Albert Einstein

“ 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. ”

“ In order to obtain Newton's theory as a first approximation we had to calculate only out of the 10 components of the gravitation-potential , for that is the only component which conies in the first approximate equations of motion of a material point in a gravitational field. ”

“ 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. ”

“ 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. ”

“ 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 mathematical apparatus useful for the general relativity theory, lay already complete in the "Absolute Differential Calculus", which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-Civita, and already applied to the problems of theoretical physics. ”

“ The right-hand side expresses the interaction of the energy of the gravitational-field on matter. The field-equations of gravitation contain thus at the same time 4 conditions which are to be satisfied by all material phenomena. We get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. We see that the appearance of a gravitation-field is connected with space-time variability of . In the general case, we can not by any suitable choice of axes, make special relativity theory valid throughout any finite region. ”

“ The result of measurement with a rod relatively at rest as regards would be a number which is greater than π. 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. Euclidean Geometry therefore does not hold for the system ; the above fixed conceptions of co-ordinates which assume the validity of Euclidean Geometry fail with regard to the system . ”

“ 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) . ”

“ 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. ”

“ 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. ”

“ They take the place of the imaginary cause . Among all the conceivable spaces and etc. moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater advantages, against which the objection which was already raised from the standpoint of the theory of knowledge cannot be again revived. ”

“ The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences. We fit to the world our space-time variables such that to any and every point-event corresponds a system of values of . ”

“ Now we see that, according to the first view of approximation the magnitudes are all small quantities of at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which .
By limiting ourselves only to terms of the lowest order we get instead of (46) , first, the equations ”