Blog Ekkehard Friebe

by A. K. T. Assis
Institute of Physics ‘Gleb Wataghin’, University of Campinas-UNICAMP
13083-859 Campinas, SP, BRAZIL  http://www.ifi.unicamp.br/~assis/

There are two competing formulations of time in physics. Newton defended in the Principia the utilization of absolute time which, according to him “flows equably without relation to anything external.” Leibniz, on the other hand, was against this concept and proposed relative time to replace it: “ As for my opinion, I have said more than once, that I hold space to be something merely relative, as time is; that I hold it to be an order of coexistences, as time is an order of successions.”

Leibniz ideas were accepted and developed by Ernst Mach in his book The Science of Mechanics. Mach proposed to replace Newton’s absolute time by the angle of rotation of the planets relative to the frame of fixed stars. In this work we consider the implementation of relational time and its consequences for physics. We concentrate our analysis in a single phenomenon, namely, the flattening of the Earth due to its diurnal rotation.

We consider the figure of the Earth in Newtonian mechanics. We point out some philosophical problems with this classical formulation. We then present the flattening of the Earth from the point of view of Relational Mechanics, which is a mathematical implementation of Mach’s principle utilizing Weber’s law for gravitation.

1. Introduction
Isaac Newton (1642-1727) presented two concepts of time in his book Mathematical Principles of Natural Philosophy, also known by its first Latin name, Principia, published originally in 1687, [1]: “Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.”

Thus, in his axioms or laws of motion, use only absolute time. Leibniz (1646-1716) never accepted the utilization in physics of Newton’s absolute time. He maintained that time depends on things, being the order of successive phenomena. There is a famous correspondence between Leibniz and Clarke (1675-1729), a disciple of Newton, which took place between 1715 and 1716.

Leibniz said the following in the fourth paragraph of his third letter to Clarke, [2]: “As for my opinion, I have said more than once, that I hold space to be something merely relative, as time is; that I hold it to be an order of coexistences, as time is an order of successions.”

Ernst Mach (1838-1916) also rejected the employment of absolute time in physics. His points of view as regards time were presented clearly on pp. 273, 287 and 295 of his book The Science of Mechanics, published originally in 1883. He proposed to replace the time which appears in Newton’s laws of motion by the angle of rotation of the planets with respect to the fixed stars.

For instance, on p. 295 of his book he wrote the following, [3]: “We measure time by the angle of rotation of the earth, but could measure it just as well by the angle of rotation of any other planet.” We agree with Leibniz and Mach as regards the time concept.

However, in this work we will utilize the expression “relational time” instead of “relative time.” There are two main reasons for this choice: (a) To avoid confusion with the time concept which appears in Einstein’s special and general theories of relativity. (b) To comply with Relational Mechanics, a formulation which implements Mach’s principle quantitatively, [4] and [5]. In this work we consider the implementation of relational time in physics. Our goal is to consider the consequences arising with this implementation.

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