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Roczniki Filozoficzne

Volume 13, Issue 3, 1965

Michał Heller
Pages 75-86

Time-Space Continuum of the Special Theory of Relativity

Some of the results of modem methodology of science have been applied to problems of time-space relativity so as to discover the dependences between the mathematical apparatus and the physical contents of the theory. 1. Time-space geometry. The paper discusses the basic ideas of the so-called „Erlangen programme” according to which a given geometry should be considered as the theory of invariants of a group of transformations. The transformations composing the so-called Lorentz group are a particular case of affine transformations, on the matrix of which the condition of pseudo- -orthogonality has been imposed. The theory of invariants respectively to the transformations of the Lorentz group is called pseudo-Euclidian geometry. 2. Time-space physics. If there exists a one-to-one correspondence between the elements s of ą physical manifold S and the systems m of n-numbers each, belonging to a set M of n-number systems of numbers, we say that the parametrization of an n-dimensional physical manifold has been performed. The set of equations (transformations) defining the passage from one parametrization to another may form a group of transformations. The theory of invariants in respect of the transformations of this group is a kind of geometry. In this way, the physical model (interpretation) of a given geometry is defined. The space as seen in classical mechanics is the model of the three-dimensional Euclidian space, while the space of events of the special theory of relativity is the model of the four-dimensional pseudo-Euclidian space. 3. Time-space philosophy. The so-called coordinative definitions do not define notions; they only attribute concrete physical reality to notions already defined, appearing in a given physical theory. The methods proposed by Einstein for the measure of time and the length of a body at rest and a body moving in relation to a given system of reference, as well as his definition of simultaneity of events postulate a number of coordinative definitions. It appears that 1. time measurement and the definition of the simultaneity of events distant from one another require antecedent space measurement and depend on them; 2. the measure of the length of a body moving in relation to a given system of reference depends in turn on the definition of simultaneity. For that reason, the space of events of the special theory of relativity is best parametrized through 4-number systems, three of these numbers define the position of the event in space, while the fourth „places” it in time. The continuous set of all possible values of the four coordinates constitutes the four-dimensional time-space continuum „subject” to pseudo-Euclidian geometry. Though the clearest geometrical picture of movement is obtained in the four dimensional time-space, yet the theory of relativity does not consider time and space as identical. Their distinctness is made apparent in 1. the difference of sign in the basic formula defining the length of the time-space interval, and 2. in a different physical definition of time and space by means of coordinative definitions.

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