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Next: The Physical Meaning of Geometrical Propositions Relativity: The Special and General Theory...
Relativity: The Special and General Theory possible relative position of practically rigid bodies.1) Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses. Of course the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the "truth" of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation....
Relativity: The Special and General Theory three perpendiculars can be determined by a series of manipulations with rigid measuring−rods performed according to the rules and methods laid down by Euclidean geometry. In practice, the rigid surfaces which constitute the system of co−ordinates are generally not available ; furthermore, the magnitudes of the co−ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be sought in accordance with the above considerations. 3) We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for "distances;" the "distance" being represented physically by means of the convention of two marks on a rigid body....
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Relativity: The Special and General Theory every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand is indisputable. That light M as for the path B M is in reality neither a requires the same time to traverse the path A supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity." It is clear that this definition can be used to give an exact meaning not only to two events, but to as many events as we care to choose, and independently of the positions of the scenes of the events with respect to the body of reference 1) (here the railway embankment). We are thus led also to a definition of " time " in physics. For this purpose we suppose that clocks of identical construction are placed at the points A, B and C of the railway line (co−ordinate system) and that they are set in such a manner that the positions of their pointers are simultaneously (in the above sense) the same. Under these conditions we understand by the " time " of an event the reading (position of the hands) of that one of these clocks which is in the immediate vicinity (in space) of the event. In this manner a time−value is associated with every event which is essentially capable of observation. This stipulation contains a further physical hypothesis, the validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go at the same rate if they are of identical construction. Stated more exactly: When two clocks arranged at rest in different places of a reference−body are set in such a manner that a particular position of the pointers of the one clock is simultaneous (in the above sense) with the same position, of the pointers of the other clock, then identical " settings " are always simultaneous (in the sense of the above definition)....
be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework. Without committing any fundamental error, 21...
But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation x1 = wt1 B)...
This, hypothesis, which is not justifiable by any electrodynamical facts, supplies us then with that 34...
Moreover, according to this equation the time difference ”t1 of two events with respect to K1 does not in general vanish, even when the time difference ”t1 of the same events with reference to K vanishes. Pure " space−distance " of two events with respect to K results in " time−distance " of the same events with respect to K. But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four−dimensional space−time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three−dimensional 37...
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where du and dv signify very small numbers. In a similar manner we may indicate the distance (line−interval) between P and P1, as measured with a little rod, by means of the very small number ds. Then according to Gauss we have ds2 = g11du2 + 2g12dudv = g22dv2 where g11, g12, g22, are magnitudes which depend in a perfectly definite way on u and v. The magnitudes g11, g12 and g22, determine the behaviour of the rods relative to the u−curves and v−curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring−rods, but only in this case, it is possible to draw the u−curves and v−curves and to attach numbers to them, in such a manner, that we simply have : ds2 = du2 + dv2 53...
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