Vielbein - Relativity and other

Vielbein

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Some notes on Relativity and other arguments

Einstein - Autobiographical Sketch

Nell'agosto del 1954 Einstein fu invitato a scrivere una breve Autobiografia scientifica in occasione dei 100 anni del Politecnico di Zurigo (ETH). Einstein invi&ograve;&nbsp;l'articolo a solo due settimane dalla morte avvenuta nel 1955. Qui di seguito riportiamo il contributo in inglese:
In 1895, aged sixteen, I arrived in Zurich from Italy after spending a year at&nbsp;my parents&rsquo; home in Milan without attending school or having any tuition at all. My goal&nbsp;was admission to the Polytechnic, but I had no clear idea for achieving it. I was a headstrong&nbsp;yet modest young person who had acquired the pertinent elements of his patchy&nbsp;knowledge mostly by studying on his own. While I was keen to delve deeper, I lacked&nbsp;the talent for absorbing knowledge and was hampered by a bad memory, so university&nbsp;studies seemed far from easy to me. I was right to feel hesitant as I registered for the&nbsp;entrance examination in the engineering department. Although the examination made&nbsp;me painfully aware of the gaps in my education, the examiners were patient and understanding.&nbsp;My failure seemed completely justified, but it was comforting that the physicist&nbsp;H. F. Weber let me know I could attend his lectures if I stayed in Zurich. Meanwhile the&nbsp;rector, Professor Albin Herzog, recommended me to the cantonal school in Aarau, where&nbsp;I studied for a year and received my high school graduation certificate. This school, with&nbsp;its liberal spirit and the unpretentious, serious attitude of the teachers, who relied on&nbsp;their own judgment rather than any outside authority, made a lasting impression on me.&nbsp;Comparing this with my six years of education at a German high school run by authoritarian&nbsp;methods made me acutely aware that it is far better to teach people to act freely and&nbsp;on their own responsibility than to educate them on principles of military drill, external&nbsp;authority, and ambition. Genuine democracy is not an empty illusion.&nbsp;During the year in Aarau, the following question occurred to me: If one chases a light&nbsp;wave at the speed of light one would arrive at a time-independent&nbsp;wave field. But nothing&nbsp;like that really seems to exist! This was the first childlike experiment in thinking about&nbsp;special relativity theory. Innovation itself is not the result of logical thought, even though&nbsp;the end product is tied to a logical structure.
From 1896 to 1900 I studied at the specialist teachers&rsquo; department at the Federal Polytechnic.&nbsp;Before long I realized that I had to be content with being an average student. To&nbsp;be a good student you must be able to grasp things easily; you must be willing to concentrate&nbsp;your energies on everything you are told in the lectures; you must enjoy writing&nbsp;down everything presented in the lectures in an orderly fashion and working on it conscientiously.&nbsp;Regretfully, I realized that I fundamentally lacked all these qualities. This&nbsp;meant that I gradually learned to live in peace with a degree of bad conscience and to&nbsp;organize my studies to suit my intellectual tastes and my own interests. I followed some of&nbsp;the lectures with interest and excitement. Otherwise, I often skipped classes and studied&nbsp;the masters of theoretical physics at home with a divine zeal. This was a good thing in&nbsp;itself and soothed my bad conscience so effectively that I avoided any serious emotional&nbsp;upsets. I resumed my earlier habit of long private study sessions, in which I was joined by&nbsp;a Serbian student, Mileva Mari&#263;, whom I later married. At the same time, I worked zealously&nbsp;and passionately in Professor H. F. Weber&rsquo;s physics laboratory. I was also fascinated&nbsp;by Professor Geiser&rsquo;s lectures on infinitesimal geometry, which were veritable masterpieces&nbsp;of the art of education and proved very helpful later when I was wrestling with the&nbsp;general theory of relativity. Aside from this, higher mathematics was of little interest to&nbsp;me during my university studies. I mistakenly believed that this was such a widespread&nbsp;and diffuse territory that one could easily waste all one&rsquo;s energy in a remote province.&nbsp;In my innocence I thought it was enough for a physicist to have grasped the elementary&nbsp;mathematical concepts clearly and to be able to use them readily, and that everything&nbsp;else involved subtleties that were unproductive for physicists&mdash;an&nbsp;error I regretfully realized&nbsp;only later. Evidently I lacked sufficient mathematical talent to be able to distinguish&nbsp;between central, fundamental matters and peripheral things of no major importance.
In my university days I became close friends with a fellow student, Marcel Grossmann.&nbsp;I had a regular weekly date with him in Caf&eacute; Metropol on Limmatquai, and we talked&nbsp;not only about our studies but also about everything else of possible interest to curious,&nbsp;open-minded&nbsp;young people. Unlike me, he was not a vagabond and a loner type; he was&nbsp;someone who was deeply rooted in his Swiss surroundings yet never lost his innate independence.&nbsp;He was also richly endowed with all the gifts I lacked: he was quick to grasp&nbsp;things and orderly in every sense. He attended all the lectures that interested us, and his&nbsp;work on them was so excellent that his notebooks were eminently fit for publication. He&nbsp;lent me those notebooks to prepare for the examinations, and they were like a life belt for&nbsp;me. I do not care to speculate how things would have turned out for me without them.
Even with this invaluable help, and although the subjects discussed in our lectures&nbsp;were intrinsically interesting, I still had to struggle to overcome my reluctance to learn all&nbsp;these things thoroughly. Academic studies are not necessarily beneficial for withdrawn,&nbsp;contemplative people like myself. Constantly being forced to eat so many good things can&nbsp;ruin your appetite and give you bellyache. It can extinguish the delicate light of divine&nbsp;curiosity forever. Fortunately, in my case my studies ended well, and this intellectual&nbsp;depression lasted only a year after I finished university.
The best thing Marcel Grossmann ever did for me as a friend happened around a year&nbsp;after I finished studying. With his father&rsquo;s help he recommended me to the director of the&nbsp;Swiss Patent Office, which was still called the Office of Intellectual Property in those days.&nbsp;After an extensive interview Mr. Haller gave me a job there. This relieved me of existential&nbsp;pressures during the years when I produced my best work. Aside from this, working to&nbsp;evaluate technical patent applications was a real blessing for me. It compelled me to think&nbsp;in many different ways and provided important stimuli for my thinking about physics.&nbsp;A practical profession is ultimately a blessing for people like myself. An academic career&nbsp;puts a young person into a coercive situation in which he is compelled to produce impressive&nbsp;quantities of scientific papers, and this creates a temptation to superficiality that&nbsp;only strong-minded&nbsp;characters can resist. What is more, in most practical professions an&nbsp;averagely gifted person can perform according to expectations. His life in society is not&nbsp;dependent on any special intellectual discoveries. If his scientific interests go deeper, he&nbsp;can get immersed in his favorite problems alongside his usual work obligations. He has&nbsp;no need to worry that his efforts might fail to yield results. Thanks to Marcel Grossmann,&nbsp;I was now in this fortunate position.
Among the scientific experiences of those happy years in Bern I shall mention one in&nbsp;particular, which turned out to be the most fruitful idea of my life. The theory of special&nbsp;relativity was already a few years old. The question was whether the principle of relativity&nbsp;was limited to inertial systems, that is, coordinate systems that move in a straight line at&nbsp;constant velocity relative to each other (linear coordinate transformations). On a formal&nbsp;level one would instinctively say, &ldquo;Probably not!&rdquo; Yet the foundation of every kind of&nbsp;mechanics until then&mdash;the&nbsp;principle of inertia&mdash;seemed&nbsp;to exclude any extension of the&nbsp;principle of relativity. In fact, if one introduces an accelerated coordinate system (relative&nbsp;to an inertial system), an &ldquo;isolated&rdquo; mass point no longer moves uniformly and in a&nbsp;straight line in relation to it. At this juncture, a mind not bound by narrow thinking habits&nbsp;would have asked, &ldquo;Does this type of motion offer me a means of differentiating between&nbsp;an inertial and a non-inertial&nbsp;system?&rdquo; The untrammeled mind would have necessarily&nbsp;have answered negatively (at least in the case of uniform acceleration in a straight line).&nbsp;For one could also interpret the mechanical behavior of bodies relative to such an accelerated&nbsp;coordinate system as the effect of a gravitational field; this is possible by virtue of&nbsp;the empirical fact that in a gravitational field, too, the acceleration of bodies is always the&nbsp;same, independent of their nature. This insight (the equivalence principle) not only made&nbsp;it probable that the natural laws must be invariant with respect to a universal group of&nbsp;transformations corresponding to the group of Lorentz transformations (extension of&nbsp;the principle of relativity), but also that this extension would lead to an advanced theory&nbsp;of the gravitational field. I had not the least doubt that this idea was correct in principle.&nbsp;But it seemed nearly impossible to establish this conclusively. To begin with, there were&nbsp;elementary considerations that the transition to a further group of transformations is not&nbsp;compatible with a direct physical interpretation of the space-time&nbsp;coordinates that had&nbsp;prepared the ground for special relativity theory. Moreover, at first it was hard to see how&nbsp;to choose the extended group of transformations. In fact, I arrived at the equivalence&nbsp;principle by way of a detour that is beyond the scope of the present account.
This problem occupied me continuously all through the period 1909&ndash;1912,&nbsp;while&nbsp;I was engaged in teaching theoretical physics at the universities of Zurich and Prague.&nbsp;By 1912, when I was appointed as a professor at Zurich Polytechnic, I had come much&nbsp;closer to solving the problem. Hermann Minkowski&rsquo;s analysis of the formal foundations&nbsp;of special relativity theory emerged as an important factor here. It can be summarized in&nbsp;the sentence: Four-dimensional&nbsp;space has an (invariant) pseudo-Euclidean&nbsp;metrics; this&nbsp;determines the experimentally verifiable metrical characteristics of the space as well as&nbsp;the inertia principle and, moreover, the form of Lorentz-invariant&nbsp;equivalence systems.&nbsp;This space contains preferred, namely, quasi-Cartesian&nbsp;coordinate systems, which are the&nbsp;only &ldquo;natural&rdquo; ones here (inertial systems).
The equivalence principle leads us to introduce nonlinear coordinate transformations&nbsp;into this type of space, that is, non-Cartesian&nbsp;(&ldquo;curvilinear&rdquo;) coordinates. In this case, the&nbsp;pseudo-Euclidean&nbsp;metrics assumes the universal form:
<img title="ds^2=\Sigma g_{ik}dx_idx_k" src="http://latex.codecogs.com/gif.latex?ds^2=\Sigma&amp;space;g_{ik}dx_idx_k" />
summed over the indices i and k (which take values from 1&ndash;4).&nbsp;These g functions are&nbsp;then functions of the four coordinates that, according to the equivalence principle,&nbsp;describe not only the metrics but also the gravitational field. The latter has a very special&nbsp;quality, of course; as we know, it can be transformed into the special form
<img title="-dx_1^2-dx_2^2-dx_3^2+dx_4^2" src="http://latex.codecogs.com/gif.latex?-dx_1^2-dx_2^2-dx_3^2+dx_4^2" />
that is, a form in which the gik functions are independent of the coordinates. In this case,&nbsp;it can be &ldquo;transformed away&rdquo; by the gik-defined&nbsp;gravitational field. In the latter special&nbsp;form, the inertial motion of isolated bodies is expressed by a (timelike) straight line. In&nbsp;the universal form, it is expressed by the &ldquo;geodetic line.&rdquo;
Although this formulation still derived from the case of pseudo-Euclidean&nbsp;space,&nbsp;it clearly showed how the transition to gravitational fields of a general type was to be&nbsp;achieved. Here, too, the gravitational field can be described by a type of metrics, that is,&nbsp;by a symmetrical tensor field, gik. The generalization merely consists in leaving out the&nbsp;precondition that this field can be transformed into a pseudo-Euclidean&nbsp;one simply by&nbsp;transforming the coordinates.
This solution reduced the problem of gravitation to a purely mathematical one. Are&nbsp;there differential equations for the gik functions that are invariant with respect to nonlinear&nbsp;coordinate transformations? Such differential equations, and only such, would come&nbsp;into consideration as field equations of the gravitational field. The law of motion of material&nbsp;points was then given by the equation of the geodetic line.
It was with this problem in mind that I went to see my old friend from university, Marcel&nbsp;Grossmann, in 1912. By then he had become a professor of mathematics at the Swiss&nbsp;Federal Polytechnic. He caught on to the problem right away, even though as a true mathematician&nbsp;he had a rather skeptical attitude toward physics. In our student days when&nbsp;we used to discuss ideas over coffee, he once made such a lovely characteristic remark&nbsp;that I cannot resist quoting it: &ldquo;I must admit that studying physics has actually benefited&nbsp;me considerably. Before, when I sat down on a chair that felt a bit warm from the person&nbsp;who had occupied it previously, I used to feel rather uncomfortable. This feeling has&nbsp;completely gone now, because physics has taught me that the heat is something totally&nbsp;impersonal.&rdquo;
In the event, he was fully prepared to collaborate on the problem with me, but only on&nbsp;condition that he would not have to take responsibility for any claims and interpretations&nbsp;related to physics. He checked out the literature and soon discovered that the mathematical&nbsp;problem in question was already solved, in particular by Riemann, Ricci, and Levi-Civita.&nbsp;The whole development connected up with the theory of the Gaussian curvature,&nbsp;which systematically used generalized coordinates for the first time. Riemann&rsquo;s achievement&nbsp;was the greatest. He showed that tensors of second-order covariant differentiation&nbsp;could be formed out of the field of gik tensors. This revealed what the gravitational field.&nbsp;equations must look like&mdash;under&nbsp;the condition that invariance is required with respect&nbsp;to the group of all continuous coordinate transformations. It was, however, not very easy&nbsp;to see that this condition was justified, especially as I believed I had found arguments&nbsp;against it. This reservation, which eventually turned out to be mistaken, resulted in the&nbsp;theory only appearing in its final form for the first time in 1916.
While I was avidly working with my old friend, none of us dreamed that this outstanding&nbsp;man would be carried off by a pernicious illness far too early. It was the urge to express&nbsp;my gratitude to Marcel Grossmann at least once in my lifetime that prompted me to write&nbsp;this rather slapdash autobiographical note.
Forty years have passed since the theory of gravitation was completed. Those years&nbsp;were almost entirely devoted to efforts to develop a unified field theory that could form&nbsp;a foundation for the whole of physics by generalizing from gravitational field theory.&nbsp;Many people worked toward the same goal. I investigated several seemingly promising&nbsp;approaches and subsequently discarded them. But the past ten years finally led to a theory&nbsp;that seems natural and promising to me. Still, I have not been able to convince myself&nbsp;about whether I should regard this theory as valuable for physics or not. This is fundamentally&nbsp;due to mathematical difficulties that are insurmountable for the time being,&nbsp;like those, incidentally, that arise in the use of any nonlinear field theory. Moreover, it&nbsp;seems doubtful altogether whether a field theory can properly account for the atomistic&nbsp;structure of matter and radiation as well as of quantum phenomena. Most physicists&nbsp;would immediately answer &ldquo;no,&rdquo; because they believe that the quantum problem has been&nbsp;solved in principle in another way. Be that as it may, we should take comfort from Lessing&rsquo;s&nbsp;dictum that the search for truth is more precious than its possession.
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