Absolute Differential Calculus Research Articles

Some remarks on the history of Ricci’s absolute differential calculus

The article offers a general account of the genesis of the absolute differential calculus (ADC), paying special attention to its links with the history of differential geometry. In relatively recent times, several historians have described the development of the ADC as a direct outgrowth either of the theory of algebraic and differential invariants or as a product of analytical investigations, thus minimizing the role of Riemann’s geometry in the process leading to its discovery. Our principal aim consists in challenging this historiographical tenet and analyzing the intimate connection between the development of Riemannian geometry and the birth of tensor calculus.

Logunov and Mestvirishvil Disprove "General Relativity"

Based on the various documents, 1989-2002, through the original texts, in addition to the author's contributions, this paper presents the refutation of the mathematicians and physicists A. Logunov and M. Mestvirishvil of A. Einstein's "general relativity", from the relativistic theory of gravitation of these authors, who applying the fundamental principle of the science of physics of the conservation of the energy-momentum and using absolute differential calculus they rigorously perform their mathematical tests. It is conclusively shown that, from the Einstein-Grossman-Hilbert equations, gravity is absurdly a metric field devoid of physical reality unlike all other fields in nature that are material fields, interrupting the chain of transformations between the different existing fields. Also, in Einstein's theory the proved "inertial mass" equal to gravitational mass has no physical meaning. Therefore, "general relativity" does not obey the correspondence principle with Newton's gravity.

CR Embedded Submanifolds of CR Manifolds

We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Cartan (or tractor) connections via a CR Gauss formula. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more naive methods. We conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.

Traditions in Collision: The Emergence of Logical Empiricism between the Riemannian and Helmholtzian Traditions

This article attempts to explain the emergence of the logical empiricist philosophy of space and time as a collision of mathematical traditions. The historical development of the “Riemannian” and “Helmholtzian” traditions in nineteenth-century mathematics is investigated. Whereas Helmholtz’s insistence on rigid bodies in geometry was developed group theoretically by Lie and philosophically by Poincaré, Riemann’s Habilitationsvotrag triggered Christoffel’s and Lipschitz’s work on quadratic differential forms, paving the way to Ricci’s absolute differential calculus. The transition from special to general relativity is briefly sketched as a process of escaping from the Helmholtzian tradition and entering the Riemannian one. Early logical empiricist conventionalism, it is argued, emerges as the failed attempt to interpret Einstein’s reflections on rods and clocks in general relativity through the conceptual resources of the Helmholtzian tradition. Einstein’s epistemology of geometry should, in spite of his rhetorical appeal to Helmholtz and Poincaré, be understood in the wake the Riemannian tradition and of its aftermath in the work of Levi-Civita, Weyl, Eddington, and others.

An Italian calculus for general relativity

The mathematical tool that allowed Albert Einstein to develop the general theory of relativity, concluded in 1915 and now considered his greatest scientific masterpiece, was tensor calculus, then known as absolute differential calculus. This algorithmic system had been created at the end of the nineteenth century by the Italian mathematician Gregorio Ricci-Curbastro who, together with his most brilliant pupil, Tullio Levi-Civita, had applied it to several problems in geometry and mathematical physics. However, before the birth of the great Einsteinian theory, the new calculus was not been sufficiently appreciated by the contemporaries, as its complexity appeared to render it superfluous. It asserted itself only with the triumph of general relativity, since it proved to be indispensable to formulate the theory and had shown its power and elegance in its construction.

On the role of virtual work in Levi-Civita’s parallel transport

The current literature on history of science reports that Levi-Civita’s parallel transport was motivated by his attempt to provide the covariant derivative of the absolute differential calculus with a geometrical interpretation (For instance, see Scholz in The intersection of history and mathematics, Birkhauser, Basel, pp 203–230, 1994, Sect. 4). Levi-Civita’s memoir on the subject was explicitly aimed at simplifying the geometrical computation of the curvature of a Riemannian manifold. In the present paper, we wish to point out the possible role implicitly played by the principle of virtual work in Levi-Civita’s conceptual reasoning to formulate parallel transport.

The role of virtual work in Levi‐Civita's parallel transport

AbstractAccording to current history of science, Levi‐Civita introduced parallel transport solely to give a geometrical interpretation to the covariant derivative of absolute differential calculus. Levi‐Civita, however, searched a simple computation of the curvature of a Riemannian manifold, basing on notions of the Italian school of mathematical physics of his time: holonomic constraints, virtual displacements and work, which so have a remarkable, if not dominant, role in the origin of parallel transport. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

크리스토펠, 리치, 레비-치비타에 의한 19세기 중반부터 20세기 초반까지 미분기하학의 발전

Contemporary differential geometry owes much to the theory of connections on the bundles over manifolds. In this paper, following the work of Gauss on surfaces in 3 dimensional space and the work of Riemann on the curvature tensors on general n dimensional Riemannian manifolds, we will investigate how differential geometry had been developed from mid 19th century to early 20th century through lives and mathematical works of Christoffel, Ricci-Curbastro and LeviCivita. Christoffel coined the Christoffel symbol and Ricci used the Christoffel symbol to define the notion of covariant derivative. Levi-Civita completed the theory of absolute differential calculus with Ricci and discovered geometric meaning of covariant derivative as parallel transport.

Outline of a dynamical inferential conception of the application of mathematics

We outline a framework for analyzing episodes from the history of science in which the application of mathematics plays a constitutive role in the conceptual development of empirical sciences. Our starting point is the inferential conception of the application of mathematics, recently advanced by Bueno and Colyvan (2011). We identify and discuss some systematic problems of this approach. We propose refinements of the inferential conception based on theoretical considerations and on the basis of a historical case study. We demonstrate the usefulness of the refined, dynamical inferential conception using the well-researched example of the genesis of general relativity. Specifically, we look at the collaboration of the physicist Einstein and the mathematician Grossmann in the years 1912–1913, which resulted in the jointly published “Outline of a Generalized Theory of Relativity and a Theory of Gravitation,” a precursor theory of the final theory of general relativity. In this episode, independently developed mathematical theories, the theory of differential invariants and the absolute differential calculus, were applied in the process of finding a relativistic theory of gravitation. The dynamical inferential conception not only provides a natural framework to describe and analyze this episode, but it also generates new questions and insights. We comment on the mathematical tradition on which Grossmann drew, and on his own contributions to mathematical theorizing. The dynamical inferential conception allows us to identify both the role of heuristics and of mathematical resources as well as the systematic role of problems and mistakes in the reconstruction of episodes of conceptual innovation and theory change.

Erich Kretschmann as a proto-logical-empiricist: Adventures and misadventures of the point-coincidence argument

The present paper attempts to show that a 1915 article by Erich Kretschmann must be credited not only for being the source of Einstein's point-coincidence, but also for having anticipated the main lines of the logical-empiricist interpretation of general relativity. Whereas Kretschmann was inspired by the work of Mach and Poincaré, Einstein inserted Kretschmann's point-coincidence parlance into the context of Ricci and Levi-Civita's absolute differential calculus. Kretschmann himself realized this and turned the point-coincidence argument against Einstein in his second and more famous 1918 paper. While Einstein had taken nothing from Kretschmann but the expression “point-coincidences”, the logical empiricists, however, instinctively dragged along with it the entire apparatus of Kretschmann's conventionalism. Disappointingly, in their interpretation of general relativity, the logical empiricists unwittingly replicated some epistemological remarks Kretschmann had written before general relativity even existed.

Ricci calculus package in REDUCE

Ricci calculus is performed in the framework of the computer algebra system reduce by the package being presented. The program called riccir is specialized in the requirements of general relativity.

General relativity and general Lorentz-covariance

The principle of general relativity means the principle of generalLorentz-covariance of the physical equations in the language of tetrads and metrical spinors. A generalLorentz-Covariant calculus and the generalLorentz-covariant generalisations of the Ricci calculus and of the spinor calculus are given. The generalLorentz-covariant representation implies theEinstein principle of space-time covariance and allows the geometrisation of gravitational fields according toEinstein's principle of equivalence.

Rotating-field theory and general analysis of synchronous and induction machines

The two-reaction theory of synchronous machines as developed by Blondel, Doherty, Nickle, Park and others is well known. The present paper attempts to develop the rotating-field theory of synchronous and induction machines as an important alternative to the two-reaction theory. The general analysis includes analysis of rotating machines under steady-state, transient-state and hunting conditions. Rotating reference frames are introduced. The new components, known as the forward and backward components, or f and b components, are then simply correlated to the direct and quadrature components, or d and q components. By use of the f and b components, it is shown that any external network and transmission line with lumped or distributed constants can be connected to a synchronous machine. This forms the basis of interconnection between synchronous and induction machines. A steady-state vector diagram of a salient-pole synchronous machine is given. Transient solution of short-circuit current and torque follows. An equivalent circuit for the same machine with direct- and quadrature-axis excitation is derived, and a 6-terminal network is developed for transient studies. The induction machine is similarly analysed with respect to its own rotor or with respect to a synchronously rotating reference frame. Interconnection of synchronous and induction machines is then possible for steady-state as well as transient studies. The hunting circuit involving one synchronous machine and a load is developed, based on Kron's uniformly rotating reference frame instead of the Blondel-Park reference frame. Hunting circuits for two rotating machines are developed, based on simple relations in absolute differential calculus. A number of instantaneous torque expressions are developed from the tensorial point of view. New expressions are given in terms of armature currents and flux linkages or their components.

Applications of the absolute differential calculus

Applications of the absolute differential calculus

Origins of the General Relativity Theory

THE Gibson foundation lecture, delivered at the University of Glasgow by Prof. A. Einstein on June 20, consisted of a first-hand account of the mental struggles that precede the establishment of new fundamental ideas in science. The special relativity theory showed that velocity was purely relative, and from one point of view the same should be true of acceleration, yet physics seemed to show evidence to the contrary. The attempt to include gravitation in the special theory had to be abandoned. Prof. Einstein came to the conclusion that the key to the real understanding of inertia and gravitation was the experimental result that all bodies in a gravitational field were subject to the same acceleration. From 1908 until 1911 he endeavoured to apply this, but a dilemma arose from which he did not escape until 1912, when he conjectured that the space-time continuum had a Riemann metric. The development of this hypothesis by the aid of the absolute differential calculus of Ricci and Levi-Civita kept Einstein and Grossmann busy from 1912 until 1914. They found the correct gravitational equations, but failed to recognise their physical validity, and thus wasted two years of hard work. Finally Einstein “returned penitentially to the Riemann curvature”. “Our final results appear almost self-evident … but the years of searching in the dark for a truth that one feels but cannot express; the intense desire, and the alternations of confidence and misgiving, until one breaks through to clarity and understanding, are only known to him who has himself experienced it.”

On Some Applications of the Absolute Differential Calculus to Physics

By means of the tensor analysis, coupled with the first fundamental theorem on the invariants of orthogonal transformations, expressions, independent of the particular type of coordinates used are derived for Hooke's law in elasticity theory and for the constitutive relations in electromagnetic theory. It is shown that these two laws are but special cases of a more general physical law connecting, in a linear way, the components of two tensors having physical significance. It is to be noted that in the case of Hooke's law for a transversely isotropic medium, six independent coefficients of elasticity are involved instead of five found in the corresponding classical expressions.

Book Review: Application of the Absolute Differential Calculus

Book Review: Application of the Absolute Differential Calculus

A complete system of tensors of linear homogeneous second-order differential equations

In an article appearing in the American Mathematical Monthlyt Lane called attention to the need in projective differential geometry of an absolute differential calculus for forms of higher degree than the quadratic. Such a calculus for forms of arbitrary degree had been published by the writer4 and it seemed that some adaptation to the theory of projective differential geometry would be desirable. However, it appeared more satisfactory to base a tensor theory directly upon the differential equations. Inasmuch as various systems of differential equations are used in the projective theory, a separate tensor theory will need to be constructed for each type. In this paper a study has been made of the differential equations upon which the ruled surface theory is based. The remarkable simplicity which has been introduced by tensor methods in the theory of Riemannian geometry and of algebraic invariants appears also in this theory. The Wilczynski theory, while general, has had the blemish of containing awkward unsymmetrical sets of equations involving cumbersome numerical constants. And with the improvement in notation there is an accompanying advantage in method, the simple formal tensor processes replacing the earlier ingenious methods. Some geometric interpretations of the tensors and invariants are given here in which the dependent variables yi(i = 1, * * * , n) are interpreted as nonhomogeneous co6rdinates of a point. This is in contrast to the convention peculiar to projective differential geometry of choosing a fundamental set of 2n independent solutions (y', * * , y), o =1, * * , 2n, and interpreting these as n points (y * * yn)), i 1, .. , n, in a homogeneous space of 2n-1 dimensions. An interpretation of our tensors in this latter space will constitute a generalization of Wilczynski's ruled surface theory.

Some differential and algebraic consequences of the Einstein field equations

Introduction. If a set of four directions in a Riemannian four-dimensional space, V4, is orthogonal, then the ds2 can be expressed in terms of their sixteen parameters, hka(XO, X1l x2, X3), as in Einstein's recent papers. The first purpose of this paper is to set up sixteen invariant linear firstorder partial differential equations in these parameters (?2). The solutions of these equations include all solutions for empty space of the Einstein field equations of 1917. There is a restriction which excludes some special cases. In addition to the ka, these sixteen equations contain four linear combinations of the components of the curvature tensor. These four combinations are to be taken as independent variables, Xk. Since only alternating tensors appear it is convenient to use Cartan's notationt for symbolic differential forms and for their derivatives and products. Covariant differentiation in the sense of the absolute differential calculus is not used, except in ?4. The components of the curvature tensor may be taken as coefficients in the equation of a quadratic line complex in a three-dimensional projective space,P34 ?. The directions hka correspond to the vertices of a tetrahedron in P3. Bivector and simple bivector correspond to linear complex and special complex. Where there is no danger of misunderstanding, the language of V4 will be used interchangeably with that of P3. The second purpose of this paper is the application of some of the theory of quadratic complexes to the study of the curvatures in V4.? The lines which lie in a plane in P3 and which belong to the quadratic complex are tangent to a conic. The envelope of planes for which this conic

The Absolute Differental at Calculus (Calculus of Tensors)

THE “Lezioni di calcolo differenziale assoluto” by Prof. Levi-Civita were published in Italian in 1925. This account of the foundations of the absolute differential calculus has now been translated into English. Miss Long has thus conferred a benefit upon British mathematicians for which they will be very grateful. The subject is of fundamental importance in the applications of mathematical methods to modern theoretical physics; the treatment is as masterly as can be expected, coming as it does from the lecture notes of one of the outNo. standing exponents of the subject and one of the outstanding contributors to its development; the translation is excellent, combining, as Prof. Levi- Civitaa says in the preface, “scrupulous respect for the text with its effective adaptation to the spirit of the English language.” The Absolute Differental at Calculus (Calculus of Tensors). By Prof. Tullio Levi-Civita. Edited by Dr. Enrico persico. Authorised translation by Miss M. Lorg. Pp. xvi + 450. (London and Glasgow: Blackie and Son, Ltd., 1927.) 21s. net.