Einstein’s “Relativity & Problem of Space”

Disturbed space
Reference: Disturbance Theory

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Einstein wrote the article Relativity and the Problem of Space  in 1952, three years before his death, as a new appendix to his book, Relativity: The Special and General Theory (1920). This article is quoted below.

In the Special theory of relativity Einstein views space as an independent physical reality that remains after all matter and field are removed. Later in General theory of relativity, Einstein reverses this view and says, “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.

My understanding is based on this latter viewpoint of space. Einstein’s article follows in in black italics. My understanding follows in bold color italics.

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Relativity & Problem of Space

Albert Einstein (1952)

It is characteristic of Newtonian physics that it has to ascribe independent and real existence to space and time as well as to matter, for in Newton’s law of motion the idea of acceleration appears. But in this theory, acceleration can only denote “acceleration with respect to space”. Newton’s space must thus be thought of as “at rest”, or at least as “unaccelerated”, in order that one can consider the acceleration, which appears in the law of motion, as being a magnitude with any meaning. Much the same holds with time, which of course likewise enters into the concept of acceleration.

Newton himself and his most critical contemporaries felt it to be disturbing that one had to ascribe physical reality both to space itself as well as to its state of motion; but there was at that time no other alternative, if one wished to ascribe to mechanics a clear meaning.

It is indeed an exacting requirement to have to ascribe physical reality to space in general, and especially to empty space. Time and again since remotest times philosophers have resisted such a presumption. Descartes argued somewhat on these lines: space is identical with extension, but extension is connected with bodies; thus there is no space without bodies and hence no empty space. The weakness of this argument lies primarily in what follows. It is certainly true that the concept extension owes its origin to our experiences of laying out or bringing into contact solid bodies. But from this it cannot be concluded that the concept of extension may not be justified in cases which have not themselves given rise to the formation of this concept. Such an enlargement of concepts can be justified indirectly by its value for the comprehension of empirical results.

The assertion that extension is confined to bodies is therefore of itself certainly unfounded. We shall see later, however, that the general theory of relativity confirms Descartes’ conception in a roundabout way.

What brought Descartes to his remarkably attractive view was certainly the feeling that, without compelling necessity, one ought not to ascribe reality to a thing like space, which is not capable of being “directly experienced”.

The psychological origin of the idea of space, or of the necessity for it, is far from being so obvious as it may appear to be on the basis of our customary habit of thought. The old geometers deal with conceptual objects (straight line, point, surface), but not really with space as such, as was done later in analytical geometry. The idea of space, however, is suggested by certain primitive experiences. 

Suppose that a box has been constructed. Objects can be arranged in a certain way inside the box, so that it becomes full. The possibility of such arrangements is a property of the material object “box”, something that is given with the box, the “space enclosed” by the box. This is something which is different for different boxes, something that is thought quite naturally as being independent of whether or not, at any moment, there are any objects at all in the box. When there are no objects in the box, its space appears to be “empty”.

So far, our concept of space has been associated with the box. It turns out, however, that the storage possibilities that make up the box-space are independent of the thickness of the walls of the box. Cannot this thickness be reduced to zero, without the “space” being lost as a result? The naturalness of such a limiting process is obvious, and now there remains for our thought the space without the box, a self-evident thing, yet it appears to be so unreal if we forget the origin of this concept. One can understand that it was repugnant to Descartes to consider space as independent of material objects, a thing that might exist without matter.  (At the same time, this does not prevent him from treating space as a fundamental concept in his analytical geometry.) The drawing of attention to the vacuum in a mercury barometer has certainly disarmed the last of the Cartesians. But it is not to be denied that, even at this primitive stage, something unsatisfactory clings to the concept of space, or to space thought of as an independent real thing.

The ways in which bodies can be packed into space (e.g. the box) are the subject of three-dimensional Euclidean geometry, whose axiomatic structure readily deceives us into forgetting that it refers to realizable situations.

If now the concept of space is formed in the manner outlined above, and following on from experience about the “filling” of the box, then this space is primarily a bounded space. This limitation does not appear to be essential, however, for apparently a larger box can always be introduced to enclose the smaller one. In this way space appears as something unbounded.

I shall not consider here how the concepts of the three-dimensional and the Euclidean nature of space can be traced back to relatively primitive experiences.

Rather, I shall consider first of all from other points of view the rôle of the concept of space in the development of physical thought.

When a smaller box s is situated, relatively at rest, inside the hollow space of a larger box S, then the hollow space of s is a part of the hollow space of S, and the same “space”, which contains both of them, belongs to each of the boxes. When s is in motion with respect to S, however, the concept is less simple. One is then inclined to think that s encloses always the same space, but a variable part of the space S. It then becomes necessary to apportion to each box its particular space, not thought of as bounded, and to assume that these two spaces are in motion with respect to each other.

Before one has become aware of this complication, space appears as an unbounded medium or container in which material objects swim around. But it must now be remembered that there is an infinite number of spaces, which are in motion with respect to each other.

The concept of space as something existing objectively and independent of things belongs to pre-scientific thought, but not so the idea of the existence of an infinite number of spaces in motion relatively to each other.

This latter idea is indeed logically unavoidable, but is far from having played a considerable rôle even in scientific thought.

“Empty space” represents the extension of field. The simplest field-substance shall have a frequency of 1 Hertz. We may identify its as “field-space”. This field-space is not abstract or mathematical; it is real. Beyond field-space is “emptiness”, which theorizes absence of all phenomena. We may identify emptiness as theoretical field of 0 Hertz.

The field-space has frequency of 1 Hertz, or a disturbance level of 0 (see The Disturbance Level). Compared to substances of higher disturbance levels the field-space is at rest.

In Newtonian mechanics we have the concept of acceleration. It refers to change in body’s existing motion. For field-substance this would be change in its disturbance level. Since the disturbance level of field-space is 0, all other disturbance levels refer to absolute acceleration. This seems to represent the inertia of field-substance.

We define the “space” of an empty box by the material it can store. When the boundaries of the box are removed, we may still imagine the “space” of the box to be there as its material storage possibilites. But the actual space is different having adjusted to its new boundary conditions. By considering material storage possibilites we are not defining the actual space.

This assumption of “material storage possibility” forms the axiomatic structure of three-dimensional Euclidean geometry. An inertial frame of reference used by Special theory of relativity is also based on this assumption.

But what about the psychological origin of the concept of time? This concept is undoubtedly associated with the fact of “calling to mind”, as well as with the differentiation between sense experiences and the recollection of these. Of itself it is doubtful whether the differentiation between sense experience and recollection (or simple re-presentation) is something psychologically directly given to us. Everyone has experienced that he has been in doubt whether he has actually experienced something with his senses or has simply dreamt about it. Probably the ability to discriminate between these alternatives first comes about as the result of an activity of the mind creating order.

An experience is associated with a “recollection”, and it is considered as being “earlier” in comparison with present “experiences”. This is a conceptual ordering principle for recollected experiences, and the possibility of its accomplishment gives rise to the subjective concept of time, i.e. that concept of time which refers to the arrangement of the experiences of the individual.

What do we mean by rendering objective the concept of time? Let us consider an example. A person A (“I”) has the experience “it is lightning”. At the same time the person A also experiences such a behaviour of the person B as brings the behaviour of B into relation with his own experience “it is lightning”. Thus it comes about that A associates with B the experience “it is lightning”. For the person A the idea arises that other persons also participate in the experience “it is lightning”. “It is lightning” is now no longer interpreted as an exclusively personal experience, but as an experience of other persons (or eventually only as a “potential experience”). In this way arises the interpretation that “it is lightning”, which originally entered into the consciousness as an “experience”, is now also interpreted as an (objective) “event”. It is just the sum total of all events that we mean when we speak of the “real external world”.

We have seen that we feel ourselves impelled to ascribe a temporal arrangement to our experiences, somewhat as follows. If b is later than a and c later than b then c is also later than a (“sequence of experiences”).

Now what is the position in this respect with the “events” which we have associated with the experiences? At first sight it seems obvious to assume that a temporal arrangement of events exists which agrees with the temporal arrangement of the experiences. In general, and unconsciously this was done, until skeptical doubts made themselves felt.  In order to arrive at the idea of an objective world, an additional constructive concept still is necessary: the event is localised not only in time, but also in space.

The concept of time is objective when it is based on a regularly repeating cycle, such as, the rotation of earth. The duration of a cycle may be subdivided ad infinitum, but it still represents time objectively.

Time based on recollection of experience is subjective. But a documented observation is not the same as experience. It remains objective even when reviewed later. 

Objectivity of an event is determined by the continuity, harmony and consistency of observations. This may be accomplished by one observer or many. Objectivity does not necessarily depend on agreement with others. Therefore, there is no such thing as “real external world”, which implies agreement. 

An event is specific in terms of space and time. It may be observed from a viewpoint localized in space and time, or as a coordination among many viewpoints distributed in space and time. Objectivity is not affected by space and time.

In the previous paragraphs we have attempted to describe how the concepts space, time and event can be put psychologically into relation with experiences. Considered logically, they are free creations of the human intelligence, tools of thought, which are to serve the purpose of bringing experiences into relation with each other, so that in this way they can be better surveyed.

The attempt to become conscious of the empirical sources of these fundamental concepts should show to what extent we are actually bound to these concepts. In this way we become aware of our freedom, of which, in case of necessity, it is always a difficult matter to make sensible use.

We still have something essential to add to this sketch concerning the psychological origin of the concepts space-time-event (we will call them more briefly “space-like”, in contrast to concepts from the psychological sphere). We have linked up the concept of space with experiences using boxes and the arrangement of material objects in them. Thus this formation of concepts already presupposes the concept of material objects (e.g. ”boxes”). In the same way persons, who had to be introduced for the formation of an objective concept of time, also play the rôle of material objects in this connection. It appears to me, therefore, that the formation of the concept of the material object must precede our concepts of time and space.

All these space-like concepts already belong to pre-scientific thought, along with concepts like pain, goal, purpose, etc. from the field of psychology. Now it is characteristic of thought in physics, as of thought in natural science generally, that it endeavors in principle to make do with “space-like” concepts alone, and strives to express with their aid all relations having the form of laws. The physicist seeks to reduce colours and tones to vibrations, the physiologist thought and pain to nerve processes, in such a way that the psychical element as such is eliminated from the causal nexus of existence, and thus nowhere occurs as an independent link in the causal associations. It is no doubt this attitude, which considers the comprehension of all relations by the exclusive use of only space-like concepts as being possible in principle, that is at the present time understood by the term “materialism” (since “matter” has lost its rôle as a fundamental concept).

Why is it necessary to drag down from the Olympian fields of Plato the fundamental ideas of thought in natural science, and to attempt to reveal their earthly lineage? Answer: in order to free these ideas from the taboo attached to them, and thus to achieve greater freedom in the formation of ideas or concepts. It is to the immortal credit of D. Hume and E. Mach that they, above all others, introduced this critical conception.

Science has taken over from pre-scientific thought the concepts space, time, and material object (with the important special case “solid body”) and has modified them and rendered them more precise. Its first significant accomplishment was the development of Euclidean geometry, whose axiomatic formulation must not be allowed to blind us to its empirical origin (the possibilities of laying out or juxtaposing solid bodies). In particular, the three-dimensional nature of space as well as its Euclidean character are of empirical origin (it can be wholly filled by like constituted “cubes”).

The subtlety of the concept of space was enhanced by the discovery that there exist no completely rigid bodies.

All bodies are elastically deformable and alter in volume with change in temperature. The structures, whose possible congruences are to be described by Euclidean geometry, cannot therefore be represented apart from physical concepts. But since physics after all must make use of geometry in the establishment of its concepts, the empirical content of geometry can be stated and tested only in the framework of the whole of physics.

In this connection atomistics must also be borne in mind, and its conception of finite divisibility; for spaces of sub-atomic extension cannot be measured up.

Atomistics also compels us to give up, in principle, the idea of sharply and statically defined bounding surfaces of solid bodies. Strictly speaking, there are no precise laws, even in the macro-region, for the possible configurations of solid bodies touching each other.

In spite of this, no one thought of giving up the concept of space, for it appeared indispensable in the eminently satisfactory whole system of natural science.

Mach, in the nineteenth century, was the only one who thought seriously of an elimination of the concept of space, in that he sought to replace it by the notion of the totality of the instantaneous distances between all material points. (He made this attempt in order to arrive at a satisfactory understanding of inertia).

Empirically, both observer and the event observed are part of the same universe. The criterion of objectivity is the perception of continuity, harmony and consistency of the universe. Therein lay the freedoms and constraints.

Here is the wisdom of Einstein as a pioneer:

Our familiarity with space and time is based on our experience with matter. Physics may, therefore, appear “materialistic” but a physicist is simply seeking to arrive at better understand by reducing the complex, multidimensional variables of life to simple concepts. Understanding of  life may improve over time. Such understanding requires freeing ideas from the taboo attached to them.

The idea of space does not have to be based on completely rigid bodies as in Euclidean geometry. The empirical basis of geometry must be consistent with the physical reality of elastic deformations and changes with temperature. It should be able to account for sub-atomic extensions. Strictly speaking, there are no precise laws, even in the macro-region, for the possible configurations of solid bodies touching each other.

Space may be described by the notion of the totality of the instantaneous distances between all material points; but to do this we need a better understanding of the nature of matter itself.

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The Field

In Newtonian mechanics, space and time play a dual rôle. First, they play the part of carrier or frame for things that happen in physics, in reference to which events are described by the space co-ordinates and the time. In principle, matter is thought of as consisting of “material points”, the motions of which constitute physical happening. When matter is thought of as being continuous, this is done as it were provisionally in those cases where one does not wish to or cannot describe the discrete structure. In this case small parts (elements of volume) of the matter are treated similarly to material points, at least in so far as we are concerned merely with motions and not with occurrences which, at the moment, it is not possible or serves no useful purpose to attribute to motions (e.g. temperature changes, chemical processes).

The second rôle of space and time was that of being an “inertial system”. From all conceivable systems of reference, inertial systems were considered to be advantageous in that, with respect to them, the law of inertia claimed validity.

In this, the essential thing is that “physical reality”, thought of as being independent of the subjects experiencing it, was conceived as consisting, at least in principle, of space and time on one hand, and of permanently existing material points, moving with respect to space and time, on the other. The idea of the independent existence of space and time can be expressed drastically in this way: If matter were to disappear, space and time alone would remain behind (as a kind of stage for physical happening).

The surmounting of this standpoint resulted from a development which, in the first place, appeared to have nothing to do with the problem of space-time, namely, the appearance of the concept of field and its final claim to replace, in principle, the idea of a particle (material point). In the framework of classical physics, the concept of field appeared as an auxiliary concept, in cases in which matter was treated as a continuum. For example, in the consideration of the heat conduction in a solid body, the state of the body is described by giving the temperature at every point of the body for every definite time. Mathematically, this means that the temperature T is represented as a mathematical expression (function) of the space co-ordinates and the time t (Temperature field). 

The law of heat conduction is represented as a local relation (differential equation), which embraces all special cases of the conduction of heat. The temperature is here a simple example of the concept of field. This is a quantity (or a complex of quantities), which is a function of the co-ordinates and the time. Another example is the description of the motion of a liquid. At every point there exists at any time a velocity, which is quantitatively described by its three “components” with respect to the axes of a co-ordinate system (vector). The components of the velocity at a point (field components), here also, are functions of the co-ordinates (x, y, z) and the time (t).

It is characteristic of the fields mentioned that they occur only within a ponderable mass; they serve only to describe a state of this matter. In accordance with the historical development of the field concept, where no matter was available there could also exist no field. But in the first quarter of the nineteenth century it was shown that the phenomena of the interference and motion of light could be explained with astonishing clearness when light was regarded as a wave-field, completely analogous to the mechanical vibration field in an elastic solid body. It was thus felt necessary to introduce a field, that could also exist in “empty space” in the absence of ponderable matter.

This state of affairs created a paradoxical situation, because, in accordance with its origin, the field concept appeared to be restricted to the description of states in the inside of a ponderable body. This seemed to be all the more certain, inasmuch as the conviction was held that every field is to be regarded as a state capable of mechanical interpretation, and this presupposed the presence of matter. One thus felt compelled, even in the space which had hitherto been regarded as empty, to assume everywhere the existence of a form of matter, which was called “aether”.

The emancipation of the field concept from the assumption of its association with a mechanical carrier finds a place among the psychologically most interesting events in the development of physical thought. During the second half of the nineteenth century, in connection with the researches of Faraday and Maxwell it became more and more clear that the description of electromagnetic processes in terms of field was vastly superior to a treatment on the basis of the mechanical concepts of material points. By the introduction of the field concept in electrodynamics, Maxwell succeeded in predicting the existence of electromagnetic waves, the essential identity of which with light waves could not be doubted because of the equality of their velocity of propagation. As a result of this, optics was, in principle, absorbed by electrodynamics. One psychological effect of this immense success was that the field concept, as opposed to the mechanistic framework of classical physics, gradually won greater independence.

Nevertheless, it was at first taken for granted that electromagnetic fields had to be interpreted as states of the aether, and it was zealously sought to explain these states as mechanical ones. But as these efforts always met with frustration, science gradually became accustomed to the idea of renouncing such a mechanical interpretation. Nevertheless, the conviction still remained that electromagnetic fields must be states of the aether, and this was the position at the turn of the century.

The material-substance seems to move within the field-substance. The material-substance is thought of as discrete “material points” that occupy the continuous field-substance.

The motion of a substance is balanced by its inertia. This results in a constant drift velocity. The drift velocity characterizes the space, time and inertia of a substance. 

The “material point” may be described as having a range of inertia. We may actually visualize a gradient of inertia from space to matter.

Matter with its mechanical properties appears at the highest range of inertia. At lower ranges we find quantum and electromagnetic characteristics.

Inertia exists all the way down to space. Field-space represents the simplest form of real substance that has almost no inertia. 

The aether-theory brought with it the question: How does the aether behave from the mechanical point of view with respect to ponderable bodies? Does it take part in the motions of the bodies, or do its parts remain at rest relatively to each other? Many ingenious experiments were undertaken to decide this question. The following important facts should be mentioned in this connection: the “aberration” of the fixed stars in consequence of the annual motion of the earth, and the “Doppler effect”, i.e. the influence of the relative motion of the fixed stars on the frequency of the light reaching us from them, for known frequencies of emission. The results of all these facts and experiments, except for one, the Michelson-Morley experiment, were explained by H. A. Lorentz on the assumption that the aether does not take part in the motions of ponderable bodies, and that the parts of the aether have no relative motions at all with respect to each other. Thus the aether appeared, as it were, as the embodiment of a space absolutely at rest. But the investigation of Lorentz accomplished still more. It explained all the electromagnetic and optical processes within ponderable bodies known at that time, on the assumption that the influence of ponderable matter on the electric field – and conversely – is due solely to the fact that the constituent particles of matter carry electrical charges, which share the motion of the particles. Concerning the experiment of Michelson and Morley, H. A. Lorentz showed that the result obtained at least does not contradict the theory of an aether at rest.

Aether cannot have inertia because it does not take part in the motions of ponderable (material) bodies. That means aether cannot have mechanical properties.

Light has negligible inertia compared to material bodies. The “aberration” of fixed stars (in consequence of the annual motion of the earth) can be explained by the ratio of the velocity of earth to that of light, which is essentially the ratio of inertia of light to that of earth.

The “Doppler effect” is how the relative motion of stars influence the frequency of light. This happens because light has a finite velocity even when it is large. This means that light has fixed inertia even when it is negligible.

On a direct measurement basis the Michelson and Morley experiment couldn’t detect the inertia of light. But if light has no inertia its velocity would have been infinite. A finite velocity implies presence of inertia even if undetectable.

In spite of all these beautiful successes the state of the theory was not yet wholly satisfactory, and for the following reasons. Classical mechanics, of which it could not be doubted that it holds with a close degree of approximation, teaches the equivalence of all inertial systems or inertial “spaces” for the formulation of natural laws, i.e. the invariance of natural laws with respect to the transition from one inertial system to another. Electromagnetic and optical experiments taught the same thing with considerable accuracy. But the foundation of electromagnetic theory taught that a particular inertial system must be given preference, namely that of the luminiferous aether at rest. This view of the theoretical foundation was much too unsatisfactory. Was there no modification that, like classical mechanics, would uphold the equivalence of inertial systems (special principle of relativity)?

The answer to this question is the special theory of relativity. This takes over from the theory of Maxwell-Lorentz the assumption of the constancy of the velocity of light in empty space. In order to bring this into harmony with the equivalence of inertial systems (special principle of relativity), the idea of the absolute character of simultaneity must be given up; in addition, the Lorentz transformations for the time and the space co-ordinates follow for the transition from one inertial system to another. The whole content of the special theory of relativity is included in the postulate: The laws of Nature are invariant with respect to the Lorentz transformations. The important thing of this requirement lies in the fact that it limits the possible natural laws in a definite manner.

Classical mechanics deals with material systems that are equivalent in terms of highest inertia. This equivalence ensures the invariance of natural laws in classical mechanics. But field systems have a range of low to high inertia. It therefore requires an understanding of inertia to establish correspondence between classical and field systems. This is what the theory of relativity attempted to do.

The theory of relativity uses light as its reference point to measure and compare drift velocities of material bodies. This works for cosmological configurations where inertia of light can be ignored. But for atomic configurations inertia of light become significant and this theory does not work.

To resolve the problem of space for atomic configurations we need an actual reference point of zero inertia. 

What is the position of the special theory of relativity in regard to the problem of space? In the first place we must guard against the opinion that the four-dimensionality of reality has been newly introduced for the first time by this theory. Even in classical physics the event is localised by four numbers, three spatial co-ordinates and a time co-ordinate; the totality of physical “events” is thus thought of as being embedded in a four-dimensional continuous manifold. But on the basis of classical mechanics this four-dimensional continuum breaks up objectively into the one-dimensional time and into three-dimensional spatial sections, only the latter of which contain simultaneous events. This resolution is the same for all inertial systems. The simultaneity of two definite events with reference to one inertial system involves the simultaneity of these events in reference to all inertial systems. This is what is meant when we say that the time of classical mechanics is absolute. According to the special theory of relativity it is otherwise.

The sum total of events which are simultaneous with a selected event exist, it is true, in relation to a particular inertial system, but no longer independently of the choice of the inertial system. The four-dimensional continuum is now no longer resolvable objectively into sections, all of which contain simultaneous events; “now” loses for the spatially extended world its objective meaning. It is because of this that space and time must be regarded as a four-dimensional continuum that is objectively unresolvable, if it is desired to express the purport of objective relations without unnecessary conventional arbitrariness.

Since the special theory of relativity revealed the physical equivalence of all inertial systems, it proved the untenability of the hypothesis of an aether at rest. It was therefore necessary to renounce the idea that the electromagnetic field is to be regarded as a state of a material carrier. The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter in the theory of Newton.

Up to now we have directed our attention to finding in what respect the concepts of space and time were modified by the special theory of relativity. Let us now focus our attention on those elements which this theory has taken over from classical mechanics. Here also, natural laws claim validity only when an inertial system is taken as the basis of space-time description. The principle of inertia and the principle of the constancy of the velocity of light are valid only with respect to an inertial system. The field-laws also can claim to have a meaning and validity only in regard to inertial systems.

The material-substance is beyong gamma radiation on the spectrum, and its corresponding “frequency” is extremely high. The number of cycles of this frequency in spatial dimension accounts for distance in material dimensions. 

Lower down the spectrum frequency reduces and less number of cycles occupy the same spatial dimension. Thus length appears to contract and time appears to slow down because space and time maintain a constant ratio for a cycle.

Thus a “point” of space-time is no longer as sharp at lower frequencies. This explains the Heisenberg’s uncertainty principle for quantum mechanics. Einstein explained this “out of focus” phenomenon as non-simultaneity.

Thus space and time are not independent but operate as a four dimensional continuum. Simultaneity exists in relation to a particular inertial system, but not in relation to systems of different inertia. 

Mechanical properties imply high inertia. They do not exist at lower part of the spectrum where inertia also reduces. Therefore, there is no such thing as mechanical aether. The field-substance then becomes an irreducible element of physical description.

Thus, as in classical mechanics, space is here also an independent component in the representation of physical reality. If we imagine matter and field to be removed, inertial-space or, more accurately, this space together with the associated time remains behind. The four-dimensional structure (Minkowski-space) is thought of as being the carrier of matter and of the field. Inertial spaces, with their associated times, are only privileged four-dimensional co-ordinate systems, that are linked together by the linear Lorentz transformations. Since there exist in this four-dimensional structure no longer any sections which represent “now” objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four-dimensional existence, instead of, as hitherto, the evolution of a three-dimensional existence.

This rigid four-dimensional space of the special theory of relativity is to some extent a four-dimensional analogue of H. A. Lorentz’s rigid three-dimensional aether. For this theory also the following statement is valid: The description of physical states postulates space as being initially given and as existing independently. Thus even this theory does not dispel Descartes’ uneasiness concerning the independent, or indeed, the a priori existence of “empty space”. The real aim of the elementary discussion given here is to show to what extent these doubts are overcome by the general theory of relativity.

When material-substance of highest inertia is removed the field-substance of low inertia still remains. The field-substance may also be removed except for the field-space of 1 Hertz, which forms the basis of the spectrum of substance. Removal of field-space means a return to theoretical emptiness (absence of all substance, space and time).

The mathematical Minkowski-space is an abstraction of the real field-space. Lorentz transformations are abstractions of space-time throughout the spectrum of substance. The concept of “now” comes from the space-time of material-substance. 

This rigid four-dimensional continuum of the special theory of relativity is the space-time of material-substance. Space and time appear to be independent in this frame because of the compactness of cycles. Going down the spectrum, we find inertia decreasing and space-time losing its compactness. At no time is space empty because it represents the extension of substance.

The application of Einstein’s special theory to atomic dimensions requires a better understanding of inertia.

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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.

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.

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.

Gravity is considered to be associated with mass, but the concept of mass simply represents a certain level of inertia. Gravity shall be present between any two systems of different inertia. A difference in disturbance levels between two field-substances shall represent the presence of a gravitational field between them.

Acceleration when applied to a substance constantly is internalized as inertia. This appears as change in drift velocity of the substance. In his general relativity, Einstein applies the following principle: Natural laws must be covariant with respect to arbitrary continuous transformations of the co-ordinates. This simply means that all relationships in this universe and the changes among them are continuous, harmonious and consistent.

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 characterized by the fact that dx1² + dx2² + dx3² is the square of the spatial separation, measured with a unit gauge, of two infinitesimally neighboring 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.

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.

The gravitational potential depends on the difference between two disturbance levels. It is not something peculiar to mass. It is just that the largest gravitational potential exists between mass and space because their disturbance levels are so far apart.

The first order field is the electromagnetic field made up of electromagnetic cycles. The gravitational potential represents a second-order field of the gradient of disturbance levels. The gradient begins field-space of DL 0 (disturbance level of 0). Minkowski-space is an abstraction of field-space.

When constant acceleration is applied to a substance its inertia increases substance until it balances the acceleration. This results in more compact space-time and lower drift velocity. Thus space-time dimension is characterized by relative compactness of the cycles of substance. These cycles are extremely compact for material-substance. A gravitational field now also exists around the substance.

The electromagnetic field implies a constant disturbance level, wheras the gravitational field implies a gradient of disturbance level. These two fields are therefore, connected. The field-space forms the basis of both fields.

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 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’.

In General relativity, space (space-time) not only provides the co-ordinates but also has the property of compactness denoted by Einstein’s gravitational constants (gik). These gravitational constants in the general theory describe not only the field but, at the same time, also the topological and metrical structural properties of the manifold.

Space-time does not claim existence on its own, but only as a structural quality of the field. Thus, space is really the field-space as it was originally visualized by Descartes. There is no such thing as “empty space”. But there can be the notion of emptiness.

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Generalized Theory of Gravitation

The theory of the pure gravitational field on the basis of the general theory of relativity is therefore readily obtainable, because we may be confident that the “field-free” Minkowski space with its metric in conformity with (1) must satisfy the general laws of field. From this special case the law of gravitation follows by a generalisation which is practically free from arbitrariness.

The further development of the theory is not so unequivocally determined by the general principle of relativity; it has been attempted in various directions during the last few decades. It is common to all these attempts, to conceive physical reality as a field, and moreover, one which is a generalisation of the gravitational field, and in which the field law is a generalisation of the law for the pure gravitational field. After long probing I believe that I have now found  the most natural form for this generalisation, but I have not yet been able to find out whether this generalised law can stand up against the facts of experience.

The question of the particular field law is secondary in the preceding general considerations. At the present time, the main question is whether a field theory of the kind here contemplated can lead to the goal at all. By this is meant a theory which describes exhaustively physical reality, including four-dimensional space, by a field. The present-day generation of physicists is inclined to answer this question in the negative. In conformity with the present form of the quantum theory, it believes that the state of a system cannot be specified directly, but only in an indirect way by a statement of the statistics of the results of measurement attainable on the system. The conviction prevails that the experimentally assured duality of nature (corpuscular and wave structure) can be realised only by such a weakening of the concept of reality. I think that such a far-reaching theoretical renunciation is not for the present justified by our actual knowledge, and that one should not desist from pursuing to the end the path of the relativistic field theory.

Arbitrariness comes from having no definite reference point. Einstein’s theory of pure gravitational field sets up Minkowski space as a reference point. Realistically, it is the reference point of field-space of frequency 1 Hertz. Beyond this reference point we simply have emptiness (absence of all phenomena).

Einstein admits that the general principle of relativity has led one into many different directions. No precise theory has come out of these efforts. But Einstein sounds optimistic. He is probably looking at revising the general principle itself.

Einstein’s question is, “Does a field theory, of the kind contemplated here, lead to the goal at all, i.e. exhaustive description of physical reality?” Quantum mechanics believes that the state of a system cannot be specified directly. It depends on the statistics of the results of measurement attainable on the system. Its basic assumption is the conviction in duality of nature (corpuscular and wave-structure).

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Final Comment

The future research is now pointed in the direction of examining the general principle of relativity more closely. This principle needs to be reformulated in a way such that it explains the duality of nature presented in corpuscular and wave-structure.

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