之前寫過有 ill 的字，原來數目有不少，但 generate 卻只有 venerate 是相近，字義當然是不同的。
Venerate – to honour or very much respect a person or thing.
之前寫過有 ill 的字，原來數目有不少，但 generate 卻只有 venerate 是相近，字義當然是不同的。
Venerate – to honour or very much respect a person or thing.
金京來博士 – 創世記
小肥問我 th 是甚麼意思，我向他解釋 1, 2 及 3 之後要加上 1st (first), 2nd (second)及 3rd (third)，21, 22 及 23 ，31, 32 及33 打後都是用這編排。特別的地方是 11, 12 及 13 則是用 th, 跟其他數目字一樣，因為英文的讀法是 eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, 跟以後的叫法例如 twenty-one, twenty-two, twenty-three 等都不同。
突然想到為何 11 – 19在讀法及寫法在編排上是與其它的不同呢？在網路找到這解釋：
The reason the nine numbers after ten are known as eleven, twelve, and the teens is clarified by looking at Roman numerals.
In other words, they are all plus or minus units of ten and were interpreted into archaic English with this in mind.
Eleven (XI) or “leave one” means ten is one less than eleven.
Twelve (XII) means ten is two less than twelve.
Thirteen is three plus ten (or “teen”), four plus ten is fourteen, and so on.
Lunar and Solar
農曆新年英文是 Lunar New Year, 中英文都很有意思。農曆在中國人的社會是代表以農耕為主的曆法，而英文 Lunar 是指月球。太陽是 Solar，Lunar & Solar 是地球的主要光體。
原來月球跟地球運行的軌跡是傾斜了5 度，就只小小的 5 度就令大家的相互關係起了變化。
地球本身傾斜了 23.5 度，24小時自轉一週，365.26日環繞太陽的軌跡一週，就造成一年四季的天氣變化。
創世記 1:14-19 記載
自從造天地以來、 神的永能和神性是明明可知的、雖是眼不能見、但藉著所造之物、就可以曉得、叫人無可推諉． (羅馬書 1:20)
For since the creation of the world God’s invisible qualities–his eternal power and divine nature–have been clearly seen, being understood from what has been made, so that men are without excuse. (Romans 1:20)
`Origin of the Universe’ by Stephen William Hawking
The problem of the origin of the universe, is a bit like the old question: Which came first, the chicken, or the egg. In other words, what agency created the universe. And what created that agency. Or perhaps, the universe, or the agency that created it, existed forever, and didn’t need to be created. Up to recently, scientists have tended to shy away from such questions, feeling that they belonged to metaphysics or religion, rather than to science. However, in the last few years, it has emerged that the Laws of Science may hold even at the beginning of the universe. In that case, the universe could be self contained, and determined completely by the Laws of Science.
The debate about whether, and how, the universe began, has been going on throughout recorded history. Basically, there were two schools of thought. Many early traditions, and the Jewish, Christian and Islamic religions, held that the universe was created in the fairly recent past. For instance, Bishop Usher calculated a date of four thousand and four BC, for the creation of the universe, by adding up the ages of people in the Old Testament. One fact that was used to support the idea of a recent origin, was that the Human race is obviously evolving in culture and technology. We remember who first performed that deed, or developed this technique. Thus, the arguement runs, we can not have been around all that long. Otherwise, we would have already progressed more than we have. In fact, the biblical date for the creation, is not that far off the date of the end of the last Ice Age, which is when modern humans seem first to have appeared.
On the other hand, some people, such as the Greek philosopher, Aristotle, did not like the idea that the universe had a beginning. They felt that would imply Divine intervention. They prefered to believe that the universe, had existed, and would exist, forever. Something that was eternal, was more perfect than something that had to be created. They had an answer to the argument about human progress, that I described. It was, that there had been periodic floods, or other natural disasters, which repeatedly set the human race right back to the beginning.
Both schools of thought held that the universe was essentially unchanging in time. Either it had been created in its present form, or it had existed forever, like it is today. This was a natural belief in those times, because human life, and, indeed the whole of recorded history, are so short that the universe has not changed significantly during them. In a static, unchanging universe, the question of whether the universe has existed forever, or whether it was created at a finite time in the past, is really a matter for metaphysics or religion: either theory could account for such a universe. Indeed, in 1781, the philosopher, Immanuel Kant, wrote a monumental, and very obscure work, The Critique of Pure Reason. In it, he concluded that there were equally valid arguements, both for believing that the universe had a beginning, and for believing that it did not. As his title suggests, his conclusions were based simply on reason. In other words, they did not take any account of observations about the universe. After all, in an unchanging universe, what was there to observe?
In the 19th century, however, evidence began to accumulate that the earth, and the rest of the universe, were in fact changing with time. On the one hand, geologists realized that the formation of the rocks, and the fossils in them, would have taken hundreds or thousands of millions of years. This was far longer than the age of the Earth, according to the Creationists. On the other hand, the German physicist, Boltzmann, discovered the so-called Second Law of Thermodynamics. It states that the total amount of disorder in the universe (which is measured by a quantity called entropy), always increases with time. This, like the argument about human progress, suggests that the universe can have been going only for a finite time. Otherwise, the universe would by now have degenerated into a state of complete disorder, in which everything would be at the same temperature.
Another difficulty with the idea of a static universe, was that according to Newton’s Law of Gravity, each star in the universe ought to be attracted towards every other star. So how could they stay at a constant distance from each other. Wouldn’t they all fall together. Newton was aware of this problem about the stars attracting each other. In a letter to Richard Bentley, a leading philosopher of the time, he agreed that a finite collection of stars could not remain motionless: they would all fall together, to some central point. However, he argued that an infinite collection of stars, would not fall together: for there would not be any central point for them to fall to. This argument is an example of the pitfalls that one can encounter when one talks about infinite systems. By using different ways to add up the forces on each star, from the infinite number of other stars in the universe, one can get different answers to the question: can they remain at constant distance from each other. We now know that the correct proceedure, is to consider the case of a finite region of stars. One then adds more stars, distributed roughly uniformly outside the region. A finite collection of stars will fall together. According to Newton’s Law of Gravity, adding more stars outside the region, will not stop the collapse. Thus, an infinite collection of stars, can not remain in a motionless state. If they are not moving relative to each other at one time, the attraction between them, will cause them to start falling towards each other. Alternatively, they can be moving away from each other, with gravity slowing down the velocity of recession.
Despite these difficulties with the idea of a static and unchanging universe, no one in the seventeenth, eighteenth, nineteenth or early twentieth centuries, suggested that the universe might be evolving with time. Newton and Einstein, both missed the chance of predicting, that the universe should be either contracting, or expanding. One can not really hold it against Newton, because he was two hundred and fifty years before the observational discovery of the expansion of the universe. But Einstein should have known better. Yet when he formulated the General Theory of Relativity to reconcile Newton’s theory with his own Special Theory of Relativity, he added a so-called, “cosmological constant”. This had a repulsive gravitational effect, which could balance the attractive effect of the matter in the universe. In this way, it was possible to have a static model of the universe.
Einstein later said: The cosmological constant was the greatest mistake of my life. That was after observations of distant galaxies, by Edwin Hubble in the 1920’s, had shown that they were moving away from us, with velocities that were roughly proportional to their distance from us. In other words, the universe is not static, as had been previously thought: it is expanding. The distance between galaxies is increasing with time.
The discovery of the expansion of the universe, completely changed the discussion about its origin. If you take the present motion of the galaxies, and run it back in time, it seems that they should all have been on top of each other, at some moment, between ten and twenty thousand million years ago. At this time, which is called the Big Bang, the density of the universe, and the curvature of spacetime, would have been infinite. Under such conditions, all the known laws of science would break down. This is a disaster for science. It would mean that science alone, could not predict how the universe began. All that science could say is that: The universe is as it is now, because it was as it was then. But Science could not explain why it was, as it was, just after the Big Bang.
Not surprisingly, many scientists were unhappy with this conclusion. There were thus several attempts to avoid the Big Bang. One was the so-called Steady State theory. The idea was that, as the galaxies moved apart from each other, new galaxies would form in the spaces inbetween, from matter that was continually being created. The universe would have existed, and would continue to exist, forever, in more or less the same state as it is today.
The Steady State model required a modification of general relativity, in order that the universe should continue to expand, and new matter be created. The rate of creation needed was very low: about one particle per cubic kilometre per year. Thus, this would not be in conflict with observation. The theory also predicted that the average density of galaxies, and similar objects, should be constant, both in space and time. However, a survey of extra-galactic sources of radio waves, was carried out by Martin Ryle and his group at Cambridge. This showed that there were many more faint sources, than strong ones. On average, one would expect that the faint sources were the more distant ones. There were thus two possibilities: Either, we were in a region of the universe, in which strong sources were less frequent than the average. Or, the density of sources was higher in the past, when the light left the more distant sources. Neither of these possibilities was compatible with the prediction of the Steady State theory, that the density of radio sources should be constant in space and time. The final blow to the Steady State theory was the discovery, in 1965, of a background of microwaves. These had the characteristic spectrum of radiation emited by a hot body, though, in this case, the term, hot, is hardly appropriate, since the temperature was only 2.7 degrees above Absolute Zero. The universe is a cold, dark place! There was no reasonable mechanism, in the Steady State theory, to generate microwaves with such a spectrum. The theory therefore had to be abandoned.
Another idea to avoid a singularity, was suggested by two Russians, Lifshitz and Khalatnikov. They said, that maybe a state of infinite density, would occur only if the galaxies were moving directly towards, or away from, each other. Only then, would the galaxies all have met up at a single point in the past. However, one might expect that the galaxies would have had some small sideways velocities, as well as their velocity towards or away from each other. This might have made it possible for there to have been an earlier contracting phase, in which the galaxies somehow managed to avoid hitting each other. The universe might then have re-expanded, without going through a state of infinite density.
When Lifshitz and Khalatnikov made their suggestion, I was a research student, looking for a problem with which to complete my PhD thesis. Two years earlier, I had been diagnosed as having ALS, or motor neuron disease. I had been given to understand that I had only two or three years to live. In this situation, it didn’t seem worth working on my PhD, because I didn’t expect to finish it. However, two years had gone by, and I was not much worse. Moreover, I had become engaged to be married. In order to get married, I had to get a job. And in order to get a job, I needed to finish my thesis.
I was interested in the question of whether there had been a Big Bang singularity, because that was crucial to an understanding of the origin of the universe. Together with Roger Penrose, I developed a new set of mathematical techniques, for dealing with this and similar problems. We showed that if General Relativity was correct, any reasonable model of the universe must start with a singularity. This would mean that science could predict that the universe must have had a beginning, but that it could not predict how the universe should begin: for that one would have to appeal to God.
It has been interesting to watch the change in the climate of opinion on singularities. When I was a graduate student, almost no one took singularities seriously. Now, as a result of the singularity theorems, nearly everyone believes that the universe began with a singularity. In the meantime, however, I have changed my mind: I still believe that the universe had a beginning, but that it was not a singularity.
The General Theory of Relativity, is what is called a classical theory. That is, it does not take into account the fact that particles do not have precisely defined positions and velocities, but are smeared out over a small region by the Uncertainty Principle of quantum mechanics. This does not matter in normal situations, because the radius of curvature of spacetime, is very large compared to the uncertainty in the position of a particle. However, the singularity theorems indicate that spacetime will be highly distorted, with a small radius of curvature, at the beginning of the present expansion phase of the universe. In this situation, the uncertainty principle will be very important. Thus, General Relativity brings about its own downfall, by predicting singularities. In order to discuss the beginning of the universe, we need a theory which combines General Relativity with quantum mechanics.
We do not yet know the exact form of the correct theory of quantum gravity. The best candidate we have at the moment, is the theory of Superstrings, but there are still a number of unresolved difficulties. However, there are certain features that we expect to be present, in any viable theory. One is Einstein’s idea, that the effects of gravity can be represented by a spacetime, that is curved or distorted by the matter and energy in it. Objects try to follow the nearest thing to a straight line, in this curved space. However, because it is curved, their paths appear to be bent, as if by a gravitational field.
Another element that we expect to be present in the ultimate theory, is Richard Feynman’s proposal that quantum theory can be formulated, as a Sum Over Histories. In it simplest form, the idea is that a particle has every possible path, or history, in space time. Each path or history has a probability that depends on its shape. For this idea to work, one has to consider histories that take place in “imaginary” time, rather than the real time in which we perceive ourselves as living. Imaginary time may sound like something out of science fiction, but it is a well defined mathematical concept. It can be thought of as a direction of time that is at right angles to real time, in some sense. One adds up the probabilities for all the particle histories with certain properties, such as passing through certain points at certain times. One then has to extrapolate the result, back to the real space time in which we live. This is not the most familiar approach to quantum theory, but it gives the same results as other methods.
In the case of quantum gravity, Feynman’s idea of a “Sum over Histories” would involve summing over different possible histories for the universe. That is, different curved space times. One has to specify what class of possible curved spaces should be included in the Sum over Histories. The choice of this class of spaces, determines what state the universe is in. If the class of curved spaces that defines the state of the universe, included spaces with singularities, the probabilities of such spaces would not be determined by the theory. Instead, they would have to be assigned in some arbitrary way. What this means, is that science could not predict the probabilities of such singular histories for spacetime. Thus, it could not predict how the universe should behave. However, it is possible that the universe is in a state defined by a sum that includes only non singular curved spaces. In this case, the laws of science would determine the universe completely: one would not have to appeal to some agency external to the universe, to determine how it began. In a way, the proposal that the state of the universe is determined by a sum over non singular histories only, is like the drunk looking for his key under the lamp post: it may not be where he lost it, but it is the only place in which he might find it. Similarly, the universe may not be in the state defined by a sum over non singular histories, but it is the only state in which science could predict how the universe should be.
In 1983, Jim Hartle and I, proposed that the state of the universe should be given by a Sum over a certain class of Histories. This class consisted of curved spaces, without singularities, and which were of finite size, but which did not have boundaries or edges. They would be like the surface of the Earth, but with two more dimensions. The surface of the Earth has a finite area, but it doesn’t have any singularities, boundaries or edges. I have tested this by experiment. I went round the world, and I didn’t fall off.
The proposal that Hartle and I made, can be paraphrased as: The boundary condition of the universe is, that it has no boundary. It is only if the universe is in this “no boundary” state, that the laws of science, on their own, determine the probabilities of each possible history. Thus, it is only in this case that the known laws would determine how the universe should behave. If the universe is in any other state, the class of curved spaces, in the “Sum over Histories”, will include spaces with singularities. In order to determine the probabilities of such singular histories, one would have to invoke some principle other than the known laws of science. This principle would be something external to our universe. We could not deduce it from within the universe. On the other hand, if the universe is in the “no boundary” state, we could, in principle, determine completely how the universe should behave, up to the limits set by the Uncertainty Principle.
It would clearly be nice for science if the universe were in the “no boundary” state, but how can we tell whether it is? The answer is, that the no boundary proposal makes definite predictions, for how the universe should behave. If these predictions were not to agree with observation, we could conclude that the universe is not in the “no boundary” state. Thus, the “no boundary” proposal is a good scientific theory, in the sense defined by the philosopher, Karl Popper: it can be falsified by observation.
If the observations do not agree with the predictions, we will know that there must be singularities in the class of possible histories. However, that is about all we would know. We would not be able to calculate the probabilities of the singular histories. Thus, we would not be able to predict how the universe should behave. One might think that this unpredictability wouldn’t matter too much, if it occurred only at the Big Bang. After all, that was ten or twenty billion years ago. But if predictability broke down in the very strong gravitational fields in the Big Bang, it could also break down whenever a star collapsed. This could happen several times a week, in our galaxy alone. Thus, our power of prediction would be poor, even by the standards of weather forecasts.
Of course, one could say that one didn’t care about a breakdown in predictability, that occurred in a distant star. However, in quantum theory, anything that is not actually forbidden, can and ~will happen. Thus, if the class of possible histories includes spaces with singularities, these singularities could occur anywhere, not just at the Big Bang and in collapsing stars. This would mean that we couldn’t predict anything. Conversely, the fact that we are able to predict events, is experimental evidence against singularities, and for the “no boundary” proposal.
So what does the no boundary proposal, predict for the universe. The first point to make, is that because all the possible histories for the universe are finite in extent, any quantity that one uses as a measure of time, will have a greatest and a least value. So the universe will have a beginning, and an end. However, the beginning will not be a singularity. Instead, it will be a bit like the North Pole of the Earth. If one takes degrees of latitude on the surface of the Earth to be the anallogue of time, one could say that the surface of the Earth began at the North Pole. Yet the North Pole is a perfectly ordinary point on the Earth. There’s nothing special about it, and the same laws hold at the North Pole, as at other places on the Earth. Similarly, the event that we might choose to label, as “the beginning of the universe”, would be an ordinary point of spacetime, much like any other, the laws of science would hold at the beginning, as elsewhere.
From the analogy with the surface of the Earth, one might expect that the end of the universe would be similar to the beginning, just as the North Pole is much like the South Pole. However, the North and South Poles correspond to the beginning and end of the history of the universe, in imaginary time, not the real time that we experience. If one extrapolates the results of the “Sum over Histories” from imaginary time to real time, one finds that the beginning of the universe in real time can be very different from its end. It is difficult to work out the details, of what the no boundary proposal predicts for the beginning and end of the universe, for two reasons. First, we don’t yet know the exact laws that govern gravity according to the Uncertainty Principle of quantum mechanics. Though we know the general form and many of the properties that they should have. Second, even if we knew the precise laws, we could not use them to make exact predictions. It would be far too difficult, to solve the equations exactly. Nevertheless, it does seem possible to get an approximate idea, of what the no boundary condition would imply. Jonathan Halliwell and I, have made such an approximate calculation. We treated the universe as a perfectly smooth and uniform background, on which there were small perturbations of density. In real time, the universe would appear to begin its expansion at a minimum radius. At first, the expansion would be what is called inflationary. That is, the universe would double in size every tiny fraction of a second, just as prices double every year in certain countries. The world record for economic inflation, was probably Germany after the First World War. The price of a loaf of bread, went from under a mark, to millions of marks in a few months. But that is nothing compared to the inflation that seems to have occurred in the early universe: an increase in size by a factor of at least a million million million million million times, in a tiny fraction of a second. Of course, that was before the present government.
This inflation was a good thing, in that it produced a universe that was smooth and uniform on a large scale, and was expanding at just the critical rate to avoid recollapse. The inflation was also a good thing in that it produced all the contents of the universe, quite literally out of nothing. When the universe was a single point, like the North Pole, it contained nothing. Yet there are now at least 10 to the 80 particles in the part of the universe that we can observe. Where did all these particles come from? The answer is, that Relativity and quantum mechanics, allow matter to be created out of energy, in the form of particle anti particle pairs. So, where did the energy come from, to create the matter? The answer is, that it was borrowed, from the gravitational energy of the universe. The universe has an enormous debt of negative gravitational energy, which exactly balances the positive energy of the matter. During the inflationary period, the universe borrowed heavily from its gravitational energy, to finance the creation of more matter. The result was a triumph for Reagan economics: a vigorous and expanding universe, filled with material objects. The debt of gravitational energy, will not have to be repaid until the end of the universe.
The early universe could not have been exactly homogeneous and uniform, because that would violate the Uncertainty Principle of quantum mechanics. Instead, there must have been departures from uniform density. The no boundary proposal, implies that these differences in density, would start off in their ground state. That is, they would be as small as possible, consistent with the Uncertainty Principle. However, during the inflationary expansion, they would be amplified. After the period of inflationary expansion was over, one would be left with a universe that was expanding slightly faster in some places, than in others. In regions of slower expansion, the gravitational attraction of the matter, would slow down the expansion still further. Eventually, the region would stop expanding, and would contract to form galaxies and stars. Thus, the no boundary proposal, can account for all the complicated structure that we see around us. However, it does not make just a single prediction for the universe. Instead, it predicts a whole family of possible histories, each with its own probability. There might be a possible history in which Walter Mondale won the last presidential election, though maybe the probability is low.
The no boundary proposal, has profound implications for the role of God in the affairs of the universe. It is now generally accepted, that the universe evolves according to well defined laws. These laws may have been ordained by God, but it seems that He does not intervene in the universe, to break the laws. However, until recently, it was thought that these laws did not apply to the beginning of the universe. It would be up to God to wind up the clockwork, and set the universe going, in any way He wanted. Thus, the present state of the universe, would be the result of God’s choice of the initial conditions. The situation would be very different, however, if something like the no boundary proposal were correct. In that case, the laws of physics would hold, even at the beginning of the universe. So God would not have the freedom to choose the initial conditions. Of course, God would still be free to choose the laws that the universe obeyed. However, this may not be much of a choice. There may only be a small number of laws, which are self consistent, and which lead to complicated beings, like ourselves, who can ask the question: What is the nature of God? Even if there is only one, unique set of possible laws, it is only a set of equations. What is it that breathes fire into the equations, and makes a universe for them to govern. Is the ultimate unified theory so compelling, that it brings about its own existence. Although Science may solve the problem of ~how the universe began, it can not answer the question: why does the universe bother to exist? Maybe only God can answer that.
Origin of the Universe – Stephen Hawking
Stephen Hawking on Religion: Science Will Win (6.7.10)