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.g.t = 0.For this value of t we thenobtain from the first of the equations (5)x' = axTwo points of the x'-axis which are separated by the distance  x' = I when measured in theK1 system are thus separated in our instantaneous photograph by the distanceBut if the snapshot be taken from K'(t' = 0), and if we eliminate t from the equations (5), taking intoaccount the expression (6), we obtainFrom this we conclude that two points on the x-axis separated by the distance I (relative to K) willbe represented on our snapshot by the distance72Relativity: The Special and General TheoryBut from what has been said, the two snapshots must be identical; hence  x in (7) must be equal to x' in (7a), so that we obtainThe equations (6) and (7b) determine the constants a and b.By inserting the values of theseconstants in (5), we obtain the first and the fourth of the equations given in Section 11.Thus we have obtained the Lorentz transformation for events on the x-axis.It satisfies theconditionx'2 - c2t'2 = x2 - c2t2.(8a).The extension of this result, to include events which take place outside the x-axis, is obtained byretaining equations (8) and supplementing them by the relationsIn this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of lightof arbitrary direction, both for the system K and for the system K'.This may be shown in thefollowing manner.We suppose a light-signal sent out from the origin of K at the time t = 0.It will be propagatedaccording to the equationor, if we square this equation, according to the equationx2 + y2 + z2 = c2t2 = 0.(10).It is required by the law of propagation of light, in conjunction with the postulate of relativity, that thetransmission of the signal in question should take place  as judged from K1  in accordance withthe corresponding formular' = ct'73Relativity: The Special and General Theoryor,x'2 + y'2 + z'2 - c2t'2 = 0.(10a).In order that equation (10a) may be a consequence of equation (10), we must havex'2 + y'2 + z'2 - c2t'2 = à (x2 + y2 + z2 - c2t2) (11).Since equation (8a) must hold for points on the x-axis, we thus have à = I.It is easily seen that theLorentz transformation really satisfies equation (11) for à = I; for (11) is a consequence of (8a) and(9), and hence also of (8) and (9).We have thus derived the Lorentz transformation.The Lorentz transformation represented by (8) and (9) still requires to be generalised.Obviously itis immaterial whether the axes of K1 be chosen so that they are spatially parallel to those of K.It isalso not essential that the velocity of translation of K1 with respect to K should be in the direction ofthe x-axis.A simple consideration shows that we are able to construct the Lorentz transformationin this general sense from two kinds of transformations, viz.from Lorentz transformations in thespecial sense and from purely spatial transformations.which corresponds to the replacement of therectangular co-ordinate system by a new system with its axes pointing in other directions.Mathematically, we can characterise the generalised Lorentz transformation thus :It expresses x', y', x', t', in terms of linear homogeneous functions of x, y, x, t, of such a kind that therelationx'2 + y'2 + z'2 - c2t'2 = x2 + y2 + z2 - c2t2 (11a).is satisficd identically.That is to say: If we substitute their expressions in x, y, x, t, in place of x', y',x', t', on the left-hand side, then the left-hand side of (11a) agrees with the right-hand side.Next: Appendix II: Minkowski's Four Dimensional SpaceRelativity: The Special and General Theory74Relativity: The Special and General TheoryAlbert Einstein: RelativityAppendixAppendix IIMinkowski's Four-Dimensional Space ("World")(supplementary to section 17)We can characterise the Lorentz transformation still more simply if we introduce the imaginaryin place of t, as time-variable.If, in accordance with this, we insertx1 = xx2 = yx3 = zx4 =and similarly for the accented system K1, then the condition which is identically satisfied by thetransformation can be expressed thus :x1'2 + x2'2 + x3'2 + x4'2 = x12 + x22 + x32 + x42 (12).That is, by the afore-mentioned choice of " coordinates," (11a) [see the end of Appendix II] istransformed into this equation.We see from (12) that the imaginary time co-ordinate x4, enters into the condition of transformationin exactly the same way as the space co-ordinates x1, x2, x3.It is due to this fact that, according tothe theory of relativity, the " time "x4, enters into natural laws in the same form as the space coordinates x1, x2, x3.A four-dimensional continuum described by the "co-ordinates" x1, x2, x3, x4, was called "world" byMinkowski, who also termed a point-event a " world-point." From a "happening" inthree-dimensional space, physics becomes, as it were, an " existence " in the four-dimensional "world."This four-dimensional " world " bears a close similarity to the three-dimensional " space " of(Euclidean) analytical geometry.If we introduce into the latter a new Cartesian co-ordinate system(x'1, x'2, x'3) with the same origin, then x'1, x'2, x'3, are linear homogeneous functions of x1, x2,x3 which identically satisfy the equationx'12 + x'22 + x'32 = x12 + x22 + x32The analogy with (12) is a complete one.We can regard Minkowski's " world " in a formal manneras a four-dimensional Euclidean space (with an imaginary time coordinate) ; the Lorentztransformation corresponds to a " rotation " of the co-ordinate system in the fourdimensional "world."Next: The Experimental Confirmation of the General Theory of Relativity75Relativity: The Special and General TheoryRelativity: The Special and General Theory76Relativity: The Special and General TheoryAlbert Einstein: RelativityAppendixAppendix IIIThe Experimental Confirmation of the General Theory ofRelativityFrom a systematic theoretical point of view, we may imagine the process of evolution of anempirical science to be a continuous process of induction.Theories are evolved and are expressedin short compass as statements of a large number of individual observations in the form ofempirical laws, from which the general laws can be ascertained by comparison.Regarded in thisway, the development of a science bears some resemblance to the compilation of a classifiedcatalogue.It is, as it were, a purely empirical enterprise.But this point of view by no means embraces the whole of the actual process ; for it slurs over theimportant part played by intuition and deductive thought in the development of an exact science.Assoon as a science has emerged from its initial stages, theoretical advances are no longer achievedmerely by a process of arrangement.Guided by empirical data, the investigator rather develops asystem of thought which, in general, is built up logically from a small number of fundamentalassumptions, the so-called axioms.We call such a system of thought a theory.The theory findsthe justification for its existence in the fact that it correlates a large number of single observations,and it is just here that the " truth " of the theory lies [ Pobierz całość w formacie PDF ]