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It would be soon shown that the general relativity theory can not hold fast to this simple physical significance of space and time. To the classical mechanics no less than to the special relativity theory, is attached an epistemological defect, which was perhaps first clearly pointed out by E. We shall illustrate it by the following example; Let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another and from all other masses that only that sort of gravitational forces are to be taken into account which the parts of any of these bodies exert upon each other.

The distance of the bodies from one another is invariable. The relative motion of the different parts of each body is not to occur. But each mass is seen to rotate by an observer at rest relative to the other mass round the connecting line of the masses with a constant angular velocity definite relative motion for both the masses.

We now ask, why is this difference between the two bodies? An answer to this question can only then be regarded as satisfactory [1] from the epistemological standpoint when the thing adduced as the cause is an observable fact of experience. The law of causality has the sense of a definite statement about the world of experience only when observable facts alone appear as causes and effects. The Newtonian mechanics does not give to this question any satisfactory answer.

The Galili ean space, which is here introduced is however only a purely imaginary cause, not an observable thing.

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The cause must thus lie outside the system. These distant masses, and their relative motion as regards the bodies under consideration are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner.

We thus arrive at an extension of the relativity postulate. Besides this momentous epistemological argument, there is also a well-known physical fact which speaks in favour of an extension of the relativity theory. Let there be a Galili ean co-ordinate system K relative to which at least in the four-dimensional region considered a mass at a sufficient distance from other masses move uniformly in a line.

This conception is feasible, because to us the experience of the existence of a field of force namely the gravitation field has shown that it possesses the remarkable property of imparting the same acceleration to all bodies. From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can "create" a gravitational field by a simple variation of the co-ordinate system.

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This conception of time and space is continually present in the mind of the physicist, though often in an unconscious way, as is clearly recognised from the role which this conception has played in physical measurements. This conception must also appear to the reader to be lying at the basis of the second consideration of the last paragraph and imparting a sense to these conceptions. But we wish to show that we are to abandon it and in general to replace it by more general conceptions in order to be able to work out thoroughly the postulate of general relativity,— the case of special relativity appearing as a limiting case when there is no gravitation.

We introduce in a space, which is free from Gravitation-field, a Galili ean Co-ordinate System K x, y, z, t and also, another system K' x', y', z', t' rotating uniformly relative to K. The origin of both the systems as well as their Z -axes might continue to coincide.


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Let us now think of measuring the circumference and the diameter of these circles, with a unit measuring rod infinitely small compared with the radius and take the quotient of both the results of measurement. This can be seen easily when we regard the whole measurement-process from the system K and remember that the rod placed on the periphery suffers a Lorentz-contraction, not however when the rod is placed along the radius.

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In order to see this we suppose that two similarly made clocks are arranged one at the centre and one at the periphery of the circle, and considered from the stationary system K. According to the well-known results of the special relativity theory it follows — as viewed from K — that the clock placed at the periphery will go slower than the second one which is at rest. The observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light will see the clock at the periphery going slower than the clock beside him.

Since he cannot allow the velocity of light to depend explicitly upon the time in the way under consideration he will interpret his observation by saying that the clock on the periphery "actually" goes slower than the clock at the origin. He cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on its position. We therefore arrive at this result.

In the general relativity theory time and space magnitudes cannot be so defined that the difference in spatial co-ordinates can be immediately measured by the unit-measuring rod, and time-like co-ordinate difference with the aid of a normal clock. The means hitherto at our disposal, for placing our co-ordinate system in the time-space continuum, in a definite way, therefore completely fail and [ ] it appears that there is no other way which will enable us to fit the co-ordinate system to the four-dimensional world in such a way, that by it we can expect to get a specially simple formulation of the laws of Nature.

So that nothing remains for us but to regard all conceivable [4] co-ordinate systems as equally suitable for the description of natural phenomena. This amounts to the following law:—. That in general, Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems, that is, which are covariant for all possible transformations. It is clear that a physics which satisfies this postulate will be unobjectionable from the standpoint of the general relativity postulate.

Because among all substitutions there are, in every case, contained those, which correspond to all relative motions of the co-ordinate system in three dimensions. This condition of general covariance which takes away the last remnants of physical objectivity from space and time, is a natural requirement, as seen from the following considerations.

All our well-substantiated space-time propositions amount to the determination of space-time coincidences. If, for example, the event consisted in the motion of material points, then, for this last case, nothing else are really observable except the encounters between two or more of these material points.

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The results of our measurements are nothing else than well-proved theorems about such coincidences of material points, of our measuring rods with other material points, coincidences between the hands of a clock, dial-marks and point-events occurring at the same position and at the same time.

The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences. As the purpose of all physical laws is to allow us to remember such coincidences, there is a priori no reason present, to prefer a certain co-ordinate system to another; i. I am not trying in this communication to deduce the general Relativity-theory as the simplest logical system possible, with a minimum of axioms. But it is my chief aim to develop the theory in such a manner that the reader perceives the psychological naturalness of the way proposed, and the fundamental assumptions appear to be most reasonable according to the light of experience.

In this sense, we shall now introduce the following supposition; that for an infinitely small four-dimensional region, the relativity theory is valid in the special sense when the axes are suitably chosen. The nature of acceleration of an infinitely small positional co-ordinate system is hereby to be so chosen, that the gravitational field does not appear; this is possible for an infinitely small region.

These coordinates have, with a given orientation of the system, an immediate physical significance in the sense of the special relativity theory when we take a rigid rod as our unit of measure , The expression.

Foundations of Special Relativity : Kinematic Axioms for Minkowski Space-Time | J. W. Schutz

Let us take ds as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region. To the line-element considered, i. We assume firstly, that in a certain four-dimensional region considered, the special relativity theory is true for some particular choice of co-ordinates. A free material point moves with reference to such a system uniformly in a straight line. At the same time, the motion of a free point-mass in the new co-ordinates, will appear as curvilinear, and not uniform, in which the law of motion, will be independent of the nature of the moving mass-points.

We can thus signify this motion as one under the influence of a gravitation field. In the general case, we can not by any suitable choice of axes, make special relativity theory valid throughout any finite region. We shall now turn our attention to these purely mathematical propositions. It will be shown that in the solution, the invariant ds , given in equation 3 plays a fundamental role, which we, following Gauss 's Theory of Surfaces, style as the "line-element".

The fundamental idea of the general covariant theory is this: — With reference to any co-ordinate system, let certain things "tensors" be defined by a number of functions of co-ordinates which are called the components of the tensor. There are now certain rules according to which the components can be calculated in a new system of co-ordinates, when these are known for the original system, and when the transformation connecting the two systems is known. The things herefrom designated as Tensors have further the property that the transformation equation of their components are linear and homogeneous; so that all the components in the new system vanish if they are all zero in the original system.

Thus a law of Nature can be formulated by putting all the components of a tensor equal to zero so that it is a general covariant equation; thus while we seek the laws of formation of the tensors, we also reach the means of establishing general Covariant laws. Contravariant Four-vector. Covariant Four-vector.

From this definition follows the law of transformation of the covariant four-vectors.

Spacetime Diagrams - Special Relativity Ch. 2

If we substitute in the right band side of the equation. Remarks on the simplification of the mode of writing the expressions. It is therefore possible, without loss of clearness, to leave off the summation sign; so that we introduce the rule: wherever the index in any term of an expression appears twice, it is to be summed over all of them except when it is not expressedly said to the contrary. The difference between the covariant and the contravariant four-vector lies in the transformation laws [ ] [ 7 and 5 ]. Both the quantities are tensors according to the above general remarks; in it lies its significance.

In accordance with Ricci and Levi-Civita , the contravariants and covariants are designated by the over and under indices.


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  • We call a thing which, with reference to any reference system is defined by 16 quantities and fulfils the transformation relation 9 , a contravariant tensor of the second rank. Not every such tensor can be built from two four-vectors, according to 8. From it, we can prove in the simplest way all laws which hold true for the tensor of the second rank defined through 9 , by proving it only for the special tensor of the type 8.

    Thus it is clear from 8 and 9 that in this sense, we can look upon contravariant four-vectors, as contravariant tensors of the first rank. Covariant tensor. By means of these transformation laws, the covariant tensor of the second rank is defined. All remarks which we have already made concerning the contravariant tensors, hold also for covariant tensors.

    Remark :— It is convenient to treat the scalar Invariant either as a contravariant or a covariant tensor of zero rank. Its transformation law is. Naturally there are mixed tensors with any number of covariant indices, and with any number of contravariant indices. The covariant and contravariant tensors can be looked upon as special cases of mixed tensors. Symmetrical tensors : — A contravariant or a covariant tensor of the second or higher rank is called symmetrical when any two components obtained by the mutual interchange of two indices are equal. It must be proved that a symmetry so defined is a property independent of the system of reference.

    It follows in fact from 9 remembering Anti-symmetrical tensor. A contravariant or covariant tensor of the 2nd, 3rd or 4th rank is called [ ] anti-symmetrical when the two components got by mutually interchanging any two indices are equal and opposite. Symmetrical tensors of ranks higher than the fourth, do not exist in a continuum of 4 dimensions. For example, we obtain the tensor T from the tensors A and B of different kinds: —.

    The proof of the tensor character of T , follows immediately from the expressions 8 , 10 or 12 , or the transformation equations 9 , 11 , 13 ; equations 8 , 10 and 12 are themselves examples of the outer multiplication of tensors of the first rank. Reduction in rank of a mixed Tensor.

    From every mixed tensor we can get a tensor which is two ranks lower, when we put an index of covariant character equal to an index of the contravariant character and [ ] sum according to these indices "Reduction". The proof that the result of reduction retains a truly tensorial character, follows either from the representation of tensor according to the generalisation of 12 in combination with 6 or out of the generalisation of Inner and mixed multiplication of Tensors.

    This consists in the combination of outer multiplication with reduction. We now prove a law, which will be often applicable for proving the tensor-character of certain quantities. From which by referring to 11 , the theorem at once follows.

    This law correspondingly holds for tensors of any rank and character. The proof is quite similar. The law can also be put in the following form. This law can easily be generalized in the case of covariant and contravariant tensors of any rank. Hence the proposition follows at once. The covariant fundamental tensor. In the invariant expression of the square of the linear element. We call it the "fundamental tensor". Afterwards we shall deduce some properties of this tensor, which will also be true for any tensor of the second rank. But the special role of the fundamental tensor in our Theory, which has its physical basis on the particularly exceptional character of gravitation makes it clear that those relations are to be developed which will be required only in the case of the fundamental tensor.

    The contravariant fundamental tensor. According to the law of multiplication of determinants, we have. Invariant of volume. According to This can never be the case; so that g can never change its sign; we would, according to the special relativity theory assume that g has a finite negative value. It is a hypothesis about the physical nature of the continuum considered, and also a pre-established rule for the choice of co-ordinates. If however - g remains positive and finite, it is clear that the choice of co-ordinates can be so made that this quantity becomes equal to one.

    We would afterwards see that such a limitation of the choice of co-ordinates would produce a significant simplification in expressions for laws of nature. It would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate.

    We do not seek those laws of nature which are covariants with regard to the transformations having the determinant 1, but we ask: what are the general covariant laws of nature? First we get the law, and then we simplify its expression by a special choice of the system of reference. Building up of new tensors with the help of the fundamental tensor. Through inner, outer and mixed multiplications of a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed. From this equation, we can in a well-known way deduce 4 total differential equations which define the geodetic line; this deduction is given here for the sake of completeness.

    Relying on the equation of the geodetic line, we can now easily deduce laws according to which new tensors can be formed from given tensors by differentiation. For this purpose, we would first establish the general covariant differential equations. We achieve this through a repeated application of the following simple law. From which follows immediately that.

    Here however we can not at once deduce the existence of any tensor. Inventory on Biblio is continually updated, but because much of our booksellers' inventory is uncommon or even one-of-a-kind, stock-outs do happen from time to time. If for any reason your order is not available to ship, you will not be charged. Your order is also backed by our In-Stock Guarantee! What makes Biblio different? Facebook Instagram Twitter. Sign In Register Help Cart. Cart items. Toggle navigation. Stock photo. Search Results Results 1 -7 of 7. Springer, Berlin: Springer, pp.

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