31 results in Theoretical Concepts in Physics
19 - Cosmology
- Malcolm S. Longair, University of Cambridge
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Cosmology and physics
By cosmology, I mean the application of the laws of physics to the Universe as a whole. As a result, the validation of theory depends upon observation rather than experiment, placing us at one stage further removed from our ‘apparatus’ than is the case in laboratory physics. Yet, throughout history, astronomical observations have played their role in establishing new physics, which has been rapidly assimilated into the mainstream of established science. In Chapter 2, Tycho Brahe's observations of the motions of the planets, which led to Newton's law of gravitation, were discussed. From observation of the eclipses of the satellites of Jupiter, Ole Rømer showed conclusively that the speed of light is finite and in 1676 estimated it from the time it takes light to travel across the Earth's orbit about the Sun.
To construct self-consistent cosmological models a relativistic theory of gravity is needed, and most of the tests of general relativity involve the use of astronomical objects. The discovery of the binary pulsar PSR 1913 + 16 has proved to be of particular importance for physics. The pulsar is a magnetised, rotating, neutron star, which has mass about 1.4 times the mass of the Sun, radius r ≈ 10 km and general relativistic parameter 2GM/(rc2) ≈ 0.3. Its companion star is another neutron star of similar mass and they orbit their common centre of mass every 7.75 hours.
Case Study IV - Thermodynamics and statistical physics
- Malcolm S. Longair, University of Cambridge
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Thermodynamics is the science of how the properties of matter and systems change with temperature. A system may be viewed on the microscopic scale, in which case we study the interactions of the constituent particles or quanta and how these change with temperature. In this approach, we need to construct physical models for these interactions. The opposite approach is to study the system on the macroscopic scale and then the unique status of classical thermodynamics becomes apparent. In this approach, the behaviour of matter and radiation in bulk is studied and in effect we deny that they have any internal structure at all. In other words, the science of classical thermodynamics is solely concerned with relations between macroscopic measurable quantities such as pressure, volume and temperature.
Now this may seem to make classical thermodynamics a rather dull subject but, in fact, it is quite the opposite. In many physical problems, we may not know in detail the correct microscopic physics, and yet the thermodynamic approach can provide answers about the macroscopic behaviour of the system which are independent of the unknown detailed microphysics. Another way of looking at it is to think of classical thermodynamics as providing the boundary conditions which any microscopic model must satisfy. The thermodynamic arguments have absolute validity independent of the model adopted to explain any particular phenomenon.
It is remarkable that these profound statements can be made on the basis of the first and second laws of thermodynamics. Let us state them immediately.
4 - Newton and the law of gravity
- Malcolm S. Longair, University of Cambridge
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Introduction
Richard Westphal's monumental biography Never at Rest was the product of a lifetime's study of Isaac Newton's life and work. In the preface, he writes:
The more I have studied him, the more Newton has receded from me. It has been my privilege at various times to know a number of brilliant men, men whom I acknowledge without hesitation to be my intellectual superiors. I have never, however, met one against whom I was unwilling to measure myself so that it seemed reasonable to say that I was half as able as the person in question, or a third or a fourth, but in every case a finite fraction. The end result of my study of Newton has served to convince me that with him there is no measure. He has become for me wholly other, one of the tiny handful of supreme geniuses who have shaped the categories of human intellect, a man not finally reducible to the criteria by which we comprehend our fellow beings.
In the next paragraph, he writes:
Had I known, when in youthful self-confidence I committed myself to the task, that I would end up in similar self-doubt, surely I would never have set out.
Newton's impact upon science is so all pervasive that it is worthwhile filling in some of the background to his character and extraordinary achievements. The chronology which follows is that adopted in the Introduction of the volume Let Newton Be.
Case Study V - The origins of the concept of quanta
- Malcolm S. Longair, University of Cambridge
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Quanta and relativity are phenomena of physics which are quite outside our everyday experience – they are also perhaps the greatest discoveries of twentieth-century physics. In the debate between those supporting the Copernican and geocentric models of the Universe, discussed in Chapter 3, one of the issues raised by opponents of the Copernican picture concerned ‘the deception of the senses’: if these phenomena are of such fundamental importance, why are we not aware of them in our everyday lives? Quanta and relativity were both discovered by very careful experiment, the results of which could not be accounted for within the context of Newtonian physics. Indeed, these phenomena are arguably ‘non-intuitive’, and yet they lie at the foundations of the whole of modern physics.
In this case study, we will study in some detail the origins of the concept of quanta. For me, this is one of the most dramatic stories in intellectual history. It is also very exciting and catches the flavour of an epoch when, within 25 years, physicists' view of nature changed totally and completely new perspectives were opened up. The story illustrates many important points about how physics and theoretical physics work in practice. We find the greatest physicists making mistakes, individuals having to struggle against the accepted views of virtually all physicists, and, most of all, a level of inspiration and scientific creativity which I find dazzling. If only everyone, and not only those who have had a number of years training as physicists or mathematicians, could appreciate the intellectual beauty of this story.
5 - The origin of Maxwell's equations
- Malcolm S. Longair, University of Cambridge
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How it all began
Electricity and magnetism have an ancient history. Magnetic materials are mentioned as early as 800 bc by the Greek writers, the word ‘magnet’ being derived from the mineral magnetite, which was known to attract iron in its natural state and which was mined in the Greek province of Magnesia in Thessaly. Magnetic materials were of special importance because of their use in compasses, and this is reflected in the English word for the mineral, lodestone, meaning leading stone. Static electricity was also known to the Greeks through the electrostatic phenomena observed when amber is rubbed with fur – the Greek work for amber is elektron. The first systematic study of magnetic and electric phenomena was published in 1600 by William Gilbert (1544–1603) in his treatise De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure. The main subject of the treatise was the Earth's magnetic field, which he showed was similar to that of a bar magnet. He also described the force between two bodies charged by friction and named it the electric force between them.
In addition to his famous experiments, in which he showed that lightning is an electrostatic discharge, Benjamin Franklin (1706–90) systematised the laws of electrostatics and defined the conventions for naming positive and negative electric charges. In the course of these studies, he also enunciated the law of conservation of electric charge.
11 - Black-body radiation up to 1895
- Malcolm S. Longair, University of Cambridge
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The state of physics in 1890
In the course of the case studies treated so far, we have been building up a picture of the state of physics and theoretical physics towards the end of the nineteenth century. The achievement had been immense. In mechanics and dynamics, the Lagrangian and Hamiltonian dynamics described briefly in Chapter 7, were well understood. In thermodynamics, the first and second laws were firmly established, largely through the efforts of Clausius and Lord Kelvin, and the full ramifications of the concept of entropy in classical thermodynamics were being elaborated. In Chapters 5 and 6, we described how Maxwell derived the equations of electromagnetism. Hertz's experiments of 1887–9 demonstrated beyond any shadow of doubt that, as predicted by Maxwell, light is a form of electromagnetic wave. This discovery provided a firm theoretical foundation for the wave theory of light, which could account for virtually all the known phenomena of optics.
The impression is sometimes given that most physicists of the 1890s believed that the combination of thermodynamics, electromagnetism and classical mechanics could account for all known physical phenomena and that all that remained to be done was to work out the consequences of these hard-won achievements. In fact, it was a period of ferment when there were still many fundamental unresolved problems which exercised the greatest minds of the period.
We have discussed the ambiguous status of the kinetic theory of gases and the equipartition theorem as expounded by Clausius, Maxwell and Boltzmann.
2 - From Ptolemy to Kepler – the Copernican revolution
- Malcolm S. Longair, University of Cambridge
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Ancient history
The first of the great astronomers of whom we have knowledge is Hipparchus, who was born in Nicaea in the second century bc. Perhaps his greatest achievement was his catalogue of the positions and brightnesses of 850 stars in the northern sky. The catalogue was completed in 127 bc and represented a quite monumental achievement. A measure of his skill as an astronomer is that he compared his positions with those of Timocharis made in Alexandria 150 years earlier and discovered the precession of the equinoxes, the very slow change in direction of the Earth's axis of rotation relative to the frame of reference of the fixed stars. We now know that this precession is caused by tidal torques due to the Sun and Moon acting upon the slightly non-spherical Earth. At that time, however, the Earth was assumed to be stationary and so the precession of the equinoxes had to be attributed to a movement of the ‘sphere of fixed stars’.
The most famous of the ancient astronomical texts is the Almagest of Claudius Ptolomeaus, or Ptolemy, who lived in the second century ad. The word ‘Almagest’ is a corruption of the Arabic translation of the title of his book, the Megelé Syntaxis or Great Composition, which in Arabic becomes al-majisti. It consisted of 13 volumes and provided a synthesis of all the achievements of the Greek astronomers and, in particular, leant heavily upon the work of Hipparchus.
12 - 1895–1900: Planck and the spectrum of black-body radiation
- Malcolm S. Longair, University of Cambridge
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Planck's early career
Max Planck was typically straightforward and honest about his early career, as can be learned from his short scientific autobiography. He studied under Helmholtz and Kirchhoff in Berlin, but in his own words
I must confess that the lectures of these men netted me no perceptible gain. It was obvious that Helmholtz never prepared his lectures properly … Kirchhoff was the very opposite … but it would sound like a memorised text, dry and monotonous.
It is reassuring to students struggling to understand physics that even the greatest of physicists are sometimes inadequate as university teachers. In my own experience, although the best physicists are very often the most inspiring lecturers, there is considerable variation in their degree of commitment to passing on the torch to the next generation. We have all had to put up with the uneven quality of the lecturers who happen to be delivering courses. In the end, however, we have to understand the material ourselves rather than be spoon-fed, and so perhaps it is not as disastrous as it might seem at first sight. Indeed, one might argue that a bad lecturer requires the student to think harder about the material, which is a good thing – let me hasten to add that this is no excuse for bad lecturing!
Planck's research interests were inspired by his reading of the works of Clausius and he set about investigating how the second law of thermodynamics could be applied to a wide variety of different physical problems, as well as elaborating as clearly as possible the basic tenets of the subject.
Case Study I - The origins of Newton's laws of motion and of gravity
- Malcolm S. Longair, University of Cambridge
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Our first case study encompasses essentially the whole of what can be considered the modern scientific process. Unlike the other case studies, it requires little mathematics but a great deal in terms of intellectual imagination. For me, it is a heroic tale of scientists of the highest genius lying the foundations of modern science. Everything is there – the rôles of brilliant experimental skill, of imagination in the interpretation of observational and experimental data and of the remarkable leaps of the imagination which were to lay the foundations for the Newtonian picture of the world. This achievement may not at first sight seem so remarkable to the twenty-first-century reader, but closer inspection shows that in fact it is immense. As expressed by Herbert Butterfield in his Origins of Modern Science, the understanding of motion was one of the most difficult steps that scientists have ever undertaken. In the quotation by Douglas Gough in Chapter 1, he expresses eloquently the ‘pain’ experienced on being forced to discard a cherished prejudice in the sciences. How much more difficult must have been the process of laying the foundations of modern science, when the concept that the laws of nature can be written in mathematical form had not yet been formulated.
How did our modern appreciation of the nature of our physical Universe come about? I make no apology for starting at the very beginning. In Chapter 2, the first of three chapters that address Case Study I, we set the scene for the subsequent triumphs, and tragedies, of two of the greatest minds of modern science – Galileo Galilei and Isaac Newton.
Case Study VII - General relativity and cosmology
- Malcolm S. Longair, University of Cambridge
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It is sometimes a matter of dispute whether general relativity and cosmology should appear in undergraduate syllabuses at all. The criticism is often made that they are included simply to add glamour to physics courses, to act as a lure to attract students into ‘real hard-core’ physics. I take a much more positive view of their inclusion as an integral part of physics.
General relativity follows on rather naturally from special relativity and results in even more profound changes to our concepts of space and time than those which follow from special relativity. The idea that the geometry of space–time is influenced by the distribution of matter, which then moves along paths in curved space–time is one of the fundamental concepts of modern physics. Indeed, the bending of space–time is now regularly observed as the gravitational lens effect in deep astronomical images (Fig. VII.1). Unfortunately, general relativity is technically complex, in the sense that the necessary mathematics goes beyond what can normally be introduced at the undergraduate level, and it would be wrong to underestimate these technical difficulties. Nonetheless, a great deal can be achieved without the use of advanced techniques, provided the reader is prepared to accept a few results which will simply be made plausible, the exact results then being quoted. This seems a small price to pay for some insight into the intimate connection between space–time geometry and gravitation and for understanding some of the more remarkable phenomena which are expected to be observed in strong gravitational fields, such as those found in the vicinity of black holes.
18 - The technology of cosmology
- Malcolm S. Longair, University of Cambridge
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Introduction
This chapter is very different from all the others in this book. I have a firmly held conviction that astrophysical cosmology is an observational, if not experimental, science and that the quality of the astrophysics is only as good as the data available to validate cosmological and astrophysical theories. The objective of this chapter is to survey some aspects of the technologies which have enabled astrophysical cosmology to be placed on a firm observational basis. In the telling of this story, we will encounter a number of heroic figures who deserve as much honour, in my view, as the rather better-known theorists and physicists who have played such a startling role in the development of cosmological understanding.
My reason for including this chapter is to bring home to even the most hard-line of theorists the essential role which experimental genius plays in the development of theory. In many ways, this chapter is an attempt to do for cosmology what Peter Galison achieved in his splendid book Image and Logic for particle physics, but at a very much more modest level. Without the imaginative development of novel technology, with which to address the challenges presented by theory, theoretical physics lacks experimental validation.
To oversimplify greatly, the revolution in twentieth-century astrophysics and cosmology can be traced to three great technical developments, which were the heritage of the nineteeth century: (i) the invention of astronomical spectroscopy, (ii) the measurement of the parallaxes, and hence the distances, of stars and (iii) the invention of the photographic process.
9 - Basic thermodynamics
- Malcolm S. Longair, University of Cambridge
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Heat and temperature
Like so many aspects of the revolution which led to the birth of modern science, the origins of the scientific study of heat and temperature can be traced to the early years of the seventeenth century. The quantitative study of heat and temperature depended upon the development of instruments which could give a quantitative measure of concepts such as ‘the degree of hotness’. The first appearance of the word thermomètre occurred in 1624 in the volume Récréaction Mathématique by the French Jesuit Jean Leurechon. This was translated into English in 1633 by William Oughtred, under the title Of the Thermometer, or an Instrument to Measure the Degrees of Heat or Cold in Aire. The Oxford English Dictionary gives 1633 as the date of the first appearance of the word ‘thermometer’ in English literature.
There had been earlier descriptions of the use of the expansion of gases to measure ‘the degree of hotness’, by Galileo and others, but the first thermometers which resemble their modern counterparts were constructed in the 1640s to 1660s. Coloured alcohol within a glass bulb was used as the expanding substance and this liquid extended into the long stem of the thermometer, which was sealed off to prevent evaporative loss of the fluid. In 1701, Daniel Fahrenheit constructed an alcohol-in-glass thermometer in which the sealed tube was evacuated. Then, in 1714, he extended the temperature range of the thermometer by using mercury rather than fluids, such as alcohol, which boil at relatively low temperatures.
7 - Approaches to mechanics and dynamics
- Malcolm S. Longair, University of Cambridge
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14 - Einstein and the quantisation of light
- Malcolm S. Longair, University of Cambridge
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1905 – Einstein's annus mirabilis
Up to 1905, Planck's work had made little impression, and he was no further forward in understanding the profound implications of what he had done. As discussed in Chapter 13, he expended a great deal of unsuccessful effort in trying to find a classical interpretation for the ‘quantum of action’ h, which he correctly recognised had fundamental significance for understanding the spectrum of black-body radiation. The next great steps were taken by Albert Einstein, and it is no exaggeration to state that he was the first person to appreciate the full significance of quantisation and the reality of quanta. He showed that this is a fundamental aspect of all physical phenomena, rather than just a ‘formal assumption’ for accounting for the Planck distribution. From 1905 onwards, he never deviated from his belief in the reality of quanta – it was some considerable time before the great figures of the day conceded that Einstein was indeed correct. He came to this conclusion in a series of brilliant papers of dazzling scientific virtuosity.
Einstein completed what we would now call his undergraduate studies in August 1900. Between 1902 and 1904, he wrote three papers on the foundations of Boltzmann's statistical mechanics. Once again, notice how a deep understanding of thermodynamics and statistical physics provided the starting point for the investigation of basic problems in theoretical physics. As was explained in Case Study IV, thermodynamics and statistical physics do not deal with specific physical processes, which might not be particularly well understood; rather, they deal with the overall properties of physical systems and provide general rules about the expected behaviour.
Case Study II - Maxwell's equations
- Malcolm S. Longair, University of Cambridge
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Each case study has a different emphasis and this one is as extreme as they get. The central theme is the origin of Maxwell's equations, which might seem a much more straightforward story than some of the other case studies. This was my opinion until I understood how Maxwell actually arrived at his great discovery of the displacement current. It turns out to be as remarkable an example of model building as I have encountered anywhere in physics and strikes to the heart of the nature of electromagnetism. It is also a wonderful example of how fruitful it can be to work by analogy, provided one is constrained by experiment. The story culminates in the discovery that electromagnetic disturbances propagate at the speed of light, leading directly to the unification of light and electromagnetism and to Hertz's beautiful experiments, which fully vindicated Maxwell's theory.
Along the way, we pay tribute to Faraday's genius as an experimenter in discovering the phenomenon of electromagnetic induction (Figure II.1) and many other aspects of electromagnetic phenomena. His invention of the concept of lines of force, what I have called ‘mathematics without mathematics’, was crucial to the mathematisation of electromagnetism, and to Maxwell's theoretical studies. The key role of vector calculus in simplifying the mathematics of electromagnetism provides an opportunity for revising some of that material, and a number of useful results are included in the appendix to Chapter 5.
6 - How to rewrite the history of electromagnetism
- Malcolm S. Longair, University of Cambridge
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Introduction
Now that we have derived Maxwell's equations as he himself derived them, let us do everything backwards, starting with Maxwell's equations and regarding them simply as a set of vector equations relating the vector fields E, D, B, H and J. Therefore, initially we ascribe no physical significance to these fields. We then make a minimum number of postulates in order to give them physical significance and so derive from them all the experimentally established laws of electromagnetism. This approach is taken by Stratton in his book Electromagnetic Theory.
We can then apply Maxwell's equations to further aspects of electromagnetic theory – the properties of electromagnetic waves, the emission of waves by accelerated charges, and so on – which provide tests of the theory that go far beyond the empirically derived laws from which Maxwell's equations were deduced. If the theory were to prove to be inconsistent with experiment then the interlocking nature of many of the results, as illustrated below, indicates how the whole edifice would have to be changed.
A number of my colleagues have objected strenuously to this approach to electromagnetism, principally on the grounds that historically it is most unlikely that anyone would have discovered Maxwell's equations by this route. I am not prepared to speculate about that. What I do know is that this procedure of starting with a mathematical structure, which is then given physical meaning, is found in other aspects of fundamental physics, for example in the theory of linear operators and quantum mechanics and in tensor calculus and the special and general theories of relativity.
Index
- Malcolm S. Longair, University of Cambridge
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3 - Galileo and the nature of the physical sciences
- Malcolm S. Longair, University of Cambridge
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Introduction
There are three separate but linked stories to be told. The first concerns Galileo as natural philosopher. Unlike Tycho Brahe the observer and Kepler the mathematician, Galileo was an experimental physicist whose prime concern was understanding the laws of nature in quantitative terms, from his earliest writings to his final great treatise Discourse and Mathematical Demonstrations concerning Two New Sciences.
The second story is astronomical, and occupies a relatively small, but crucial, period of Galileo's career, from 1609 to 1612, during which time he made a number of fundamental astronomical discoveries which had a direct impact upon his understanding of the physics of motion.
The third story, and the most famous of all, is his trial and subsequent house arrest, which continues to be the subject of considerable controversy. The scientific aspects of his censure and subsequent trial are of the greatest interest and strike right at the heart of the nature of the physical sciences. The widespread view is to regard Galileo as the hero and the Catholic Church as the villain of the piece, a source of conservative reaction and bigoted authority. From the methodological point of view Galileo made an logical error, but the church authorities made a much more disastrous blunder, which has resonated through science and religion ever since, and which was only officially acknowledged by Pope John Paul II in the 1980s.
My reasons for devoting a whole chapter to Galileo, his science and his tribulations are that it is a story which needs to be better known and which has resonances for the way in which physics as a scientific discipline is carried out today.
17 - An introduction to general relativity
- Malcolm S. Longair, University of Cambridge
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Introduction
Einstein's great paper of 1905 on the special theory of relativity appears effortless, and as discussed in Chapter 16, he did not regard the formulation of the theory as a particularly ‘revolutionary act’. The route to the discovery of the general theory of relativity was very different. Whereas others had come close to elucidating the essential features of special relativity, Einstein was on his own and went far beyond all his contemporaries in his discovery of the general theory. How he arrived at the theory is one of the great stories of theoretical physics and involved the very deepest physical insight, imagination, intuition and sheer doggedness. It would lead to concepts barely conceivable even by a genius like Einstein – the phenomenon of black holes and the possibility of testing theories of gravity in the strong-field limit through the observation of relativistic stars. General relativity also provided for the first time a relativistic theory of gravity which could be used to construct fully self-consistent models of the Universe as a whole.
The history of the discovery of general relativity is admirably told by Abraham Pais in his scientific biography of Einstein, Subtle is the Lord … the Science and Life of Albert Einstein, which discusses many of the technical details of the papers published in the period 1907 to 1915. Equally to be recommended is the survey by John Stachel of the history of the discovery of both theories of relativity.
8 - Dimensional analysis, chaos and self-organised criticality
- Malcolm S. Longair, University of Cambridge
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Introduction
The increasingly powerful mathematical tools described in Chapter 7 provided the means for tackling complex dynamical problems in classical physics. Despite these successes, in many areas of physics problems can become rapidly very complex and, although we may be able to write down the differential or integral equations which describe the behaviour of the system, often it is not possible to find analytic solutions.
The objective of this chapter is to study techniques developed to tackle these complex problems, some of them so non-linear that they seem quite beyond the scope of traditional analysis. First, we review the techniques of dimensional analysis. Used with care and insight, this approach is powerful and finds many applications in pure and applied physics. We will give as examples the non-linear pendulum, fluid flow, explosions, turbulence and so on.
Next, we briefly study chaos, the analysis of which became feasible only with the development of high-speed computers. The equations of motion are deterministic and yet the outcome is extremely sensitive to the precise initial conditions. Beyond these examples are even more extreme systems, in which so many non-linear effects come into play that it is impossible to predict the outcome of an experiment, in any conventional sense. And yet regularities are found in the form of scaling laws. There must be some underlying simplicity in the way in which the system behaves, despite the horrifying complexity of the many processes involved. These topics involve fractals and the burgeoning field of self-organised criticality.