V.6 No 1


The problem of physical time in today physics

5. Discrepancies of 4-D geometrisation in the relativistic conception

The above shown contradiction of relativistic postulatory basis to the physical reality has reflected not only on the way, how the SRT and its structure were formed with multiple paradoxes, but on the Minkowski 4-D world on the whole.

From the very beginning, relativists were sure that their formalism is in a strong agreement with physical reality: “ “The views on space and time which I would like to tell you have originated from the experimental physics – this is their power. These views are radical. Since now, space as such and time as such have to become merely shadows, only some kind of their unification has to remain independent”. So the mathematician Herman Minkowski began his speech at the 80th Assemblage of German natural scientists and doctors in Koln on 21 September, 1908” [24, p. 72]. With it, “The further application of these formally equal rights of space and time coordinates brought a perfectly distinctive statement of the theory of relativity that made its applications considerably easier. The physical events are drawn in the 4-D world and space-time relationships between them are in this 4-D world the geometric theorems” [29, p. 186].

At the same time, we already showed: as opposite to the Newtonian separation of time into relative and absolute, in the relativistic conception just the relative time has been taken as that absolute and geometrised through attributing the time scale to the light beam propagation between the points of space. The problem that arises in this connection and produces paradoxes, just as the very issue of time geometrisation, is an exceptional ‘achievement’ of relativists. It was absent in classical physics, as the approach to the problem of space and time did not assume joining of these very different forms of description of material bodies’ location and motion in some joint characteristic. As Einstein thought, “in Newton’s mechanics, space and time take a dual part. First of all they are for the objects of physics the carriers or frame relative to which the events are described with the spatial coordinates and time. Basically, the substance is thought consisting of the ‘material points’ whose motion produces the physical event. If the substance is thought continuous, this is done only in cases when we do not want or cannot describe its discrete structure. In this case the small parts (elements of volume) of substance are interpreted like material points, at least until we are interesting only in motions, not in phenomena which we at the moment cannot or may not attribute to motions (e.g., changes of temperature, chemical processes). The second part of space and time was that they served ‘the inertial system’. Of all thought reference systems, the inertial systems were thought privileged because concerning them the law of inertia is true. With it, an essential circumstance is that ‘the physical reality’ existed independently of subjects cognising it was thought consisting, at least basically, of space and time, on one hand, and of permanently existing material points moving relative to the space and time – on the other. The idea of independent existence of space and time can be expressed as follows: should the substance disappeared, there would remain only space and time (some kind of scene on which the physical phenomena play)” [16, p. 750].

In this citation Einstein on one hand correctly pointed the essence of epistemology of classical physics, though the point of scene where space and time play in absence of weighty matter is surely exaggerated. The relativistic understanding that only in classical mechanics the substance is thought as material points is incorrect, the more – relativistic confidence that their approach gives the field description of material substance: “Which new ideas developing the foundations of physics after Newton allowed to overcome the exceptional type of inertial systems? First of all, involving the idea of field to Faraday’s and Maxwell’s theory of electromagnetic phenomena (? – Authors) or, rather, involving the field as an independent, non-reducible basic concept. As far as we are able to judge now, we can think the general theory of relativity only as the field theory. We would be unable to create it adhering the point that the real world consists of material points moving under affection of forces of their interaction” [25, p. 854].

At the same time, we should not identify the generality of classical phenomenology with the particular modelling of processes, the more reduce the classical approach exceptionally to the meaning the material substance as an amount of points. In particular, Maxwell whose formulas, separated from the physical meaning, relativists used specially emphasised that he based on the idea of material spaces, not on the amount of points: “Still it was done to begin studying the laws of electric and magnetic forces with an assumption (! – Authors) that the cause of these phenomena are attractive and repulsive forces between the definite points. We would like to consider this issue from another point, more suitable to our study – namely, determining the value and direction of forces under discussion with the help of speed and direction of motion of non-compressed liquid” [10, p. 17]. Furthermore, relativists in their studies use the same concept of material point, to distract from the parameters that excessively complicate the problem: “Let in the electromagnetic field moves a point particle with an electric charge gepsilon.gif (832 bytes) …” [4, p. 32]. Or so: “We yield in the frame K the shape of body moving relative to this system, determining the points in the frame K with which at definite time t the material points of moving body coincide” [30, p. 187]. Or so: “In the diagram we can show also the motion of body whose size we may not neglect; then the world lines of all points of such body will occupy the space within the tube that we call the world tube [24, p. 73–74].

Moreover, as if avoiding to model the material bodies as points, Einstein in the same work “Relativistic theory of asymmetric field” finishes his study with an admission which not simply reduces the modelling to the system of points but introduces the concept of field as of the system of discrete entities; so he finally discredits the merits which Relativity voluntarily took for itself: “We can convincingly prove that the reality cannot at all be represented as a continuous field. It seemingly follows from quantum phenomena that the finite system with the finite energy can be completely described by the finite set of numbers (quantum numbers). This, seemingly, is incompatible with the theory of continuum and requires a mere algebraic theory to describe the reality ” [25, p. 873]. We can add to this: in the light of results yielded above, the principle of relativity on whose basis relativists have built their conception is applicable to the description of field processes neither directly nor indirectly. Then the modelling of processes through joining the space and time into some geometric abstraction reduces more to the question, whether such geometrisation is legal, and of its form, and whether this form describes the reality correctly.

To answer these questions, we have, following Newton, to understand first the difference between the particular modelling and phenomenology, between the particular and general, to determine correctly the place and part of absolute space and time in classical physics (and in the relativistic formalism, too). It follows from Newton’s citation in the introduction that the absolute space and time are philosophical categories used to generalise the results of particular experiments. They as such, without relative space and time, are senseless, at least until the scientists are able to attribute the reference frame to the aether. But even in this case, the absolute time, due to its specific, will remain some abstraction distracted from real errors of particular measurement of time, and so being convenient when generalising the experimental results. Despite convenience of relative concept of time and specific conception of absolute time, the time itself does not lose its main feature described as long ago as by Aristotle: “That time either does not exist at all or barely [exists], we can premise, basing on the following. One part of it was, and already it is not, another – will be, and it still is not; the infinite time is composed also of these parts, and every time the selected [interval of] time [is composed so]. This what is composed of non-existent is seemed to be unable to have relation to the existence. Furthermore, for each divisible entity, if only it exists, it is necessary, until it exists, either all or several its parts to exist, while in time which [also is] divisible some parts already were, other – will be, and nothing exists. But ‘now’ is not a part, as a part does not measure the whole which has to be composed of parts; and time, to all appearances, is not composed of ‘now’. Then, it is not easy to see, whether ‘nows’ that apparently separate the past and future, remain always integer and identical either [become] each time other. If it was always other and other and in time no one part does not exist with the other (except of covering and covered, as smaller time is covered by bigger), and non-existing now but existed before disappeared once by necessity, ‘nows’ will not [exist] with each other but the previous always must annihilate” [33, p. 145–146].

Proceeding from this, if material bodies are absent in space, space does not disappear as such, though the orientation is fully lost and any measurements, even abstractly-absolute, are senseless. So we can think space without material substance only in philosophical categories and in generalisation of some concepts and regularities in physics. While in reality, as soon as we say, we intend to measure, we necessarily have to introduce to the space at least four mutually resting material points that do not belong to any three crossing planes at the same time, to relate to them some reference frame. Introducing this frame, we already make the space not empty and not homogeneous. But time, by its features, yet is fully absent, it appears only when events change. While, if in our space nothing moves, the change of events that could give us a sense of time is absent, too. So the statement of relativists that “the word ‘time’ is usually used for two different values that in the language of mathematics are denoted by symbols ds and dt. These values essentially differ from each other; for example, ds is invariant, while dt is not, dt is a full differential, ds is not” [7, p. 48] is basically incorrect from the point of basic properties of time. It is even not the matter, what is a full differential, what is not. The spatial coordinates are involved to the definition of ds, which makes time tangible in the past and future in their parts, and from the philosophical point we always can select the values of spatial coordinates so that the 4-D interval physically appeared in the past and this past co-existed with the present and future. Time disallows it. It appears only when at least one material point appears whose location relative to four points will change. And though some periodical process with which we will compare the relative time yet can be absent, none the less, from the point of epistemology of classical physics, the absolute time is already present, as the change of location of this moving point in space already changes relative to the physical points of metric. Consequently, we can establish some sequence of locations of this moving point relative to the reference frame related to the resting points. Neglecting this approach and substituting the concept of time by its sector that is necessary, the light to pass the given interval, relativists unavoidably come to the fact that in their constructions “ discrepancies arise naturally, when we try to find an answer to such indefinite (from the relativistic point – Authors) questions as, e.g., whether the time is absolute, or whether for two observers between two their meetings the same time passes” [7, p. 48]. It is corroborated by our above results, where we showed that through the very fact of use of mathematical tool of classical physics, relativists involuntarily and independently of their interpretations retained the absolute space in which the equiphase surfaces of field excitations do not transform. In this way they came to the basic contradiction with their basic postulates and destroyed their own ‘logical’ constructions and thought out experiments. And should Einstein, before publishing his work of 1905, plotted for himself these equiphase surfaces, not building the conception on the ‘conventional opinion’ and guesses, the theory of relativity in its known appearance would never be built even not as the theory but as the hypothesis, as the absolute space and absolute time have been put into the basis by the very mathematical regularities describing the light propagation in space, and we cannot run it around either escape otherwise.

Here we meet an interesting question, whether the frame taken in classical physics as the absolute really is absolutely resting? As we showed in one of previous works [14], generally – of course, no. The absolute frame also is introduced not for sake of mere abstraction but as the frame with which we can conveniently interrelate the motions of other frames which we think relative. So, dependently on the problems we solve, the surface of Earth, or Sun, or the centre of our Galaxy, etc. can serve as the absolute frame. While, being absolute for definite types of problems, these frames are at the same time the relative frames for other types of problems. Though among multitude of such frames there exists one about which Newton could not suspect when stated so: “For it may be that there is no body really at rest, to which the places and motions of others may be referred” [2, p. 32]. His frame relates to the aether which Newton premised, but having no experimental data, had no possibility to study. The advantage of this frame is not the mechanical properties of the aether with which the founders of Relativity fought, as if zealously: “We can assume the ether existing, we only should not care to attribute to it a definite state of motion; in other words, abstracting, we have to take from it the last mechanical sign which Lorentz still retained for it” [34, p. 685]. The advantage of such frame is in the feature of which Newton said – that the aether is responsible for the transmission of interaction between bodies. But even in this case we cannot be finally sure that the aether or its parts do not move. So even in frames of Newtonian space we may not surely state the frame to be absolutely at rest in the universe but may speak only of the frame which we take as absolutely resting. We refer the absolute frame to the aether because, admitting all physical fields existing and propagating because of aether, we naturally refer the absolute frame to the location resting in relation to this material substance, since the propagation of all excitations in this luminiferous medium will be isotropic relative to this frame; due to this, we can conveniently study the anisotropy that arises in all moving frames.

In the relativistic conception, as we showed above, in absence of material bodies, the gravity and EM fields are absent, too; hence, the matter, space and time are absent also. As opposite to the classical conception in which some static material space can be thought without time, “the main what the theory of relativity teaches us is perhaps that the concepts of space and time cannot be thought independently of each other but only in their amount giving the 4-D description of phenomena, i.e. description in the language of space-time[35, p. 18]. With it, despite the claims of relativists that “since the reference frames to which we can refer things do not exist in nature, we will refer the 4-D multiplicity first to the fully arbitrary coordinates (corresponding to the Gaussian coordinates in the theory of surfaces) and limit the choice of frames only when the considered problem impels us” [36, p. 305], their limitation has the exceptional cause: “the choice of privileged frame is possible so that to provide the principle of constant speed of light” [36, p. 305]. This last, as we have revealed in the previous section of this paper, from the physical point is realisable only in the frame resting relative to the aether, which is admitted the absolute in classical physics just because in this frame the speed of light is isotropic. True, as we showed above, such frame is the only, not multiple, as relativists say. This is just the reason, why their formalism is abundant of paradoxes and corruption of formal logic: they took as an underpinning of their theory the erroneous statement that such frames are multiple, so they have to agree with this mistake all their following theses, making them same mistaken.

And though it seems that now the difference between conceptions is seen more clear, it appears that the four-dimensionality as such still is not the salient feature of such or other competing conceptions.

“Consider five types of space-time, denoting them as follows:

Aristotle’s space-time, (1)

Galilean space-time, (2)

Newtonian space-time, (3)

Minkowski’s space-time, (4)

Einsteinian space-time. (5)

In each case the space-time will be a smooth 4-D multiplicity, but some additional geometric structure reflecting a typical aspect of dynamics will be attributed to it” [35, p. 18].

It follows from this last that not the 4-dimensionality of continuum as such is the salient feature of competing conceptions, not variations of the concept of time flow but the presence of some geometry changing the description of processes in space and time, dependently on the basis on which such or other geometry was developed. “The basic difference of space-times of Minkowski and Einstein from three previous is, no additional concept of time difference between the events is involved in them. Instead, in the space-time there has been determined the pseudo-Riemann metric form ds2 with the hyperbolic normal signature (+, –, –, –). Then the time difference (?! – Authors) between two points A and B of space-time depends on the choice of world line connecting these points and is given through the integral along the world line:


Everywhere along the admissible (i.e., timelike or light) world line ds2 gequaiitymore.gif (834 bytes) 0 , so that gtaucut.gif (827 bytes) appears to be a real parameter. The value gtaucut.gif (827 bytes) determines the time interval (of natural time) between the events A and B measured by (an ideal) clock whose world line coincides with this curve. As the natural time became now the concept dependent on the path, we can return to the definition of ‘geodetic’ as an extreme path. The yielded so system of (timelike) geodetics determines (according to the theory) motion of particles by inertia. As opposite to … the Galilean and Newtonian cases, not the motion by inertia is determined immediately, as soon as we specify the behaviour of (ideal) clock” [35, p. 22–23].

If now we join Penrose’s definition with the definition of Einstein who premised so: “according to the classical mechanics, this 4-D continuum is decomposed objectively into the 1-D temporal and 3-D spatial sections, and only this last contains the simultaneous events. This ‘splitting’ is one and the same for all inertial frames. The simultaneity of two definite events relative to one inertial frame involves the simultaneity of these events relative to all inertial frames. This is what is meant when saying of absolute time in classical mechanics. But according to the special theory of relativity, this is already not so. Though relative to some definite inertial frame there exists a set of events simultaneous with some observed event, this set already will not be independent of the choice of inertial frame. The 4-D continuum does not decompose objectively into the sections among which there would be the sections containing all simultaneous events; for the spatial continuous world the concept ‘now’ loses its objective sense. In this connection the space and time have to be considered as the objectively non-decomposing 4-D continuum, if one wants to express the contents of objective relationships without unnecessary arbitrariness” [16, p. 753], – then we can see the basic difference in two interpretations of relativistic understanding of time. Penrose states that in the relativistic conception no additional concept of time difference between the events is involved in them”, and with it, pointing the features of world line along which (5.1) is true makes quite concrete sections along the timelike world lines. And these quite concrete sections we could see in [11]. True, they were inclined to the time axis, as the expression under the integral sign in (5.1), together with the change of time, contains the change of coordinates of this frame, so tau.gif (827 bytes) already will not determine the time as such. Because of the presence of such regularity, tau.gif (827 bytes) will determine also the spatial arrangement of bodies, as different values of coordinates in the expression under the integral sign expression will give different tau.gif (827 bytes) in one and the same reference frame. Gaining its independence of coordinates, time becomes as tangible as space – this means, separate parts of the past and future can co-exist. But if we think the spatial structure obeying (5.1), the cross-sections are formed in full accordance with this idea of perverted ‘non-simultaneity’, and these cross-sections are in parallel to each other, just as in case of plane of events in classical physics. It is important that the cross-sections remain in general theory of relativity, as without them it would be impossible to determine the curvature of Riemann’s multiplicity: “The radius of curvature of any normal (orthogonal) cross-section of the surface is determined in a well known way” [7, p. 282].

On the other hand, Einstein’s citation emphasises the non-simultaneity of events in passing from one inertial frame to another. As we showed in [11], this is because the Lorentz transforms incline the plane of events. This is an other approach to the problem of time in the relativistic conception, though on the other hand both these approaches are common in the meaning that using the techniques analysed in [11], relativists rejected the phenomenological basis worked out during centuries by the thought exercises, observations and experimentally, and operating only with the mathematical symbols cut away from their physical meaning that substantiated namely this, not other form of regularities, they formed some abstract geometric nucleus, forcing the laws of nature to submit this abstract geometry that is based on the following principle: “If we reveal the difference between the theory and experience, it would be easier to agree with the change of physical laws than with the change of axiomatic Euclidean geometry” [37, p. 86]. Or so: “Thus, if we, doing not knowing the common covariant equations of the gravity field, specialise the frame and compose the gravity field equations only for this special frame, the theory can raise no objections except one, namely – that the composed equations, possibly, have no physical meaning. But in the considered case no one will support this objection seriously” [38, p. 322].

At the same time, geometry is known to be true only and exceptionally because its axioms are some prototypes of natural phenomena. We do not permanently repeat it only because this relation has been initially well-grounded and does not need such repetition. Relativists perfectly know this rule and often use it, doing not proving every time what they think already grounded. So their statement “Can the human mind without any experience, only by way of pondering, to grasp the properties of real things? As far as I can see, the answer is briefly such: if the theorems of mathematics are applied to reflect the real world, they are not exact; they are exact until they don’t refer to the reality. As seems to me, we can make this issue fully clear only with the area of mathematics which is known as ‘axiomatics’ ” [37, p. 83], or “Geometry treats the objects denoted by words ‘direct line’, ‘point’ etc. With it not the knowledge or idea of these things is premised but only the truth of axioms, the same merely formal, i.e. hawing no visual and natural contents” [37, p. 84] – have the only aim: intentionally to break the interrelation between the geometry and experience and to insert into this breakage the abstract postulates of their relativistic geometry based not on the experience but on Fitzgerald’s successful manipulation with the formulas of interference calculation of Michelson’s experiment.

We would mark, the idea to cut the geometry and its axiomatics from reality is far from being new, it was debunked as long ago as at times of Galiley. He answered to it quite simply and specifically: “Wishing to show me that the material sphere is in touch with the material plane not at one point, you use the sphere that is not a sphere and plane that is not a plane, because, as you said, either these things do not exist in the world or, if they are, they deteriorate when applied to the subject. This means, it would be more correct to admit, at least tentatively – namely, should there in nature existed and remained unchanged some perfect spheres and planes, they would touch each other at only one point, and then reject such possibility in reality.

Simplicio. – I think, we have to take the statement of philosophers namely in this sense, as, undoubtedly, the imperfect matter causes things taken specifically do not relate to the things considered abstractly.

Salviati. – Why do not they relate? On the contrary, this what you are saying yourself now proves, they exactly do relate.

Simplicio. – In which way?

Salviati. – Do not you say that, because of imperfect matter, the body which would have to be absolutely spherical and the plane that would have to be absolutely plane specifically appear to be not such as you think them in abstraction?

Simplicio. – Just so I am saying.

Salviati. – It means, each time when you specifically apply the material sphere to the material plane, you apply the imperfect sphere to the imperfect plane and say, they touch each other not at one point. And I say you that in the abstraction, too, the immaterial sphere which is the imperfect sphere can touch an immaterial, as well as an imperfect plane not by one point but by a part of surface. Thus, this what occurs specifically takes place in the abstraction. It would be greatly unexpected if calculations and actions made on the numbers abstractly would then be irrelevant specifically to the silver and gold coins and goods. But do not you know, signor Simplicio, what occurs in fact and how, when calculating sugar, silk and linen, we have to knock off the weight of boxes, wrapper and other tare; the same the philosopher-geometer, wishing to check specifically the results yielded from the abstract proof, has to throw away the obstacle of matter, and if he can do so, I can assure you, it all will tally no less exactly than in the arithmetical calculation. Thus, errors are not in the abstract, not in the specific, not in geometry, not in physics but in the calculator which cannot calculate correctly. So, if you have some perfect sphere and plane, even material, do not doubt, they touch each other at one point. And if you cannot obtain them, none the less, the statement that sphaera aenea non tangit in puncto is quite far from the point. But I can tell you more, signor Simplicio; if I cede you admitting that there cannot exist a perfect material spherical figure or perfect plane – well, how do you think, can there exist two material bodies some part of whose surface will be in some way curved, if you will – even in irregular way?

Simplicio. – I think, we will not feel lack of such.

Salviati. – Whatever would they be, they also can touch at one point, as to touch at one point is not at all an exclusive privilege of absolutely spherical body and perfect plane. On the contrary, he who will consider this issue more attentively will find that it is much more difficult to find two bodies which would touch each other by a part of their surfaces than by a single point, because in order two surfaces to coincide enough with each other, it is necessary, they both to be absolutely plane or, if one is convex, another has to be concave and to have such curvature which would fully correspond to the convexity of this first. It is much more difficult to find such conditions, because they are too definite than others which are infinite by their wide indefiniteness” [39, p. 161–162]. In other words, “Maybe, these mathematical statements that are true in the abstraction will not exactly fit when applied to the physical material circles. But it seems to me that coopers, to find the half-diameter of the bottom that they have to do for a barrel, use an abstract rule of mathematicians, although the bottoms are things quite concrete and material” [39, p. 178].

This follows from the fact that “geometry operates with the concepts that arise from the experience due to the known abstraction of the subjects of material world” [31, p. 37]. And the founder of Relativity E. Mach agreed with it: “experience was admitted as the source of our geometrical concepts” [40, p. 83]. Consequently, in geometry we can deal only with those properties of figures and regularities of their relations which have been multiply corroborated by the experience.

As the geometry as such is unable to cover the whole complex of properties inherent in real objects, it looks natural when we distract from some properties, such as the colour, material, roughness etc. that are not the subject of theorems of geometry. From this, “all other properties which we used to imagine hearing the words ‘point’, ‘direct line’, ‘plane’ take no part in the logic construction of geometry and we have not to mention them in the basic statements of geometry” [31, p. 37]. This limitation of properties of real things by their geometric shape imposes some limitations onto the area where the geometric constructions are true. “In the rigorous logic considerations, when proving the theorems, we have to deal only with these properties of things, – and they have to be marked in the axioms and definitions” [31, p. 37]. “So we have to think the premises of used Geometry as if rigorously proven, with it we have to be sure that irrespectively of the experience, it would be in vain to seek the proof for such truth which yet is not involved per se in our idea of bodies” [41, p. 33]. “The mathematical 3-D figure is called a body, corpus solidum in Latin, which means even a palpable body, thus, its name has been taken not from a free imagination of our mind but from the rough reality” [42, p. 39]. “Thus, geometry is an application of mathematics to the experience concerning the space. Like mathematical physics, it becomes an exact deductive science only because it images the objects of experience as schematic, idealised concepts. Like mechanics can state the constancy of masses or reduce the interaction of a body only to the accelerations in limits of observational errors, we can state with the same reservation the existence of direct lines, planes, the value of sum of the triangle angles etc.” [40, p. 76]. Exceeding the limits in which the axioms agree with the experience, geometry degenerates into a paradoxical abstraction. Some examples of such abstraction are shown in Fig. 5.1.


fig10a.jpg (16165 bytes)

a) Are all horizontal lines parallel to each other?


fig10b.jpg (34874 bytes)

b) Is it a regular spiral? No spiral, these are circles


fig10c.jpg (6108 bytes)

c) Do you see a cube in the corner?

Or a little cube in a larger cube?


fig10d.jpg (17371 bytes)

d) An excessive quadrate in the lower triangle – where from it appeared?


fig10e.jpg (21034 bytes)

e) Is it a regular quadrate?


fig10f.jpg (9728 bytes)

f) Up by stairs that go down

  Fig. 5.1. Examples of paradoxical constructions in geometry



As we could see from citations above, Einstein used just this way of paradoxical constructions when formulated his conception of relativity. He described this approach himself in the paper “Geometry and experience”: “Geometry (G) says nothing of the behaviour of real things; only geometry describes this behaviour, together with the amount of physical laws (P). To express it symbolically, we can say that only the sum (G) + (P) is the subject to check experimentally. Thus, we can arbitrarily choice both (G) and separate parts of (P): all these laws are conventions. To avoid such discrepancies, we have to choose the rest parts of (P) so that (G) and whole (P) together were experimentally valid. With such opinion, the axiomatic geometry, from the view of cognition, is an equivalent of a part of laws of nature elevated to the rank of convention” [37, p. 86].

From this point, nothing of surprise when first Poincare, then Minkowski and Einstein determined the length in the 4-D complex space as


fully ignoring the regularities of the theory of complex variable, as we described it in [12]; when Einstein voluntarily substituted the classical principle of equivalence of frames and added to it the constancy of light speed in all frames [11]; when he equalised the physical laws in inertial and non-inertial frames [12]; when Schwarzschild and other relativists after him ‘forgot’ to return to the initial variables when deriving the Schwarzschild metric [13], and so on.

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