Cosmology_A Very Short Introduction

by Peter Coles

  Universal gravitation

  Gravity is one of the fundamental forces of nature. It represents the universal tendency of all matter to attract all other matter. There are in fact four fundamental forces (gravity, electromagnetism, and the ‘weak’ and ‘strong’ nuclear forces). The universality of gravity sets it apart from, for example, the electrical forces between charged bodies. Electrical charges can be of two different kinds, positive or negative. While electrical forces can lead either to attraction (between unlike charges) or repulsion (between like charges), gravity is always attractive. That is why it is so important for cosmology.

  In many ways, the force of gravity is extremely weak. Most material bodies are held together by electrical forces between atoms which are many orders of magnitude stronger than the gravitational forces between them. But, despite its weakness, gravity is the driving force in astronomical situations because astronomical bodies, with very few exceptions, always contain exactly the same amount of positive and negative charge and therefore never exert forces of an electrical nature on each other.

  One of the first great achievements of theoretical physics was Isaac Newton’s theory of universal gravitation, which unified what, at the time, seemed to be many disparate physical phenomena. Newton’s theory of mechanics is encoded in three simple laws:

  1. Every body continues in a state of rest or uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.

  2. Rate of change of momentum is proportional to the impressed force, and is in the direction in which this force acts.

  3. To every action, there is always opposed an equal reaction.

  These three laws of motion are general, applying just as accurately to the behaviour of balls on a billiard table as to the motion of the heavenly bodies. All that Newton needed to do was to figure out how to describe the force of gravity. Newton realized that a body orbiting in a circle, like the Moon going around the Earth, is experiencing a force in the direction of the centre of motion (just as a weight tied to the end of a piece of string does when it is twirled around one’s head). Gravity could cause this motion in the same way as it could cause apples to fall to Earth from trees. In both these situations, the force has to be towards the centre of the Earth. Newton realized that the right form of mathematical equation was an ‘inverse-square’ law: ‘the attractive force between any two bodies depends on the product of the masses of the bodies and upon the square of the distance between them.’

  It was a triumph of Newton’s theory, based on the inverse-square law of universal gravitation, that it could explain the laws of planetary motion obtained by Johannes Kepler more than a century earlier. So spectacular was this success that the idea of a Universe guided by Newton’s laws of motion was to dominate scientific thinking for more than two centuries, until the arrival on the scene of Albert Einstein.

  The Einstein revolution

  Albert Einstein was born in Ulm (Germany) on 14 March 1879, but his family soon moved to Munich, where he spent his school years. The young Einstein was not a particularly good student, and in 1894 he dropped out of school entirely when his family moved to Italy. After failing the entrance examination once, he was eventually admitted to the Swiss Institute of Technology in Zurich in 1896. Although he did fairly well as a student in Zurich, he was unable to get a job in any Swiss university, as he was held to be extremely lazy. He left academia to work in the Patent Office at Bern in 1902. This gave him a good wage and, since the tasks given to a junior patent clerk were not exactly onerous, it also gave him plenty of spare time to think about physics.

  Einstein’s special theory of relativity was published in 1905. It stands as one of the greatest intellectual achievements in the history of human thought. It is made even more remarkable by the fact that Einstein was still working as a patent clerk at the time, and was doing physics as a particularly demanding hobby. What’s more, he also published seminal works that year on the photoelectric effect (which was to inspire many developments in quantum theory) and on the phenomenon of Brownian motion (the jiggling of microscopic particles as they are buffeted by atomic collisions). But the reason why the special theory of relativity stands head and shoulders above his own work of this time, and that of his colleagues in the world of mainstream physics, is that Einstein managed to break away completely from the concept of time as an absolute property that marches on at the same rate for everyone and everything. This idea is built into the Newtonian picture of the world, and most of us regard it as being so obviously true that it does not bear discussion. It takes a genius to break down conceptual barriers of such magnitude.

  The idea of relativity did not originate with Einstein. Galileo had articulated the basic principle nearly three centuries earlier. Galileo claimed that only relative motion matters, so there could be no such thing as absolute motion. He argued that if you were travelling in a boat at constant speed on a smooth lake, then there would be no experiment that you could do in a sealed cabin on the boat that would indicate to you that you were moving at all. Of course, not much was known about physics in Galileo’s time, so the kinds of experiment he could envisage were rather limited.

  Einstein’s version of the principle of relativity simply turned it into the statement that all laws of nature have to be exactly the same for all observers in relative motion. In particular, Einstein decided that this principle must apply to the theory of electromagnetism, constructed by James Clerk Maxwell, which describes amongst other things the forces between charged bodies mentioned above. One of the consequences of Maxwell’s theory is that the speed of light (in vacuum) appears as a universal constant (usually given the symbol ‘c’). Taking the principle of relativity seriously means that all observers have to measure the same value of c, whatever their state of motion. This seems straightforward enough, but the consequences are nothing short of revolutionary.

  Einstein decided to ask himself specific questions about what would be observed in particular kinds of experiments involving the exchange of light signals. He worked a great deal with gedanken (thought) experiments of this kind. For example, imagine there is a flash bulb in the centre of a railway carriage moving along a track. At each end of the carriage there is a clock, so that when the flash illuminates it we can see the time. If the flash goes off, then the light signal reaches both ends of the carriage simultaneously, from the point of view of passengers sitting in the carriage. The same time is seen on each clock.

  Now picture what happens from the point of view of an observer at rest who is watching the train from the track. The light flash travels with the same speed in our reference frame as it did for the passengers. But the passengers at the back of the carriage are moving into the signal, while those at the front are moving away from it. This observer therefore sees the clock at the back of the train light up before the clock at the front does. But when the clock at the front does light up, it reads the same time as the clock at the back did! This observer has to conclude that something is wrong with the clocks on the train.

  This example demonstrates that the concept of simultaneity is relative. The arrivals of the two light flashes are simultaneous in the frame of the carriage, but occur at different times in the frame of the track. Other examples of strange relativistic phenomena include time dilation (moving clocks appear to run slow) and length contraction (moving rulers appear shorter). These are all consequences of the assumption that the speed of light must be the same as measured by all observers. Of course, the examples given above are a little unrealistic. In order to show noticeable effects, the velocities concerned must be a sizeable fraction of c. Such speeds are unlikely to be reached in railway carriages. Nevertheless, experiments have been done that show that time dilation effects are real. The decay rate of radioactive particles is much slower when they are moving at high velocities because their internal clock runs slowly.

  Special relativity also spawned the most famous equation in all of physics: E = mc2, expres
sing the equivalence between matter and energy. This has also been tested experimentally; amongst other things it is the principle behind both atomic and chemical explosives.

  Remarkable though the special theory undoubtedly is, it is seriously incomplete because it deals only with bodies moving with constant velocity with respect to each other. Even chapter 1 of the laws of nature, written by Newton, had been built around the causes and consequences of velocities that change with time. Newton’s second law is about the rate of change of momentum of an object, which in layman’s terms is its acceleration. Special relativity is restricted to so-called inertial motions, i.e. the motions of particles that are not acted upon by any external forces. This means that special relativity cannot describe accelerated motion of any kind and, in particular, cannot describe motion under the influence of gravity.

  The equivalence principle

  Einstein had deep insights into how to incorporate gravitation into relativity theory. For a start, consider Newton’s theory of gravity. In this theory, the force on a particle of mass m due to another particle of mass M depends on the product of these masses and the square of the distance between the particles. According to Newton’s laws of motion, this induces an acceleration in the first particle given by F = ma. The m in this equation is called the inertial mass of the particle, and it determines the particle’s resistance to being accelerated. In the inverse-square law of gravity, however, the mass m measures the reaction of one particle to the gravitational force produced by the other particle. It is therefore called the passive gravitational mass. But Newton’s third law of motion also states that if body A exerts a force on body B then body B exerts a force on body A which is equal and opposite. This means that m must also be the active gravitational mass (if you like, the gravitational charge) produced by the particle. In Newton’s theory, all three of these masses – the inertial mass, the active and passive gravitational masses – are equivalent. But there seems to be no reason, on the face of it, why this should be the case. Couldn’t they be different?

  Einstein decided that this equivalence must be the consequence of a deeper principle called the principle of equivalence. In his own words, this means that ‘all local, freely-falling laboratories are equivalent for the performance of all physical experiments’. What this means is essentially that one can do away with gravity as a separate force of nature and regard it instead as a consequence of moving between accelerated frames of reference.

  To see how this is possible, imagine a lift equipped with a physics laboratory. If the lift is at rest on the ground floor, experiments will reveal the presence of gravity to the occupants. For example, if we attach a weight on a spring to the ceiling of the lift, the weight will extend the spring downwards. Next, imagine that we take the lift to the top of a building and let it fall freely. Inside the freely falling lift there is no perceptible gravity. The spring does not extend, as the weight is always falling at the same rate as the rest of the lift, even though the lift’s speed might be changing. This is what would happen if we took the lift out into space, far away from the gravitational field of the Earth. The absence of gravity therefore looks very much like the state of free-fall in response to a gravitational force. Moreover, imagine that our lift was actually in space (and out of gravity’s reach), but there was a rocket attached to it. Firing the rocket would make the lift accelerate. There is no up or down in free space, but let us assume that the rocket is attached so that the lift would accelerate in the opposite direction from before, i.e. in the direction of the ceiling.

  What happens to the spring? The answer is that the acceleration makes the weight move in the reverse direction relative to the lift, thus extending the spring towards the floor. (This is like what happens when a car suddenly accelerates – the passenger’s head is flung backwards.) But this is just like what happened when there was a gravitational field pulling the spring down. If the lift carried on accelerating, the spring would remain extended, just as if it were not accelerating but placed in a gravitational field. Einstein’s idea was that these situations do not merely appear similar: they are completely indistinguishable. Any experiment performed in an accelerated lift in space would give exactly the same results as one performed in a lift upon which gravity is acting. To complete the picture, now consider a lift placed inside a region

  2. Thought-experiment illustrating the equivalence principle. A weight is attached to a spring, which is attached to the ceiling of a lift. In (a) the lift is stationary, but a gravitational force acts downwards; the spring is extended by the weight. In (b) the lift is in deep space, away from any sources of gravity, and is not accelerated; the spring does not extend. In (c) there is no gravitational field, but the lift is accelerated upwards by a rocket; the spring is extended. The acceleration in (c) produces the same effect as the gravitational force in (a). In (d) the lift is freely falling in a gravitational field, accelerating downwards so no gravity is felt inside; the spring does not extend because in this case the weight is weightless and the situation is equivalent to (b).

  where gravity is acting, but which is allowed to fall freely in the gravitational field. Everything inside becomes weightless, and the spring is not extended. This is equivalent to the situation in which the lift is at rest and where no gravitational forces are acting. A freely falling observer has every reason to consider himself to be in a state of inertial motion.

  The general theory of relativity

  Einstein now knew how he should construct the general theory of relativity. But it would take him another ten years to produce the theory in its final form. What he had to find was a set of laws that could deal with any form of accelerated motion and any form of gravitational effect. To do this he had to learn about sophisticated mathematical techniques, such as tensor analysis and Riemannian geometry, and to invent a formalism that was truly general enough to describe all possible states of motion. He got there, but clearly it wasn’t easy. While his classic papers of 1905 were characterized by brilliant clarity of thought and economy of mathematical calculation, his later work is mired in technical difficulty. People have argued that Einstein grew up as a scientist while he was developing the general theory. If so, it was obviously a difficult process for him.

  Understanding the technicalities of the general theory of relativity is a truly daunting task. Even on a conceptual level, the theory is difficult to grasp. The relativity of time embodied in the special theory is present in the general theory, but there are additional effects of time dilation and length contraction due to gravitational effects. And the problems don’t end with time! In the special theory, space at least is well behaved. In the general theory, even this goes out of the window. Space is curved.

  The curvature of space

  The idea that space could be warped is so difficult to grasp that even physicists don’t really try to visualize such a thing. Our understanding of the geometrical properties of the natural world is based on the achievements of generations of Greek mathematicians, notably the formalized system of Euclid – Pythagoras’ theorem, parallel lines never meeting, the sum of the angles of a triangle adding up to 180 degrees, and so on. All of these rules find their place in the canon of Euclidean geometry. But these laws and theorems are not just abstract mathematics. We know from everyday experience that they describe the properties of the physical world extremely well. Euclid’s laws are used every day by architects, surveyors, designers, and cartographers – anyone, in fact, who is concerned with the properties of shape, and the positioning of objects in space. Geometry is real.

  It seems self-evident, therefore, that these properties of space that we have grown up with should apply beyond the confines of our buildings and the lands we survey. They should apply to the Universe as a whole. Euclid’s laws must be built into the fabric of the world. Or must they? Although Euclid’s laws are mathematically elegant and logically compelling, they are not the only set of rules that can be used to build a system of geometry. Mathematicians of the ninet
eenth century, such as Gauss and Riemann, realized that Euclid’s laws represent only a special case of geometry wherein space is flat. Different systems can be constructed in which these laws are violated.

  Consider, for example, a triangle drawn on a flat sheet of paper. Euclid’s theorems apply here, so the sum of the internal angles of this triangle must be 180 degrees (equivalent to two right-angles). But now think about what happens if you draw a triangle on a sphere instead. It is quite possible to draw a triangle on a sphere that has three right angles in it. For example, draw one point at the ‘north pole’ and two on the ‘equator’ separated by one quarter of the circumference. These three points form a triangle with three right angles that violates Euclidean geometry.

  Thinking this way works fine for two-dimensional geometry, but our world has three dimensions of space. Imagining a three-dimensional curved surface is much more difficult. But in any case it is probably a mistake to think of ‘space’ at all. After all, one can’t measure space. What one can measure are distances between objects located in space using rulers or, more realistically in an astronomical context, light beams. Thinking of space as a flat or curved piece of paper encourages one to think of it as a tangible thing in itself, rather than simply as where the tangible things are not. Einstein always tried to avoid dealing with entities such as ‘space’ whose category of existence was unclear. He preferred to reason instead about what an observer could actually expect to measure with a given experiment.