Cartesian tensors: an invitation to indices

LOCAL DIFFERENTIAL GEOMETRY consists in the first instance of an amplification and refinement of tensorial methods. In particular, the use of an index notation is the key to a great conceptual and geometrical simplification. We begin therefore with a transcription of elementary vector algebra in three dimensions. The ideas will be familiar but the notation new. It will be seen how the index notation gives one insight into the character of relations that otherwise might seem obscure, and at the same time provides a powerful computational tool.

The standard Cartesian coordinates of 3-dimensional space with respect to a fixed origin will be denoted xi (i = 1,2,3) and we shall write A = Ai to indicate that the components of a vector A with respect to this coordinate system are Ai . The magnitude of A is given by A · A = AiAi . Here we use the Einstein summation convention, whereby in a given term of an expression if an index appears twice an automatic summation is performed: no index may appear more than twice in a given term, and any ‘free’ (i.e. non-repeated) index is understood to run over the whole range.

Enlightenment is man's emergence from his self-incurred immaturity. Immaturity is the inability to use one's own understanding without the guidance of another. This immaturity is self-incurred if its cause is not lack of understanding, but lack of resolution and courage to use it without the guidance of another. The motto of enlightenment is therefore : Sapere aude! Have courage to use your own understanding!

Laziness and cowardice are the reasons why such a large proportion of men, even when nature has long emancipated them from alien guidance (naturaliter maiorennes, nevertheless gladly remain immature for life. For the same reasons, it is all too easy for others to set themselves up as their guardians. It is so convenient to be immature! If I have a book to have understanding in place of me, a spiritual adviser to have a conscience for me, a doctor to judge my diet for me, and so on, I need not make any efforts at all. I need not think, so long as I can pay ; others will soon enough take the tiresome job over for me. The guardians who have kindly taken upon themselves the work of supervision will soon see to it that by far the largest part of mankind (including the entire fair sex) should consider the step forward to maturity not only as difficult but also as highly dangerous. Having first infatuated their domesticated animals, and carefully prevented the docile creatures from daring to take a single step without the leading-strings to which they are tied, they next show them the danger which threatens them if they try to walk unaided. Now this danger is not in fact so very great, for they would certainly learn to walk eventually after a few falls. But an example of this kind is intimidating, and usually frightens them off from further attempts.

Thus it is difficult for each separate individual to work his way out of the immaturity which has become almost second nature to him. He has even grown fond of it and is really incapable for the time being of using his own understanding, because he was never allowed to make the attempt.

NO SINGLE theoretical development in the last three decades has had more influence on gravitational theory than the discovery of the Kerr solution in 1963. The Kerr metric is a solution of the vacuum field equations. It is a generalization of the Schwarzschild solution, and represents the gravitational field of a special configuration of rotating mass, much as the external Schwarzschild solution represents the gravitational field of a spherical distribution of matter.

However, unlike the Schwarzschild case, no simple non-singular fluid ‘interior’ solution is known to match onto the Kerr solution. There is, nevertheless, no reason a priori why such a solution shouldn't exist.

Fortunately such speculations are in some respects beside the point, since the real interest in the Kerr solution for many purposes is its characterization of the final state of a black hole, after the hole has had the opportunity to ‘settle down’ and shed away (via gravitational radiation and other processes) eccentricities arising from the structure of the original body that formed the black hole.

To put the matter another way, suppose someone succeeded in exhibiting a good fluid interior for the Kerr metric. Well, that would be in principle very interesting; but there is no reason to believe that naturally occurring bodies (e.g. stars, galaxies, etc.) would tend to fall in line with that particular configuration.

However exalted we may wish our concepts to be, and however abstract we may make them in relation to the realm of the senses, they will continue to be associated with figurative notions. The proper function of these is to make such concepts, which are not in other respects derived from experience, suitable for use in the experiential world. For how else could we endow our concepts with sense and significance if we did not attach them to some intuition (which must ultimately always be an example derived from some possible experience)? If we then subtract the figurative associations from this concrete act of the understanding— first those of fortuitous sense-perception, and then the pure sensuous intuition itself—we are left with the pure concept of the understanding, but with its scope now enlarged so as to constitute a complete rule of thought. This is the way in which even universal logic came into being; and in the application of our understanding and reason to experience, there may still lie hidden certain heuristic methods of thought which, if we could carefully extract them from experience, might well enrich philosophy with useful maxims, even in abstract thought.

To this category belongs that principle to which the late Moses Mendelssohn expressly declared his allegiance—but only, so far as I know, in his last writings (see his Morgenstunden ﹛Morning Hours), pp. 164f. and his letter An die Freunde Lessings (To Lessing's Friends), pp. 33 and 67): namely the maxim that it is necessary to orientate oneself in the speculative use of reason (which Mendelssohn, on other occasions, credited with considerable powers in the cognition of supra-sensory objects, and even with the power of conclusive proof) by means of a certain guideline which he sometimes described as common sense (in his Morgenstunden), sometimes as healthy reason, and sometimes as plain understanding (in An die Freunde Lessings). Who would have thought that this admission would not only have such disastrous effects on his favourable opinion of the power of speculative reasoning in theological matters (which was in fact inevitable), but also that even ordinary healthy reason, given the ambiguous position to which he relegated the use of this faculty in contrast to speculation, would risk becoming the basic principle of zealotry and of the complete subversion of reason?

IT IS INEVITABLE that with the passage of time Einstein's general relativity theory, his theory of gravitation, will be taught more frequently at an undergraduate level. It is a difficult theory—but just as some athletic records fifty years ago might have been deemed nearly impossible to achieve, and today will be surpassed regularly by well-trained university sportsmen, likewise Einstein's theory, now over seventy-five years since creation, is after a lengthy gestation making its way into the world of undergraduate mathematics and physics courses, and finding a more or less permanent place in the syllabus of such courses. The theory can now be considered both an accessible and a worthy, serious object of study by mathematics and physics students alike who may be rather above average in their aptitude for these subjects, but who are not necessarily proposing, say, to embark on an academic career in the mathematical sciences. This is an excellent state of affairs, and can be regarded, perhaps, as yet another aspect of the overall success of the theory.

A fresh look at anti-symmetric tensors

WE have introduced local differential geometry in a notation that makes great use of indices. This is the classical route and it does have a great deal of merit. There is a parallel development in an index free notation that is more generally used by pure mathematicians. The different approaches have their separate advantages and drawbacks: a calculation with indices may be cumbersome and sprawling; conversely an index-free notation may labour what is easily written with indices.

Kant's standing as a political thinker has been substantially enhanced in the English-speaking world since this volume went to the printers just over two decades ago. More and more scholars are willing to rank him among the leading figures in the history of political thought. John Rawls's important and much discussed treatise A Theory of Justice is indebted to him, and that has certainly made an impact. (Even legal historians and jurists have taken note of his writings, but to discuss their findings would go beyond the scope of this edition.) The secondary literature on his political thought has grown appreciably, and not only in Germany where research on Kant flourishes as always. Yet much of this writing, perhaps inevitably, covers well-tilled ground; there has been no revolution in the interpretation of Kant's political thought. Nevertheless, in view of this growing interest, it is perhaps justifiable not only to raise some new issues but also to elaborate some of the features which were mentioned only briefly, or merely alluded to, in my original introduction. Some of the following remarks are of a general nature, and others refer to specific issues. For ease of reference, they are grouped under the following headings: ‘the nature of rational discourse in polities’; “the nature of mature political judgement”; ‘property as the basis of the legal order’; ‘morality and polities’; ‘the republican constitution: representation and the separation of powers’; ‘Kant and the French Revolution’; ‘Kant's rejection of the right of rebellion’; ‘the rejection of the right of rebellion and twentieth-century totalitarianism’; ‘the limits of obedience to goverment’; ‘the Prussian context’; and ‘Kant's argument against world government’.

THE NATURE OF RATIONAL DISCOURSE IN POLITICS

To tackle the more general issues first: Hans Saner, in his challenging, and, on the whole, well-received study Kants Weg vom Krieg zum Frieden. Widerstreit und Einheit. Wege zu Kants politischem Denken (which has been translated into English with the somewhat misleading title Kant's Political Thought. Its Origins and Development), has argued that Kant's metaphors indicate a profound interest in politics from the very beginnings of his academic career; he points out that, from 1755 onwards, Kant continuously uses images of war and peace in his writings. Saner overstates a good case; for metaphors have to be interpreted with much care.

'THE PERPETUAL PEACE'

A Dutch innkeeper once put this satirical inscription on his signboard, along with the picture of a graveyard. We shall not trouble to ask whether it applies to men in general, or particularly to heads of state (who can never have enough of war), or only to the philosophers who blissfully dream of perpetual peace. The author of the present essay does, however, make one reservation in advance. The practical politician tends to look down with great complacency upon the political theorist as a mere academic. The theorist's abstract ideas, the practitioner believes, cannot endanger the state, since the state must be founded upon principles of experience ; it thus seems safe to let him fire off his whole broadside, and the worldly-wise statesman need not turn a hair. It thus follows that if the practical politician is to be consistent, he must not claim, in the event of a dispute with the theorist, to scent any danger to the state in the opinions which the theorist has randomly uttered in public. By this saving clause, the author of this essay will consider himself expressly safeguarded, in correct and proper style, against all malicious interpretation.

FIRST SECTION

Which Contains the Preliminary Articles of a Perpetual Peace Between States

1. ‘No conclusion of peace shall be considered valid as such if it was made with a secret reservation of the material for a future war.’

For if this were the case, it would be a mere truce, a suspension of hostilities, not a peace. Peace means an end to all hostilities, and to attach the adjective ‘perpetual’ to it is already suspiciously close to pleonasm. A conclusion of peace nullifies all existing reasons for a future war, even if these are not yet known to the contracting parties, and no matter how acutely and carefully they may later be pieced together out of old documents. It is possible that either party may make a mental reservation with a view to reviving its old pretensions in the future. Such reservations will not be mentioned explicitly, since both parties may simply be too exhausted to continue the war, although they may nonetheless possess sufficient ill will to seize the first favourable opportunity of attaining their end.

TRANSCENDENTAL LOGIC II, DIALECTIC, I, I: OF IDEAS IN GENERAL

A constitution allowing the greatest possible human freedom in accordance with laws which ensure that the freedom of each can co-exist with the freedom of all the others (not one designed to provide the greatest possible happiness, as this will in any case follow automatically), is at all events a necessary idea which must be made the basis not only of the first outline of a political constitution but of all laws as well. It requires that we should abstract at the outset from present hindrances, which perhaps do not arise inevitably out of human nature, but are rather occasioned by neglect of genuine ideas in the process of legislation. For there is nothing more harmful, or more unworthy of a philosopher, than the vulgar appeal to an allegedly contrary experience, which would not have existed at all if the above measures had been taken at the right time in accordance with ideas, and if crude concepts, for the very reason that they were derived from experience, had not instead vitiated every good intention. The more closely the legislation and government were made to harmonise with this idea, the rarer punishments would become, and it is thus quite rational to maintain (as Plato does) that none would be necessary at all in a perfect state. Even if the latter should never come about, the idea which sets up this maximum as an archetype, in order to bring the legal constitution of mankind nearer and nearer to its greatest possible perfection, still remains correct. For no-one can or ought to decide what the highest degree may be at which mankind may have to stop progressing, and hence how wide a gap may still of necessity remain between the idea and its execution. For this will depend on freedom, which can transcend any limit we care to impose.

A collection of rules, even of practical rules, is termed a theory if the rules concerned are envisaged as principles of a fairly general nature, and if they are abstracted from numerous conditions which, nonetheless, necessarily influence their practical application. Conversely, not all activities are called practice, but only those realisations of a particular purpose which are considered to comply with certain generally conceived principles of procedure.

It is obvious that no matter how complete the theory may be, a middle term is required between theory and practice, providing a link and a transition from one to the other. For a concept of the understanding, which contains the general rule, must be supplemented by an act of judgement whereby the practitioner distinguishes instances where the rule applies from those where it does not. And since rules cannot in turn be provided on every occasion to direct the judgement in subsuming each instance under the previous rule (for this would involve an infinité regress), theoreticians will be found who can never in all their lives become practical, since they lack judgement. There are, for example, doctors or lawyers who did well during their schooling but who do not know how to act when asked to give advice. But even where a natural talent for judgement is present, there may still be a lack of premises. In other words, the theory may be incomplete, and can perhaps be perfected only by future experiments and experiences from which the newly qualified doctor, agriculturalist or economist can and ought to abstract new rules for himself to complete his theory. It is therefore not the fault of the theory if it is of little practical use in such cases. The fault is that there is not enough theory; the person concerned ought to have learnt from experience. What he learnt from experience might well be true theory, even if he were unable to impart it to others and to expound it as a teacher in systematic general propositions, and were consequently unable to claim the title of a theoretical physician, agriculturalist or the like. Thus no-one can pretend to be practically versed in a branch of knowledge and yet treat theory with scorn, without exposing the fact that he is an ignoramus in his subject.

Kant's reviews (1785) of the first instalments of Johann Gottfried Herder's Ideas on the Philosophy of the History of Mankind (1784—91) and his Conjectures on the Beginning of Human History (1786) (which is virtually another reply to Herder's Ideas and also to his Oldest Document of Mankind1 are not specifically concerned with political questions—they do not deal with the theory of right which for Kant sets the framework within which politics ought to be conducted—but they deal with, and amplify, his conception of history, which is an integral part of his political thought. They also strikingly bring out the manner of his reasoning when interpreting history philosophically, thus revealing the boundaries between those interpretations and theories which can, in his view, be justified in relation to empirical or rational enquiries and those which must be dismissed as untenable speculations.

To appreciate these writings properly it is necessary to set them in the context of Kant's attitude to Herder (1744-1803), one of the seminal thinkers of eighteenth-century Germany. Herder's influence has been immense. His impact on literature, above all through his friendship with Goethe, whose mentor he was in his youth (they met in Strasbourg in 1770 where Goethe was a student of law and Herder had just undergone an eye operation) can hardly be overestimated. His contributions to aesthetics and poetics are also important. He proclaimed the inner coherence and historical uniqueness of works of literature which should not, in his view, be judged by an appeal to universal aesthetic criteria. He thus furthered, perhaps more than anyone else, the historical and genetic approach to the study of literature and history. He also put folk poetry on the map of literary taste. His writings engendered the belief in the nation as a cultural entity and promoted cultural and eventually political nationalism, although he himself never abandoned his eighteenth-century cosmopolitanism; that step was taken only by the German Romantics.

Herder was born in Morungen in East Prussia on 25 August 1744, the son of an impecunious school-teacher. By a stroke of good fortune he was able in 1762 to go to Königsberg (now Kaliningrad) to study at the university, earning his living as a tutor at the local grammar school, the Fridericianum.

Light-cone geometry: the key to special relativity

WE HAVE SEEN how an index notation is strikingly helpful in the development of physical formulae for flat three-dimensional space. We found it convenient to work with a fixed Cartesian coordinate system, expressing the components of vectors and tensors with respect to that system. We know, nevertheless, as a matter of principle, that the general conclusions we draw are independent of the particular coordinatization chosen for the underlying space.

We now propose to formulate special relativity in essentially the same spirit. We shall regard space-time as a flat four-dimensional continuum with coordinates xa (a = 0,1,2,3). The points of space-time are called ‘events’, and we are interested in the relations of events to one another. Our purpose here is two-fold: first, to review some aspects of special relativity pertinent to that which follows later; and second, to develop further a number of index-calculus tools which are very useful in general relativity as well as special relativity.

This volume, to the best of my knowledge, is the first in English to contain all the political writings of Kant which the author himself had published. There have been earlier translations of almost all the pieces which make up this volume; Dr Nisbet has asked me to acknowledge his debt to these, particularly to Professor John Ladd's translation of The Metaphysical Elements of Rights (The Metaphysical Elements of Justice, Indianapolis, New York, Kansas City, 1965). The aim of this volume is to introduce English-speaking readers in general and students of political theory in particular to Kant's Political writings. The bibliography in the present volume may serve as a guide for further reading. For a general introduction to Kant, the student can do no better than read Stephan Körner's Kant (Penguin Books, Harmondsworth, Middlesex, 1955), easily available in a pocket edition.

Only those writings which deal explicitly with the theory of politics and which were published by him have been included. I have omitted other essays, such as the Conjectures on the Beginning of Human History (Mutmasslicher Anfang des Menschengeschlechts), The End of All Things (Das Ende aller Dinge) and Kant's review of J. G. Herder's Ideen, which touch only marginally on politics. I have, however, included a brief but essential passage from the Critique of Pure Reason (Kritik der reinen Vernunft). In accordance with the aims of the series, I have not included any extracts, unless they form self-contained wholes. A few passages in other writings published by Kant are excluded, since they do not add anything of substance to his theory of politics. I decided to include the first part of Theory and Practice (Über den Gemeinspruch: Das mag in der Theorie richtig sein, taugt aber nicht für die Praxis), which is devoted to ethics. Since this volume does not set out to be a definitive critical edition of Kant's political writings I did not follow this precedent in the case of The Metaphysics of Morals (Die Metaphysik der Sitten) and The Contest of Faculties (Der Streit der Fakultäten). To print both works in full would inevitably have distracted attention from the main purpose of this volume.