A Brief history of Time 10, 11, and conclusion

 

Chapter  11

The Unification of Physics

 Einstein refused to believe in the reality of quantum mechanics, despite the important role he had played in its development.

The main difficulty in finding a theory that unifies gravity with the other forces is that general relativity is a “classical” theory; that is, it does not incorporate the uncertainty principle of quantum mechanics.

 So this is the first step to be taken. 

This can produce some remarkable consequences, such as black holes not being black, and the universe not having any singularities but being completely self-contained and without boundary.

 The trouble is that the uncertainty principle means that even “empty” space is filled with pairs of virtual particles and antiparticles. These pairs would have infinite amount of energy and, therefore, by Einstein’s famous equation E= mc² they would have infinite amount of mass. Their gravitational attraction would thus curve up the universe to infinitely small size.

But can there really be a unified theory? Or are we perhaps chasing a mirage? There seem to be three possibilities:

1. There really is a complete unified theory and we will discover it one day.

2. There is no ultimate theory of the universe, just an infinite sequence of theories that describe the universe more and more accurately.

3. There is no theory of the universe: events cannot be predicted beyond certain extend but occur in a random and arbitrary manner.

The third choice leaves place for God.

·         With the advent of quantum mechanics, we have come to recognize that events cannot be predicted with complete accuracy but there is always a degree of uncertainty.

 

Chapter 12

Conclusion

In this book “I have given special prominence to the laws that govern gravity, because it is gravity that shapes the large-scale structure of the universe, even though it is the weakest of the four categories of forces.

According to the theory of relativity, there must have been a state of infinite density in the past, the big bang, which would have been an effective beginning of time.  Similarly if the whole universe recollapsed, there must be another state of infinite density in the future, the big crunch, which would be an end of time.

 Even if the whole universe did not recolapse, there would be singularities in any localized regions that collapsed to form black holes. These singularities would be the end of time for anyone who fell into the black hole. At the big bang and other singularities, all the laws would have broken down, so God would still have had complete freedom to choose what happened and how the universe began”.

When we combine quantum mechanics with general relativity, there seems to be a new possibility, that did not arise before: that space and time together might form a finite, four-dimensional space without singularities or boundaries, like the surface of the earth but with more dimensions. But this has profound implications for the role of God as Creator.

There may be one or a small number of complete unified theories such as the heterotic string theory, that are self-consistent and allow the existence of structures as complicated as human beings who can investigate the laws of the universe and ask about the nature of God.

·         Up to now, most scientists are too occupied with the development of new theories that describe what the universe is to ask the question why.

 

NEW DISCOVERIES AFTER THE LAST 1996 EDITION OF THE BOOK

1. In 1998 our picture of the universe was radically revised: the expansion of our universe is accelerating. This means the eventual recollapse of the universe no longer appears to be an option. Space it seems will expand forever.

2. We can now understand the origins of structure in our universe using measurements of cosmic microwave background radiation.

3. Eternal inflation and the no boundary proposal together predict that our universe is not unique. Instead, from the quantum fuzz at the big bang many different universes emerge, possibly with different local laws of physics and Chemistry.

4. Gravitational waves can be produced in the modern universe. Exactly a century after Einstein first predicted the existence of gravitational waves, a worldwide consortium of scientists known as the LIGO Scientific Collaboration announced in 2016 that these waves had been detected for the first time.

5. What falls into the black holes is not lost forever.

A brief history of time 10

 Chapter 10

The wormholes and time travel

Observations of the microwave background and of the abundances of the light elements in universe indicate that the early universe did not have the kind of curvature required to allow time travel.

If you can travel faster than light, the theory of relativity implies you can also travel back in time. There is a problem with breaking the speed-of-light barrier.

There were experiments in Fermilab or CERN (European Centre for Nuclear Research) which showed that we can accelerate particles to 99.99 percent of the speed light, but however much power we feed in we can’t get them beyond the speed-of-light barrier. Similarly with spaceships: no matter how much rocket power they have, they can’t accelerate beyond the speed of light.

That might seem to rule out both rapid space travel and travel back in time.

However, there is a possible way out. If there was a shortcut between A and B. One way to do this would be to create a wormhole between A and B. This is a thin tube of space-time which can connect two nearly flat regions far apart. So if one could travel faster than light and through a wormhole from Earth to our nearest star A Centauri and then come back to Earth also through a wormhole, he would travel into past!

In 1935, Einstein and Nathan Rosen called these “bridges”. An advanced civilization could keep a wormhole open!

So what one needs to allow travel into the past is matter with negative energy density. The quantum theory allows the energy to be negative in some places, provided that this is made up for by positive energy densities in other places, so that the total energy remains positive.

We have experimental evidence that space can be curved in the way necessary to allow time travel. One might hope therefore that as we advance in science and technology, we would eventually manage to build a time machine.

Why hasn’t anyone come back from the future and told us how to do it?

Of course someone might claim that sightings of UFOs are evidence that we have been visited either from aliens or by people from the future.

But why do they reveal themselves to those who are not regarded as reliable witnesses?

The radiation by black holes shows that quantum theory allows travel back in time on a microscopic scale

The possibility of time travel remains open.

A Brief History of Time 9, by Stephen Hawking

 Chapter 9

The arrow of time

Our views of the nature of time have changed over the years.

Up to the beginning of the 20^th century people believed in an absolute time. However, the discovery that the speed of light appeared the same to every observer, no matter how he was moving, led to the theory of relativity and so we had to abandon the idea of absolute time.

When one tried to unify gravity with quantum mechanics, one had to introduce the idea of “imaginary” time.

 Imaginary time is indistinguishable from directions in space.

 If one can go forward in imaginary time, one ought to be able to turn round and go backward.  This means that there can be no important difference between the forward and backward directions of imaginary time.

 On the other hand, there is a big difference between the forward and the backward directions, in “real” time, as we know.

 Where does this difference between the past and the future come from?

 Why do we remember the past but not the future?

 

The second law of thermodynamics says that in any closed system disorder or entropy, always increases with time: “things always tend to go wrong”.

 The increase of disorder or entropy with time is one example of what is called an arrow of time, something that distinguishes the past from the future, giving a direction to time.

The second law of thermodynamics results from the fact that there are always more disordered states than there are ordered ones.

The laws of science do not distinguish between the forward and backward directions of time- between the past and the future, in the imaginary time.

However, there are at least three arrows of time that do distinguish the past from the future:

·         The thermodynamic arrow of time, the direction of time in which disorder increases

·         The psychological arrow of time, this is the direction in which we feel time passes, the direction in which we remember the past but not the future

·         Finally there is the cosmological arrow of time; this is the direction of time in which the universe is expanding rather than contracting.

 

The psychological arrow is essentially the same as the thermodynamic arrow, so that the two would always point in the same direction.

The “no boundary” proposal for the universe predicts the existence of a well-defined thermodynamic arrow of time because the universe must start in a smooth and ordered state. And the reason we observe this thermodynamic arrow to agree with the cosmological arrow is that intelligent beings can exist only in the expanding phase.

The contracting phase will be unsuitable because it has no strong thermodynamic arrow of time.

 The progress of human race in understanding the universe has established a small corner of order in an increasingly disordered universe.

A brief history of time 8

 

Chapter 8

The origin and the fate of the universe

 

As the universe expands any matter or radiation in it gets cooler.

When it doubles its size, its temperature falls by half.

As the temperature is a measure of the average energy -or speed- of the particles, this cooling of the universe would have a major effect on the matter in it.

At very high temperatures, particles would be moving around so fast that they would escape any attraction toward each other due to nuclear or electromagnetic forces, but as they cooled off one would expect particles that attract each other to start to clump together.

Moreover, even the types of particles that exist in the universe would depend on the temperature. At high enough temperatures, particles have so much energy that whenever they collide many different particle/antiparticle pairs would be produced-and although these particles would annihilate on hitting antiparticles, they would be produced more rapidly than they could annihilate.

At lower temperatures, however, when colliding particles have less energy, particle/antiparticle pairs would be produced less quickly-and annihilation would become faster than production.

 

One second after the big bang, the temperature of the universe would fall to 10 thousand million (ten billion) degrees. At this time the universe would have contained mostly, photons, electrons, neutrinos and their antiparticles, together with some protons and neutrons.

As the universe continued to expand and the temperature to drop, most of the electrons would have annihilated with each other to produce more photons, leaving only a few electrons left over.

The neutrinos and antineutrinos however, would not have annihilated with each other. So they should still be around today. We might be able to detect them only indirectly: they could be a form of “dark matter”, with sufficient gravitational attraction to stop the expansion of the universe and cause it to collapse again.

One hundred seconds after the big bang, the temperature would have fallen to one thousand million (one billion) degrees, the temperature inside the hottest star. At this temperature protons and neutrons would have started to combine together to produce the nuclei of atoms of deuterium (heavy hydrogen) which contain one proton and one neutron.

The deuterium nuclei would then have combined with more protons and neutrons to make helium nuclei, which contain two protons, and two neutrons, and also small amounts of a couple of heavier elements, lithium and beryllium.

The remaining neutrons would have decayed into protons, which are the nuclei of ordinary hydrogen atoms.

 Within a few hours from the big bang , the production of helium and other elements would have stopped and it would continue to expand for about the next million years, without anything much happening.

Eventually, once the temperature had dropped to a few thousand degrees, the electrons and nuclei would not have much energy to overcome the electromagnetic forces between them, and they would have started combining to form atoms.

 The universe would have continued expanding and cooling, but in regions that were slightly denser than average, the expansion would have been slowed down by the extra gravitational attraction. This would eventually stop expansion in some areas and cause them to start to recollapse. As the collapsing region got smaller, it would spin faster and balance the attraction of gravity, and in this way disklike rotating galaxies were born. Other regions which did not happen to pick up a rotation, would become oval shaped objects called elliptical galaxies.

As time went on, hydrogen burning into helium and radiating the resulting energy as heat and light would form stars like our sun.

The outer regions of a star may sometimes get blown off in a tremendous explosion (supernova), which would outshine all the other stars in its galaxy.

Some of the heavier elements produced near the end of the life of a star would be flung back into the gas in the galaxy, and would provide some of the raw materials for the next generation of stars.

 Our own sun contains 2 percent of these heavier elements, because it is a second or third-generation star, formed some five thousand million years ago from a cloud of rotating gas containing the debris of earlier supernovas. Most of the gas in that cloud went to form the sun or got blown away, but a small amount of the heavier elements collected together to form the bodies that now orbit the sun as planets like the earth.

The earth was initially hot and without atmosphere. In the course of time it cooled and acquired an atmosphere from the emission of gases from the rocks. This early atmosphere was not one in which we could have survived. It contained no oxygen, but a lot of other gases that are poisonous to us but not to other primitive forms of life. They are thought to have developed in the oceans, possibly as a result of chance combinations of atoms into large structures, called macromolecules, which were capable of assembling other atoms in the ocean into similar structures. They would thus have reproduced themselves and multiplied.

 In some cases there would be errors in the reproduction. Mostly these errors would have been such that the new macromolecule could not reproduce itself and eventually would have been destroyed.

However, a few of the errors would have reproduced new macromolecules that were even better at reproducing themselves. They would have therefore an advantage and would have tended to replace the original molecules. In this way a process of evolution was started that led to the development of more and more complicated and self-producing organisms. These primitive organisms consumed various materials and released oxygen. This gradually changed the atmosphere to the composition that it has today, and allowed the development of higher forms of life such as fish, reptiles, mammals, and ultimately human race.

 This picture of the universe which started off very hot and cooled as it expanded is in agreement with all the observational evidence that we have today. Nevertheless, it leaves a number of important questions unanswered:

·         Why was the early universe so hot?

·         Why is universe so uniform on a large scale? Why does it look the same at all points of space and in all directions?

·         Why did the universe start out with so nearly the critical rate of expansion that separates models that recollapse from those that go on expanding forever, that even now, ten thousand million years later, it is still expanding at nearly the critical rate?

·         Despite the universe is so uniform and homogeneous on a large scale, it contains irregularities such as the stars and galaxies which are thought to have developed from small differences in the density of the early universe from one region to another. What was the origin of these density fluctuations?

 The general theory of relativity, on its own, cannot explain these features or answer these questions.

  How did God start the initial state of the universe? One possible answer could be that God chose the initial configuration of the universe for reasons that we cannot hope to understand. This would certainly have been within the power of an omnipotent being. BUT if he had started it off in such an incomprehensible way, why did he choose to let it evolve according to laws that we COULD understand?

The whole history of science has been the gradual realization that events do not happen in an arbitrary manner but there is an underlying order, which may or may not be divinely inspired. Why shouldn’t this order apply to the initial state of the universe?

There may be different initial conditions of the universe but there ought to be a principle that picks out one initial state.

 One such possibility is what is called the chaotic boundary conditions. These assume either that the universe is spatially infinite or there are infinitely many universes. In any case the initial state of the universe, according to this principle, is chosen purely randomly.

 But it is difficult to see how such chaotic initial conditions could have given rise to a universe that is so smooth and regular on a large scale as ours is today.

 Another principle is the anthropic principle.

The first version of it that is called the weak anthropic principle states that in a universe which is large or infinite in space and/or time, the conditions necessary for the development of intelligent life will be met only in certain regions that are limited in space and time.

The strong version of the principle states that there are either many different universes or many different regions of a single universe, each with its own initial configuration, and perhaps with its own set of laws of science.

 

 The laws of science as we know them at present contain many fundamental numbers. For example the size of the electric charge of the electron. If the electric charge of the electron had been slightly different, stars either would have been unable to burn hydrogen and helium, or else they would not have exploded. Of course there might be other forms of intelligent life that did not require the light of the sun or the heavier chemical elements that are made in stars, and are flung back into space when the stars explode.

   In order to predict how the universe should have started off, one needs laws that hold at the beginning of time.

 If the classical theory of relativity was correct, the singularity theorems that Roger Penrose and I (says Hawking) proved show that the beginning of time would have been a point of infinite density and infinite curvature of space-time. All the known laws of science would break down at such a point. One would suppose that there were new laws that held at singularities, but it would be very difficult even to formulate such laws and we would have no guide from observations as to what those laws might be.

 So, classical theory is no longer a good description of the universe.

 One has to use a quantum theory of gravity to discuss the very early stages of the universe.

 It is possible in the quantum theory for the ordinary laws of science to hold everywhere, including at the beginning of time: it is not necessary to postulate new laws for singularities, because there need not be any singularities in the quantum theory.

 We don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. But we are fairly certain of some features that such a unified theory should have.

 The idea that place and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people have come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws.

 So as long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place then for a creator?

 

Η αρχή και το τέλος του Σύμπαντος

 

Το big bang, η τροποποίηση της θεωρίας του σχετικά με τη μεγάλη έκρηξη, και η προσπάθειά του να συνδυάσει τη γενική θεωρία της σχετικότητας με την κβαντική μηχανική , στο 8^ο κεφάλαιο  του “Brief History of Time” του Stephen Hawking  με τίτλο “The origin and fate of the universe”.

«Αν η κλασική θεωρία της σχετικότητας ήταν σωστή, τα θεωρήματα «μοναδικότητας» (singularity theorems) που ο Roger Penrose και εγώ αποδείξαμε» υποδείκνυαν πως το σύμπαν ξεκίνησε με μια Μεγάλη Έκρηξη, όπου ολόκληρο το σύμπαν και τα πάντα μέσα του ήταν συμπιεσμένα σε ένα και μοναδικό σημείο άπειρης πυκνότητας, μια ιδιομορφία του χωροχρόνου. Υπήρξαν πολλές αντιρρήσεις στη δουλειά μας αλλά τελικά την αποδέχτηκαν και έτσι σήμερα σχεδόν όλοι υποθέτουν ότι το σύμπαν άρχισε με την ιδιομορφία του Big Bang.

«Αποτελεί ίσως ειρωνεία ότι, έχοντας αλλάξει γνώμη, προσπαθώ τώρα να πείσω τους άλλους φυσικούς  ότι στην πραγματικότητα δεν υπήρξε μοναδικότητα (singularity) στην αρχή του σύμπαντος. Αυτή μπορεί να εξαφανιστεί αν λάβουμε υπ’ όψιν  την κβαντική μηχανική».

The Big Bang theory

·         Ένα δευτερόλεπτο μετά τη μεγάλη έκρηξη, η θερμοκρασία του σύμπαντος έπεσε στους 10 δις βαθμούς. Εκείνη τη στιγμή, το σύμπαν θα περιείχε κυρίως φωτόνια, ηλεκτρόνια, νετρίνα και τα αντισωματίδιά τους καθώς και μερικά πρωτόνια και νετρόνια.

Καθώς το σύμπαν συνέχισε να επεκτείνεται η θερμοκρασία του συνέχιζε να πέφτει.

 Αρχικά σχηματίστηκαν τα πρώτα αστέρια. Κάποια από αυτά εξερράγησαν (supernova)και από τα συντρίμμια τους δημιουργήθηκαν καινούρια αστέρια (ήλιοι) και κάποιοι πλανήτες.

 Ο δικός μας ήλιος θεωρείται ότι δημιουργήθηκε από δεύτερη ή τρίτη έκρηξη ενός supernova και μαζί με αυτόν και οι πλανήτες του ηλιακού μας συστήματος. Αυτό πριν 5 δισεκατομμύρια χρόνια.

 Η γη αρχικά είχε πολύ υψηλή θερμοκρασία και καθόλου ατμόσφαιρα. Με τον καιρό άρχισε να ψύχεται και απέκτησε ατμόσφαιρα από την εκπομπή αερίων από τους βράχους.

 Αυτή η αρχική ατμόσφαιρα δεν ήταν τέτοια στην οποία θα μπορούσαμε να επιβιώσουμε εμείς σήμερα.

 Δεν περιείχε οξυγόνο αλλά άλλα αέρια πολλά από τα οποία θα ήταν δηλητηριώδη για μας αλλά όχι για κάποιες πρωτόγονες μορφές ζωής τότε, που πιστεύεται ότι είχαν αναπτυχθεί στους ωκεανούς, πιθανόν σαν αποτέλεσμα τυχαίων συνδυασμών ατόμων σε μεγαλύτερες δομές, που ονομάζονται μακρομόρια τα οποία ήταν ικανά να μαζεύουν άλλα άτομα στον ωκεανό με παρόμοιες κατασκευές και έτσι πολλαπλασιάζονταν.

 Σε μερικές περιπτώσεις θα υπήρχαν λάθη στην αναπαραγωγή. Κυρίως αυτά τα λάθη θα ήταν τέτοια ώστε το καινούριο μακρομόριο δεν θα μπορούσε να αναπαραχτεί και τελικά θα καταστρέφονταν.

Κάποια όμως από τα μακρομόρια θα μπορούσαν να αναπαράγουν καινούρια μακρομόρια που θα ήταν ακόμη καλύτερα στο να αναπαράγονται. Έτσι θα είχαν ένα πλεονέκτημα και θα είχαν την τάση να αντικαταστήσουν τα αρχικά μακρομόρια.

 Με αυτόν τον τρόπο άρχιζε μια διαδικασία εξέλιξης που οδήγησε στην ανάπτυξη όλο και πιο πολύπλοκων και αυτοαναπαραγόμενων οργανισμών.

 Αυτοί οι πρωτόγονοι οργανισμοί κατανάλωναν διάφορα υλικά και απελευθέρωναν οξυγόνο.

 Αυτό σταδιακά άλλαξε την ατμόσφαιρα στη σύνθεση που έχει σήμερα  και επέτρεψε την ανάπτυξη ανώτερων μορφών ζωής όπως ψάρια, ερπετά, θηλαστικά και τελικά το ανθρώπινο γένος.

 

 

·         Όμως δεν «είμαστε ακόμη σίγουροι ποια θεωρία συνδυάζει επιτυχώς την γενική σχετικότητα και την κβαντική μηχανική, αν και γνωρίζουμε πολλά από τη μορφή που μια τέτοια θεωρία πρέπει να έχει», ώστε να το αποδείξουμε.

 

 Όσο θεωρούμε ότι το σύμπαν είχε μια αρχή, μπορούμε να υποθέσουμε ότι είχε και έναν δημιουργό.Αλλά αν το σύμπαν δεν έχει όρια , δεν θα έχει ούτε αρχή ούτε τέλος: απλώς θα υπάρχει. Και τότε δεν θα υπήρχε θέση για έναν δημιουργό.

 

 

·         Το ενδιαφέρον του  για το θέμα της καταγωγής και της μοίρας του διαστήματος, αναφέρει ο Στήβεν Χώκινγκ, αφυπνίστηκε εκ νέου όταν το 1981 η καθολική εκκλησία κάλεσε τους επιστήμονες να ενημερώσουν τους “Jesuits in the Vatican” επί θεμάτων κοσμολογίας, ενθυμούμενοι «ότι τρεις αιώνες πριν είχαν κάνει το λάθος να απορρίψουν τις επιστημονικές θεωρίες του Γαλιλαίου». Ανάμεσά τους και ο ίδιος. Αφού ενημερώθηκε ο Πάπας για τη θεωρία της Μεγάλης έκρηξης (Big bang) όσον αφορά την αρχή του σύμπαντος, συμβούλεψε να ασχοληθούν οι επιστήμονες με τα θέματα μετά την έκρηξη αλλά όχι με τη στιγμή της έκρηξης, γιατί εκείνη η ώρα ήταν η ώρα του Θεού.

A short history of time 7

 

Chapter 7

Black holes ain’t so black

 

According to Hawking’s theory, a black hole ought to emit particles and radiation as if it were a hot body with a temperature which depends only on the black hole’s mass: the higher the mass, the lower the temperature.

 But how is it possible that a black hole appears to emit particles when we know that nothing can escape from within its event horizon (=the boundary of the black hole)?

 The answer that quantum theory tells us is that the particles do not come from within the black holes but from the “empty” space just outside the black hole’s event horizon, its boundary. So it’s not actually empty. There must be quantum fluctuations, which one can think them as pairs of particles of light or gravity that appear together at some time, move apart, and then come together again and annihilate each other.

These particles are virtual and cannot be observed directly by a particle detector but their indirect effects can be measured. 

So black holes must radiate like hot bodies if our other ideas about general relativity and quantum mechanics are correct.

 The existence of radiation from black holes seems to imply that gravitational collapse is not as final and irreversible as we once thought.

If a body falls into a black hole, energy equivalent its mass will return to the universe.

 When the mass of the black hole becomes very small, it will more likely disappear from our part of the universe and with it any singularity there might be inside it.

This was the first indication that quantum mechanics might remove the singularities that were predicted by general relativity.

 

Black holes, chapter 6

 

Chapter 6

Black holes

·         Two hundred years ago, there were two theories about light. One which Newton favored was that light was composed of particles. The other one that it was made of waves. Now by the wave/particle duality of quantum mechanics, light can be regarded as both a wave and a particle.

·         A star that was sufficiently massive and compact would have such a strong gravitational field that light could not escape: any light emitted from the surface of the star would be dragged back by the star’s gravitational attraction before it could get very far. Although we would not be able to see these stars because the light from them would never reach us, we would still feel their gravitational attraction.

Such objects are what we now call black holes.

A consistent theory of how gravity affects light did not come along until Einstein proposed general relativity in 1915.

·         To understand how a black hole might be formed, we first need an understanding of the life cycle of a star: A star is formed when a large amount of gas (mostly hydrogen) starts to collapse in on itself due to its gravitational attraction. As it contracts the atoms of the gas collide with each other and the gas heats up. Eventually the gas will be so hot that when the hydrogen atoms collide they no longer bounce off each other but they coalesce to form helium. The heat released in this reaction, which is like controlled hydrogen bomb explosion, is what makes the star shine. This additional heat also increases the pressure of the gas until it is sufficient to balance the gravitational attraction, and the gas stops contracting.

·         When a star runs out of fuel, it starts to cool off and so to contract. If its mass is under a certain limit, it stops contracting and settles down to a possible final state as a “white dwarf”.

But stars with masses above the Chandrasekhar limit, when they come to the end of their fuel, in some cases they may explode or manage –rarely – to throw off enough matter to reduce the mass below the limit.

·         The gravitational field of a star changes the paths of light rays in space-time and eventually when the gravitational field at the surface of the star becomes very strong, the light cones are bent inward so much that light can no longer escape. So, if light cannot escape, neither can anything else. Everything is dragged back by the gravitational field. The light rays are trapped in and no light can be seen.

 A black hole is a region of space-time (where a star collapsed) from which nothing, not even light, can escape, because gravity is so strong.

·         In the long history of the universe, many stars must have burned all their nuclear fuel and have had to collapse. The number of black holes may well be greater than the number of visible stars, which totals about one hundred thousand million in our galaxy alone.

 Black holes are not really black after all: they glow like a hot body, and the smaller they are, the more they glow!

(continued in chapter 7)