need be changed when we are talking about objects that are moving extremely fast. However, soon after that Einstein and other physicists realized that one of the main concepts of Special Relativity, which tells us that nothing can move faster than the speed of light, is in contradiction with Newton’s theory of gravity. By solving the first conflict Special Relativity lead to the other. 10 years later Einstein solved this conundrum in his General theory of Relativity (I shall denote it GR later on) whereby he dramatically altered our understanding of reality once again. In this article we shall be concerned with the main principles and consequences of GR. You can find a more detailed and a very clear explanation of these matters in Brian Greene’s book “The Elegant Universe”.

**The Inconsistency between Newton’s Mechanics and Special Relativity**

Newton was an absolutely astonishing physicist. The starting point of modern science dates back to his works in the XVII century. He had such a mighty intellect that when he realized that his works require some mathematical apparatus which had not been discovered yet, he found it! We now call this apparatus “Differential Calculus”. (Apart from Newton it was independently discovered by Gottfried Leibniz.) And what’s remarkable about Newton’s theory of gravity is that it is still used by scientists in many situations, such as the calculation of satellites’ trajectories in order to reach any point in our Solar system.

Newton perfectly described how gravity works but what he left unanswered in his theory is the question as to what gravity actually is. According to Newton, gravity depended on just two parameters – the mass of the object which exerts the gravitational pull on you and the distance between you and that object. However, this is the force which makes Earth orbit the Sun at the average distance of 93 million miles. How could this force be exerted on Earth without any direct interaction? In Newton’s picture gravity was some mysterious force which acted *instantaneously* through immense distances. For example, if, all of a sudden, the Sun exploded, the Earth would immediately go off its orbit despite the fact that we would still see the Sun shining in the sky, and only in roughly 8.3 minutes we would obtain the visual information about the catastrophe.

And this is exactly where the profound inconsistency between this extraordinarily precise theory and Einstein’s Special Relativity emerges. Einstein’s theory, as we have seen in the previous article, restricts everything from travelling faster than the speed of light. And not only is an object not capable of travelling faster than light, but so is any interaction or perturbation. That is to say, any sort of information cannot propagate faster than light. *Nothing* can surpass a photon, *including gravity*.

Therefore at the start of the XX century Einstein realized that there was a direct contradiction between his theory of Special Relativity and Newton’s theory of gravity which almost nobody had doubted in for several centuries. Being absolutely confident in the correctness of Special Relativity, Einstein started looking for a new theory of gravity which would be consistent with his former theory. This searching has eventually led him to the General theory of Relativity which again forced us to reassess our views on the working of the Universe.

**The Equivalence Principle**

In 1907, while working on a new model of gravity, Einstein managed to feel around for a way to do that. It is based on the physical concept which had been known since the times of Newton. In the early 1900s Einstein applied this concept to formulate his “equivalence principle”. To get an idea of what it means we should first take the question of the different sorts of mass into consideration. (Sorry, a bit of maths is coming out!)

The first kind of mass is called the inertial mass, and it is given by the second law of Newton with the famous formula **F = ma**, where **F** stands for the force which you need exert on an object to make it accelerate, **m** – for the mass of that object, and **a** – for the acceleration vector of the object. What this means is that the force that you must exert on an object to make it accelerate depends on the mass of the object and on how much you want it to be accelerated. That is, the more mass an object has the more force you have to exert on it (the harder you have to push it) for a similar acceleration. Imagine two things, e.g. a smartphone and an empty cup, on your table and let us say that the smartphone weighs 1 kg, whilst the cup weighs 2 kg (those are a pretty heavy smartphone and cup, aren’t they?). So to make them both accelerate at the same rate you have to push your cup twice as hard as the smartphone. Note that we are ignoring the friction against the table and those kinds of things here.

The other sort of mass has to do with gravity. As you might have heard, any object exerts a gravitational pull on other objects. As I have mentioned in the previous section, the force of gravity is given by Newton’s law of universal gravitation and is expressed mathematically as **F = Gm1m2/r^2**. Here **F** refers to the force of gravity, **G** stands for the universal gravitational constant, **m1** and **m2** – for the respective masses of two bodies under consideration, and finally **r** – for the distance between these bodies. Since the masses of both objects appear in the numerator, bodies of higher mass exert a stronger gravitational pull. And the squared distance in the denominator tells us that the force of gravity weakens with the squared rule when the distance increases. That is, if you measure the gravitational force between your smartphone and the cup which are located, say, one meter apart, and you obtain some value, then this value will be 4 times less when you increase the distance between them twice (make them 2 meters apart), and 9 times less if you put them 3 meters apart.

The interesting thing about all this emerges when you are combining these equations together to calculate the acceleration caused by the force of gravity. In this case we get **Gm1m2/r^2 = m2a. **Here we are assuming that M1 represents a heavier body and M2 a lighter body (e.g. an apple falling down from a tree under the force of gravity exerted by the Earth). So in our example **G** stands for the universal gravitational constant, **m1** – for the Earth’s mass, **m2** – for the apple’s mass, **r** – for the distance between the barycenters of the Earth and the apple, and finally **a** – for the acceleration rate at which the apple falls down. What’s particularly interesting for us here is that we now have the mass of our apple on both sides of the equation, which means that we can divide through it such that this parameter has no effect on the resulting value. What this tells us is that *whatever mass a falling object has, it will accelerate at the exact same rate under the force of gravity*!

I hope that you have been following this mathematical description because if you did, you now conceptually understand the equivalence between inertial and gravitational masses. What we shall see next is how Einstein applied this concept and showed that acceleration and gravitational pull are actually two sides of the same coin!

It is conceptually well presented by the famous thought experiment. Suppose you are in a spaceship floating in a completely empty space. It neither moves in any direction nor experiences the force of gravity. You hold your smartphone and your empty cup, whose weights are different (1 kg and 2 kg), and you let them go at the same time. Since you are in weightlessness, it should not be very surprising for you that both weights will just go around next you and will not move in any particular direction. Now what happens when we introduce acceleration? The smartphone and the cup are still floating next you, therefore the spaceship’s floor has to meet them both at the same time irrespective of their masses. And this is exactly the same situation that we’ve got when we considered an object falling down in the presence of the gravitational field. You can see it shown on the picture below. Although here we have only one weight, I think you get the idea.

What Einstein showed in his theory is that these situations are actually completely equivalent. If you are placed in a sealed box, instead of being in a spacecraft, you will have no chance to tell whether you have been pulled towards the floor because the force of gravity is pulling you towards the floor or because the box itself is accelerating upwards. And there is no experiment that would allow you to distinguish between these two things. All the experiments which you carry out in an accelerated reference frame will give you the exact same results as the experiments conducted in a stationary reference frame with the presence of the gravitational field.

In the previous article we were talking about the physical indistinguishability between the points of view of two different observers moving with a constant speed relative to each other. Particularly, you might recall that if you were moving inside a train that moves with a constant speed with no acceleration, you won’t be able to define the speed at which the train is moving, or even whether or not it is moving at all. But if we introduced acceleration, you will immediately recognize that the train is moving. What Einstein showed in General Relativity is that if you take into account a corresponding gravitational field you can see that the laws of physics are completely invariant not only for objects moving with a constant speed, but for *any object irrespective of its motion*! Thus, GR has completed the work started in 1905 by Special Relativity.

**Accelerated Motion and the Curvature of Space-Time**

The equivalence principle was a very important step towards the formulation of GR, but in order to achieve his goal Einstein had to go further and explore how gravity actually works. Luckily for him, a piece of mathematical apparatus had already been developed by an astonishing XIX century German mathematician Bernhard Riemann, which allowed Einstein to work on his new theory explicitly. Based on the concepts provided by Riemannian geometry Einstein was able to build his notion of curved space and time which I shall try to explain next.

For this we shall consider a carousel which is an amusement ride consisting of a rotating circular platform with seats for riders.

Suppose we’ve got the specification for the carousel in question in which we can see its circumference and radius. Using basic Euclidean geometry we obtain the ratio of the circumference to the radius as being equal to **2π**. However, this specification gives us the measure of these parameters while the carousel is stationary. Now suppose we see it only in motion, such that it never stops. If we now measure its circumference and radius, would our results be consistent with the specification and would the ratio of the circumference to the radius still equal **2π**?

What I should emphasize is that the motion of our carousel is *accelerated*, because its direction constantly changes, so it differs from motion with a constant speed which we were concerned with in the previous article.

If we ask our friend Jim to perform this measurement, and he starts with the circumference by putting his tape measure to it, we would immediately see (if we happened to observe this act from above) that his result would not be the same as in the specification (and as we see it from above as well). This is due to the Lorentz contraction which we talked about in the previous article. Since the carousel is in motion, Jim’s tape measure *is getting contracted* *along the line of the motion*, but Jim himself is certain that it is still of the same length since there is no relative motion between him and the tape measure. But because the tape measure is contracted Jim would obtain a greater value for the circumference than we had in the specification.

Now you might think that when Jim measures the radius of the carousel he will also obtain a greater value so that the ratio would still be **2π**. But not that fast. When Jim measures the radius with the same tape measure, its length *is not contracted since it is now placed perpendicular to the direction of motion*! Thus, the value for the radius obtained by Jim would not change. What this means is that the ratio of the circumference to the radius of our carousel is *greater* than **2π** according to Jim’s measurement.

But how could this be true? According to the geometry of The Ancient Greeks, *any* circle must have this ratio being equal to **2π**. Have we just found the case where Nature avoids mathematical description? Well, not actually. As Einstein explained, Euclidean geometry still holds good in any situation where we depict a figure on a flat plane (e.g. a piece of paper or a table), but the shape of this figure will be distorted if it is depicted on a curved plane, such that the aforementioned ratio for the circle would not be **2π**. That is shown on the picture below. Here we can assume that the radii on every circle are the same, but since the radial lines on the circle depicted on the spherical surface are converging, the resulting circumference would be less than **2πr** which is the case for the standard Euclidean surface. Likewise, the circumference measure of the circle depicted on the hyperbolic surface would not be equal to **2πr** either, but here this value is greater than **2πr**, which was the case in our example with the carousel.

Similarly, our millennial notions of triangles need be altered when we are talking about curved surfaces. The standard rule according to which the sum of three angles in a triangle always equals 180° does not hold in a triangle depicted on a curved surface. A triangle on a positively curved (spherical) surface has angles whose sum is greater than 180°, whereas a surface with a negative (hyperbolic) curvature would have triangles whose angles sum up to less than 180°.

These ideas led Einstein to conclude that such violations of Euclidean geometry are due to the curvature of the very space-time fabric. When one moves with acceleration, Euclidean geometry rules no longer hold from their perspective.

Okay, now we’ve got an idea of what space curvature is, but what do we mean by saying that time is also curved when we are moving with acceleration? As we previously saw, Special Relativity declares the unity of space and time. Consequently, if we say that something is true for space, it must also be true for time. But how could we make conceptual sense out of the curvature of time? To achieve this let us join Jim in our experiment on the carousel. Let’s ask him to stand at the edge of the platform and we shall stand at the platform’s center. Then we shall start slowly converging him and compare the rate at which time is passing for ourselves and for Jim. Here we should take notice of the fact that the greater someone’s distance from the center, the greater their speed. This is because they must pass a greater distance to finish one full cycle of the carousel. In this sense, Jim is moving faster than we do if he is standing further from the center. And as we know from the concepts of Special Relativity, the greater the speed of your motion through space, the slower time elapses for you! This tells us that our clock is ticking quicker than Jim’s clock does, and unless we approach him this will be true. This is what we mean by the curvature of time due to accelerated motion. Time is curved when its rate is being changed.

We have considered two main ideas which allowed Einstein to make his final step to the new theory of gravity. After he showed that gravity and accelerated motion are, in a sense, two sides of the same coin, and that accelerated motion is directly related to the curvature of space and time, it was obvious to conclude that gravity itself represents nothing but the curvature of space-time. Let’s see how we can make intuitive sense of this.

**The Curvature of Space due to Gravity**

Let us start again with the curvature of space. According to GR, empty space represents a completely flat surface on which a pictured triangle will be perfectly well described by Euclidean geometry; hence its angles will sum up to 180 degrees.

Now if we ask what happens when a massive body is present on that surface, Newton’s theory of gravity would say that nothing happens at all, since space is just a background on which events take place. As Einstein showed, however, the presence of mass curves the very structure of space.

In the two images above we see the 2 dimensional representation of both flat and curved space. We can also get a good visual representation with a rubber band analogy. Imagine such a rubber band. With no objects on it, it represents empty space, and if we introduce a very light spherical object (e.g. a ping-pong ball) and give it a push, it will follow a straight path until it reaches the edge of the surface of our band. Now what would happen if we placed a massive body at the center of the rubber band? Imagine we put a billiard ball there. The rubber band now curves due to the presence of this ball, and if we now place our ping-pong ball and give it an appropriate push – not too weak and not too strong – it will follow a circular path around the billiard ball. Moreover, if we could ignore the friction against the rubber band, our ping-pong ball would settle into orbit around the billiard ball. And this is exactly what happens in space, such that in our Solar System. The Sun, being a body of a huge mass, bends space around it, so that objects of less mass – such as the Earth, Venus, Jupiter and all the other planets, and asteroids as well – settle into orbit around the Sun.

What’s also important is that all the planets and asteroids are massive objects as well. Correspondingly, they bend space around them also. And if we recall, the force of gravity between two bodies depends upon their mass and the distance between them. That is why planets, possessing much less mass than the Sun, can capture objects which get too close to them. This is why the Moon has settled into the orbit around the Earth, just as a great number of moons of both Jupiter and Saturn have settled into orbits around those planets. In this sense, when a parachutist jumps off their air vehicle, they glide down into the well in space caused by the presence of the Earth.

Now we see that Einstein did explain how gravity really acts. It does that through the curvature of space. Fascinating!

The last thing that I want to consider in this section is that our analogies with the pictures shown above and with the rubber band are incomplete, albeit very helpful. For one reason, both of them are 2 dimensional, but space has three dimensions (according to String theory it has even more, but we won’t consider this in this article). So our analogy is somewhat limited, because the mass of the Sun actually bends space in three spatial dimensions, not just two.

The second problem with the above analogy is that the rubber band is bent because something pulls it down. As we have seen above, this is not what actually happens. The reason why objects settle into orbit around massive bodies such as the Sun is simply that the motion through the path *around* this object fulfills the principle of stationary action. Or simply speaking, when a celestial body follows this trajectory, it experiences the least resistance; hence this trajectory is the most stable.

The last problem with the rubber band analogy is that it takes into account only space and leaves time behind. Although it is good for a construction of a conceptual picture of the curvature of space, it is fundamentally incomplete, because as we already know, space and time are indispensable to each other. Thus, in order to encompass the entire picture we must consider time, which we shall do in a moment.

**The Resolution of the Contradiction**

Okay, this is all well and good, you might say, but what about the contradiction which we started this article with? Does GR resolve it? It does! Let’s consider the rubber band analogy once again. If our band is flat with no massive objects on it, and we introduce the ping-pong ball, it follows a straight path. If we suddenly put our billiard ball there, the direction of the ping-pong ball’s motion will change, but *not instantaneously*. If we, for example, take a video of this and watch it in slow-motion, we shall see that it takes some time for the perturbation of the rubber band structure to reach the ping-pong ball and change the direction of its motion. That perturbation resembles waves in a pond after you drop a stone into it. The rate of the perturbation’s propagation depends on the particular material of which our rubber band is made.

The same is true for the structure of space. If an object moves in empty space, and a huge mass suddenly appears at some distance, the gravitational perturbations will reach the object after some time has passed. And what’s important, Einstein calculated the speed at which these perturbations propagate, and this speed exactly equals the speed of light! In the situation with the Sun’s explosion, which we considered at the start of the article, the gravitational perturbation caused by this catastrophic event would reach us at the exact same moment as light, roughly 8.3 minutes. Thus, in this case we shall not obtain the information about the catastrophe until we see it with our eyes. This showed that the main feature of Special Relativity, which tells us that nothing can move faster than light, is correct, and Einstein proved it once again in GR.

**Gravitational Time-Dilation**

From the pictures shown above, we can intuitively understand how space warps due to the presence of mass, and now we need concern the curvature of time due to gravity. This question is not that trivial, since we do not have a picture of time in our heads. But we can approach it by looking at an example.

Let us assume that we with Jim use two spaceships both with very accurate clocks precisely synchronized in advance. Jim will approach the Sun, while we will stay sufficiently far away from it, and we both will compare the rate at which time elapses for us and for Jim. When he starts moving away, the rate would be the same for both clocks. However, as he gradually approaches the Sun, we could notice that his clock is ticking more slowly than our clock does. But for an ordinary star such as our Sun, the time dilation effect is very tiny. If our spaceship is located, say, 1 billion km away from the Sun, and Jim’s one is very close to its surface, his clock would tick just 0.0002 % slower than ours. But if he was near the surface of a neutron star, the time dilation would be very noticeable, up to 76 %. And for a black hole it is even more extreme. This is saying that the stronger the gravitational field, the more it curves the very structure of both space and time.

What I should also mention is that we are already familiar with the concept of time dilation from the previous article, and here I am also talking about time dilation, but the effects are slightly different. If you recall, previously we compared two observers moving relative to each other at a constant speed, and we concluded that both these observers could say that their clocks tick with a normal rate and that the other’s tick more slowly. That is to say, there is a symmetry between their points of view, so that both of them are correct. However, when Jim approaches the Sun he certainly feels its gravitational pull, so that he feels that he is exposed to gravity. What this tells us is that we’ve lost the symmetry in this situation, and now the passengers on *both* spaceships are certain that Jim’s clock is ticking slower than ours. Both of these effects are described in a similar manner, but they slightly differ in details as we’ve just seen.

**Experimental Confirmation of GR**

As we’ve seen, GR provides an amazingly elegant description of gravity. It shows that the very unity of space-time and gravity is far more dynamic than in Newton’s picture. But regardless of its elegancy, we want an experimental confirmation of any theory. If a theory in question eludes experimental confirmation then it must be thrown away despite its beauty.

Newton’s theory of gravity had been experimentally confirmed again and again, but in the XIX century a French mathematician Urbain Le Verrier established that the orbit of the planet Mercury is slightly shifted relative to the Newtonian theory’s predictions. This anomaly was titled “the anomalous precession of the perihelion of Mercury” and there were various proposals to the solution of this contradiction, such as the gravitational influence of an unknown planet close to the Sun, the flattening of the Sun and several others. However, none of them were particularly robust. In 1915 Einstein calculated the precession using the equations of his new theory and obtained the result which *exactly* matched the observed value of the precession. For Einstein this was a fantastic success of his new theory. Most other physicists, however, wanted a prediction of yet unknown phenomena, rather than the explanation of an existing anomaly. This is not a surprise because it is how science actually works. A new theory will not be very robust until it makes some testable predictions, and they are verified by experiment. And Einstein made such a prediction.

Sometimes we are fascinated while looking at the starlit sky. We cannot usually see stars in the skies in the daytime, because their light is too dim in comparison with sunlight. However, during a solar eclipse this can be done, since the Moon blocks a sufficient amount of sunlight for stars to be visible in the sky. And this is where Einstein’s prediction comes about. GR predicts that mass bends space, and the higher an object’s mass the greater curvature it causes in the space-time structure. And when the light from a distant object passes very close to the Sun’s surface, it must be deflected from its straight path by the magnitude which is easily derivable by applying relevant mathematical concepts.

This causes a star to appear slightly shifted from our perspective here on the Earth. This shift then can be compared to the actual position of a star (obtained when its light does not pass close to the Sun). In 1915 Einstein calculated the magnitude of such a shift being equal to 0.00049 degrees (1.75 arcseconds). This is an extraordinarily tiny angle, but our tools at that time could already obtain such a precise measurement. On May 29, 1919 two British teams of astronomers made their observations of an apparent position of a star, whose light passed very close to the Sun, during a solar eclipse. One of these teams, which made their observation at Principe, the island in West Africa, was led by Sir Arthur Eddington, and the other group, led by Charles Davidson, observed the apparent position of a star from Sobral, Brazil.

After these observations were made, the data was analyzed for roughly 5 months, and eventually it was declared, at the summit of the Royal Society on November 6, 1919, that Einstein’s prediction is confirmed. The Sun, as well as other stars, deflects the trajectory of light which happens to pass nearby. This was Einstein’s moment of glory. The news of this success spread all over the world in a short time, and on the next day after the summit the main article in London “The Times” read about a new revolution in science and about the downfall of Newton’s ideas. Since that time, there was no single experiment giving a result inconsistent with GR. But some of GR consequences were too extreme even for Einstein himself. Nevertheless, even they have been shown to be true. This will be the topic of the last two sections of this article.

**GR Consequences: Black Holes**

Soon after Einstein finished his works on GR, German physicist and astronomer Karl Schwarzschild managed to derive a very precise picture of the space curvature in the vicinity of an ideally spherical star. What’s important is that Schwarzschild’s solution of the GR equations showed that if the mass of a star is squeezed to a very small volume, the gravitational pull of this star becomes too strong for even light to escape the region around the star which we now call event horizon. And as we know from Special Relativity, nothing can move faster than light, so if even light is not capable of escaping such a region then nothing can escape it at all. At first, such theoretical entities were called “Dark Stars”, or sometimes “Frozen Stars”, but then John Wheeler suggested the name “Black Holes” which has persisted.

We can start by looking at the image above. Here we can see a dramatic distortion of space-time fabric caused by a black hole. The orange circle here shows the event horizon, a point of no-return. When anything crosses it, there is no way for it to get back out of the black hole. But although the gravitational influence of a black hole is enormous in the regions close to the event horizon, you won’t feel any gravitational difference between a regular star and a similar mass black hole if you are at a safe distance from it. That is, if our Sun turns into a black hole, our planet would orbit it the way it usually does, because what matters, as we saw earlier, is the mass of the body exerting a gravitational pull on you, and your distance from it. So the only reason why the replacement of the Sun with a similar mass black hole would be bad for us is because it will stop shining. If you are not familiar with the physics of black holes, and try to cross an event horizon, then you will be exposed to extremely strong tidal forces. If you are falling into a black hole and your feet are, say, 2 meters closer to the BH than your head, then your feet will be experiencing much stronger gravitational force than your head does. Because of this, your body will have been being stretched until it is torn apart. Physicists even gave this process a humorous name, “spaghettification”.

If you know the principles of GR and do not cross the event horizon, you could use a black hole as a time-travel machine. Suppose the BH has a mass of 1,000 solar masses, and you come very close to its event horizon but don’t cross it. As we have previously seen, the gravitational field of an object with a huge mass curves time in such a way that it passes more slowly if you are close to that object. So let us say you get extremely close to the event horizon, 3 cm above it. In this case the passage of time for you will be slowed down *very much*. In fact your clock would tick roughly 10,000 times slower than the clock of your friend on Earth. That is to say, if you stay there for only one minute, the Earth will count roughly 7 days at the same time! And if you stay there for a year, you will return to the Earth more than 10 thousand years later.

To give you a sense of scales involved, I shall provide a couple of examples. To turn a star into a black hole we have to squeeze it such that its radius reaches its “Schwarzschild radius” value. Our Sun, for example, has to be squeezed into a sphere with a radius of roughly 3 km. You can get a sense of the density involved by looking at the actual radius of the Sun, which is roughly 695,800 km. So if you somehow managed to squeeze all of its mass to the size of Manhattan, then it will have become a black hole. An object of Earth’s mass has to be squeezed to the sphere with a radius less than centimeter. A lot of physicists were skeptical to the possibility of such extreme configurations of matter, but the existence of BHs has now been observationally tested to an extraordinary level of confidence. This can be done by observing a star, typically a red giant, orbiting some invisible companion. When the companion is much denser, it strips off the material from the red giant. This material then spirals on the companion and heats up to enormous temperatures emitting very bright X-rays and visible light. Observing such a binary we can gather the necessary data to calculate the mass of the companion and its size. Some of these companions turn out to have their radius less than Schwarzschild radius, which is a direct indication of the black hole nature of the object in question.

Apart from this, we can observe the behavior of stars close to the center of our galaxy. They turn out to move so fast that we can calculate that there is an object out there whose mass is roughly 4 million solar masses. But even this fades out in comparison with quasars, the objects so bright that they easily outshine their entire galaxies. These quasars are powered by black holes at the centers of galaxies, and the masses of such black holes are *billions* that of the Sun! This was a very short description of black holes, and I encourage those who are interested to read Peter Cooper’s article on this topic.

**GR Consequences: The Big Bang and the Expansion of the Universe**

The most profound consequence of GR was in showing that the Universe is not static as had been thought for ages. At the start of 1920s Russian physicist and mathematician Alexander Friedmann used Einstein’s equations to draw this conclusion. As he showed, the Universe as a whole cannot be static; according to the GR equations, it must either contract or expand. This was too much even for Einstein himself, so that he slightly modified his equations to return ourselves to the comfortable conditions of a static Universe. He added a new parameter to the equations which is now called the cosmological constant. However, several years later American astronomer Edwin Hubble experimentally established that the Universe is indeed expanding. Despite Einstein’s reluctance to accept this conclusion, his theory *predicted* it! And it was confirmed again and again, so there is no doubt at all that the Universe is indeed expanding.

Since we now know that the Universe is getting bigger with time, we can imaginatively reverse the flow of time to study its origins. When we go back into the past, the Universe is contracting, galaxies gradually come closer, the density of the Universe increases. If we go back in time for roughly 13.8 billion years, any complex structures could not exist at that time, and all the matter was in the form of hot plasma with an unimaginable density. If we go even further, the entire Universe is squeezed to the size of a planet, then to the size of an apple, and eventually becomes just a dimensionless point which we call a singularity. (We shall look at the models that slightly reassess this picture, such as Inflationary cosmology and String theory, in later articles, but the Big Bang theory itself will in no way be reassessed, only some of its details.) According to our modern understanding of the Universe, this was a starting point, the Big Bang. And this theory is now confirmed experimentally, since it predicted such things as the Cosmic Microwave Background and explained the abundance of various chemical elements in the Universe. Both these predictions and several others give the expected values which match the observed results extraordinarily well.

And apart from everything else, GR’s concepts could tell us about the ultimate fate of the Universe. I shall not dig deep into this right now, but the shape of the whole Universe could shed light on what is going to happen with it. In a closed universe, which is positively curved – an ellipse – its expansion eventually stops and changes to contraction. This will lead to the collapse of the universe to a final singularity, termed the “Big Crunch”. An open universe, which is negatively curved – a saddle – expands forever and ends up with either “Big Freeze”, in which the Universe cools down until it reaches the state of maximum entropy, or “Big Rip”, in which the repulsive force of Dark Energy eventually becomes so powerful that even atoms cannot be held together by strong nuclear forces. Finally, if the average density of the Universe exactly equals the critical density, its shape is flat and it expands forever at a continually decelerated rate, with expansion asymptotically approaching zero. However, with the presence of dark energy even a flat universe could share the fate of an open one. Our latest observations with WMAP and Planck show that the average density is very close to the critical, and that our Universe is most probably flat. We shall talk about this later.

This year marks the 100^{th} anniversary of General theory of Relativity. It is an extraordinary theory which familiarized us with a lot of very subtle concepts behind the nature of the Universe. And as we saw at the start, it was constructed upon an inconsistency between Newton’s mechanics and Special Relativity. But GR led to another contradiction itself, which I mentioned in the first article of this series. Some of GR notions are contradicted by another extraordinarily successful theory of the XX century, namely Quantum Mechanics. To understand where this contradiction takes place, we need to look at the main principles of QM, which we will do in the next article.

Thanks everybody.

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