The relations must be so chosen that the law of the transmission of light in vacuo is satisﬁed for one and the same ray of light (and of course for every ray) with respect to K and K′. The following statement corresponds to the fundamental idea of the general principle of relativity: “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.” We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible than it is 98 the general theory of relativity when in the form of the natural extension of the special principle of relativity. It is for this reason that we have been in the habit of treating time as an independent continuum. Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab. 2, the x-axes of both systems permanently coincide. When there are no objects in the box, its space appears to be “empty”. 'Theory of Relativity' is a generalized term used for two different classes of theories given by Albert Einstein namely special relativity and general relativity. From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body. This can be interpreted in regard to our present knowledge only in the sense of Doppler’s principle, as an expansive motion of the system of stars in the large—as required, according to Friedman, by the ﬁeld equations of gravitation. The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the inﬁniteness of our space without coming into conﬂict with the laws of thought or with experience (Riemann, Helmholtz). They will call the line joining the free ends of these lines a “circle.” For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured 111 112 considerations on the universe with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value π, which is independent of the diameter of the circle. It requires the idea of the ﬁeld as the representative of reality, in combination with the general principle of relativity, to show the true kernel of Descartes’ idea; there exists no space “empty of ﬁeld”. But the experiment gave a negative result—a fact very perplexing to physicists. The following statements hold generally: Every physical description resolves itself into a number of statements, each of which refers to the space-time coincidence of two events A and B. Our partners will collect data and use cookies for ad personalization and measurement. However, already in the ’twenties, the Russian mathematician Friedman showed that a diﬀerent hypothesis was natural from a purely theoretical point of view. In spite of all these beautiful successes the state of the theory was not yet wholly satisfactory, and for the following reasons. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. Thus this formation of concepts already presupposes the concept of material objects (e.g. . The point will be both to recall what SR is all about, and to introduce tensors and related However, the Gauss treatment for ds2 which we have given above is not always possible. from Lorentz transformations in the special sense and from purely spatial transformations, which corresponds to the replacement of the rectangular co-ordinate system by a new system with its axes pointing in other directions. We assign to every point of the continuum (event) four numbers, x1, x2, x3, x4 (co-ordinates), which have not the least direct physical signiﬁcance, but only serve the purpose of numbering the points of the continuum in a deﬁnite but arbitrary manner. In course of time their velocity will reach unheard-of values—provided that we are viewing all this from another reference-body which is not being pulled with a rope. It is still less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into inﬁnite space, never to return, and without ever again coming into interaction with other objects of nature. Since equation (8a) must hold for points on the x-axis, we thus have σ=1. It was at all times clear that, from the point of view of the idea it conveys to us, every motion must be considered only as a relative motion. We start out from an inertial system S1 whose space is, from the physical point of view, empty. Some of the techniques listed in Relativity: The Special and the General Theory may require a sound knowledge of Hypnosis, users are advised to either leave those sections or must have a basic understanding of the subject before practicing them. The only reason why we must wait for a total eclipse is because at every other time the atmosphere is so strongly illuminated by the light from the sun that the stars situated near the sun’s disc are invisible. But a simple and apparently quite reliable consideration seems to suggest that, for the present at any rate, there is little hope of success in such an attempt. This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational ﬁeld, at least if we attribute the length 1 to the rod in all positions and in every orientation. We call such a system of thought a theory. If, then, the observer ﬁrst measures the circumference of the disc with his measuring-rod and then the diameter of the disc, on dividing the one by the other, he will not obtain as quotient the familiar number π = 3.14 . The relative discrepancies to be expected between the stellar photographs obtained during the eclipse and the comparison photographs amounted to a few hundredths of a millimetre only. After some time perhaps the following answer would occur to you: “The signiﬁcance of the statement is clear in itself and needs no further explanation; of course it would require some consideration if I were to be commissioned to determine by observations whether in the actual case the two events took place simultaneously or not.” I cannot be satisﬁed with this answer for the following reason. Namely, the original ﬁeld equations admit a solution in which the “world radius” depends on time (expanding space). 1 relativity and problem of space type (1), but absolutely nothing, and also no “topological space”. What is the reason for this preference? If we call A′ and B′ the two points on the train whose distance 1 e.g. In the same way, any such lines in our continuum correspond to many points in motion. the relativity of distance apart is required, then both of these points are moving with the velocity v along the embankment. The first edition of the novel was published in 1916, and was written by Albert Einstein. This non-rigid reference-body, which might appropriately be termed a “reference-mollusc,” is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. 55 17 MINKOWSKI’S FOURDIMENSIONAL SPACE The non-mathematician is seized by a mysterious shuddering when he hears of “four-dimensional” things, by a feeling not unlike that awakened by thoughts of the occult. Nevertheless, there are two general facts which at the outset speak very much in favour of the validity of the principle of relativity. A similar diﬃculty presents itself when we attempt to apply our earlier deﬁnition of simultaneity in such a case, but I do not wish to go any farther into this question. But our construction of squares must necessarily come into disorder during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do not. 2 is too small in c comparison with the mass m, which was present before the alteration of the energy. The form there used, “All bodies of reference K, K′, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion,” cannot be maintained, because the use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. Under these conditions we understand by the “time” of an event the reading (position of the hands) of that of one of these clocks which is in the immediate vicinity (in space) of the event. Mean displacements of lines towards the less refrangible end of the spectrum are certainly revealed by statistical investigations of the ﬁxed stars; but up to the present the examination of the experimental confirmation available data does not allow of any deﬁnite decision being arrived at, as to whether or not these displacements are to be referred in reality to the eﬀect of gravitation. By this is meant a theory which describes exhaustively physical reality, including fourdimensional space, by a ﬁeld. geometers 140 geometrical ideas 4, 5 propositions 1 truth of 1–5 geometry, empirical content of 146 gravitation 65, 70, 78, 101, 157 generalised theory of 157 gravitational equations 156 ﬁeld 65, 68, 75, 78, 94, 98, 99, 100, 115, 153, 154 potential of 132–5 mass 66, 70, 101, 153 Grebe 134 group-density of stars 107 “Happening” 152 heat conduction 146 Helmholtz 110 heuristic value of relativity 43 Hubble 137 Hume 145 induction 125 inertia 66, 146, 152 inertial mass 48, 66, 70, 100, 102, 153 space 151 system 146, 151, 153, 156 instantaneous photograph (snapshot) 120 intensity of gravitational ﬁeld 108 interference of light 148 intuition 126 invariance of natural laws 150 ions 146 Kant 141 Kepler 128 kinetic energy 46, 100 lattice 111 law of gravitation 157 inertia 12, 63, 64, 98, 146 laws of Galilei-Newton 15 Nature 52, 72, 99, 150, 154, 155 lengths, in Euclidean geometry 154 Leverrier 102, 129 light, as a wave ﬁeld 148 163 164 index light-signal 35, 117, 121 light-stimulus 35 light waves 149 limiting velocity (c) 38, 39 lines of force 108 Lorentz, H. A. The translated version of this book is available in Spanish, English, Chinese, Russian, Hindi, Bengali, Arabic, Portuguese, Indonesian / Malaysian, French, Japanese, German and many others for free download. Newton himself and his most critical contemporaries felt it to be disturbing that one had to ascribe physical reality both to space itself as well as to its state of motion; but there was at that time no other alternative, if one wished to ascribe to mechanics a clear meaning. 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. Let us consider how these diﬃculties arise. Sections 20 and 23). At every point there exists at any time a velocity, which is quantitatively described by its three “components” with respect to the axes of a co-ordinate system (vector).

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