NOTE: Einstein’s statements are in black italics. My understanding follows in bold color italics.
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The Concept of Space in the General Theory of Relativity
This theory arose primarily from the endeavour to understand the equality of inertial and gravitational mass. We start out from an inertial system S1, whose space is, from the physical point of view, empty. In other words, there exists in the part of space contemplated neither matter (in the usual sense) nor a field (in the sense of the special theory of relativity). With reference to S1 let there be a second system of reference S2 in uniform acceleration. Then S2 is thus not an inertial system. With respect to S2 every test mass would move with an acceleration, which is independent of its physical and chemical nature. Relative to S2, therefore, there exists a state which, at least to a first approximation, cannot be distinguished from a gravitational field. The following concept is thus compatible with the observable facts: S2 is also equivalent to an “inertial system”; but with respect to S2 a (homogeneous) gravitational field is present (about the origin of which one does not worry in this connection). Thus when the gravitational field is included in the framework of the consideration, the inertial system loses its objective significance, assuming that this “principle of equivalence” can be extended to any relative motion whatsoever of the systems of reference. If it is possible to base a consistent theory on these fundamental ideas, it will satisfy of itself the fact of the equality of inertial and gravitational mass, which is strongly confirmed empirically.
An inertial system consists of constant inertia of objects. This may be extended to constant frequency in case of fields. The inertial system represents “constant space and time”, which appears as constant velocity among objects. When a system consists of uniformly increasing frequency and inertia it is said to be “accelerating”. This acceleration represent changing characteristics of space and time. It manifests as a gravitational field.
Considered four-dimensionally, a non-linear transformation of the four co-ordinates corresponds to the transition from S1 to S2. The question now arises: What kind of non-linear transformations are to be permitted, or, how is the Lorentz transformation to be generalised? In order to answer this question, the following consideration is decisive.
Einstein is working with pure abstraction of space and time. His approach is completely mathematical. The gravitation is essentially a manifestation of changes in frequency and inertia. Here we need to understand the mathematical interpretation of frequency and inertia fully.
Constant frequency seems to represent a field of constant wavelength and period. It represents a static inertia. Gravitation seems to represent a uniformly increasing frequency The gravitational field has uniformly decreasing wavelength and period, and a uniformly increasing inertia.
We ascribe to the inertial system of the earlier theory this property: Differences in co-ordinates are measured by stationary “rigid” measuring rods, and differences in time by clocks at rest. The first assumption is supplemented by another, namely, that for the relative laying out and fitting together of measuring rods at rest, the theorems on “lengths” in Euclidean geometry hold. From the results of the special theory of relativity it is then concluded, by elementary considerations, that this direct physical interpretation of the co-ordinates is lost for systems of reference (S2) accelerated relatively to inertial systems (S1). But if this is the case, the co-ordinates now express only the order or rank of the “contiguity” and hence also the dimensional grade of the space, but do not express any of its metrical properties. We are thus led to extend the transformations to arbitrary continuous transformations. This implies the general principle of relativity: Natural laws must be covariant with respect to arbitrary continuous transformations of the co-ordinates. This requirement (combined with that of the greatest possible logical simplicity of the laws) limits the natural laws concerned incomparably more strongly than the special principle of relativity.
The characteristics of wavelength and period may actually represent the space and time of the field; and the characteristics of frequency may represent the inertia of the field. It tells one how “dense” the field is. The nucleus of an atom is simply an extremely “dense” field. Inertia represents this “density”.
There are very high gradients of “density” or inertia at the surface of particles like electrons, protons and neutrons. These gradients manifest as gravitational force. In this context, the charge on particles may relate to how this gradient “curves” to form the particle, but this is simply a conjecture.
The “acceleration” of the field results in the wavelength (space) and the period (time) becoming more dense (increase in inertia). A continuity is maintained in the sense that the ratio of wavelength to period is maintained as ‘c’. This ratio is referred to as the “speed of light”.
This train of ideas is based essentially on the field as an independent concept. For the conditions prevailing with respect to S2 are interpreted as a gravitational field, without the question of the existence of masses which produce this field being raised. By virtue of this train of ideas it can also be grasped why the laws of the pure gravitational field are more directly linked with the idea of general relativity than the laws for fields of a general kind (when, for instance, an electromagnetic field is present). We have, namely, good ground for the assumption that the “field-free” Minkowski-space represents a special case possible in natural law, in fact, the simplest conceivable special case. With respect to its metrical character, such a space is characterised by the fact that dx1² + dx2² + dx3² is the square of the spatial separation, measured with a unit gauge, of two infinitesimally neighbouring points of a three-dimensional “space-like” cross section (Pythagorean theorem), whereas dx4 is the temporal separation, measured with a suitable time gauge, of two events with common (x1, x2,x3). All this simply means that an objective metrical significance is attached to the quantity
ds² = dx1² + dx2² + dx3² – dx4² (1)
as is readily shown with the aid of the Lorentz transformations. Mathematically, this fact corresponds to the condition that ds² is invariant with respect to Lorentz transformations.
The background SPACE simply provides a reference point of zero frequency. An electromagnetic field is a field of constant frequency. The inertial or gravitational field is a field of uniformly accelerating frequency. The reality is made up of a disturbance that expresses itself as field and matter.
The Minkowski-space is an abstraction of the electromagnetic field. The space element (ds) is represented by the “three-dimensional wavelength” of the electromagnetic field. The time element (dt) is represented by the period of the electromagnetic field. The ratio (ds/dt) is maintained as the constant ‘c’, which represents the continuity between space dimensions and the changes in those dimensions.
As frequency increases on a gradient in the gravitational field, the space element (ds) shrinks making the space more dense. Correspondingly, the time element (dt) also shrinks maintaining the continuity as ratio ‘c’. This mathematics does not make subjective assumptions as those made for Lorentz transformations. The invariance of the ratio ‘c’ simply depicts the natural continuity of changes.
If now, in the sense of the general principle of relativity, this space (cf. eq. (1) ) is subjected to an arbitrary continuous transformation of the co-ordinates, then the objectively significant quantity ds is expressed in the new system of co-ordinates by the relation
ds² = gik dxi dxk (1a)
which has to be summed up over the indices i and k for all combinations 11, 12, . . . up to 44 . The terms gik now are not constants, but functions of the co-ordinates, which are determined by the arbitrarily chosen transformation. Nevertheless, the terms gik are not arbitrary functions of the new co-ordinates, but just functions of such a kind that the form (1a) can be transformed back again into the form (1) by a continuous transformation of the four co-ordinates. In order that this may be possible, the functions gik must satisfy certain general covariant equations of condition, which were derived by B. Riemann more than half a century before the formulation of the general theory of relativity (“Riemann condition”). According to the principle of equivalence, (1a) describes in general covariant form a gravitational field of a special kind, when the functions gik satisfy the Riemann condition.
It follows that the law for the pure gravitational field of a general kind must be satisfied when the Riemann condition is satisfied; but it must be weaker or less restricting than the Riemann condition. In this way the field law of pure gravitation is practically completely determined, a result which will not be justified in greater detail here.
In short, when the wavelength of the electromagnetic field is subjected to continuous change, a gravitational field is produced. The continuous change is constrained through the ratio ‘c’, which results in a curvature forming the particle. The details of the gravitational field can be worked out from this.
We are now in a position to see how far the transition to the general theory of relativity modifies the concept of space. In accordance with classical mechanics and according to the special theory of relativity, space (space-time) has an existence independent of matter or field. In order to be able to describe at all that which fills up space and is dependent on the co-ordinates, space-time or the inertial system with its metrical properties must be thought of at once as existing, for otherwise the description of “that which fills up space” would have no meaning. On the basis of the general theory of relativity, on the other hand, space as opposed to “what fills space”, which is dependent on the co-ordinates, has no separate existence. Thus a pure gravitational field might have been described in terms of the gik (as functions of the co-ordinates), by solution of the gravitational equations. If we imagine the gravitational field, i.e. the functions gik, to be removed, there does not remain a space of the type (1), but absolutely nothing, and also no “topological space”. For the functions gik describe not only the field, but at the same time also the topological and metrical structural properties of the manifold. A space of the type (1), judged from the standpoint of the general theory of relativity, is not a space without field, but a special case of the gik field, for which – for the co-ordinate system used, which in itself has no objective significance – the functions gik have values that do not depend on the co-ordinates. There is no such thing as an empty space, i.e. a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field.
Thus we start from a background of no dimensions, no frequency, no inertia, and no change. The introduction of frequency brings about the dimensions and inertia. The constant frequency establishes an electromagnetic field. The uniformly increasing frequency establishes the gravitational field. The general theory of relativity recognizes that space-time is not independent of the disturbance that fills it.
The gravitational field can thus exist within the electromagnetic field This is the case with the presence of electrons, protons and neutrons within the atom. The particles are in continuity with the field, which is in continuity with the background SPACE of no dimensions and no inertia.
Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space. The notion indeed appears absurd, as long as physical reality is seen exclusively in ponderable bodies. 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’.
Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space.
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Earlier notes by Vinaire:
According to classical mechanics “space” and “what fills the space” has existence independent of each other, and we think of them as existing simultaneously. On the basis of the general theory of relativity, on the other hand, space as opposed to “what fills space”, has no separate existence.
In this generalization, we maintain continuity of transformation between “space” and “what fills the space”, and we adjust the natural laws to accommodate this.
We start with a more general idea of “space-time” as an independent concept. We then impose on it a condition of continuity using “Riemann condition” as we accelerate the system. This brings about a gravitational field that fills the space. If we then imagine the gravitational field to be removed, there does not remain a space either. There seem to be no such thing as an empty space, i.e. a space without field.
The Disturbance Theory looks at “empty space” as theoretical “undisturbed space”. As it is disturbed it starts to get “filled” with disturbance. The undisturbed space simply provides a framework for the disturbed space.
The disturbance has frequency. This frequency varies over a large spectrum but the ratio of wavelength to period is always a constant “c”. This establishes continuity among inertial systems. When the frequency is increasing or decreasing, the continuity of the gradient change further ensures continuity among inertial systems.
Region of disturbed space having a constant frequency appears as having a constant velocity or uniform motion. Region of disturbed space with changing frequency appears as having acceleration, deceleration or a gravitational field.
The imposition of continuity means that no matter how sharp some boundaries or interfaces may appear there is still no abrupt break at atomic dimensions. This changes the way we look at Quantum Mechanics.
Medium gradient of changing frequency produces the electronic region of an atom. Very high gradient of changing frequency produces the nuclear mass. So gravity and mass go together.
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