SCIENCE PHOTO LIBRARY
EINSTEIN’S THEORY OF GENERAL RELATIVITY
German-born physicist Albert Einstein (left) created the famous Theory of General Relativity. British astrophysicist Sir Arthur Eddington (right), later put this theory to the test and confirmed its accuracy.
ROYAL ASTRONOMICAL SOCIETY / SCIENCE PHOTO LIBRARY
Einstein would have loved the Vomit Comet. The fact that the effects of gravity can be completely removed by falling freely in a gravitational field was, for him, the thought experiment that led to his theory of General Relativity. How wonderful it would have been for him to experience it as I did! The reason I say this is that, as I floated next to my little plastic Albert in the Vomit Comet, I understood very deeply why Einstein was so interested in freefall. The point is this; inside the plane, falling towards Earth, it is absolutely impossible to tell that you are moving. It is impossible to tell that you are near a planet. It is impossible to tell that, according to someone stood on the ground, you are accelerating at 9.81 m/s2 towards the ground. You are simply floating, along with everything else in the plane. I let some little drops of water out of a bottle and they floated in front of my face; the cameraman and director floated next to the water droplets, little plastic Albert and me. There was self-evidently no force acting on anything at all, otherwise things would have moved around.
And yet, from the point of view of someone on the ground, we were flying in a parabolic arc, moving forwards through the air at hundreds of miles an hour and accelerating violently towards the ground. The force of gravity is very much present in this description. Einstein’s theory takes the view that the two ways of looking at the Vomit Comet – from inside and outside – should be treated as equivalent. No one inside the plane or out has the right to claim that they are right and the other is wrong! If, inside the plane, there is no experiment you can do to prove that you are accelerating towards the ground, you are well within your rights to claim that you are not. Acceleration has cancelled out gravity. Of course, you could look out of the windows, but even then you could claim that Earth is accelerating up towards you and that you are simply floating. From this perspective, everyone on Earth feels a gravitational force pulling them onto the ground because they are being accelerated upwards at a rate of 9.81 m/s2. Acceleration is therefore equivalent to gravity; this is known as the equivalence principle, and it was very important to Einstein.
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So what, then, is gravity? The explanation in Einstein’s theory is beautifully simple: gravity is the curvature of spacetime.
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In technical language, Einstein would have defined the Vomit Comet, during its time in freefall, as an inertial frame of reference – which is to say that it can be legitimately considered to be at rest, with no forces acting on it.
The assertion that sitting in a falling aircraft should be considered as being absolutely equivalent to floating around in space, far beyond the gravitational pull of any planet or moon, can be used to explain why all objects fall at the same rate.
Why? Simply because there are two equally valid ways of looking at what is happening. From the point of view inside the plane, nothing at all is happening; everything is simply floating, untouched by any forces of any kind. If no forces are acting, then everything naturally stays where it is put. Shift outside the plane, however, and things appear different; everything is falling towards Earth, accelerating under the action of the force of gravity. But, very importantly, the reality of the situation cannot change depending on which point of view we adopt – everything has to behave in the same way in reality, irrespective of how you look at it. If plastic Albert and the globules of water float in front of my face when viewed from my vantage point inside the plane, then plastic Albert and the globules of water had better float in front of my face when viewed from a vantage point outside the plane. In other words, we had all better be accelerating towards the ground at exactly the same rate! Notice that we’ve made no assumptions about the equivalence between gravitational and inertial masses here; we’ve just said that a freely falling box in Earth’s gravitational field is indistinguishable from a freely falling box in space, or indeed any freely falling box anywhere in the Universe, around any planet, any star, or any moon.
So what, then, is gravity? The explanation in Einstein’s theory is beautifully simple: gravity is the curvature of spacetime. What is spacetime? Spacetime is the fabric of the Universe itself.
A good way to picture spacetime, and what it means to curve it, is to think about a simpler surface; the surface of Earth. Our planet has a two-dimensional surface, which is to say that you only need two numbers to identify any point on it: latitude and longitude. Earth’s surface is curved into a sphere, but you don’t need to know that to move around on it and navigate from place to place. The reason we can picture the curvature is that we are happy to think in three dimensions, so we can actually see that Earth’s surface is curved. But imagine that we were two-dimensional beings, confined to move on the surface of Earth with absolutely no concept of a third dimension. We would know nothing about up and down, only about latitude and longitude. It would be very difficult indeed for us to picture in our mind’s eye the curvature of our planet’s surface.
Now let’s extend our analogy to see how the curvature of something can give rise to a force. Imagine that a pair of two-dimensional friends are standing on the Equator and decide to take a journey due north. They decide to walk parallel to each other, with the intention of never bumping into each other. If they both keep walking, they will walk up parallel lines of longitude, and they will find that as they get closer and closer to the North Pole they will get closer and closer together. Eventually, when they reach the North Pole, they will bump into each other! As three-dimensional beings, we can see what happened; Earth’s surface is curved, so all the lines of longitude meet at the poles. However, from the perspective of our two-dimensional friends, even though they kept assiduously to their parallel lines they still were mysteriously drawn together. They may well conclude from this that a force was acting between them, attracting them towards one another. In Einstein’s theory, that force is gravity.
The complicated bit about Einstein’s Theory of General Relativity is that the surface we need to think about, spacetime, is not two-dimensional but four-dimensional. It is a mixture of the familiar three dimensions of space, plus an additional dimension of time mixed in. It will take us too far from our story to explore spacetime in detail, but it was found to be necessary by Einstein and others at the turn of the twentieth century to explain, in particular, the behaviour of light and the form of Maxwell’s equations that we met in Chapter 1. Suffice to say that the surface of our universe, on which we all live our lives, is four-dimensional. What Einstein showed is that the presence of matter and energy – in the form of stars, planets and moons– curves the surface of spacetime, distorting it into hills and valleys. His equations describe exactly what shape spacetime should be around any particular object, such as the Sun, for example, and they also describe how things move over the curved surface. And here is the key point: just like our two-dimensional friends, things move in straight lines; but just like our two-dimensional friends, this isn’t what it looks like if you don’t know that spacetime is curved. When you’re moving across the curved surface, it appears that a force is acting on you, distorting your path. One of the first things Einstein did with his new, geometric theory of gravity was to calculate what Mercury’s straight-line path through the curved spacetime around the Sun would look like to us, trapped on the surface of spacetime. To his delight, he found that Mercury would orbit the Sun, and in precisely the way that had been observed over the centuries of transit observations. Where Newton failed, Einstein succeeded.
Einstein had found a completely geometrical way of describing the force of gravity, and it is quite wonderfully elegant. Not only does it predict the orbit of Mercury, but it also provides a very appealing explanation for the
equivalence principle. Why do all objects fall at the same rate in a gravitational field, irrespective of their mass or composition? Because the path they take has nothing to do with them at all – they are simply following straight-line paths through the curved spacetime.
Perhaps the most startling demonstration of this is the bending of light by gravity. Light has no mass, and so in Newton’s theory it shouldn’t be affected by gravity at all. However, according to Einstein’s theory, it doesn’t matter that it has no mass, it will still be following a straight line through the curved spacetime, so it will appear to follow exactly the same path as everything else. Let’s do a thought experiment to see how strange this is. Stand on the ground (on a very, very big planet – I’ll explain why I said this in a moment!) with a rock in one hand and a laser beam in the other. Point the laser beam horizontally, drop the rock and fire the laser. Which one hits the ground first? The answer is that they both hit the ground at the same time, because they both move through the same curved space. Light falls at the same rate in a gravitational field as everything else. Now, there is a caveat here. Why did I say a very very big planet? Because light travels at almost 300,000 kilometres per second, so if the rock takes a second to hit the ground, so will the light. But it will have flown 300,000 kilometres in the horizontal direction by the time it reaches the ground, and on Earth that would mean the surface of the planet had long since curved away! However, the principle still holds.
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In the language of General Relativity, we might say that the presence of Earth bends spacetime near it such that time passes more slowly than it does far away.
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As an interesting aside, what would happen if you fired the laser beam directly at the ground? Light must always travel at the same speed, it can’t speed up, so it will travel towards the ground at exactly 299,792,458 metres per second. But shouldn’t it accelerate at 9.81 m/s2 as it drops? No, it can’t, because it always travels at exactly 299,792,458 metres per second. So what happens? Well, the energy of the light can change, although the speed cannot, so the light gets shifted towards the blue end of the spectrum as it flies towards the ground and gains energy from its fall. That is to say that its wavelength gets shorter and its frequency increases. This is very interesting because the second is defined as the length of time it takes a fixed number of wavelengths of a particular colour of light to pass by an observer. Let’s say that you use the frequency of the laser beam held in your hand to synchronize a clock, then you fire the laser at the ground; when the light hits the ground, its frequency will have increased. This means that the peaks and troughs of the laser light beam are arriving more frequently than they did when they set off. So, from the point of view of someone on the ground, the clock above the ground will be running slightly fast. Is this true? Yes, it is. The effect is known as gravitational time dilation; gravity slows down time, so clocks close to the ground run slower than those in orbit. In the language of General Relativity, we might say that the presence of Earth bends spacetime near it such that time passes more slowly than it does far away. This is a very real effect and is one that has to be taken into account in the GPS satellite navigation system, which relies on precise timekeeping to measure distances. The GPS satellites orbit at an altitude of 20,000 kilometres (12,500 miles), which means that their clocks run faster than they do on the ground by 45 microseconds per day, because they are in a weaker gravitational field. The fact that they are moving relative to the ground also affects the rate of their clocks, and when everything is taken into account the timeshift reduces to 38 microseconds per day. This would be equivalent to a distance error of over 10 kilometres (6 miles) per day, which would make the system useless. So, every time we get into our cars and use satellite navigation, we are using Einstein’s theory of gravity in order to correctly ascertain our position on the surface of Earth.
As light has no mass, Newton’s theory states that it is not affected by gravity, although observation does infact show that it is.
NASA
To summarise, then, had Einstein experienced the Vomit Comet, he would have described it, during its time in freefall, as following a straight-line path through spacetime. As long as it continues on this path, the plane and its passengers will not feel the force of gravity at all; it is only when something stops the plane following its straight-line path through spacetime that a force is felt. If the plane didn’t stop itself falling, this obstacle would be the ground!
It is worth making a final brief aside here, which also serves to underline what we’ve just learnt. The experimental fact that triggered all this discussion is that the gravitational and inertial masses of objects are the same. Einstein provides a natural explanation for this: gravity is simply a result of the fact that there is such a thing as spacetime, and that it is curved, and that things move in straight lines through this curved spacetime. It is also possible to take a different view; there could be some deep reason why the gravitational and inertial masses of things are equal – a reason that we have yet to discover. The fact that they are equal allows us to build a geometric theory of gravity. In that case, Einstein’s theory might more properly be considered to be a model, in the same way that Newton’s theory is a model. At the moment we have no way of deciding between these two possibilities, but it’s worth being aware that they are both valid ways of looking at the situation.
Einstein’s Theory of General Relativity is rightly considered to be one of the great intellectual achievements of all time. It is conceptually elegant and probably the theory that physicists most often attach the word ‘beautiful’ to. Ultimately, though, it doesn’t matter how beautiful a theory is, the only thing that matters is that its predictions are in accord with our observations of the natural world. The orbit of Mercury is one such observation; the slowing down of time in gravitational fields is another; but to really test Einstein’s theory to the limit, we have to journey far out into space and visit the most exotic and massive objects in the known Universe – places where the force of gravity becomes exceptionally strong
This coloured X-ray image shows the area around the supermassive black hole, known as Sagittarius A*, which sits at the centre of the Milky Way Galaxy.
NASA / SCIENCE PHOTO LIBRARY
INTO THE DARKNESS
The success of Einstein’s Theory of General Relativity is one of the greatest of human achievements, and in my view it will be remembered as such for as long as there is anything worth calling a civilisation. But there is a final twist to the story of gravitation, because Einstein’s remarkable theory predicts its own demise.
The collapse of a neutron star is prevented by neutron degeneracy pressure. Neutrons are fermions, as are electrons, but because they are more massive than electrons, they can be packed much more tightly together before the Pauli exclusion principle steps in once more and forbids further contraction. Another stable staging post against gravity should be provided by quark degeneracy pressure, because quarks too are fermions, but ultimately, if the star is too massive gravity will overwhelm even these fantastically dense objects. It is believed that the limit above which no known law of physics can intervene to stop gravity is around three times the mass of the Sun. This is known as the Tolman-Oppenheimer-Volkoff limit. For the remnants of stars with masses beyond this limit, gravity will win.
In 1915, only one month after Einstein published the Theory of General Relativity, the physicist Karl Schwarzschild found a solution to Einstein’s equations which is now known as the Schwarzschild metric. The Schwarzschild metric describes the structure of spacetime around a perfectly spherical object. There are two interesting features of Schwarzschild’s spacetime: one occurs at a particular distance from the object, known as the Schwarzschild radius, but for distances less than the Schwarzschild radius, space and time are distorted in such a way that the entire future of anything that falls in will point inwards. This sounds weird, but remember that space and time are mixed up together in Einstein’s theory. I
n more technical language, we say that the future light cones inside the Schwarzschild radius all point towards the centre. This means that, as inexorably as we here on Earth march into the future, if you were to cross the line defined by the Schwarzschild radius, you would inexorably march inwards towards the object that is bending spacetime. There would be no escape, not even for light itself, in the same way that you cannot escape your future. This surface, defined by the Schwarzschild radius surrounding the object, is known as the event horizon. But what has happened to the object itself? This is the second interesting feature of the Schwarzschild metric. Let’s first think about the Sun. If you asked what the Schwarszchild radius for a star with the mass of the Sun is, it would be 3 kilometres (1 mile). This is inside the Sun! So there is no problem here, because you can’t get that close to the Sun without actually being inside it, at which point all the mass outside you doesn’t count any more.
But what about an object like a collapsing neutron star, getting smaller and smaller and denser and denser? What if you could have an object that was dense enough to have the mass of the Sun and yet be physically smaller than the Schwarszchild radius? It seems that there are such objects in the Universe; the stars for which even neutron degeneracy pressure will not suffice to resist the force of gravity. These objects are called black holes. At the very centre of the black hole, at r=0, the Schwarzschild metric has another surprise in store; the spacetime curvature becomes infinite. In other words, the gravitational field becomes infinite. This is known as a singularity. In physical theories, the existence of singularities signals the edge of the applicability of the theory; in simple language, there must be more to it! This has led many physicists to search for a new theory of gravity. Quantum theories of gravity such as string theory may be able to avoid the appearance of singularities, by effectively setting a minimum distance scale below which spacetime does not behave in the manner described by Einstein’s equations.
Wonders of the Universe Page 18
