Albert Einstein Relativity (2)

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.The Gauss co-ordinate system has to take theplace of the body of reference.The following statement corresponds to the fundamental idea of thegeneral principle of relativity: "All Gaussian co-ordinate systems are essentially equivalent for theformulation of the general laws of nature."We can state this general principle of relativity in still another form, which renders it yet more clearlyintelligible than it is when in the form of the natural extension of the special principle of relativity.According to the special theory of relativity, the equations which express the general laws of naturepass over into equations of the same form when, by making use of the Lorentz transformation, wereplace the space-time variables x, y, z, t, of a (Galileian) reference-body K by the space-timevariables x1, y1, z1, t1, of a new reference-body K1.According to the general theory of relativity, onthe other hand, by application of arbitrary substitutions of the Gauss variables x1, x2, x3, x4, theequations must pass over into equations of the same form; for every transformation (not only theLorentz transformation) corresponds to the transition of one Gauss co-ordinate system intoanother.If we desire to adhere to our "old-time" three-dimensional view of things, then we can characterisethe development which is being undergone by the fundamental idea of the general theory ofrelativity as follows : The special theory of relativity has reference to Galileian domains, i.e.to thosein which no gravitational field exists.In this connection a Galileian reference-body serves as bodyof reference, i.e.a rigid body the state of motion of which is so chosen that the Galileian law of theuniform rectilinear motion of "isolated" material points holds relatively to it.Certain considerations suggest that we should refer the same Galileian domains tonon-Galileian reference-bodies also.A gravitational field of a special kind is then present withrespect to these bodies (cf.Sections 20 and 23).In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus thefictitious rigid body of reference is of no avail in the general theory of relativity.The motion of clocksis also influenced by gravitational fields, and in such a way that a physical definition of time which ismade directly with the aid of clocks has by no means the same degree of plausibility as in thespecial theory of relativity.For this reason non-rigid reference-bodies are used, which are as a whole not only moving in anyway whatsoever, but which also suffer alterations in form ad lib.during their motion.Clocks, forwhich the law of motion is of any kind, however irregular, serve for the definition of time.We haveto imagine each of these clocks fixed at a point on the non-rigid reference-body.These clockssatisfy only the one condition, that the "readings" which are observed simultaneously on adjacentclocks (in space) differ from each other by an indefinitely small amount.This non-rigid59Relativity: The Special and General Theoryreference-body, which might appropriately be termed a "reference-mollusc", is in the mainequivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily.That which givesthe "mollusc" a certain comprehensibility as compared with the Gauss co-ordinate system is the(really unjustified) formal retention of the separate existence of the space co-ordinates as opposedto the time co-ordinate.Every point on the mollusc is treated as a space-point, and every materialpoint which is at rest relatively to it as at rest, so long as the mollusc is considered asreference-body.The general principle of relativity requires that all these molluscs can be used asreference-bodies with equal right and equal success in the formulation of the general laws ofnature; the laws themselves must be quite independent of the choice of mollusc.The great power possessed by the general principle of relativity lies in the comprehensive limitationwhich is imposed on the laws of nature in consequence of what we have seen above.Next: The Solution of the Problem of Gravitation on the Basis of the General Principle of RelativityRelativity: The Special and General Theory60Relativity: The Special and General TheoryAlbert Einstein: RelativityPart II: The General Theory of RelativityThe Solution of the Problem of Gravitation on the Basis of the GeneralPrinciple of RelativityIf the reader has followed all our previous considerations, he will have no further difficulty inunderstanding the methods leading to the solution of the problem of gravitation.We start off on a consideration of a Galileian domain, i.e.a domain in which there is nogravitational field relative to the Galileian reference-body K.The behaviour of measuring-rods andclocks with reference to K is known from the special theory of relativity, likewise the behaviour of"isolated" material points; the latter move uniformly and in straight lines.Now let us refer this domain to a random Gauss coordinate system or to a "mollusc" asreference-body K1.Then with respect to K1 there is a gravitational field G (of a particular kind).Welearn the behaviour of measuring-rods and clocks and also of freely-moving material points withreference to K1 simply by mathematical transformation.We interpret this behaviour as thebehaviour of measuring-rods, docks and material points tinder the influence of the gravitationalfield G.Hereupon we introduce a hypothesis: that the influence of the gravitational field onmeasuringrods, clocks and freely-moving material points continues to take place according to thesame laws, even in the case where the prevailing gravitational field is not derivable from theGalfleian special care, simply by means of a transformation of co-ordinates.The next step is to investigate the space-time behaviour of the gravitational field G, which wasderived from the Galileian special case simply by transformation of the coordinates.This behaviouris formulated in a law, which is always valid, no matter how the reference-body (mollusc) used inthe description may be chosen.This law is not yet the general law of the gravitational field, since the gravitational field underconsideration is of a special kind.In order to find out the general law-of-field of gravitation we stillrequire to obtain a generalisation of the law as found above.This can be obtained without caprice,however, by taking into consideration the following demands:(a) The required generalisation must likewise satisfy the general postulate of relativity.(b) If there is any matter in the domain under consideration, only its inertial mass, and thusaccording to Section 15 only its energy is of importance for its etfect in exciting a field.(c) Gravitational field and matter together must satisfy the law of the conservation of energy (and ofimpulse).Finally, the general principle of relativity permits us to determine the influence of the gravitationalfield on the course of all those processes which take place according to known laws when agravitational field is absent i.e.which have already been fitted into the frame of the special theory ofrelativity [ Pobierz całość w formacie PDF ]

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