cphil_bh
by Ray Shelton
INTRODUCTION
The earliest anticipation of anything like a black hole came in the late eighteenth century, when the French physicist Pierre-Simon, Marquis de Laplace (1749-1827) and the English cleric John Michell had the same remarkable thought. All physicists in those days were intensely interest in astronomy. Everything that was known about astronomical bodies was known by the light that they emitted or, in the case of the Moon and the planets, the light that they reflected. In the time of Michell and Laplace, Isaac Newton, though dead for half a century, was by far the most powerful influence in physics. Newton believed that light was composed of tiny particles that he called corpuscles. And if so, why wouldn’t light be affected by gravity? Laplace and Michell wondered whether there could be stars so massive and dense that light could not escape their gravitational pull. Wouldn’t such stars, if they existed, be completely dark and therefore invisible? This question that Michell and Laplace asked was whether a star could have such large mass and small size that the escape velocity would exceed the speed of light. On 27 November 1783, John Michell, who was Rector of Thornhill in Yorkshire, England, read a paper before the Royal Society of London suggesting that if a heavenly body got big enough, its gravity would become so powerful that not even light could escape from its surface. This was the first reference to the objects that John Wheeler would later call black holes. As the black-hole expert Kip Throne of the California Institute of Technology said,
“They had the wrong theory of light and the wrong theory of gravity. Yet, when they combined the two, they got the right prediction.”
“Thirteen years later, the great French mathematician Pierre Simon, Marquis de Laplace, arrived at a similar conclusion. He popularized the same prediction in the first edition of his famous work Le Systeme du Monde, without reference to Michell’s earlier work. Laplace kept his dark star prediction in the second (1799) edition, but by the time of the third (1808) edition, Thomas Young’s discovery of the interference of light with itself which forced natural philosophers to abandon the corpuscular explanation of light in favor of a wave theory devised by Christian Huygens. Of course, the wave theory could not be meshed with Newton’s law of gravity so as to compute the effect of a star’s gravity on the light it emits. For this reason, presumably, Laplace deleted the concept of a dark star from the third and subsequent editions of his book.” [1]
FOOTNOTES
The discovery of black holes was predicted by the German astronomer Karl Schwarzschild (1874-1916) while studying Einstein’s theory of gravity, the general theory of relativity, in between his own calculations of artillery trajectories at the Russian front during World War I in 1916. He read Einstein’s formulation of general relativity in the 25 November 1915 issue of the Proceedings of the Prussian Academy of Science. Remarkably, just months after Einstein had put the finishing touches on his general relativity, Schwarzschild was able to use the theory to gain a complete and exact understanding of the way space and time warp in the vicinity of a perfectly spherical star. Within a few days, from Einstein’s new field equations, Schwarzschild had calculated the curvature of space-time outside of any spherical, non-spinning star. His calculation was elegant and beautiful, and the curved space-time geometry that it predicted, the Schwarzschild’s geometry as it soon came to be known, was destined to have enormous impact on our understanding of gravity and the universe. Schwarzschild sent the results of his calculations from the Russian front to Einstein, who presented them on Schwarzschild’s behalf to the Prussian Academy in Berlin on 13 January 1916.
Beyond confirming and making mathematically precise the warping of space-time, Schwarzschild’s work, which has now come to be known as “Schwarzschild’s solution”, revealed a stunning implication of general relativity. Schwarzschild showed that if a star is concentrated in a small enough spherical region, so that its mass divided by its radius exceeds a particular critical value, the resulting space-time warp is so radical that anything, including light, that gets too close to the star will be unable to escape its gravitational grip. Since not even light can escape such “compressed stars,” these stars were initially called dark or frozen stars. A more catchy name was coined years later by John Wheeler, who called them black holes; black because they cannot emit light, holes because anything getting too close falls into them, never to return. The name stuck.
Although black holes have a reputation of rapacity, objects that pass by them at a “safe” distance are deflected in much the same way that they would be by an ordinary star, and would proceed upon their merry way. But objects of any composition whatsoever gets too close, closer than what has been termed the black hole’s event horizon, are doomed: they will be drawn inexorably toward the center of the black hole and subject to an ever increasing and ultimately destructive gravitational pull. For example, if you dropped feet-first through the event horizon, as you approached the black hole’s center, you would find yourself getting increasing uncomfortable. The gravitational force of the black hole would increase so dramatically that its pull on your feet would be much stronger than its pull on your head (since in a feet-first fall, your feet would always be closer to the black hole’s center than your head); so much stronger, in fact, that you would be stretched with a force that would quickly tear your body to sherds. In other words, a black hole warps the surrounding space-time fabric so severely, that anything that comes within its “event horizon” can not escape from its gravitational grip. No one knows exactly what happens at the deepest point of a black hole.
If, on the contrary, you were more wise in your movement about the black hole and took care not to cross the event horizon, then you could make use of the black hole for another rather amazing feat. Imagine that you had discovered a black hole whose mass is 1000 times the mass of the sun, and that you were to lower yourself on a cable to about an inch above the black hole’s event horizon. If you hover there just above the event horizon for a year, and then climb back up the cable to your waiting starship for a short trip home. And upon your arrival to earth you would find that more than ten thousand years had passed since your initial departure. You would have successfully used the black hole as a kind of time machine, allowing you to travel into the earth’s distance future. This happens because gravitational fields cause a warping of time, and this means that the passage of time would slow way down. Your watch would tick about ten thousand more slowly than your friends back on earth. Thus you would find upon arrival back on earth that more than ten thousand years had passed since your initial departure. This warping of time near a black hole was an implication of Einstein’s general theory of relativity as revealed by Karl Schwarzschild’s calculations.
Schwarzschild died only a few months after finding his solution, from a skin disease he contracted at the Russian front. He was 42. On 19 June 1916, Einstein had the sad task of reporting to the Academy that Karl Schwarzschild had died of an illness he contracted on the Russian front. His brief encounter with Einstein’s theory of gravity uncovered one of the striking and mysterious facts of the natural world.
Do black holes exist? For a long time physicists have been skeptical about whether such extreme configurations of matter in a black hole could ever actually occur, and many thought that black holes were merely the reflection of an overworked theoretical imagination. But during the last decade, an increasingly convincing body of experimental evidence for the existence of black holes has accumulated. Of course, since they are black, they cannot be observed directly by scanning the sky with telescopes. Instead, astronomers search for black holes by seeking anomalous behavior of other more ordinary light-emitting stars, that may be positioned just outside a black hole’s event horizon. For instance, as dust and gas from the outer layers of nearby ordinary stars fall toward the event horizon of a black hole, they will be accelerated to nearly the speed of light. At such speeds, friction within the maelstrom of downward-swirling material will generate an enormous amount of heat, causing the dust-gas mixture to “glow,” giving off both ordinary light and X-rays. Since this radiation is produced outside the event horizon, it can escape the black hole and travel through space to be observed and studied directly. General relativity makes detailed predictions about the properties that such X ray emissions will have; observations of these predicted properties gives strong, albeit indirect, evidence for the existence of black holes. One likely candidate is the star Cygnus X-1, about six thousand light-years away, which is a massive generator of X-ray radiation. In fact, it is hard to imagine any other physical force besides gravitational collapse that can explain the enormous energy output of stars like Cygnus X-1. Another example, mounting evidence indicates that there is a very massive black hole, some two and half million times as massive as our sun, sitting in the center of our own Milky Way galaxy. It is almost impossible to see directly these black holes, but the matter near these black holes gives them away. The stars at the center our galaxy whip around at such a speed that they must have to be pulled by a mass equal to three million suns, packed into a volume smaller than our solar system. No conceivable type of star or group of stars measures up, so the culprit must be a massive black hole. And even this gargantuan black hole pales in comparison to what astronomers believe to reside in the core of the astonishingly luminous quasars that are scattered throughout the cosmos: black holes whose masses may well be billion times that of the sun.
Black holes come three basic types:
1. Stellar black holes. These black holes form when a large star runs out nuclear fuel and collapses under their own mass.
2. Supermassive black holes. These black holes are found in the cores of galaxies, and have the mass of millions or billions of suns. Astronomers don’t quite know how they are formed, but one possibility is that smaller black holes merge together.
3. Mini black holes. These black holes have the mass of an asteroid or less. None has ever been detected, but physicists speculate that they form under the extreme condition of the big bang or in particle collisions.
A further implication of the superstring theory of gravity concerns the nature of black holes. Einstein’s general theory of relativity includes the possibility of singularities, points in space at which the whole structure of space-time breaks down. But what will happen in a superstring universe in which points are no longer of primary importance? As a star collapses, its radius becomes smaller and smaller, and in the process, enormous gravitational energy is released. But such energy has the potential for the curving space-time (remember that both energy and matter will curve the fabric of space-time). The conventional theory has it that the gravitational attraction will cause the star to collapse right down to a dimensionless point and along the way create a black hole. Einstein’s general theory of relativity permits such singular points, called singularities, at which the fabric of space-time breaks down, and indeed all the laws of physics vanish. Surrounding the space-time singularity is an event horizon; the horizon may be typically 1 km in diameter. Nothing can escape from within the event horizon of a black hole, not even light itself. Therefore anything that crosses this event horizon is doomed to be swallowed up by the black hole.
But what will happen when a collapsing star reaches 10-33 centimeters (the dimensions of the superstring)? No one really knows, but some theoreticians have speculated that vibrating superstrings, while not eliminating black holes, could help to avoid the creation of singularities. Shrunk down to 10-33 centimeters, the star will occupy the space of a superstring, and the tremendous energy that is released in gravitational collapse can now be used to excite the vibration and rotation modes of the string. Since there are an infinite number of modes of the string, they will soak up even the vast energy of a collapsing star. The star need shrink no further. Although it alredy occupies an unimaginably small distance in space, it will never collapse to a dimensionless point. While the black hole itself will still exist, it need not contain a singularity at its heart.
Many physicists will be please with this, for this means that the laws of physics need never break down at the point of the singularity. What John Wheeler calls “the crsis in physics” would be averted. (Note that this does not mean that black holes will not exist, only that they will no longer contain pointlike singularities at which the laws of physics break down. Even though the enormous energy of a collapsing star is used to excite the various vibrations of a superstring, this energy can never be radiated away; light simply cannot move fast enough to escape from the event horizon of a black hole.)
Quantum processes are also related to space-time curvature in the immediate neighborhood of black holes. It was Stephen Hawking, the brilliant Cambridge physicist, who first argued that close to a black hole, the extreme degree of curvature of space-time actually creates elementary particles. Matter, according to Hawking, is created out of the fabric of space-time itself. But if geometry is the source of quantum matter, then clearly relativity and quantum theory have to be unified at some deeper level.
Stephen Hawking was born on January 8, 1942, as he delights to tell people, precisely three hundred years to the day after Galileo died. Hawking’s father worked on research into tropical diseases at the National Institute of Medical Research and encouraged Stephen to follow an academic path aimed at entrance to Oxford University. But the encouragement did not extend to Hawking’s decision to study mathematics; his father tried to talk him out of this, arguing that there were no jobs for mathematicians. Even so, Stephen entered Oxford University in 1959 to study mathematics and physics. His contemporarties and tutors recall today that he was a remarkable student with a mind unlike that of anyone else. He passed examinations with almost contemptuous ease, obtained a First Class degree, and moved to Cambridge University to begin research in cosmology.
At this time, in the early 1960s, Hawking began his involvement with singularities, an involvement that lies at the heart of all his major contributions to science and that is the key to understanding the moment of creation itself. He was fascinated by the idea of a mathematical singularity, a point where not only matter but space and time as well are either crushed out of existence or, in the case of the Big Bang, created. The standard equations of general relativity theory predict the existence singularities, but in the early 1960s hardly anyone took this prediction seriously. Singularities were assumed to be an indication that the simplest version of Einstein’s theory, with a smooth distribution of matter through space-time, was not a realistic way to describe the confusion of a superdense state, and that probably a better understanding of the equations would show that as a collapsing object approached a singularity, at some stage there would be a “bounce”, making it expand again, or some other effect that halted the collapse short of a point of infinite density. Either that, or Einstein’s theory was incomplete and would break down at very high densities; that is, in very strong gravitational fields. Hawking determined to find out if this were true. But it was to be several years before this determination bore fruit, because it was in his first year of graduate work, 1962, that the first symptoms of his illness appeared and were diagnosed. Hawking’s body is afflicted with the dreaded Lou Gehrig’s disease (amyotrophic lateral sclerosis, ALS), a disease that gradually destroys the nerves which control the body’s muscles and leaves the muscles, one after another, to waste away. Given only a few years to live, Hawking became depressed, took to drink, and virtually gave up his work, because he thought that he would be dead in a few years.
But as the months passed it became clear that the progression of the disease had halted and stabilized. Hawking was slightly incapacitated physically, but he wasn’t getting any worse. And at the same time, he realized first that his intellect had been totally unaffected by the disease and his intellect would be unaffected whatever happened to his body. His work was entirely brainwork, which could be carried on regardless of the deterioration of his physical condition. since then, as far as a casual acquaintance can tell, Hawking has never looked back, either in his private life or in his work. He married in 1967, has two sons and a daughter, and leads as normal a life as possible. It was also in the late 1960s that he began to achieve recognition for his scientific work.
One of Hawking’s major achievements at the time was carried out in collaboration with mathematicisn Rogen Penrose, who was then working at the University of London. Penrose had developed a theorem that since gravity is always attractive, a star collapsing under its own gravity is trapped in a region whose surface shrinks to zero size; that is, any body undergoing gravitational collapse must eventually form a singularity of infinite density and space-time curvature. A singularity is a point in space-time at which space-time curvature becomes infinite; it is a point of infinite density. Some theories predict that a singularity will be found at the center of a black hole or at the beginning or end of the universe. In 1965, when Hawking read about Penrose’s theorem, he realized that if one reversed the direction of time in Penrose’s theorem, so that the collapse became an expansion, the conditions of of Penrose’s theorem would still hold, provided the universe were roughly like a Friedmann model on large scales at present time. Penrose’s theorem had shown that any collapsing star must end in a singularity; the time-reversed argument showed that any Friedmann-like expanding universe must have begun with a singularity. In next few years, Hawking developed new mathematical methods to remove technical conditions from the theorem and prove that singularities must occur. In 1970, together Hawking and Penrose published a joint paper in which they proved that the equations of General Relativity in their classical form (that is, without allowing for quantum effects) absolutely require that there was a singularity at the birth of the universe, a point at which time began. This theorem proved that there must be a big bang singularity provided only that the general theory of relativity is correct and that the universe contains the amount of matter as was observed.
There are four basic assumptions that underlie this singularity theorem:
1. Gravity is always attractive.
2. Time moves forward only, never backward.
3. The universe contains enough matter to generate at least one black hole.
4. The equations of General Relativity accurately describe the universe.
The first two assumptions appear obvious. Indeed, their rejection would be worse than the singularity. The third was the subject of a paper by George Ellis and Stephen Hawking in which they sought to determine that the gravitational attraction exerted by the microwave background radiation by itself is always sufficient to create a “trapped” region, a black hole. According to the fourth, there was no way around the singularity problem within the framework of classical General Relativity. If singularities are to be avoided in the real universe, the only hope is to improve relativity theory by bringing in the effects of the quantum theory and developing a quantum theory of gravity.
In the 1970s, Hawking’s investigations of the mathematics of black holes led, through the introduction of quantum effects, to the startling conclusion that black holes can “evaporate” and must eventually explode. This work brought him into the popular limelight, at least in the science magazines. And in 1974, at the very young age of thirty-two, he was elected a Fellow of the Royal Society.
By this time, he was confined to a wheelchair following a further progression of the disease. But for the next ten years he seemed to have remained much the same physically. He has only very limited control over the muscles of his body, and slumps rather than sits in his wheelchair; his speech is labored and almost incomprehensible to anyone who does not know him well and has not become familiar with his voice. The honors that Hawking has received include the Albert Einstein Award in 1978, and in 1980 he became the Lucasian Professor of Mathematics in the University of Cambridge; the chair previously occupied by Isaac Newton and Paul Dirac, among others. These honors, and honorary degrees heaped upon Hawking by universities around the world, are the sort of thing usually associated with a scientist who has completed his greatest work and can now settle down to a comfortable position of eminence as an administrator and teacher. Few mathematicians achieve much in the way of new work after they reach an age of thirty; new ideas come from young minds that are not hidebound by convention, or so we are told. But Hawking’s mind is as sharp as ever, and he has now put forward a model of the universe that attempts to combine the ideas of General Relativity and quantum physics and that not only removes the uncomfortable singularity at the moment of creation but that, in principle, explain everything in one package.
General Relativity tells that there must be a singularity at the beginning of time: the moment of creation. But General Relativity, like all of our theories of physics, breaks down for times earlier than the Planck-Wheeler time: approximately 10-44 seconds. Although a variation on the steady state idea (that the universe is eternal and unchanging), with either a smooth meta-universe, or some sort of overall chaos (that the universe in the past was in a state of high temperature and density), plus inflation, could provide a way to produce a local region of expanding space-time rather like the one we live in, it would be much more satisfying if we could develop a mathematical model, a set of equations, to describe our universe in a self-contained way; especially if that model could avoid the embarrassment of a singularity at t = 0. This is the basis of Hawking’s approach to the puzzle of our origins, and of his attempts to combine General Relativity and Quantum Theory, at least partially, in a good working model of the universe. Very few physicists are entirely happy with Hawking’s approach. Hawking has made a lot of simplifying assumptions, and they don’t always approve of the way he handles equations. But the underlying physical principles of the model are very clear and straightforward, and it is this that persuades some that Hawking is on the right track.
According to Stephen Hawking, Heisenberg’s principle allows for small regions of space-time to borrow quanta of energy (virtual particles and photons of energy) provided that they are quickly paid back. At the length scale of the elementary particles, these energy fluctuations do not pose a serious problem. But in much smaller regions, the borrowed energy can act to curve space-time. (In Einstein’s theory, energy and matter both will distort the fabric of space-time.) Indeed, at about 10-33 centimeters; that is, 1/1,000,000,000,000,000,000,000,000,000,000,000 centimeters, called Planck’s-Wheeler length, this borrowed energy is large enough to curve space-time right around itself and, in effect, create a mini black holes. Space-time loses its smooth, continuous nature and breaks apart into violently fluctuating foam. These fluctuations may be thought of as a pair of a virtual particle and antiparticle which constantly materialize out of the vacuum, separate, and then annihilate. In the vicinity of a black hole, the strong tidal forces may lead to one member of a virtual particle-antiparticle pair falling into the black hole, leaving the other without a partner with which to annihilate. If the latter does not experience the same fate as its partner, it becomes a real particle and appears to be emitted by the black hole.
In the relativity theory of gravity, which is based on real space-time, there are only two possible ways the universe can behave: either it has existed for an infinite time, or else it had a beginning as a singularity at some finite time in the past. In the quantum theory of gravity, on the other hand, a third possibility arises. Because the quantum theory is using an Euclidean space-time, in which the direction of time is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that form a boundary or edge. It would be like the surface of the earth that is finite in extent but it doesn’t have a boundary or edge: if you sail off into the sunset, you don’t fall off the edge or run into a singularity.
The quantum theory has opened a new possibility, in which there would be no boundary to space-time and there would be no need to specify the behavior at the boundary. There would be no singularity at which the laws of science break down and no edge of space-time at which one would have to appeal to God or some new law to set the boundary conditions for space-time.
In Hawking’s fourth book, A Brief History of Time (1988), he presents this third possibility. It was his first book aimed at a popular audience. It has sold very well; it is the best selling science book of all time with more than seven million copies sold. Recently it has been made into a feature length film.
Most of the book is about the history of universe and the latest discoveries about the theories of gravity. The chapters on black holes are probably the clearest ever written. It covers the research on the application of gravitational theories to the origin and the development of the universe. But the book is more than a popular-level text on gravitational theories; it makes many controversial philosophical and theological pronouncements.
In its final chapter, Hawking declares the goal of his life and work. He directs all of his efforts toward answering the following fundamental questions: “What is the nature of the universe? What is our place in it and where did it and we come from? Why is it the way it is?” [1] Hawking attempts to answer these questions through physics alone. He gives no reason for his refusal to accept, or acknowledge, answers already given elsewhere, especially in the pages of the Bible. From the close contact with Christians, including his ex-wife, Jane, and physics colleague, Don Page, we can assume that he was aware, at lease, that the Bible presented answers to these questions. Yet he apparently choose to ignore its answers.
In the “Introduction” to Hawking’s book A Brief History of Time, Carl Sagan says,
“This is also a book about God …or perhaps about the absence of God. The word God fills these pages. Hawking embarks on a quest to answer Einstein’s famous question about whether God had any choice in creating the universe. Hawking is attempting, as he explicitly states, to understand the mind of God. And this makes all the more unexpected the conclusion of the effort, at least so far: a universe with no edge in space, no beginning or end in time, and nothing for a Creator to do.” [2]
It is this idea that time and space is finite without a boundary, “no edge in space, no beginning or end in time,” in which “The universe would be completely self-contained and not affected by anything outside itself. It would neither be created nor destroyed. It would just BE.” [3] Hawking presents in his book this idea of the universe as finite without boundary as “just a proposal” since “it cannot be deduced from some other principle.” [4] This theory is difficult to prove by observations, for two reasons: first, it is a theory of quantum gravity and “we are not yet sure which theory successfully combines the general relativity and quantum mechanics…. Second, any model that described the whole universe in detail would be much too complicated mathematically for us to be able to calculate exact predictions. One therefor has to make simplifying assumptions and approximations — and even then, the problem of extracting predictions remains a formidable one.” [5]
In 1981, Hawking attended a conference on cosmology organized by the Jesuits in the Vatican. The Roman Catholic Church had with Galileo tried to lay down the law on questions of science, declaring that the sun went around the earth. Now, centuries later, the Roman Church decided to invite a number of experts, including Hawking, to advise them on cosmology. The Roman Catholic Church had seized on the big bang model and in 1951 had officially pronounced it to be in accordance with the Bible. At the end of the conference in 1981, the participants were granted an audience with the pope. The pope told them that it was all right for them to study the evolution of the universe after the big bang, but they should not inquire into the big bang itself, because that was the moment of Creation and therefore the work of God. Hawking had given during the conference a lecture in which he presented the possibility that space-time was finite but had no boundary, which meant that it had no beginning, no moment of Creation. Hawking wrote,
“I had no desire to share the fate of Galileo, with whom I had a strong sense of identity, partly because of the coincidence of having been born exactly 300 years after his death!” [6]
His paper was rather mathematical so that it implications for the role of God in the Creation of the universe were not generally recognized at the time. At the time of the conference, Hawking did not know how to use the “no boudary” idea to make predictions about the universe. But when he returned from the conference, Hawkings spent the following summer at the University of California, Santa Barbara, where a friend and colleage of his, Jim Hartle, worked with him on what conditions the universe must satisfy if space-time had no boundary. When he returned to Cambridge, Hawkings continued this work with two of his research assistants, Julian Lutrel and Jonathan Halliwell. [7] Hawking writes:
“If the universe really is in such a quantum state, there would be no singularities in the history of the universe in imaginary time. It might seem therefore that my more recent work had completely undone the results of my earlier work on singularities. But, as indicated above, the real importance of the singularity theorems was that they showed that the gravitational field must become so strong that quantum gravitational effects could not be ignored. This in turn led to the idea that the universe could be finite in imaginary time but without boundaries or singularities. When one goes back to real time in which we live, however, there will still appear to be singularities. The poor astronaut who falls into a black hole still will come to a sticky end; only if he lived in imaginary time would he encounter no singularities.
“This might suggest that the so-called imaginary time is real time; and that what we call real time is just a figment of our imaginationa. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic; and what we call real is just an idea that we invent to help us describe what we think the universe like. But, according to the approach I described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: Which is real, ‘real’ or ‘imaginary’ time? It is simply a matter of which is the more useful description.” [8]
So science does not provide us with the truth, but only with that which is useful. Although Hawking believes that knowledge of the origin of universe is unattainable, he is not an atheist. He emphatically rejects the label “atheist.” His view of God comes closer to the position that is called “deism”; that is, the view that after God created the universe, he left it to operate according to the laws that He established. In his A Brief History of Time, Hawking writes,
“These laws [of physics] may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them and does not now intervene in it.” [9]
Hawking goes on to conclude at the end of chapter 8 of his book:
“The idea that space 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 those laws. However, those laws do not tell us what the universe should have looked like when it started — it would still be up to God to wind up the clockwork and choose how to start it off. So 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?” [10]
Hawking rejects this conclusion and believes that the universe had a creator.
“It would be very difficult to explain why the universe should have begun in just this way, except as the act of a God who intended to create beings like us.” [11]
ENDNOTES
INTRODUCTION
A fully unified description of the Universe and all that it contains (called a “theory of everything”, TOE) would also have to describe gravity and spacetime at the quantum level. This implies that spacetime itself must be, on a very short-range scale, quantized into discrete lumps, not smoothly continuous.
Modern physics is built on twin foundations: quantum theory and general relativity. Yet inspite of a half a century of hard work by some of the world’s leading physicists, these two theories have stubbornly refused to be reconciled, but continue to coexist in paradoxical and incompatible ways. On the one hand, general relativity and quantum theory are irreconcilable, yet on the other, they are mutually dependent. General relativity is a theory about the structure of space-time, curved geometry being determined by the amount of energy and matter present. But matter and energy are quantum mechanical in nature, so a complete account of space-time geometry cannot ignore the quantum nature of matter and energy which creates its very form.
Cosmology and physics has three fundamental constants that control the scale of physical phenomena. They are: the velocity of light, c = 2.998 × 1010 centimeter per second, which is the scale of relativistic effects; Newton’s gravitational constant, G = 6.670 × 10-8 dyne-centimeter2/gram2, which is the measure of the strength of gravitational effects; and Planck’s constant, h = 6.6261 × 10-27 erg-seconds, which controls the scale of quantum effects and represents the elementary quantum of action, action being defined as energy multiplied by time.
Around the beginning of the twentieth century (1900), Max Planck, the forefather of the quantum theory, introduced this constant in deriving his radiation law; h multiplied by the frequency of the radiation represents a bundle of energy, that is, a quantum of energy. Radiant energy at any wavelength can occur only as multiples of this quantum of energy; thus energy is quantized. In 1955, John Wheeler, by combining the laws of quantum mechanics and the laws of general relativity in a tentative and crude way, showed that these three constants may be combined to produce three fundamental quantum units of length, time, and mass. They are known, respectively, as the Planck-Wheeler length, denoted by lp = √(Gh/2πc3), with a value of approximately 10-33 centimeters; the Planck-Wheeler time, denoted by tp = lp/c and is approximately 10-44 seconds; and the Planck-Wheeler mass, mp = √(h/2πGc) and is approximately 10-5 grams.
The Planck-Wheeler length and time are almost unbelievably small; smaller than the atomic scale which is smaller than the laboratory scale. The Planck-Wheeler mass may seem to be an unremarkable number (it is approximately the mass of a cell) but it must be compared to the typical mass scale of ordinary particle physics: It is some 1019 times greater. The scale indicated by these Planck-Wheeler units are therefore very extreme indeed. Their significance is that they are the length, time, and mass scales at which relativitic, gravitational, and quantum effects become simultaneously comparable. Such scales could never be achieved in any laboratory situation, not even in the most powerful particle accelerator. But these scales are approached in the neighborhood of the initial singularity of the universe. This suggests that the physics in the neighborhood of the initial singularity is best described using a theory in which the relativistic, gravitational, and quantum effects are combined. General relativity is a theory in which the relativistic and gravitational effects are already combined. What is needed, therefore, is a quantizied version of general relativity; that is, a quantum theory of gravity.
Whenever physicists have attempted to bring general relativity and quantum theory together, they have failed. Some physicists have wondered if this basic incompatibility arises in the very definition of space-time itself. Indeed a closer analysis suggests that relativity incorporates a limited paradigm about space-time structure that stretches back for 300 years.
When Einstein revolutionized the Newtonian concept of space and time, he nevertheless continued to assume that space and time is continuous. That is, the properties of space continue unchanged to smaller and smaller scales; space can be divided and subdivided right down to the dimensionless point, with all the properties changing smoothly from point to point. In mathematical terms, space-time and the theory of relativity use the language of calculus and differential equations, the basic grammar of science that has been employed since the eighteenth century. Could it be that Einstein’s revolution did not go far enough?
The quantum theory shows that the infinite divisibility of space must be limited. The reason can be seen in Heisenberg’s uncertainty principle, which states that the energy confined within smaller and smaller regions of space-time becomes increasingly uncertain. The reason is not too difficult to see. Suppose you as a physicist try to measure the exact energy of a quantum system within a given short time interval. Heisenberg’s principle dictates that the smaller this time interval, the more uncertain the energy.
Heisenberg’s uncertainty principle was essentially formulated to describe nature at the scale of atoms and elementary particles. To use it to talk about space-time foam requires an extrapolation that is as great as the change of scale between our own bodies and the elementary particles! Who knows that physics may not change radically within that region? But most physicists seem perfectly happy to apply Heisenberg’s principle right down to the domain of 10-33 centimeters, Planck-Wheeler length, and at such small distances, it is clearly necessary to abandon a space-time that is built upon the notion of dimensionaless points and continuity. The theory of relativity at this scale of things demands a totally new mathematical formulation, not yet discovered.
Yet despite this clear indication that space-time cannot be subdivided without limit, even quantum theory itself continues to make use of the mathematics of continuity. Erwin Schrodinger’s wave equation, for example, is a differential equation, relating what is happening at one point to what happens at another point an infinitesimally short distance away. Likewise, its solution, the wave function, is defined at each infinitesimal point in space.
On the one hand, quantum theory denies continuity and the ultimate reality of the dimensionless point, yet on the other hand, quantum theory and relativity both continue to make use of such notions in their mathematical underpinnings. Clearly physicists are being forced toward a new intuition. What is called for is a totally new way of thinking. Manly physicists believe that superstrings or twistors contain the seeds of this mathematical revolution.
Quantum mechanics and general relativity were the major developments in theoretical physics in the twentieth century. Unifying them into a single theoretical theory has proven to be extremely challeging, if not impossible. This is because the resulting quantum theories are plague by infinities that result from the fact that interactions take place at a single mathematical point (zero distance scale). By spreading out the interactions, string theory offers the hope of developing not only a unified theory of particle physics, but a finite theory of quantum gravity.
String theory attempts to get rid of the problem of infinities by getting rid of particle interactions that occur at a single point. Take a look at the Heisenberg’s Uncertainty Principle:
ΔpxΔx ≥ h/2π,
where Δpx is the uncertainty of the momentum of a particle moving in the x direction, and Δx is the uncertainty in the position of the particle in the x direction, and h is Planck’s constant, the atom of action (h = 6.625 × 10-34 joule-sec). This principle says that it is impossible to determine simultaneously the velocity (or momentum) and the position of an electron or any other microphysical particle; that is, the more accurate the determination of its velocity (or momentum) is, the hazier its position becomes and vice versa.
Now if the uncertainty of the momentum px blows up, that is, Δpx → ∞, this has been interpreted to imply that Δx → 0 and to mean that if the uncertainty of the momentum p in the x direction is very large (infinite), then the uncertainty of the position in x direction will be very small (zero) distance. Or to put it another way, pointlike interactions (zero distances) imply infinite momentum. This leads in Quantum Field Theory to loop integrals and infinities in calculations. The existence of these infinities caused some physicists to wonder if there was a basic flaw in the foundations of Quantum Field Theory. Could these infinities be somehow related to the prevailing idea of infinite divisibility of space-time and the use of dimensionless points as the building blocks of geometry? So in string theory, a point particle is replaced by a one-dimensional string. That is, in the old quantum theory, where a particle is a mathematical point, with no extension, in string theory, the particles are strings, with extension in one dimension. This gets rid of infinities. That is, the Δx does not go all the way to zero but instead cuts off at some small, but nonzero value. This means that there will be a large, but finite value of the momentum and hence Δpx does not become infinitely large. Instead the uncertainty of the momentum goes to a large, but finite value and the loop integrals can be gotten rid of. Now in order to get a cutoff by the length of the string, the uncertainty relation must be modified.
But the uncertainty relation does not need to be modified and the particles need not be replaced with one-dimensional strings. The Δx cannot be zero in the uncertainty relation, because the product of Δx and Δpx is always greater than or equal to h/2π and Planck’s constant h is never zero
(h = 6.625 × 10-34 joule-sec).
In fact, the uncertainty relation implies that the x dimension of space has a finite quantum value. Neither Δx nor Δpx can have zero value since their product is equal to h/2π which is non-zero. Thus the uncertainty relation does not need to be modified to include a term which can serve to fix a minimum distance for Δx. And strings are not needed to get rid of the infinities.
A similar relation holds for all physical quantities whose products have the same dimensions as Planck’s constant, h. We can obtain another equally important version of the uncertainty principle that was proposed by Einstein, if we multiply Δpx by vx and divide Δx by vx
(since vx = Δx/Δt):
ΔEΔt ≥ h/2π.
where ΔE and Δt represent the uncertainty of energy and of time. This shows that the uncertainty of an energy measurement depends on the time available to make it. Thus if an atom remains in an excited state for a very short time, the precise energy of the state may be determined very accurately.
It may be objected that Heisenberg’s Principle does not necessarily lead to the concept of the minimum spatiotemporal atomicity, although it is compatible with it. It is theoretically conceivable that while Δpx is increased without limit, Δx will approach zero, similarly, for ΔE → ∞, Δt would be equal to zero. Although many physicists have claimed that the Heisenberg’s Principle is mathematically compatible with the existence of pointlike positions and mathematical instants which the principle of spatiotemporal continuity requires, the principle in its mathematical formulation does not allow for that. Planck’s constant being non-zero does not allow for that Δx or Δt to be equal to zero. The minimum possible temporal interval Δt would be equal to l0/c, where l0 is the minimum length and c is velocity of light. The estimated numerical values of chronon and hodon was found by Levi, Pokrovski, Beck, and others during the period between the two World Wars. In that period, the new names “chronon” and “hodon” were invented for designating the atoms of time and of space, respectively. The value computed for the chronon was naturally extremely small. According to J. J. Thomson, it is of the order of 10-21 seconds, while according R. Levi it is 4.48 × 10-24 seconds. The computed magnitude of the hodon is of the same order as the radius of the classical electron which is 10-13 centimeters.
For practical purposes, and when considered macroscopically, space and time are continuous: the duration of chronons is so insignificant that they may safely be equated with durationless instants; similarly, the difference between mathematical points and spatial regions of the radius of 10-13 centimeters is entirely negligible on our macroscopic scale. This, however, does not make the difference between the classical continuous space and time and its modern atomistic counteparts less radical.
But speculation about the nature of discrete “chronon” and “hodon” on the part of physicists have been contradictory, or at least stated in a self-contradictory language. When they claim that time consists of chronons succeeding each other, and when they claim that the duration of each individual chronon is 4.48 × 10-24 seconds, what do they assert except that the minimum intervals of time are bounded by two successive instants, one of which succeeds the other after the time interval specified? The concept of chronon seems to imply its own boundaries; and as these boundaries are instantaneous in nature, the concept of instant is surreptitiously introduced by the very theory which purports to eliminate it. A similar consideration can be said about the atomization of space.
In answer to the above argument, of course nothing is gained if a theory introduces in a disguised way the very concept which it overtly eliminates. But it needs to be recognized that it is almost impossible to discuss concepts in which the language involved assumes the concepts that it attempts to refute and replace. What is needed is an extensive and systematic revision of our intellectual habits associated with the traditional ideas of space and time.
The first thing that must be recognized is that these early theories of atomistic space and time spoke separately of chronons and hodons, as if they if they were atoms of space and atoms of time, betraying a prerelativistic state of mind. Before the theories of relativity, it seemed legitimate to treat space and time separately because their separation was one of the basic assumptions of classical physics. The impossibility of separating space and time in the special theory of relativity is the reason for giving up the concept of absolute simultaneity or, what is the same, of purely spatial distance. By asserting the existence of the hodon, they were claiming that there is a purely spatial distance approximately equal to 10-13 centimeters; in other words, they were separating space from time on the microscopic level, although it was precisely on this level that the consequences of relativity were so spectacularly confirmed.
But if we accept the fusion of space and time even on the subatomic level, then it is evident that no separation of hodon and chronon is possible; they are complementary aspects of a single elementary entity which may be called a pulsation of time-space. Thus there is no chronon without a hodon and vice versa. To postulate a timeless (that is, a chrononless) hodon would mean that instantaneous cuts of four-dimensional processes are possible, at least on the atomic level, that there are absolutely simultaneous events within atoms. More specifically, it would mean that there are within the atoms couples of events interacting with infinite speed; for we have seen that absolutely simultaneous events would lie on a world line of any instantaneous physical action. All these assumptions (which are really one assumption in several forms) are contrary to the special theory of relativity and thus their plausibility is very small.
On the other hand, the assumption of hodonless or spaceless chronons does not seem to contradict directly the relativity theory, in which the existence of the infinitely tenuous world lines (that is, world lines without any spatial extent) was freely assumed. However, on closer inspection, even this assumption is incompatible, if not with the letter, then at least with the spirit, of relativity. The assumption of extensionless points, whose infinite continuous aggregates would constitute space, was merely another way of saying that space is infinitely divisible.
But there is no static space in the relativity theory. We have seen that the theory admits only successive timelike connections between events; there are no purely spacelike world lines as long as we take the relativity of simultaneity seriously. Thus the assertion of the spatially extensionless world lines is equivalent to the assertion either that there are purely spatial distances which are infinitely divisible or that the spatiotemporal distances themselves are infinitely divisible. As the first assertion is excluded, we have to consider only the second one. But to postulate the mathematical continuity of timelike world lines is contrary to the chronon theory. For this theory assumes that all timelike world lines, whether those of material particles or those of photons, are not divisible ad infinitum. We shall now see how the probability of this theory is strengthened by the converging empirical evidence which necessated the wave-mechanical theory of matter.
Thus in the light of the foregoing considerations, the assumption of the atomicity of space is superfluous because the existence of hodon is merely a certain aspect of the reality of the chronotopic (spatiotemporal) pulsations. While the chronon measure the minimum duration of events constituting a single world line, the hodon measures the minimum time necessary for the interaction of two independent world lines. Everything which had been said about the necessity of redefining spatiality can be repeated here; the only difference is that it is now being applied on the microcosmic scale. There are no instantaneous purely geometrical connections either in the macrocosm or in the microcosm; on either scale these connections should be replaced by chronogeometrical ones. On either scale, the concept of spatial distance is redefined in terms of causal independence. But while the interval of independence between, for instance, the world line of earth and that of Neptune is eight hours, it is equal to the duration of two chronons in the case of two microscopic “particles” when their “distance” is minimum.
For all practical purposes, this tiny interval may be disregarded; in other words, the temporal link between microphysical events can be regarded as instantaneous and the corresponding world lines as infinitely close. The relativistic picture of the world as a four-dimensional continuum of pointlike events is appoximately true on a macroscopic scale, but becomes seriously inadequate when microscopic relations are considered. But while the “pulsatinal” character of the world lines is incompatible only with what may be called “textbook relativity,” it is entirely consistent with the basic assumptions of the theory.
In order to avoid a self-contradictory formulation of the pulsational character of space-time, we have to make a serious effort to get rid of all spatial associations with which our classical concept of time is tinged. The theory of chronons, though outwardly denying the existence of instants, really assumes their existence. What does the alleged existence of chronons mean if not the assertion that two successive instants are separated by an interval of the order of 10-24 seconds?
But the self-contradictory statement in the chronon theory is due to the fact that we are trying to translate the pulsational character of world lines into visual and geometrical terms. In our imagination, we represent the flux of time by an already drawn geometrical line on which we may distinguish an unlimited number of points; hence our belief in the infinite divisibility of time. The chronon theory does not basically depart from this habit of spatialization; it merely substitutes, for the zero intevals, intervals of finite length. But again these intervals are imaginatively represented by geometrical segments; and as the concept of linear segments naturally implies the existence of its pointlike boundaries, the concept of the instant, verbally eliminated, reappears in the very act by which it is denied. What is overlooked by both those who assert and those who deny the existence of chronons is that it is impossible to reconstruct any temporal process out of static geometrical elements, whether these elements are dimensionless points or segments of finite length.
The spatial picture of time is inadequate in a triple sense:
(1) because of the essential incompleteness of time,
(2) because it is wrongly suggested that time, like a geometrical line, is without transversal extension, and
(3) because it is wrongly suggests the infinite divisibilty of time.
The last two errors led respectively to the concepts of extensionless and infinitely divisible world lines, infinitely close to each other. To such a view the idea of chronotopic pulsation is radically opposed, but we have to be on guard not to slip back into spatializing fallacies when we try to state this theory.
The difficulty which the chronon and hodon theory faces is analogous: it is extremely difficult to formulate this theory without surreptitiously introducing the concept of extensionless boundaries. Our language is so thoroughly molded by the intellectual habits created by infinitesimal calculus that we continue to speak of instants and points even when we are trying to deny them. Yet even some outstanding mathematicians have now begun to realize that the very concepts of point and instants may not be legitimate because the infinite divisibility of space and time, which two concepts presuppose, may be an unwarranted extrapolation of our limited macroscopic experience.
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