Tensor

“ We assume, that in any field of physics there is one of those world tensors, whose components have the corresponding meaning, as the components of the mentioned electrodynamic tensor; i.e., that for example also in dynamics, the energy density is defined by , the momentum density and the energy current according to (1a) by etc., while is related to the elastic stresses. That we assume tensor as being symmetric, can appear to be arbitrary at this place; because also the divergence of an unsymmetrical world tensor would be a four-vector. ”

“ But our investigations of the special theory of relativity have shown that in place of the scalar density of matter we have the tensor of energy per unit volume. In the latter is included not only the tensor of the energy of ponderable matter, but also that of the electromagnetic energy. We have seen, indeed, that in a more complete analysis the energy tensor can be regarded only as a provisional means of representing matter. In reality, matter consists of electrically charged particles, and is to be regarded itself as a part, in fact, the principal part, of the electromagnetic field. ”

“ All these deductions obtained mathematically are in a certain sense self-evident physically when we recall that a tensor, a vector and an invariant represent magnitudes to which an objective existence may be conceded. It is obvious, indeed, that if two such magnitudes are equal to each other at a point in space, a mere change of our co-ordinate system can never disturb this intrinsic equality. In the same way, if a concrete entity, such as a tensor, has vanished at a point in space, a mere change of our co-ordinate system is incapable of bringing it into existence. ”

“ We must show that this equation holds independently of the special choice of the Cartesian system of co-ordinates. is a vector, and therefore is also a vector, is a tensor of rank 2, is a tensor of rank 3. The second term on the left results from contraction in the indices . The vector character of the second term on the right is obvious. In order that the first term on the right may also be a vector it is necessary for to be a tensor. Then by differentiation and contraction results, and is therefore a vector, as it also is after multiplication by the reciprocal scalar . ”

“ Every tensor, when multiplied by , changes into a tensor density. We employ capital Gothic letters for tensor densities. ↑ The direction vector at a neighbouring point of the curve results, by a parallel displacement along the line element from the direction vector of each point considered. ”

“ In modern theory—I allude to the theory of relativity—one has found good reasons for combining into one unity, which we call the stress-energy-tensor, all these quantities of which I have spoken, viz. the energy, the flow of energy, Maxwell's stresses and the electromagnetic momentum. In Einstein's theory of gravitation this tensor determines the gravitation field that is produced by an electromagnetic system and in virtue of which such a system has an influence on the motion of material particles, unfortunately much too small to be observed. ”

“ Action and the Einsteinian interval are representative of invariants, while the majority of types of space-time curvature, the electromagnetic-field tensors, and the matter tensor are representative of tensors. It is because gravitational force and potential energy are not represented by tensors that we can attribute no absolute significance to them; and as a result, potential energy being discarded as a relative, conservation of energy falls to the ground. ”

“ The special relativity theory has led to the conception that the inertial mass is no other than energy. It can also be fully expressed mathematically by a symmetrical tensor of the second rank, the energy-tensor. We have therefore to introduce in our generalised theory an energy-tensor associated with matter, which like the energy components of the gravitation-field (equations 49, and 50) have a mixed character but which however can be connected with symmetrical covariant tensors. ”

“ In order to get nearer to the behaviour of real matter we must add to the energy tensor a term which corresponds to the pressures. The simplest case is that of a perfect fluid in which the pressure is determined by a scalar . ”

“ When we change frames, the tensor remains unaffected in the objective space-time world, but its six components in the space-and-time frame we happen to occupy are subjected to definite variations when our frame is changed; and for this reason fields that appear purely electrical when we are moving with a certain relative velocity will appear electromagnetic when our relative velocity is modified. ”

“ If we substitute the right-hand side for the left-hand side in the former integral, and put outside the sign of integration, we obtain, We see, therefore, that the generalized conception of the energy tensor is in agreement with our former result. ”

“ If Bμ and Cν are any two vectors, and if for every choice of them the inner product Aμν Bμ Cν is a scalar, then Aμν is a co-variant tensor. The last law holds even when there is the more special formulation, that with any arbitrary choice of the four-vector Bμ alone the scalar product Aμν Bμ Bν is a scalar, in which case we have the additional condition that Aμν satisfies the symmetry condition. ”

“ 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 (13) .
Inner and mixed multiplication of Tensors. This consists in the combination of outer multiplication with reduction. Examples: — From the covariant tensor of the second rank and the contravariant tensor of the first rank we get by outer multiplication the mixed tensor
Through reduction according to indices , the covariant four vector
is generated. ”

“ We must realise that the tensor , whose components represent those physical magnitudes which our senses perceive and our instruments measure, can at best be considered to portray effects observed only in a macroscopic way. Phenomena such as radioactive explosions and all we have learnt indirectly from the exploration of the atom go to show that matter in the final analysis is constituted by whirling electrons around which intense electromagnetic fields are present. ”

“ From these formulæ we shall deduce some others which are of interest for the physical applications of the theory. In case is skew-symmetrical, we obtain the tensor
which is skew-symmetrical in all pairs of indices, by cyclic interchange and addition.
If, in (78) , we replace by the fundamental tensor, , then the right-hand side vanishes identically; an analogous statement holds for (80) with respect to ; that is, the co-variant derivatives of the fundamental tensor vanish. That this must be so we see directly in the local system of co-ordinates. ”

“ Recently I find now, that one is able to dispense with hypothesis concerning the energy tensor of matter, if one fills in the energy tensor of matter into the field equations in a somehow different way than it was done in my two earlier reports. ”

“ This, censor, censer, gensor, chencer, and other variations, is the most usual form of the word, but occasionally it is found as tenser, tensor, tensur, and tensure. Tenser and tensor are used at Shrewsbury; at Worcester the same word appears as tensure or tensar (English Gilds, pp. 382, 394) .
It is difficult to say whether or no tenser is a confusion of censer. Etymologically the words seem akin, cense being a tax or toll (cess) , and tensare meaning to lay under toll or tribute. In the Iter of 1164 enquiry is directed to be made “de prisis et tenseriis omnium ballivorum domini regis ... ”

“ To get an idea of the significance of these tensor triples, we first consider the mechanical momentum and its corresponding mechanical momentum flux. The total momentum flux into the interior of a room, i.e. the increase in momentum in the interior per unit time, is equal to the resulting mechanical force which acts on the entire mass located within that space. ”

“ The only faculties of significance remaining in his neutral somatic state were those which were absolutely necessary for civilized life—levitation to avoid being disturbed by gravity, the focus for personal privacy, the construction of food. Communication was not one of those, so the itch would just have to remain. Tensor contemplated an eternity with the medulary itch without the slightest concern. ”

“ It is only the circumstance that we have not sufficient knowledge of the electromagnetic field of concentrated charges that compels us, provisionally, to leave undetermined in presenting the theory, the true form of this tensor. ”

“ After a moment of speculative silence, he said, "You haven't told me what you savages consider yourselves."
"Don't call me a savage, you naked beast," she snapped back.
"I beg your pardon," he murmured politely. "Merely a semantic difficulty. I'm sure. I assume that you consider yourselves human beings, then. Where do you come from?"
"Earth—the third planet."
"I see. And you used mechanical devices such as this little metal egg to get here. Most curious." Tensor contemplated the thought with great interest, for obviously they used mechanical skill to compensate for lack of direct control. ”

“ However, we may assume that the Einsteinian tensor laws, in the form in which they were given by Einstein, are in harmony with some particular gauge-system; and this Weyl calls the natural gauge. We shall have to conceive of this natural-gauge system as one imposed upon us by nature and defined by certain intrinsic properties of the universe. But the only determinate magnitude which is offered us by nature at every point appears to be the radius of curvature of the universe at every point. ”

“ Only when every one of these twenty components of the tensor vanishes at every point is it possible to assert that the four-dimensional space is Euclidean, or at least flat. Thus, whereas for two dimensions Euclideanism was ensured when one condition was satisfied at every point, namely, the vanishing of the Gaussian curvature, , on the other hand, when we step up to four dimensions, twenty conditions are needed, and these are given by the vanishing at every point of the twenty expressions which in their aggregate constitute the Riemann-Christoffel tensor . ”

“ Transformation of momentum, energy and stresses. The transformation formulas for momentum, energy density and the stresses in the passage from one valid reference system to another system , is definitely determined by deriving them from the components of the world tensor . ”

“ The order of the three elongations is immaterial, since the original dyadic is equal to the product of the three dyadics taken in any order. Def.—A dyadic which is reducible to this form we shall call a right tensor. The displacement represented by a right tensor is called a pure strain. A right tensor is evidently self-conjugate.
149. We have seen (No. 135) that eveiy dyadic may be expressed in the form
where are positive scalars. This is equivalent to and to Hence every dyadic may be expressed as the product of a versor and a right tensor with the scalar factor ± 1. ”

“ The right tensor already described constitutes a particular case, distinguished by perpendicular axes and positive values of the coefficients The value of the dyadic is evidently not affected by substituting vectors of different lengths but the same or opposite directions for with the necessaiy changes in the values of defined as reciprocals of But, except this change, if are unequal, the dyadic can be expressed only in one way in the above form. ”

“ From our additional term it follows, that these equations contain no new condition in respect to (9) , so that concerning the energy tensor of matter, no other presupposition has to be made than the one, that it has to be in agreement with the momentum-energy theorem. ”

“ Since the quantity that we have formed has a tensor character, its vanishing is proved for every other system of co-ordinates also, and naturally for every other four-dimensional point. The energy principle of matter (97) is thus a mathematical consequence of the field equations (96) .
In order to learn whether the equations (96) are consistent with experience, we must, above all else, find out whether they [Pg 90] lead to the Newtonian theory as a first approximation. ”

“ But if, on the other hand, there is no unique direction in space it is not logical to formulate the laws of nature in such a way as to conceal the equivalence of systems of co-ordinates that are oriented differently. We shall meet with this point of view again in the theories of special and general relativity. ↑ These considerations will make the reader familiar with tensor operations without the special difficulties of the four-dimensional treatment ”

“ Now, assuming space-time to be a Weylian manifold, a change of gauge-system will modify the expression of the fundamental tensors of Einstein’s theory (except ) , hence will modify the aspect of the mechanical laws and of that of gravitation. A natural extension compels us to assume that all the Einsteinian laws should be generalised and expressed in terms of in-tensors and in-invariants, so as to yield in-tensor equations instead of tensor equations. ”

“ In classical science these were entirely distinct magnitudes. But when we represent these intensities in terms of space-time, we find that they are given by the various components of one same space-time tensor ; the time components give the electric force, while the space components yield the magnetic force. The advantage of the new point of view is chiefly to show us that what we once considered different entities now reveal themselves as the different components of the same space-time existent. ”