L E C T U R E S A T U N I V E R S I T Y O F B U E N O S A I R E S 9 6 1 No report on the seventh lecture can be found in La Prensa. The lecture was summarized by Julio Castiñerías as follows: “He went on to discuss the elements of tensorial calculus, defining covariance, counter variance, etc., differentiation operations on tensors, demonstrating the existence of a fun- damental tensor that allows us to determine gravitational potentials he also defined the in- variant of volume in Euclidean space. He then delved into the study of absolute tensorial calculus and presented studies of parallel movement by eminent mathematicians Levi-Ci- vita and Ricci, going into detail about the differentiation of tensors and its application to Maxwell’s groups of equations.” Castiñerías 1925, p. 13. Professor Einstein Gave the Last Lecture in Series on Relativity Yesterday [20 April 1925] Riemann-Christoffel Tensors Gravitational Equations in Matter-Free Space In the report of Dr. Einstein’s sixth lecture, it was said that the g values expressed metric and gravitational properties that is, once these values are known, it is possible to resolve all geometrical problems that may arise and the movements that are attributed to gravity. Those magnitudes, which have an obviously physical character, are called gravitational po- tentials. In classical mechanics, solving problems of motion in a gravitational field requires, in addition to mechanical equations of motion, knowledge of the law of force between masses, a law established by Newton based on observations of planetary movement but in spite of this, it was just a hypothesis whose support would depend on how well it was reflected by experimental results. The metrics of space, which is absolute in classical mechanics, are posited by Euclidean geometry. In the theory that concerns us now, both geometry and the movement of bodies are de- termined by gravitational potentials, which, as in the previous case, should be defined by a hypothesis. It is obvious, moreover, that by studying four-dimensional types of geometries, one can reach a clearer understanding of mathematical entities that are applicable to the definition of g values. That is why Dr. Einstein, in the previous lecture, the seventh in his series, dealt with certain geometrical concepts, tensors and operations using them, which allow us to es- tablish similar entities that will satisfy the demands of all systems of equations attempting to define gravitational potentials. Dr. Einstein established the so-called Riemann-Christoffel tensor, whose decomposition is necessary and sufficient for the fundamental invariant to be transformed into a sum ds2
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