Elie Joseph Cartan (1869-1951) was a French mathematician and physicist, outstanding especially in the area of differential geometry and group theory. He worked in various French universities, including in the period 1912-1942 as professor at the Sorbonne in Paris. He is among the principal architects of the modern rebuilding of differential geometry, which allows the expression of its concepts and laws without the use of coordinates. Among others, he introduced the general notion of differential form in the manner used up to the present day.
A significant contribution of Cartan to the theory of time is his geometric formulation of the Newtonian theory of gravity (1923). The Cartan theory is based on four-dimensional spacetime with Newtonian absolute time (see entries on Relativity) and with Euclidean geometry in the threedimensional sections connecting simultaneous events in spacetime. Space—in contrast to time—is here not absolute, and the principle of relativity is extended to the arbitrary translatory motions of systems of reference. In Cartan’s formulation, the functioning of gravity is expressed by help of some connection with the curvature of spacetime. Consequently, spacetime with gravity is curved by course of connection, and not by course of metrics.
The Cartan connection is bound to mass density in such a way that the validity of Newton’s law of gravity is assured, and it is bound to space metrics and time metrics in such a way that the validity of Euclidean geometry of space and the existence of absolute time are assured. The necessary axioms form a somewhat complicated system (discussed in detail in the first volume of Charles W. Misner, Kip S. Thorne, and John A. Wheeler’s monograph Gravitation). They make it possible to compare the essential features of Newton’s and Einstein’s gravity theories directly. Whereas Newton’s theory is much simpler than that of Einstein in the mathematical language of the 19th century, the situation is quite opposite in modern geometric language.
The other contribution of Cartan to physics is presented by the Einstein-Cartan theory. It is a generalization of Einstein’s general theory of relativity including in its equations not only the curvature but also the torsion of spacetime related to the spin (the inner angular momentum) of matter. It cannot be excluded that the ideas of this theory will find their use in anticipated unifying theories.
See also Albert Einstein; Einstein and Newton; Isaac Newton; Space; Curvature of Spacetime; Absolute Time
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