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Everyone knows that Einstein’s special relativity contains a theory of time measurements, which are no longer conceived as absolute, but are related to the state of motion of the clock and to the point of view of the observer, and the same happens to space measurements. Everyone also knows that the theory contains the deduction that a small material mass can be converted into a huge amount of energy according to a precise quantitative relationship.But many who have tried to study the theory have failed to understand it; yet, to fully understand the part of Einstein’s theory about time and space measurements, readers just need to know what speed and square root are, and to obtain a simplified but clear idea of the part regarding the concepts of mass and energy they need just to remember elementary high-school physics. Apparently something is missing in all the many books that describe relativity in a simple or higher level.This book is written in a different way from any other. A rigorous but clear exposition will show all readers, provided they know what speed and square root are, that they can understand fully and perfectly the space-time theory and can judge it with their own intelligence. In addition, readers will have a clear idea of the equivalence between mass and energy and its logical relationship with space-time theory.This book was written for beginners and for perplexed people who have unsuccessfully attempted to study special relativity: both will understand the exact meaning of the famous and difficult essay in which Einstein expounded the theory in 1905, which is examined word by word in this book. And all readers will have a clearer idea of the relevance of relativity for the twentieth (and twenty-first) century culture.
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Relativity from Lorentz to Einstein
A Guide for Beginners, Perplexed and Experimental Scientists
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This book is a translation from Italian of:
La relatività da Lorentz a Einstein
Una guida per principianti, perplessi e scienziati sperimentali
© il glifo ebooks, 2017
ISBN of ebook: 9788897527398
ISBN of paper edition: 9788897527404
1.1Special relativity and general relativity
1.2Space measures: the sample ruler problem
1.3Time measures: the sample clock problem
1.4Simultaneity and succession of phenomena in time
1.5Measurements of the speed of light
2.Ether, electromagnetic theory and Lorentz’s hypothesis
2.1Electromagnetic Theory and law of inertia
2.2The problem of speed as physical constant
2.3Motion with respect to the ether and the Michelson and Morley experiment
2.4Propagation of electromagnetic waves
2.5Lorentz’s “physical” relativity theory
2.6Lorentz contraction factor
2.7Galilean and Lorentz transformations
2.8Formula for speed composition
2.9Length contraction and time dilatation: a recurring misunderstanding
2.10Relativity of simultaneity and time measurements
2.11Derivation of Lorentz transformations
3.1Two Einstein’s postulates
3.2Synchronization of clocks in distant systems
3.3Synchronization of distant clocks with other means than light
3.4Lorentz’s transformations come into play
3.5A question to the reader, and instructions
3.6Synchronization of distant clocks by means of light
3.7ED1905 Chapter 1, “Definition of simultaneity”
3.8ED1905 Chapter 2, “On the relativity of lengths and times”
3.9ED1905 Chapters 3, 4 and 5, Lorentz transformations and consequences
3.10Review of previous sections
3.11ED1905 Chapter 6, application to electrodynamics and invariance of Maxwell equations
3.12ED1905 Chapter 6, a detail about electrodynamics
3.13Assumptions and consequences of applying Lorentz’s transformations to electrodynamics
3.14Conventional interpretation of dilatation
3.15Apparent interpretation of dilatation
3.16Realist interpretation of dilatation
3.17Balance of the three interpretations
4.The relativistic concept of mass: E=mc²
4.1Concept of photon, or electromagnetic quantum
4.2Basics of elementary physics needed to understand E=mc²
4.3Convertibility between mass and energy
4.4E=mc²: non-relativistic derivation by Rohrlich (didactic)
4.5E=mc² and increase of mass with speed
4.6Mass, energy and speed: non-relativistic derivation by Lewis (1908)
4.7Mass, energy and speed: problems revealed by non-relativistic conception
5.Time and space dilatation and law of inertia
5.1Rewording the problem
5.2Nature of contradiction
5.3The conflict between apparent and realist interpretation of relativity
5.4Interpretation of special relativity shortly after 1905
5.5Transition from apparent to realist interpretation (1911)
5.6Realist interpretation of dilatation through general relativity (1918)
6.Discussions of relativity paradoxes
6.2The µ meson
6.3 Minkowski geometry, space-time and world lines
6.4Minkowski geometry and clocks problem
6.5Fizeau’s experiment with water
6.6The Hafele-Keating experiment and GPS system
6.7Problem of the relativistic deduction of E=mc²
7.Success of special relativity and opinions
7.1Success of the theory
7.2(Perhaps) naive considerations about the unity of time
7.3Connections with twentieth century culture
7.4Rational criticism of relativity
7.5Counterintuitive geometry of space-time
7.6Experience of the author of this book
There are thousands of books that expose and make popular Einstein’s relativity theory with different levels of complexity. This literature has developed since 1919, when the theory of general relativity unexpectedly exploded in popularity worldwide, uninterrupted to this day. Within a couple of years, hundreds of popularizations were published concerning both the theory of special relativity, which dates back to 1905, and that of general relativity, developed by Einstein in the following ten years. The production of literature continued uninterrupted always exhibiting the same theory with the same arguments and the same expressions. The standard way of describing special relativity was defined by different authors at a time roughly between 1911 and 1925, and has been maintained to date. If at a distance of so many years new books are written, that are very similar to the precedents, there must be a problem. In this discussion we adopt a different perspective in order to highlight the exact meaning of the theory of special relativity, to allow the reader to judge in full the kinematic part of theory, i.e. the one that speaks of space, time, and clocks, and to help the reader understand how the formula E=mc2 can be derived.
In relativistic literature we find three kinds of books: introductory, popular, and specialist. In each kind there are many versions that are similar among them: all books converge towards the same results and all proceed with the same criterion. Those who have consulted many of them recognize familiar issues in each book and look for a single page or a single observation, hoping to shed light on a controversial point and to find help in grasping the whole of the theory. This is a common experience because many people have tried to study relativity and have grasped the pattern of reasoning and the elements of theory, but are still perplexed about the overall connection between them. However, especially with regard to special relativity, the whole literature can be divided into three sets of introductory, popular and specialized books. In the group of introductory treatises there are the simplest ones, which are merely descriptive and serve to give a first idea of the theory, but they can neither explain nor make anything understood. They only serve to create expectations and provoke curiosity in the reader, making the conclusions of the theory known but leaving them unexplained. The books of this group are all superfluous, because what they can teach is already known through movies, television broadcasts, encyclopaedia entries and newspaper articles. In short, they repeat commonplaces and common knowledge.
The second group of treatises develops the reasoning process of the theory by using simple terms and a little algebra, but when such books are well written, they are able to give the readers an exact knowledge of the theory. This is because special relativity, also in its discussion by Einstein, makes almost no use of superior maths, except in the electrodynamic part where this is indispensable.
The third group of theory exposures is aimed at university education and at the specialist public and therefore it cannot be used by the reader who approaches it without specific training in physics, mathematics or engineering. However, this limit is far less significant than what one would think for readers who are not able to go beyond the level of the books of the second group, because the main issue of special relativity is not mathematics: it is logic. There is a logical process to follow, which is not easy, but the necessary math is the one of the school, not of the university.
This takes us to the heart of the problem: the whole relativistic literature has the defect of not fully explaining the premises and assumptions that must be made to understand the deduction of the consequences, and in this fault lies the great difficulty that almost everyone experiences in the theory. Indeed, it has often been found, and teachers confirm it, that the smartest students, those who want to go deep into things, are those who find more difficulty in understanding the deep connection between the elements of special relativity, while those who are content to learn how to repeat certain judgments and how to apply some not particularly complex mathematical formulas are usually satisfied both by the theory and by themselves. Before beginning the study of relativity, those who know just the most general topics will probably start from the view that the difficulty of theory lies in the acceptance of the non-Euclidean description of space-time, going against habits consolidated for centuries. Given what is known before acquiring accurate knowledge of the theory this suspicion is reasonable and justified. But by carefully looking at the details we shall see that the concepts of the theory present no difficulty in themselves, and that the lack of understanding depends on a completely different and unexpected factor.
This book was written for two kinds of readers: those who know all the famous images and suggestions associated with special relativity and are curious about it, but who know nothing, except for having heard something or read some article or introductory book, and those who know the fundamentals, having studied them at the popular or specialist level, but are not convinced that they have understood and assimilated the concatenation of the elements of the theory. Many readers and scholars are in this condition even after going deep into the technical elements of the theory: they know the theory of Lorentz’s transformations, Minkowski geometry, the derivation of the “World’s most renowned formula”, E=mc2, from the space-time theory, but they recognize their tenuous grasp of the connection between the parts of the theory and the consistency of the demonstrations.
So this book describes the theory of special relativity with the simplicity necessary for it to be understood by non-specialist readers, who are only required to read it carefully. At the same time it analyzes the logic implied by the typical arguments of relativistic literature, and sheds light on the premises that for long tradition are not expressed with the necessary clarity, and therefore are at the root of the difficulty of understanding the theory. This book was written taking into account both the beginner’s and the perplexed people’s needs, the needs of those who know the elements of special relativity but feel they have not understood it. Therefore, readers already familiar with fundamentals of the theory will recognize many concepts they already understood and will read quickly or skip the elementary expositions of the things they already know, to dwell on the discussions of the difficulties.
Obviously, this book was not written for those who know they have understood and assimilated the theory and do not feel the need for any clarification.
This book is, in a certain sense, a practical manual that gives the readers tips to guide them into a subject about which it cannot be the only source. As the readers assimilate this book, they will feel the need to compare it with relativistic literature, in which at that point they will feel comfortable. As for the reading method, note that in this book you will find few figures and few formulas. Better read it holding paper and pencil at hand to write down formulas and draw a few figures following the text instructions. Then, since relativistic literature is huge and full of repetitions of the same things, I have chosen to minimize the space dedicated to the most famous topics. This book should serve as an introductory guide to reading any other source about special relativity, of medium or specialist level. You can read other sources right away after finishing this, or even at the same time, by going to the specialist discussion of the single arguments after assimilating the introduction to the background ideas you find clearly stated here in a way that you will not find elsewhere. In particular, I advise to read, in parallel with this book, the popular account of the theory written by Einstein himself in 1916, widely available in many economic editions.
With regard to the logical structure of the deductive process of special relativity, there are many things yet to be said and therefore this book devotes the least possible space to dealing with what is known or is explained extensively elsewhere. This book explicitly states all that is necessary to know before entering into the deductive mechanism of special relativity, but for the simplest and most common notions, mostly historical and factual data on which there are no difficulties or discussions, you will often find the indication “see Encyclopaedia”. It means that in order to deepen what may not be known or enough clear you are advised to stop the reading of this book and take a look at the subject matter in an encyclopaedia. Even Wikipedia will be fine, but with the warning to prioritize the entries in English rather than those in other languages.
Finally, for those who want to go back to school physics before getting into the study of special relativity, I recommend a basic English textbook, Understand Physics: Teach Yourself by Jim Breithaupt, a quick, pleasant, and effective read. I also recommend reading the book written and published by Einstein in 1938 in collaboration with Leopold Infeld: The Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. This book by Einstein introduces the issue of relativity with total clarity as far as the antecedents are concerned. But as it moves to special relativity, it proposes reflections and considerations on it that are relevant to those who already know the theory, without being able to be a useful textbook for those who come to it for the first time. It is therefore advisable to read the first and second part, where the preconditions of special relativity are exposed, before going to study, and the third and fourth part after acquiring knowledge of the theory. The book of Einstein and Infeld is complete, it also deals with general relativity and refers to quantum physics from the point of view of Einstein, who, as is well known, was not favourable toward it.
In general, the biography of the author of a theory is not relevant for the judgment of her or his work; and as a rule, one should concentrate on studying a theory by remaining ignorant of the biography of the author, in order to avoid giving trivial psychological labels to what is not understood. But in Einstein’s case, it is imperative to keep in mind the context of the theory, and therefore also the biography, for which I refer to a recent and somewhat voluminous book, Einstein. His Life and Universe by Walter Isaacson. We shall see that in the case of Einstein the history of the theory is particularly relevant for understanding the history of its fame and its acceptance by the mainstream of the scientific community. The theory of special relativity was born to answer a very specific question that in most of the books is completely ignored, and in some other treatise is relegated to the background. We shall note that this fact undermines the understanding of the deductive chain of the theory, but we shall also understand why this change of perspective occurred, which at first sight should appear curious. The theory of special relativity quickly assumed a very different meaning than the original purpose, and we shall see why.
Some final practical information: the footnotes in the text contain only references to the bibliography, so you do not have to read them if you do not want to trace the sources of information. Any marginal remarks are not in footnotes, but are incorporated in the text. In the English translation of this book the concept of velocity has been referred to with the simple word “speed” instead of “velocity”, that means a vector speed having a given direction and sense. The popular character of this book allows this simplification. Only in literal quotations of English texts the term “velocity” will occur.
I keep the promise made in the introduction and refer the reader to the Encyclopaedia (any Encyclopaedia, or any other source about the subject) for the preliminary historical notions. I just mention here what is indispensable to know.
Albert Einstein was born in 1879 in Ulm, in southern Germany, and because of family vagaries and vicissitudes, he lived in Italy and Switzerland the years of his youth. Legend has it that he was a bad student, but the study of his biography reveals that he was sometimes a brilliant student, but always rebellious and inclined to contradict teachers. It is interesting to note that, since his father and uncle were pioneer entrepreneurs in the field of electrical installations, Einstein was familiar with the properties of electrical equipment since he was young and could understand their peculiarities with great and innate ease. At sixteen he wrote an essay, conventional in content, about ether and electromagnetic theory, about what would be the central issue of special relativity, and at the same age he was able to help adults in calculations for the sizing up of electrical equipment:
Einstein spent the spring and summer of 1895 living with his parents in their Pavia apartment and helping at the family firm. In the process, he was able to get a good feel for the workings of magnets, coils, and generated electricity. Einstein’s work impressed his family. On one occasion, Uncle Jakob was having problems with some calculations for a new machine, so Einstein went to work on it. “After my assistant engineer and I had been racking our brain for days, that young sprig had got the whole thing in just fifteen minutes,” Jakob reported to a friend. “You will hear of him yet.”
Einstein got a diploma with a mediocre score, because of controversies with his teachers at the Federal Polytechnic of Zurich, and it was not easy for him to find an occupation. His temper led him to experience intellectual unemployment, at that time more rare than today. Finally, he found a modest job, which anyway was paid about twice the usual salary, at the Swiss Federal Bureau of Patents and in 1905 he published four papers in a major journal, Annalen der Physik, directed by Max Planck. Of the four papers the fourth is an appendix of the third and is only three pages long.
The year 1905, for this reason, is known as Einstein’s “annus mirabilis”, and the four papers have the titles:
·“On a heuristic point of view concerning the production and transformation of light”, on the photoelectric effect. This paper is not about relativity, but we shall refer to its content later on, because we shall use the concept of photon, or “electromagnetic quantum”. This essay (and not the theory of relativity) later was the motivation of the Nobel Prize conferred on Einstein in 1921. To understand special relativity, it is useful, though not strictly necessary, to have knowledge of the notion of photon, of which there is a clear explanation in Chapter 8 of Breithaupt’s book.
·“On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat”, a paper concerning a particular phenomenon, Brownian motion (see Encyclopaedia), from which Einstein draws important consequences for atomic theory. This paper is not concerned with relativity and we shall not deal with it.
·“On the electrodynamics of moving bodies”, the paper that is the primary source of special relativity, and which we shall always refer to with the abbreviation ED1905. Note that the title does not correspond to the idea that we have about the contents of special relativity: there is nothing that seems to concern space, time and clocks.
·“Does the inertia of a body depend upon its energy content?”, a three-page appendix to the previous paper on electrodynamics, where is developed what was hinted in the main paper about the equivalence between mass and energy. It is easy to understand that here we shall find the famous equation expressing the amount of energy released as a result of mass loss in a chemical transformation and in a nuclear reaction.
So in 1905, through the paper “On the Electrodynamics of moving bodies” and the appendix “Does the inertia of a body depend upon its energy content?” the theory of special relativity was published. It includes these three elements:
·Einstein’s completely unedited theory of space and time with respect to systems in uniform rectilinear motion;
·the explicit expression of a new conception of the relationship between mass and energy, which was emerging from others’ research of that time;
·a proposed solution to a very specific theoretical problem related to the theory of electromagnetic phenomena. This latter issue gives the title to the research, but it is the least known of all in the common cognition of special relativity. We shall see that this omission is the fundamental cause of the misunderstanding of the whole theory.
Special relativity does not require superior math tools, but only algebraic ones. It is limited to the condition of uniform rectilinear motion, and to overcome this limitation in the ten years after 1905 Einstein developed the theory of general relativity, which concerns accelerated motion and gravity. The theory of general relativity uses superior, highly sophisticated mathematical tools, and we shall not deal with it at all, not only because of the greater difficulty, but because one cannot commit a greater error than dealing with general relativity before focusing on special relativity problems and its specific issues. Without fully understanding special relativity, the study of general relativity may only be mnemonic, just a superficial knowledge of the conclusions of the theory.
Both relativities were described by Einstein in papers that make up their fundamental text: we have already encountered those relating to special relativity, those on general relativity were published in 1915 and 1917 (see Encyclopaedia). Of both theories Einstein wrote in 1916 a version he considered popular, which is a useful and highly advisable reading, but that is not easy and not exactly popular, and is often quoted by relativistic literature as an important source.
Before Einstein, scientists had reached the awareness that human knowledge cannot come to absolutely valid determinations of the unit of measure of space, the unit of measure of time, and the attributions of simultaneity to events. At the dawn of the twentieth century this subject was particularly in vogue, because at that time Henri Bergson’s philosophy was spreading and becoming popular: this philosopher reflected widely on the difference and the relationship between the concrete time actually experienced by the consciousness (which lives in the present, and there accumulates memories and mixes them with expectations of the future) and the abstract time of the scientific object construction processes. Henri Poincaré’s methodological studies were also widely known: he was an important physicist, a specialist in measure-related problems, and for a long time he was also an active member of international institutions for the unification of measurement methods (the Bureau des longitudes). But besides this, Poincaré was a best-selling author. His 1902 essay, La Science et l’Hypothèse, came out in a series destined for the general public and exerted considerable influence on the whole culture of the time. In Poincaré’s book, there are ideas anticipating relativity that influenced Einstein, who read it in 1904, but among others also influenced was the eccentric figure of Alfred Jarry, father of Ubu’s character.
Let us see what the meaning of the impossibility of finding an absolute criterion for space measurements is. Let us imagine being members of a community that survived a nuclear catastrophe and having to rebuild technology with our hands, having no more available the countless technical achievements we are used to: we would soon feel the need of a sample of length to use as a primary unit to measure everything else. We could cut it in wood, but after a while the sample could shrink or bend or dilate as a result of seasoning, so a wall we measured 20 metres long might look like 19 or 21 metres some time later. This could trigger practical difficulties and conflicts with the other members of the survivors’ community. Then we could replace the wood sample with one of metal, but after some time we should notice that the sample marks different lengths in summer than in winter because the metal is subject to thermal dilatation. Moreover, different metals are subject to different thermal dilatation, each having its own coefficient. If two samples, one of copper and one of iron, were built in winter, the next summer the two samples would be in conflict with the measures of length, because both would be dilated by heat, but according to different coefficients, so each of the two owners would claim to be in possession of the “right” metric sample. Then, having noted this problem, we could look for a material more stable than metal, and experimentation would advise us to build a stone sample ruler: on the walls of churches, in fact, we often find sculpted samples of local measures that were in use in past times. However, we could never know that spatial measures of stone are absolutely stable: different types of stone may have a lower instability than metals (and in fact they have), but the instability of stone would be revealed by further comparisons, similar to those which informed us of the different behaviour of a ruler of copper and one of iron. Then we could choose a different fundamental sample, built not with stone but with a material that experience has shown us to be more appropriate. However, the process would not end there. When we stop in this research, what we choose as a fundamental measure is never absolutely stable: it will always be only the relatively least unstable we have found, given the complexity of the scientific knowledge we have acquired. We choose as a metric sample what seems to be the best choice for practical and theoretical purposes, without being able to know whether our choice has an absolute value. The best choice is the one that allows us to build the most coherent possible image of our experience considered as a whole. If a man chooses in winter to take a sample of ice as his ruler, it would be a crazy choice because it is obvious that in spring he could no longer solve any measurement problems, neither theoretical nor practical. Yet every choice has a gambling component which cannot be avoided.
We can always choose the length sample with rational considerations; however, the fundamental sample we have chosen through time-evolved criteria, never guarantees us an absolute value: for example, our world, the solar system and the Milky Way could be measured with margins of error which seem to us tiny, and yet with respect to the lengths measured with a sample found on some star of another galaxy they may be subject to a macroscopic phenomenon of general dilatation of their lengths for e.g. two thousand years at a time, and then to a contraction phenomenon for another two thousand years. The space lengths measured at Julius Caesar’s time could be twice as long as those measured in the year 2000, from the point of view of an observer who is at a star away from us: but living in our world we would never know this phenomenon, because the fundamental metric sample we use would contract and dilate together with everything else, so we would judge stable the miles measured along the Roman consular roads and every other thing measured two thousand years ago.
So here we have the first concept of relativity, well ahead of Einstein’s: we do not have any criteria for the absolute measure of space, but we always assume some fundamental sample of space that changes together with the change of scientific knowledge as a whole. So the metre of the metric decimal system, as everyone knows, was defined in the early nineteenth century as the forty millionth part of the average terrestrial meridian. But then the metre has had more than one redefinition, and today it is defined as a fraction of the distance travelled by light in a second, having measured the speed of light with a device designed according to a very rigid criterion.
The same happens with time measurement. Before we think about it, we have the impression that the absolute measure of time is possible, but soon the consideration of concrete problems leads us to the opposite conclusion. To measure time we need a repeating phenomenon and a counter of the repetitions of this phenomenon. Thus, the fundamental clock is always based on a repetitive phenomenon (a phenomenon that has a frequency), that we take as a sample because through it the time measurements give rise to the most consistent or least inconsistent image of experience in its entirety that is possible. Also in this case we have no other criterion than the practical one.
If we imagine again ourselves as survivors of a catastrophe, we could try to use the solar day from dawn to sunset as a fundamental clock, but we would soon know it would be a bad choice: we could do it well, we could define the day from dawn to dusk as our main unit of time and then find some way to divide it into equal parts; but immediately afterwards we would realize that using the fractions of the daylight hours as time units, the time needed to cook an egg would be significantly different between summer and winter.
Then we would try to build better clocks, such as sundials and hourglasses. Apart from the obvious fact that a sundial can be used only if the sun shines in the sky, we would soon see that the times marked by a sundial and an hourglass, though both built in the best way, often disagree. In fact, due to the different speed of Earth’s revolution around the Sun and the obliquity of the ecliptic, the duration of the day is not constant: every day of the year does not last twenty-four hours as the average day, but is a little longer or shorter. Differences, summing up, create a gap between true time and the time marked by the sundial that comes to more or less 15 or 16 minutes. This phenomenon is described by an equation called the equation of time. Knowing this, we would prefer the hourglass, which at this point teaches us what the character of every clock is: since we cannot do otherwise, we take as a fundamental clock some phenomenon that not only repeats itself, but in which we can neither perceive nor conceive differences of quality between a repetition and the other. It is not enough to have the repetition of a phenomenon to define a clock: it is indispensable, too, that all repetitions appear identical and indistinguishable to our judgment. Therefore, for a long time, the fundamental clock was defined on the basis of the cyclical repetition of astronomical phenomena, until in recent times it was preferred to assume a completely different repetitive phenomenon: the frequency of the microwave spectral line emitted by atoms of the metallic element caesium, which is the basis of atomic clocks. The time second was defined, up to a few decades ago, as the average solar day divided by 86,400, while today it is defined as the duration of a certain atomic phenomenon that corresponds with very high approximation, but not exactly, to the previous definition. The fundamental clock is based on a repetitive phenomenon whose repetitions appear indistinguishable without it being possible to know if they are absolutely identical in and of themselves. This phenomenon can be the Earth’s rotation on itself, the Earth’s revolution around the Sun, or the frequency of the caesium microwaves in an apparatus built according to a strictly defined design: there is no absolute and objective reason to prefer one choice to the other, but only the bet that we shall get a more coherent image of our entire experience.
This is exactly what is happening today: the time measured with atomic clocks was adopted as an international standard because it was considered advantageous for practical problems such as aerial navigation, but sometimes a difference is observed between the atomic time and the time measured by the rotation of the Earth, and since 1970 onward every few years, as needed, atomic time is conventionally corrected a second at a time so as to keep in sync with astronomical time. For example, a second was added to atomic time before midnight on June 30, 2012, and between 1972 and 2016 the corrections of a second were altogether 37. The correction is triggered when atomic time and astronomical time diverge by 0.9 seconds, but the problem is that nobody can predict when there will be a need for the next fix. After 2012 there was one in 2015 and another in 2016, while in 2013 and 2014 no correction was necessary. If the corrections were predictable, it would mean that we would have a better clock than both the astronomical and atomic clocks; and on the contrary, the fact that the correction is unpredictable shows us how inconceivable the idea of an absolutely correct clock is.
So here we reach the same valid consideration we reached for space: all the phenomena in our galaxy could be subject to a long cycle in which they first slow down and then accelerate with respect to some clock outside it, and we would not notice it, nor could we ever say if the “right” clock is ours or the one outside the galaxy. There is no “right” clock, but only the clock that returns the least inconsistent possible interpretation of the whole of the accumulated experience at a given time of our history. One could build a ruler with ice; in the same way, a man might observe that his canary is singing each day exactly one hundred times with an interval between one and the other tweet that seems constant to his ear, and thus he could define the division of his days based on the behaviour of the canary. The choice would be as legitimate as any other, and for a lonely man it might be the excellent one: a man who is not interested in anything outside his own world might feel at ease by choosing to use his pet’s habits as a clock to decide his own actions, and cook his lunch at the fiftieth tweet, and so on. But, of course, such choice would encounter numerous practical problems in all relationships with third parties.
When we think of two objects of any kind in relation to each other, we always represent the events relevant to the two things by placing them inside the same time. The necessity of this logical operation generates as its consequence the instinctive idea that the attribution of both simultaneity and order of succession of two events is a trivial problem. For example, let us say that a red and a white car have passed in front of our house at the same time, or that the red car has passed before the other, and we believe to “see” that the two events occurred at the same time, or which occurred before the other in a given order, with event A before event B and not event B before A. It is not so; we do not “see” simultaneity and succession, but we always decide to attribute them through a judgment that involves more elements than the simple mathematical representation of time as a line and the location of events in the same coordinate of time or in different coordinates. The mathematical definition of simultaneity and succession is very simple: given the representation of time as a Cartesian axis, simultaneous events have the same value as the coordinate t, whereas successive events have different values t1 and t2, which are not interchangeable. But this simple formal aspect is not enough to recognize simultaneity and succession to be the physical conditions of two events.
To become aware of the complexity of judgments about simultaneity and succession, let us first consider an example of two people, two communities or two social systems living in different worlds, knowing nothing about each other. These two systems each have their own time, which refers to the course of events in their world, and they do not have a common time. Common time arises only when the two different worlds come into a relationship, or when they are related by the thought of a third subject. In the Middle Ages it was believed that the Earth, apart from being the absolute centre of the Universe, was divided into three continents, Europe, Asia and Africa, in the centre of which there was the city of Jerusalem and around which there was only the Ocean, impossible to navigate and to cross. In this world, and in the system of geographical and temporal references of this world, in 1325 Alfonso IV of Burgundy became king of Portugal. Meanwhile in Mexico, the Aztecs lived their lives and did things in the geography of their world and in the temporal references of their chronology. It is clear that when these events happened the Portuguese and the Aztecs not only knew nothing of each other, but did not even imagine anything of each other. Someone in Portugal might have wondered, if there were men on other continents, as today we ask ourselves if there are on other planets, and the same could have been done by someone else in Mexico: but this imaginative work could lead neither the Portuguese to have a concept of Aztecs, nor Aztecs to have a concept of the Portuguese. But we know today, or we believe we know, that in the year 1325, while in Portugal the political event of Alfonso IV ascending to the throne occurred, the Aztecs founded the town of Tenochtitlán. How do we know? Obviously we get the result from comparative analysis between the European chronology and archaeological and documentary testimonies left by the Aztecs which survived the destruction of the European conquest. This, however, tells us two things: first, attributing simultaneity to two events requires that they be known by someone who can place and think them in the same reference system for time measurements. For King Alfonso IV, the foundation of Tenochtitlán is neither simultaneous, nor before nor after its ascension to the throne: it simply is nothing. Secondly, the mere representation of things that take place and change over time is not enough to attribute the character of simultaneity to two events: it is always necessary to place the events in relation to spatial facts and then correlate them reciprocally also in physical relations and not just in the geometric ones.
Let us try to remember two remote events of our early childhood: two events that have impressed us and have surprised us pleasantly or painfully and therefore are never erased from our memory, but from time to time re-emerge, making us live again the mood we experienced when they occurred. The two events are in our memory, sometimes they re-emerge solicited by other events of the present, then they return to the latent state, then they reappear. Which happened first and which later? The memory only of our mood is never capable of answering this question. If for some reason we feel the need to place the two events in exact coordinates on the line of time, we must begin by asking ourselves: what events in the outer space are associated with those two inner events? What fact generated emotional quality? The concomitant events that happened in my childhood were the cause or were the effect of my emotions? Did I make a choice because I felt some emotion, or did I feel some emotion because before I had made a certain choice? And so on: by answering these questions we can reconstruct the order of time, which the memory only of the emotional qualities is unable to return.
The objective image of the world of experience is a construction in which we need the help of distinctions in space and of the knowledge of physical relationships between events, in order to properly place distinct events in different coordinates of the time line. So, for example, while thinking of childhood memories we could say: event A must have happened before event B, because now I remember that event A happened when I lived in the first of the homes where I lived in my childhood, while event B occurred in the second home. Event C occurred after event B, because when event B was happening, Grandpa was still alive, while the memory of event C also carries the sense of strangeness I felt when I was told that Grandpa had died, and so on.
Generalizing: the time lived by consciousness is nothing more than a magma of memory contents that emerge, vanish, re-emerge, and to put order in it, we need to represent time as a spatial dimension that we could safely call the fourth dimension in a very simple sense (with little or nothing to do with Einstein). Then it is necessary to correlate the events with the spatial coordinates where they occurred, and finally it is also necessary to correlate the events among them according to physical relationships: I know something happened before another when I know the first is the cause of the second. If I do not know the physical relationship, then I can always doubt the temporal relationship.
Let us look at another example to clarify this last statement. Let us suppose we see (without audio) a movie, a comedy, in which we are alternately presented images of the country running through a window and pictures of a group of people quarrelling in a small, long but rather narrow room. The representation of the quarrel is interrupted from time to time and leaves place for short moments in which we see the country flowing out of the window, then the scene returns to the quarrelling characters. People today understand this sort of comedy scene without having to think about it, immediately telling themselves: “While a train runs through the country, some characters squabble inside the train.” But to make this interpretation possible, and to pronounce that “while”, we must know that there is a planet called Earth, which often we see with the aspect of the countryside, with cultivated fields and trees. That there is an invented means of transportation, said train, which travels at such a speed that the landscape seen from their interior takes a typical blurred aspect, and inside where it is possible to concentrate on different activities and to distract attention from travelling. That there are animals called men who sometimes become ridiculous by investing all their energies in futile squabbles. Without these premises we would not understand the scene, and we are able to give the judgment on simultaneity, “while,” only because we are able to judge what we are seeing by knowing the fundamental physical properties that we have just stated.
We can take the opposite example and come to the same conclusion. Let’s imagine seeing (always without audio) a technical documentary created to explain to technicians the operation of a kind of machinery we do not know anything about. I see a sequence in which two different viewpoints alternate. Because I know I’m watching a movie and know what a movie is, I pronounce the judgment: “The pictures I’m seeing are mounted in the film in a certain sequence.” But as to the objects represented by the images, since I do not know anything about them, I do not even understand their relationships of simultaneity or succession: when the viewpoint of the camera changes, I do not know if the filmmaker intends to indicate that he or she wants to represent disjoined and unconnected events, or events occurring in the same time coordinates, or events occurring in different time coordinates with a given succession order that cannot be reversed: assigning time coordinates requires a theory of things to which the coordinates are attributed, and this theory always contains information about physical properties. If I do not know anything, then perception will only give rise to an accumulation of images in memory, which sometimes can come back if solicited by certain states of mind; but I shall never attribute simultaneity or succession unless I have a physical theory with which to interpret the accumulated images.
Light, like all electromagnetic waves, spreads into the empty space travelling 300 million metres every second. In the Earth’s atmosphere this speed is slightly lower, and in water it is significantly lower, about 2/3 of the value that is recorded in the vacuum. How can this measurement be so accurate, and what technology is needed to perform it? The answer is simpler than we would think. Already in 1677, a Danish astronomer, Ole Rømer, was able to attribute to the speed of light a value of the correct order of magnitude, something like 220,000 kilometres per second. He got this result by simply measuring for years the duration of the eclipses of one of Jupiter’s satellites, Io, in the nights when Jupiter was visible from the Earth, and interpreting these observations on the basis of Kepler’s laws and astronomical observations available at that time, shortly before the publication of the great Newton’s treatise (1687). Rømer’s reasoning was this: Jupiter has satellites, one of which is Io, orbiting around it, and we assume that the duration of the revolutions of these satellites around Jupiter is constant as well as the duration of the Lunar revolutions around the Earth. Jupiter and the Earth both perform a revolutionary movement around the Sun, but Jupiter’s year is much longer than that of the Earth, and therefore the Earth and Jupiter continue to move away and converge on one another. As the Earth and Jupiter move one with respect to the other, the satellite Io enters and exits from the shadow of Jupiter: first Io is illuminated by the Sun, then it is immersed in the shadow of Jupiter, then it re-emerges and is again illuminated. Rømer observed that the duration of the eclipses of Io seen from the Earth was not constant: assuming that the duration of Io’s revolution was constant, and knowing (through the laws of Kepler) the distances between the Earth and Jupiter at the time of various observations, he attributed the different duration of the eclipses of Io to the time which the light reflected by Io needed to reach the Earth. On this basis he could calculate the value we said for the speed of light, which later proved to be of the correct order of magnitude. The error of 25%, for which Rømer reached the value of 220,000 kilometres instead of 300,000, was determined by the approximate measurement of the time at night with the clocks available at that time and by other factors related to the technical resources of that era: observations were made in different places in Europe and the results were exchanged by mail. Acting more sophisticatedly, Rømer’s method could have yielded a much more accurate result. For precision, it should also be said that the calculation was performed by Christiaan Huygens, with whom Rømer was in correspondence: the whole story shows the rudimentary character of the method used as a whole (see Encyclopaedia).
Other astronomical observations came to the determination of a value similar to today’s accepted value, until as early as the mid-nineteenth century devices were constructed to measure the speed of a beam of light emitted by a terrestrial source and reflected by one or more mirrors to an instrument capable of measuring the duration of the travel of the beam. The necessary technology is more rudimentary than one would guess. The first and simplest of these devices was built by Hippolyte Fizeau in 1849 and yielded a value of 313,000 kilometres per second, then quickly rectified by more sophisticated devices: for the accuracy, today’s accepted value is 299,792,458 metres per second and therefore the error of Fizeau’s first measurement was considerable. It is worth knowing how Fizeau’s first device worked to see how simple the principle was. Obviously, Fizeau knew he had to find a value of the order of magnitude of 300,000 kilometres per second, given the pre-existing astronomical determinations of this figure. So he built a large cogged wheel that had 720 teeth spaced by so many empty spaces. A projector sent a beam of light in the night to a mirror located 8.5 kilometres away, so the beam travelled a total of 17 kilometres going and coming back. The projector was placed behind the wheel teeth, so the turning wheel acted as a shutter now allowing the light to go to the mirror, now stopping it. Always behind the wheel, but on the left side of it if the projector was to the right of the wheel, an observer stared at the mirror through the interstices between the wheel teeth that passed in front of his or her eye. What happened? While the wheel did not spin, the observer saw the reflected beam in the mirror. When the wheel began to spin slowly, the observer still saw the beam. When the wheel had reached a certain speed, the observer no longer saw the beam, because it, coming back from the mirror, hit the tooth that was now standing in front of the observer’s eye. By increasing the rotation speed, the beam returned visible, reaching the observer’s eye through the interstice following the tooth that had passed in front of his or her eye as the beam travelled back and forth.
Since the journey time was about 1/20,000 of a second, and since the wheel had 720 teeth, to perform the experiment a small rotation speed was sufficient. It was needed that the disk could rotate at a speed of a few dozen turns per second; the first eclipse of the beam was produced at 12.6 rounds per second, so the apparatus could be constructed with the mechanical means available with the technology of the mid-nineteenth century. The calculations, however, gave a coarse value, which was corrected by about 5% by a more sophisticated apparatus soon built by Foucault.
Subsequent advanced measuring instruments determined the value of light speed more accurately, always measuring, and we shall see that this detail is important, the time of the journey of a beam that was emitted by a component of the measuring apparatus, or a source inside the apparatus, and reflected on the apparatus. That is, the instruments for measuring the speed of light in general do not measure the speed of beams emitted by external sources to them.
2. Ether, electromagnetic theory and Lorentz’s hypothesis
Once again, I warn that all notions below are simplified until being trivialized, and yet (as well as being indispensable) they are all accurate and sufficient to understand special relativity. Starting from any Encyclopaedia, you can find the necessary information to go deeper on every issue, but our exposure will make relativity comprehensible even without further in-depth knowledge.
Over the course of just a century, the theory of electrical and magnetic phenomena, whose manifestations were known since antiquity (the word “electric” comes from the Greek name for amber, the word “magnetic” comes from the name of the Greek city of Magnesia, where iron minerals were extracted) had immensely evolved, passing from Galvani’s experiments with his frogs, which he mistakenly believed to be generators of electricity, to Voltaic piles and from there to electrical technology that in the first years of the twentieth century allowed the operation of electric trams and locomotives, electrical distribution networks in cities, telegraphs, telephones, Marconi’s radio stations, and so on.
The theory of electrical and magnetic phenomena has been unified since, in 1820, the Danish physicist Oersted noticed, perhaps by chance, that electric current in a circuit was diverting the needles of some compasses laid on his desk. Electric charges exert forces acting in the space surrounding them, and the same does any material having magnetic properties. There are therefore, in space, electric fields and magnetic fields that interact closely: under certain conditions a current can be exploited to produce a magnetic field, under other conditions a magnetic field generates a current, and the interaction between an electric and a magnetic field is what causes the rotation of an electric motor.
The correlation between electricity and magnetism was determined quantitatively by very complex mathematical equations describing their interaction by interpreting it as a wave, with properties similar to liquid waves. During the search for these equations, in 1864, James Maxwell came to the conclusion that light is not a phenomenon of special nature, but is a part of the whole of electromagnetic phenomena, being nothing but a wave that has certain lengths: there is visible light that has wavelengths between a minimum and a maximum, there are waves of greater length than light (including those that are used for radio broadcasting) and waves of smaller length (for example, X-rays). From the study of electromagnetic phenomena, Maxwell came to determine the propagation speed of any electromagnetic wave using experimental elements and reasoning independent of those used to measure the speed of light, and found that all electromagnetic waves propagate at the same speed as light in vacuum, about 300 million metres per second. Since that time, visible light has been considered an electromagnetic wave that can have frequencies in a certain range and which generates the colours that the human eye can see.
During the development of these ideas, for which many decades were needed, a problem came to light: if electromagnetic waves are waves similar to those we observe in liquids, something must exist that moves with a wavy motion. It does not make sense to talk about sea waves without water: the sea waves are nothing more than the phenomenon of water mass movement. Something like this had to happen to allow the existence of electromagnetic waves: this consideration led to the hypothesis that wherever there is light or there are electromagnetic waves, there must also be the material medium that allows the waves to propagate, or rather, to exist. This material, hypothetical and never observed by anyone, was called the ether: the hypothesis dates back to the time when science was still written in Latin, and therefore the medium of light propagation took the name of aether luminiferus, in English luminiferous ether, which suggests the connotation of something archaic, immature and inadequate to the contemporary state of science. Moreover, the electromagnetic ether, “luminiferous”, was a concomitant creation with recent discoveries, but the generic concept of ether was not a new hypothesis: the history of the ether hypothesis is ancient and dates back to Aristotle, who did not limit himself to the hypothesis, but was quite certain that the ether constituted the substance of the celestial world, different from the four terrestrial substances, and therefore it was the “quintessence”, i.e. the “fifth substance.”
But the nineteenth-century electromagnetic ether was neither a coarse concept, nor out of date; at first, the hypothesis arose only from the need to reduce the phenomenon of light to a familiar interpretation scheme, but then the concept of ether was hypothetically determined in very precise ways, according to various models, in order to attribute to it the physical characteristics that would be consistent with the properties of electromagnetic waves tested experimentally. The word chosen to give ether a name suggests a kinship with the ancient prescientific concept, but nothing more. If a modern term instead of the word of Aristotelian origin had been chosen in the 1820s or ‘30s, many subsequent misunderstandings would have been avoided in the divulgation of the theory of special relativity which, precisely because of the misunderstanding induced by terminology, appears to be representative of a modern paradigm of science opposed to a prescientific one.
A detail to be taken into account is that the various hypotheses attributed to the ether the ability to be everywhere, not only in the vacuum, but to share the space with each species of matter, infiltrating into the interstices between atoms. This happened because the description of electromagnetic waves suggested that they needed the same homogeneous transmission medium everywhere, unlike what happens to sound waves, which propagate in different media and cause e.g. first the air to vibrate, then the wall between two rooms, than air again, until our eardrums are reached by a sound.
The conceptual problem associated with the hypothesis of ether is that it conflicts with a conceptual stronghold of modern physics, that is, the law of inertia of Galileo and Newton. The ancients imagined, as even today many continue to imagine before receiving the first physics teachings at school, that the state of rest and that of motion can be distinguished by their intrinsic character, as solid state is distinguished from the liquid or gaseous state. Instead, Galileo and Newton taught us the principle, and laid the foundations of all mechanics on it, that rest and rectilinear motion at constant speed cannot be distinguished, if not conventionally, on the basis of a reference point chosen arbitrarily. Two bodies A and B move away from each other on a straight line and at constant speed v: we can say that A is stationary and B moves at speed v, or that B is stationary and A moves at speed -v, or that both move at speed v/2, and so on: given two bodies in rectilinear motion with respect to each other, mechanics elaborated in modern age does not possess and does not admit any criterion to decide which of them is in motion, as its absolute objective property, and which is at rest. Two people sitting in a train move both with respect to the Earth; in the meantime, with respect to the train one can be sitting quietly and the other can be moving while walking. But considering only the two persons, we say that they are simply moving the one with respect to the other, and nobody is allowed to attribute them the state of motion or rest as an absolute character. More important, nobody is allowed to say if their speed with respect to the train, or to the Earth, or to any other reference, is their “true” speed. No “true” speed, objective feature of anything, exists.
Hence follows the first principle of Newton’s dynamics, the principle of inertia, for which “a body maintains its state of rest or uniform rectilinear motion, until a force does not act upon it.” Because rest and rectilinear motion at a constant speed cannot be distinguished and do not differ intrinsically in anything, it is necessary to apply a force to stop or divert a body in rectilinear motion. A car switching off the engine will stop after tens or hundreds of metres from air resistance and resistance of its elastic rubber wheels, which are deformed continuously absorbing a lot of energy. A railway vehicle left without traction force will stop not after hundreds of metres but after miles, because the air resistance is less significant because it is much heavier than a car, and because the rolling resistance of the wheels is negligible, being both the wheels and rails very rigid. But a body running straight at constant speed in the empty space can travel billions of miles; it will not stop until it enters an area where a celestial body will exert a sensible gravitational influence on it.
Descartes, trying to explain this principle to one of his acquaintances, received an ironic reply: “you say that if I sit in an armchair while another moves away walking under the sun, I can say that he is at rest while I am moving?” But this witty man did not catch the point, and modern mechanics thinks differently: the condition of uniform rectilinear motion is preserved, infinitely, as long as a force does not intervene to modify it, and the condition of uniform rectilinear motion cannot be distinguished from that of rest for any intrinsic quality. Entities of any kind that are in a relative rectilinear motion with respect to each other, without anyone being able to say which is in motion and which at rest, are called “inertial systems”: this expression will be used in the following.
So mechanics said, and continues to say: there is no absolute reference for the motion condition, but all references are arbitrarily chosen, given the need to study a phenomenon. If a person is in a railway carriage, I shall consider him or her moving with respect to the ground (attributing him or her a speed) or at rest with respect to the carriage (attributing him or her zero speed) as it will be better with respect to the problem I want to solve. Instead, the electromagnetic theory of the nineteenth century said (today it does not say it anymore, as we shall see below): there must be a medium of propagation of electromagnetic waves that is spread uniformly wherever our observations come, maybe anywhere in the Universe, and therefore the movement of electromagnetic waves has an absolute and unique reference system, the ether, unlike mechanical movements, who are not subject to anything like this. It is with respect to the ether that electromagnetic waves move at the speed of about 300 million metres per second, a speed always abbreviated with the letter c (from Latin celeritas).
The difference is not just descriptive. In equations describing electromagnetic phenomena, a constant, c, which is 300 million metres per second, represents a speed, while in the equations describing mechanical phenomena a constant speed will never appear. A man walking at the speed of 5 kilometres per hour in a train that travels at 100 kilometres per hour moves at 105 kilometres per hour on the ground: both speeds, 5 km/h and 105 km/h, are “true” because speed is calculated in a reference system chosen always for convenience and never for intrinsic reasons. Therefore mechanics cannot have a constant that represents the value of a speed. Acceleration can be a constant, and for example the acceleration of gravity on the Earth is: it is about 9.8 m/s2 on average, and precisely 9.823 m/s2 to the poles and 9.789 m/s2 to the equator.
This elementary issue must be completely clear: anyone who is completely new to the subject should not continue reading without reflecting on it. Acceleration is the ratio between a speed variation and the time it takes place. If a body changes from a speed of 2 m/s to a speed of 6 m/s in a time of 2 seconds, the acceleration is equal to the difference in speed divided by the time elapsed, that is, in our case:
Acceleration is not relative, unlike speed. Suppose that the body that accelerates from 2 m/s to 6 m/s is a pedestrian walking in a train that runs at 100 m/s. From the point of view of the ground, the pedestrian passes from the speed of 102 m/s to that of 106 m/s always in 2 seconds, so we have:
Acceleration is always 2 m/s2. That is, acceleration is invariant, while speed is related to the reference system chosen to determine it. For this reason a physical law may contain the numeric value of an acceleration and consider it a constant — for example, the numeric value of gravity acceleration — but it cannot, or rather, it should not contain the numeric value of a speed, because the “true” speed of anything can never be defined.
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