Metric tensor

“ 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. ”

“ 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 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. ”

“ The special relativity theory has led to the conception that the inertial mass (Träge Masse) 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 energy-tensor τασ associated with matter, which like the energy components tασ of the gravitation-field (equations 49, and 50) have a mixed character but which however can be connected with symmetrical covariant tensors. ”

“ In the general theory of relativity, we shall have to assume as valid the corresponding general co-variant equation. If () denotes the co-variant energy tensor of matter, the corresponding mixed tensor density, then, in accordance with (83) , we must require that be satisfied. It must be remembered that besides the energy density of the matter there must also be given an energy density of the gravitational field, so that there can be no talk of principles of conservation of energy and momentum for matter alone. ”

“ He has succeeded in proving experimentally the concept of tensors. A tensor is a mathematical expression for the fact that space is smooth and flat, in three dimensions, only at an infinite distance from matter; in the neighborhood of a particle of matter, there is a pucker or a wrinkle in space. ”

“ We are thus led to consider how it would be necessary to modify these equations for a finite universe to be realised.
Now we mentioned, when discussing the law of gravitation, that the inclusion of an additional tensor ( being any arbitrary constant) in the equation of curvature would not destroy the tensor nature of the law; hence, so far as the requirements of all general laws are concerned, this modified law of space-time curvature is acceptable. If it is accepted, the universe can no longer be quasi-Euclidean and infinite ”

“ 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 . ”

“ 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. ”

“ A four-vector – the six-vector is not needed in the following – has the direction sense of a directed distance; i.e. its four components are transformed with a Lorentz transformation like the corresponding coordinates. By a world tensor, however, we understand the totality of 16 components , which are transformed as the squares and products of . Between them, there are always six symmetry relations . Analogous to the fact that (with respect to tensor in three dimensions) the divergence becomes a space vector, here the divergence is a four-vector with components ”

“ With this choice of co-ordinates, only substitutions with determinant 1, are allowable.
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. ”

“ 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. ”

“ The six components of T can be most conveniently, geometrically interpreted as coefficients of the equation of a surface of second degree.
In the four-dimensional case we accordingly start with a six-vector f, and define a tensor (ff) by all possible scalar products of the special four-vectors (derived by equation (6a) from f) for h, j = x, y, z, l. From that, at first 16 tensor components follow
which, however, are reduced to 10 because of . ”

“ 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. ”

“ The last equation is a skew-symmetrical tensor equation of the third rank (the skew-symmetry of the left-hand side with respect to every pair of indices may easily be proved, if attention is paid to the skew-symmetry of ) . This notation is more natural than the usual one, because, in contrast to the latter, it is applicable to Cartesian left- handed systems as well as to right-handed systems without change of sign. ”

“ Formation of Tensors through Differentiation. 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. If a certain curve be given in our continuum whose points are characterised by the arc-distances s measured from a fixed point on the curve, and if further , be an invariant space function, then is also an invariant. ”

“ 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 ”

“ There exist certain interesting features about Eddington’s theory which we may mention briefly. Inasmuch as the intrinsic magnitudes of the universe must be in-tensors and in-invariants, instead of tensors and invariants, the ’s lose their position of pre-eminence; for the ’s, while transcending our mesh-system, fail to transcend our choice of a gauge-system. Accordingly, in the expression of a distance, we must replace by some in-tensor of a similar nature. ”