We have thus far discovered that time can flow very differently. The river of time has estuaries and sinks. However, does it have a source?

Now that it is clear that the properties of time are a function of the physical processes that go on in nature, this question no longer seems to be so absurd. Philosophers pondered this problem for quite a long time. However, the striking successes of Newtonian mechanics and, as a result, the universally accepted Newtonian concept of eternal and unchanged time accustomed them to thinking that the source of time was in the infinite past.

Time was thought of as a uniform river or a never changing road which stretched from the past to the future. In fact, scientists had to face the problem of the beginning of time again, and quite dramatically, in the 20th century. This happened after the discovery of the expansion of the Universe. A detailed description of this achievement is given in the book Edwin Hubble: the Discoverer of the Big Bang Universe, that A. Sharov and I wrote in 1989. (An expanded version in English translation was published in 1993 by Cambridge University Press.) Here I will only trace the main points on this road.

It all began at the end of the 19th century. A rich American astronomer Percival Lowell had a private observatory built for him in the Arizona desert.

When we were discussing the vacuum - the emptiness - in the chapter on ‘Energy extracted from black holes’, we emphasized that virtual particles are constantly created and annihilated in it. The emptiness proved to be a complex entity. The vacuum is a very complicated state of ‘boiling’ virtual particles of most different species.

The reader may not be too surprised by the statement that the properties of this state - the vacuum - depend on the recipe of its preparation. This implies that different vacua are possible; different types of emptiness!

In what follows, we will see examples of possible vacua. Now we will try to answer the following question: can the activity of the vacuum (its ‘boiling’) result in the formation of some energy density owing to the interaction of virtual particles?

Energy density can indeed appear. Zeldovich emphasized this fact in the 1960s. Each energy corresponds to a certain mass. Therefore, mass density will arise together with vacuum energy density. The reader may ask here: does it mean that some sort of universal medium, a new ‘ether’ is emerging in our notions? If this is true, such a medium can restore the concept of absolute rest and absolute motion. Indeed, the motion relative to this medium would be the motion with respect to the emptiness, in other words, with respect to the absolute space. It may seem that if we move relative to this new ‘ether’, we should feel the flow going against us, the ‘ether wind’ blowing in our face.

The Renaissance that came to replace the somber Medieval centuries brought outstanding discoveries in natural sciences. This was the time when Nicolaus Copernicus (1473–1543) developed his theory which was to produce a dramatic transformation in people's view of the world. First of all, this new concept eliminated the impenetrable barrier between the terrestrial and the celestial. Before, everything celestial was a symbol of perfection, of eternity, and of ideals. Heavenly bodies were ideal, as was their uniform motion along circular orbits. This perfection was in opposition with the rough terrestrial matter and its chaotic irregular motion. Copernicus' model showed the Earth to be an ordinary planet which revolves, just as other planets, around the Sun.

Nicolaus Copernicus became a canon of a Catholic church in Frauenberg [Frombork], a small town on the banks of the Vistula in Poland, in 1510. In quiet solitude, he worked on his astronomy. In fact, he spent his free hours on other things as well. He treated patients for no fee. A new monetary system was introduced in Poland following his proposal. He designed and constructed a hydraulic machine to supply water to households.

Copernicus was very careful about publishing his results; he clearly recognized the contradiction with the church's teaching of the singular position of the Earth and man in the Universe. His treatise, On the Revolution of Celestial Spheres, dedicated to Pope Paul III (this was agreed upon with the Holy See) was printed in 1543, not long before Copernicus' death.

When I began to study general relativity seriously (which was in the late fifties), no one knew well what a black hole would be. Even the term itself did not appear in either strictly scientific or popular science publications. This is a stark contrast with what we see today, when almost everyone has read or at least heard about them. The black hole is a product of gigantic gravitational forces. Black holes are born when the gravitational field, growing in the course of catastrophic contraction of a very large mass of matter, becomes so strong that it ceases to let out anything, even light. An object can only fall into a black hole, pulled by its huge gravitational force, but there is simply no way out.

I first read a description of very strong gravitational fields in the Landau and Lifshitz monograph that I have already mentioned. I studied it while still a student, under Zelmanov's guidance. The book gave a very brief but extremely clear presentation of the properties of the gravitational field of a strongly compressed spherical mass. The solution of Einstein's equations for this case was found by the German astronomer Karl Schwarzschild; consequently, this gravitational field is known as the Schwarzschild field.

I remember that this subsection left me rather indifferent. Nevertheless, I did make some evaluations, using the formulas in the book and the knowledge gained from talking to Abram Zelmanov.

We are now departing on a voyage to the very sources of the river of time. What was it that happened at the very beginning of time? What triggered the expansion of the Universe?

We have seen in the chapter ‘Towards the sources of the river of time’ that the huge pressure of hot matter at time zero cannot be the cause of the high velocities of recession of matter, because the uniform Universe has no pressure drop, which is the only cause of force driving an expansion. What then was the cause of the expansion?

The key to understanding the ‘primeval push’ lies in the existence of the special vacuum-like state of matter at high densities and temperatures.

We have already looked at several vacuum-like states in the chapter dealing with Grand Unification. Theorists believe that a unique vacuum-like state with enormous energy density and the corresponding gigantic mass density is formed at the temperature of ‘superunification’. This density in grams per cubic centimeter is written as unity with ninety four zeros (!). The enormity of this number defies imagination. We have already mentioned in the preceding chapter that any vacuum possessing non-zero mass density must have huge negative pressure.

In accordance with Einstein's theory of gravitation, gravitation is produced not only by mass but by pressure as well. Pressure is usually not high and so the gravitation connected with it is negligibly small.

I found this epigraph in Paul Nahin's book Time Machines (New York: AIP) published in 1993 and kindly mailed to me. Another quotation from this book that impressed me with its precision of analysis is:

Indeed, if a chance to visit the past is available, it seems that by modifying this past we could modify the lot of some individuals, the fate of mankind or even the evolution of the entire Universe. Is this true?

The argument that is especially popular in debates of this sort is the so–called ‘grandfather paradox’. It goes roughly like this: ‘If I could go back into the past in which my grandfather was very young, I could kill him and thereby make my own birth impossible’. Or another version of the same paradox: ‘I return into my own past, meet myself in my youth and kill my younger version.’

In both cases this unnatural homicide generates complete nonsense. Should we infer that such an event is impossible? But why? I have my ‘free will’, don't I? Hence I can realize this ‘free will’, at least in principle.

The person to whom I owe my fate was my grandmother. My parents were not there to take part in bringing me up, so my first consciously made steps in life grew from her love and care. Once she found for me an exciting book: Brer Rabbit's Adventures, translated into Russian. I learnt to read with this book. It was my grandmother again who bought for me, on a flea–market,my first popular book about science. It was a very difficult time, the Second World War was raging and the family was evacuated to the town of Krasnokamsk on the Volga. People thought about food first, books were very secondary. But my grandmother — mind you, she had no education whatsoever — felt, perhaps, that food for thought was just as necessary for kids as food for the stomach. The book that she bought (or swapped?) was marvelous; I will never forget it. It was Children's Encyclopaedia, a pre–1917 book, with wonderful color prints. As far as I can remember, their quality was far superior to the often smeared and bleak illustrations that I find nowadays in some editions of books that I write.

That book had a chapter about astronomy. Browsing for the first time through the volume (as for any other kid, this was the first thing to do with a new book), I was amazed by a drawing of a gigantic fountain of fire, with a small globe of our Earth alongside.

I was not quite correct when saying that only motion at relatively modest velocities was known in Isaac Newton's time. Of course, this would be true if only the motion of physical bodies was meant. However, from time immemorial mankind knew a process which propagates at a truly fantastic speed. I mean light. What is it?

Suggestions that light consists of particles which are emitted by a glowing body were made in ancient Greece. Aristotle held this opinion and Newton also shared this point of view. Aristotle assumed the velocity of light propagation to be infinitely high. The same point of view was prevalent until the middle of the 17th century. This belief was shared by the great scientists Johannes Kepler, René Déscartes and others. Galileo was the first to attempt an experimental determination of the speed of light in 1688. He placed two torches on top of two hills at a distance of less than one mile from each other. First the shutter of one torch was opened and when the beam of light reached the observer at the other hill, the latter opened the shutter of his torch. The observer with the first torch was to measure the time between the opening of its shutter and the moment when he saw the flash of the second torch. This was meant to measure the time of travel of light to the second hill and back again.

Everyone knows that the space of the Universe is three-dimensional. This means that space is characterized by length, width and height. The same is true for any body. Somewhat differently, the position of a point in space is characterized by three numbers known as coordinates. If we draw straight lines or planes or complicated curves through space, their properties are described by the laws of geometry. These laws have been known to man since ancient times and were compiled by Euclid in the 3rd century bc. Euclidean geometry is studied in schools as a harmonious system of axioms and theorems that describe all properties of lines, surfaces and solids.

If we wish to study not only the spatial position but also processes occurring in three-dimensional space, we need to add time as well. An event taking place at some point is characterized by the position of this point, that is, by indicating three numbers, and by a fourth number, that is, the moment of time at which the event occurred. For the event the time is its fourth coordinate. In this sense we say that our world is four-dimensional.

All this is well known, of course. Then why wasn't this formulation of four-dimensionality treated as serious and fraught with new knowledge before the theory of relativity was born? The catch lay in the fact that the properties of space and time seemed to be too dissimilar.

Our story of holes in space and time would not be complete if we failed to mention their wonderful property of continuously releasing energy. This feature is one of the manifestations of the as yet undeciphered relationship between time and energy. This relationship manifests itself clearly when quantum properties of matter begin to dominate.

However, I should start very briefly with empty space and its quantum properties.

According to current notions, the vacuum is not absolute emptiness, the ‘perfect nothingness’. It is a sea of so-called virtual particles and antiparticles which do not emerge as real particles. However, the vacuum is the place where pairs of virtual particles and antiparticles are constantly created for a very short moment, only to disappear immediately. They cannot transform into real particles because this would mean the creation of real energy from emptiness. The so-called uncertainty relation of quantum physics allows these particles to appear for a fleeting moment; this relation states that the product of the lifetime of a pair of virtual particles and their energy is of the order of Planck's constant. Real particles can always be removed from a volume while virtual particles cannot be removed - in principle.

Such are the properties of the vacuum. If some strong field is applied to the vacuum, then some virtual particles may ‘pick up’ sufficient energy in this field to become real; they extract the energy for that from the external field.

Albert Einstein created general relativity theory using a minimum number of experimental data on gravitation; he selected this set of data with the intuition of a genius. Over the many decades since the creation of the theory, all its predictions that allowed observational or experimental verification were invariably proved correct.

Tiny corrections to the motion of the planets of the Solar System, predicted by the theory, were detected and then carefully measured. In 1919 Arthur Eddington discovered the bending of light rays in the gravitational field of the Sun, in agreement with Einstein's prediction.

Then the reddening of light emerging from higher gravitational fields was discovered, which again confirmed Einstein's prediction.

Finally, black holes, those exotic objects that are like nothing else in nature, were discovered - with a high degree of certainty - in the 1970s. In this case, relativity theory manifests itself not in some small corrections to well-known processes but in full-blown effects that drastically change the geometry of space and the properties of time.

Not a single fact that would throw a shadow of doubt on relativity theory was found in all these years. Taken together, the entire experience of science in the 20th century makes one treat seriously the other predictions of the theory, those that have not yet been confirmed by experiment or astrophysical observations. We have seen that modern physics, which describes the most profound structure of matter, evolves in the direction outlined by Albert Einstein.

Ever since I started reading popular science books on physics, I have regarded it as self–evident that time is synonymous with empty duration, that it flows like a river and carries in this flow all events without exception. This stream is unalterable and unstoppable, going in a never–changing direction: from the past to the future.

It seemed that this interpretation, given our knowledge about the surrounding world, was unavoidable.

I learnt only many years later that people had not always held such or similar intuitive notions - far from it.

Heraclitus of Ephesus, a philosopher in ancient Greece who lived at the end of the 6th century bc, appears to have been one of the first thinkers of antiquity who set forth a belief that everything in the world changes and that this changeability is the highest law of nature (all things are in process and nothing stays still). Heraclitus set out his view in the book About Nature, of which only a few fragments survived and reached us (Cosmic Fragments).

Heraclitus taught that the world is full of contradictions and variability. All things undergo changes. Time flows relentlessly, and everything that exists moves with this unstoppable stream. The skies move, physical bodies move, a human's feelings and consciencemove as well. ‘You cannot enter twice into one and the same river’ said he, ‘because its water is constantly renewed.’ Things come to replace other things.

Introduction

This chapter deals with the kinematics and the dynamics of systems containing large-number (above 106) stars called globular clusters. It draws heavily on Chap. 10 of Vol. I and the basic ideas of stellar evolution described in the earlier chapters of this volume.

Globular clusters play an important role as systems in which many aspects of stellar-evolution theory can be directly tested and, in the process, can be used to provide significant information about the age and the mass function of stars in our galaxy. They also are examples of systems dominated by gravitational many-body interactions that hence undergo evolution in a manner quite different from other – simpler – systems. Finally, their dynamical evolution provides important points of similarity with central regions of galaxies, elliptical galaxies, and galaxy clusters. The kinematical aspects are discussed in the next section, and the rest of the chapter is devoted to dynamical issues.

Stellar Distribution and Ages of Globular Clusters

We have seen in the earlier chapters that the formation of stars from gaseous clouds is a fairly complex process that is not completely understood. Irrespective of the details, it seems reasonable to expect that, when stars form in a cloud, there will be a tendency for a large number of them to be close together, thereby forming a cluster of stars. Two broad categories of such star clusters have been seen in our galaxy, usually called open clusters and globular clusters.

Introduction

Stellar physics provides a natural starting point for the study of astrophysics for several reasons. To begin with, this is probably the best understood area of astrophysics. Second, there is a vast amount of reliable data dealing with stellar physics. This observational input motivates accurate and sophisticated theoretical modelling as well as provides the opportunity for a detailed comparison between models and observations. Finally, stellar physics also forms the basis for the study of several other related areas, even in the domain of extragalactic astronomy and cosmology. For example, measurements of cosmic distances and the ages of different structures – which are very important in cosmology – cannot be done without accurate modelling of the stellar phenomena that are used as tools; the study of formation and evolution of galactic systems requires an understanding of star formation and stellar evolution, etc. This volume deals with different aspects of the astrophysics of stellar systems.

The evolution of stars differs significantly, depending on whether the star is isolated or is a member of a binary system. The bulk of the chapters in the book (from Chap. 3 to Chap. 6) deal with stellar evolution and stellar remnants in isolated contexts. Chapter 7 is devoted to the study of evolution of binary stars, and Chap. 10 covers the dynamics of systems like globular clusters that have a very large number of stars.

Introduction

This chapter discusses the structure of stars that are in steady state. Concepts described in Vol. I, Chaps. 5–7, will be used extensively here. The models described here will be needed in several subsequent chapters dealing with stellar evolution, compact remnants, and binary stars.

Equations of Stellar Structure

A self-gravitating body of mass M and radius R will have gravitational potential energy of U ≈ −(GM2/R). If such a body is in equilibrium with the gas pressure balancing the gravity, the virial theorem implies that it will have temperature T such that NkBT ≈ (GM2/R that is, T ≈ (GMmp/kBR). For a sufficiently large value of M/R, this temperature can be high enough to ignite nuclear reactions at the centre of the body. The nuclear energy generated near the centre will be transported by radiation and convection towards the outer regions and will eventually escape from the body. This will establish a temperature gradient inside the body such that, in steady state, the energy produced by nuclear reactions is equal to the energy radiated away from the outer surface. Such a steady-state situation can last as long as the conditions in the body allow the generation of nuclear energy inside it. Observations suggest that the stars belong to such a category of self-gravitating bodies that are essentially powered by the nuclear reactions.