Presented at
Physics Beyond Relativity 2019 conference
Prague, Czech Republic, October 20, 2019
(invited presentation)
Logical Analysis of Relativity Theory

Through analysing Kepler’s massive data on the observed motion of planets, Newton reached the conclusion that our planets are orbiting around the Sun and the cause of such circular motion is the gravitational pull (centripetal force) exerted by the Sun on the planets. Upon this discovery, Newton developed the first theory of dynamics in human history. This should be understood as the first major example of how empiricism contributed to building a most important theory of physics.

Interestingly, this success also gives a stern warning to the popularized thesis that this process is one-way only. Indeed, as we will show in what follows, that Newton’s dynamics refutes Galileo’s famous reclining tower experiment.

Since the famous reclining tower experiment of Galileo, students of physics have been told that contrary to the ancient claim of Aristotle, the speed of a falling light object and that of a heavy object were the same when dropped from the same height.

In what follows, we will see that, according to Newtonian mechanics, Aristotle was correct, namely that the heavier the mass the faster it falls.

Assume $m$ and $m\prime$ are two masses with $m>m\prime$. Let $r$ be the distance from the centre of the Earth to both $m$ and $m\prime$. Let $M$ be the mass of the Earth. The gravitational force between the Earth and $m$ is $F=GMm/r^{2}.$ So,

where $a_{m}$ is the absolute acceleration of $m$ due to the gravitational pull by $M$. The absolute acceleration for $M$ in the absolute space is $-a_{M}$. Therefore the relative acceleration between $M$ and $m$ is $a_{M}+a_{m}=GM/r^{2}+Gm/r^{2}.$ The time required for $m$ and $m\prime$ to reach $M$ is not the same unless $m=m\prime$. If $m>m\prime$, we have

This means “The heavier the mass the faster it falls.”

This is a clear example of how inaccurate experiment could readily lead us to the wrong conclusion. Physics community believed in this wrong prediction by Galileo for four centuries until we brought up this problem just a few years ago.

Galileo assumed that each observer in this universe has his/her own reference frame in which physical phenomena take place. He further assumed that the observer is stationary in his/her own reference frame and that some objects are stationary in his/her frame while others are in motion in his/her frame. This means that the entire frame of another observer moving in his frame is moving inside his frame.

All of this makes the observed phenomena appear different in different reference frames. To resolve this problem, Galileo assumed that every reference frame has its own independent status and that reference frames are connected to each other through the so-called Principle of Relativity which asserts that all valid laws of physics must be shared by all reference frames.

Newton objected to this view of Galileo and presented a strikingly different view of relativism (relative motion). Within the absolute theory of dynamics as discussed above, Newton defined “relative motion” as the difference between two absolute motions in the absolute frame of the universe. So, unlike Galileo’s view, relative motions are to be observed only by an observer standing on the shoulder of God from outside of the physical universe. For Galileo, the relative motions are to be observed by the moving observer.

Without this absolute perspective, Galileo’s relativity theory suffered from a fundamental obscurity. It has always been a controversial issue in relativity theory as to whether if A observes B moving with speed $v$, B should observe that A is moving with speed $-v$. Some relativists agree and some disagree. After all, assuming this is violating the most important assumption of “symmetry” which is essential to relativism. Why a difference in the sign? To begin with, this issue raises a fundamental question on the validity of Galilean relativity theory and its extension, Einsteinian relativity theory. This certainly gives rise to a most fundamental question of what do we mean by “observation”. As an “empiricism”, relativism has a duty to make this point clear.

In the fast developing phase of the theoretical physics this fundamental difference between these views of the two giants have been overlooked. A major purpose of this monograph is to make the relation between them articulate so as to clear the existing confusion.

Galileo had not articulated laws of physics as axioms. He experimentally proved that regardless of the magnitude of “masses”, two masses take the same time to reach the ground when released from the same height. Naturally, he had no theory to precisely prove this experimental fact. So, it was impossible for his contemporaries to evaluate the validity (or the effectiveness) of the principle of relativity. A century later, Newton’s dynamics presented legitimate laws of physics and it was discovered that when we consider two Galilean reference frames that are in acceleration with respect to each other the principle of relativity fails. The standard argument goes as follows: Assume a train is in acceleration on a track. On the embankment, there is a tree. An observer on the embankment will observe that the train is under force. Another observer on the embankment will observe that the tree is under acceleration. So, Post-Newtonian era Galilean relativists resolved this problem by rejecting reference frames under acceleration. In this way they abandoned most of Newton’s dynamics. From this point on, researchers stopped paying much attention to this problem until it resurfaced in the late 19th century when the Michelson-Morley experiment threw physics into a serious crisis from which Einstein’s special theory of relativity emerged.

What is not clear is whether this argument involves the applicability of the Newtonian laws of physics to Galilean relativity theory. Newton assumed these laws only for the absolute frame. This problem can be more clearly explicated as follows: Assume we allow moving reference frames inside Newton’s absolute frame. Assume $m$ and $M$ are attracting each other with the gravitational force. If we consider a reference frame for $M$ that moves inside the absolute frame of Newton, we end up with violating the third law of Newton’s as $M$ is not moving in this frame. So, there is no hope for expecting Newton’s law of physics to be shared by all reference frames.

The Newtonian interpretation above is based upon the observation from the outside of the universe. Galileo was observing it from the frame of himself and the reclining tower, which is the frame of $M$. This will give us an entirely different conclusion. According to Galileo, $M$ does not move and so the acceleration $a_{m}$ between $m$ and $M$ is $GM/r^{2}.$ Similarly, the acceleration $a_{m^{\prime}}$ between $m^{\prime}$ and $M$ is $GM/r^{2}.$ So both $m$ and $m\prime$ will reach the ground within the same time interval. This result is false as the reasoning behind it ignored the action-reaction law in between $m$ and $M$. In classical dynamics we accept the assumption of point mass.

This deep philosophical problem will resurface in the modern setting when Einsteinian relativists present the light bend observation as the empirical verification of the claim of the general theory of relativity.

The theory of electromagnetism was started by Coulomb as an action at a distance theory of electromagnetic forces. The first systematic theorization was done by Gauss and Weber. As Faraday’s work, such as Faraday induction, made the theory more complex than Newton’s gravitational theory, Heaviside and Hertz moved towards a theory that was based upon electromagnetic force field diverting from the action at a distance theory. Having been influenced by Heaviside and Hertz, Maxwell reluctantly formalized the electromagnetic theory as a field theory. However, together with Lorentz, Maxwell was trying to deploy an action-reaction-based interpretation of the field-based theory to compensate for some fundamental deficiencies of the electromagnetic field theory. This is known as “electromagnetic aether theory”. Maxwell’s electromagnetic field equations are

Here $\mathbf{J}$ is the current vector that transcends electromagnetic force field and $\rho$ is the charge density of the current.

The first major impact of the introduction of the force field concept into the formulation of the electromagnetic theory was the apparent increase of the mass of a charge in the field in terms of acceleration. This problem already appeared in the fluid dynamics of Stokes.

(1) J. J. Thomson discovered that “energy of electromagnetic field” $E_{em}$ gives raise to an additional mass $m=4E_{em}/3c^{2}$ to charged bodies and called it electromagnetic mass (renormalized mass).

(2) Using the “momentum of electromagnetic fields”, Poincaré showed that these fields contribute an additional mass $E_{em}/c^{2}$ to a charged body.

(3) J.J. Thomson further observed that electromagnetic mass (renormalized mass) increases as the velocity of the mass increases.

(4) Summing up the above, Lorentz concluded that the ratio of the electron’s mass in the moving frame and that of the ether frame is $k^{3}\epsilon$ parallel to the direction of motion (longitudinal), and $k\varepsilon$ perpendicular to the direction of motion (transverse) where $k=\sqrt{1-v^{2}/c^{2}}$ and $\epsilon$ is a constant. Setting $\epsilon=1,$ Lorentz calculated the expressions for the electromagnetic masses in these directions as

This means that well before Einstein’s relativity theory, Thomson and Lorentz concluded that the “speed of a charged mass” could not exceed $c$. 

Indeed, what is astounding here is that this work on renormalized mass put electromagnetic theory in direct conflict with classical dynamics. In the latter, mass will not be affected by its speed. The direct link between speed and mass immediately violates the second law of Newton where force is determined only by acceleration as mass is invariant. This manifests itself further as the violation of the second law of Newton as in the so-called Lorentz force we have $\mathbf{F}=q(\mathbf{E+v\times}\ \mathbf{B}).$

All of this implies that the issue of “electromagnetic mass” is still alive and to be investigated. The most urgent problem is that Lorentz force makes the electromagnetic field theory inconsistent as it also uses Newton’s second law.

Regarding Poincarè’s electromagnetic mass, what is outstanding is that he obtained it from the electromagnetic momentum instead of the electromagnetic energy. J. J. Thomson obtained it from electromagnetic energy. We know that energy is a modal concept and not a physical reality.

As proponents of Newton’s dynamics, Michelson and Morley felt urged to find out the absolute speed of our planet in the absolute space. Their idea was that when we measure the speed of light in all directions possible, then in the direction of the motion of our planet the speed of light will be maximum. To their astonishment they found out that the measured speed of light is the same in all directions. They subsequently concluded that the speed of light is constant $c$ regardless of the speed of the emitter of light, in symbols $c+v=c$. Regarding this apparent contradiction, there are at least two major issues to be discussed before coming up with hasty conclusions.

(1) Experiment in general is not as simple a matter as we tend to think. Indeed, the great mathematician/philosopher Russell already warned us that experimental verification is viciously circular as to design an experiment, we use a theory whose validity is to be verified by the experiment. In the same sense, he pointed out that experimental refutation of a theory is also viciously circular. To design an experiment to refute a theory requires the use of the theory to be refuted. This gives rise to the questioning on the most fundamental role of experiments in physics, which, unlike engineering, is a theory. Logical coherence is a most important issue of physics.

(2) It is not quite clear what Michelson and Morley really did. Were they measuring the speed of light in the absolute frame or in the frame of the apparatus? All we can say at this stage is that, as Newton implied, the absolute speed of anything cannot be empirically known unless we stand on the shoulder of God. Was this not what made Michelson and Morley want to measure our absolute speed in the universe?

All of these are rather philosophical and conceptual questions. One of the major deficiencies of the current practice of physics is the failure to consider these conceptually important questions. The Michelson-Morley experiment is one of the main examples of this deficiency. Despite its overvalued importance, the more we think the more we get lost.

The first group of physicists who took the result of the Michelson-Morley experiment very seriously were researchers in the electromagnetic field theory, which was expected as the theory of light was a part of the EM field theory of Maxwell. Fitzgerald prematurely concluded that a moving body shrinks in the direction of the motion. Theoretically, there is no such thing as physical bodies as Newton reduced all of them to point masses in order to make the mathematics work to form a theory of dynamics. This is to say that a body is not the subject of theoretical study in dynamics. (In modern day term what Fitzgerald called body is a massively complex system of particles, each of which obeys the laws of quantum mechanics.) It was Lorentz who embraced this idea of Fitzgerald and developed it into the concept of what we now call Lorentz transformation. For Lorentz, who was not a relativist, this transformation however was limited only to between the absolute electromagnetic field and a frame moving inside it.

Einstein thought that Lorentz transformation, which was limited just to the absolute electromagnetic field frame and a moving frame, can be generalized to any two reference frames, thus removing the concept of the absolute frame. As discussed above, after Newton’s dynamics, relativists removed accelerating reference frames to avoid violating the principle of relativity, the action-reaction law in particular. Einstein was not an exception. Now the Galilean relativity theory is limited only to inertial reference frames.

To this Galilean relativity theory of inertial frames, Einstein added the axiom of the Constancy of the Speed of Light (CSL) reflecting the Michelson-Morley experiment. The CSL says that in all inertial frames the speed of light is constant $c$. So, if light in a frame F is observed by an observer A in an inertial frame F and by another observer A’ in an inertial frame F’, both A and A’ will observe that the same light moves with speed $c$.

Notwithstanding, based upon the axiom of CSL, Einstein went on to obtain time dilation (TD) and length contraction (LC) through a thought experiment. Now, the simplest way to explicate Einstein’s argument for TD and LC is to present a thought experiment that exposes the fundamental inconsistency of the theory of inertial reference frames with the CSL axiom. We call this thought experiment “the (power pole)-(power line) paradox.

It goes like this. Assume a train runs on a track. When the tip of the power pole of the train touches the power line at point P a spark occurs at P. An observer located in the train straight down the point P will observe that the light comes straight down to him/her from the point P, which is the tip of the power pole. Also the observer will see that the same light comes to him/her diagonally from the point P, which also is a stationary point of the power line. This is a contradiction. and it is consistent with Aristotle’s warning that a point on a line may not be a part of the line. the same is held by contemporary topologists but in a more modern way, namely that real numbers (even rational numbers on the real line) are defined through a limit. Therefore there is no finite access to any real number on a real line. If we cannot access it, how can we move it? If we cannot move even a single point in our 3D space, how can we move the entire 3D space inside another 3D space? In short, topology says that a point does not exist on a topological space.

So what about the issue of TD and LC? The simplest way to refute these two claims is such that topology refutes the possibility of moving 3D space inside another space. So, the setting of the thought experiment for proving TD and LC is invalid. It is no wonder that these two concepts caused all kinds of contradictions.

We tend to take mathematics we use in physics lightly just as a language. This is a perfect example of the price we pay for our ignorance and arrogance. Mathematical results at the level of topology, etc., are obtained with utmost care and precision. So often, unless we pay due attention and effort to understand, we take the results wrongly and end up with this kind of devastating mistakes.

Without knowing of the falsity of TD and LC, Einstein showed that from TD and LC alone, without relating to electromagnetic field theory, we can obtain Lorentz transformation. From time dilation

and length contraction

Lorentz transformation is obtained as

The proof goes as follows: By applying the effect of length contraction on Galilean transformation we obtain $x^{\prime}=(x-vt)/\sqrt{1-(v/c)^{2}}.$ Length contraction in the opposite direction is

Solving these two equations for $t^{\prime}$, we have

A common argument for proving TD from LT

goes as follows: Set $x=0$, then we have

A more careful logical analysis shows that what this really showed was that transformed time depends upon the location of the clock! It did not prove that LT implies TD. To the contrary, it refuted this claim. To be precise, it showed that when observed at $x=0$, time dilates with the gamma factor. TD states that, observed from anywhere on the $x$-axis, time dilates with the gamma factor. This is an interesting instance of the same formula meaning entirely different things depending upon the context it was obtained in. This is possible because there is more going on in physics that is visible on the surface of mathematical symbols.

This puts us in a delicate situation where we have to question the equivalence between Minkowski’s special theory of relativity, which does not use TD and LC, and Einstein’s special theory of relativity, which uses them. This further makes us wonder about the validity of the currently held belief that the general theory of relativity is a generalization of Einstein’s special theory of relativity. The general theory of relativity includes not Einstein’s special theory of relativity but Minkowski’s special theory (tangentially).

The Lorentz transformation plays yet other questionable roles. We can shown that this transformation fails to respect Newton’s law of gravitation, Coulombs’ law, Newton’s second law and wave equations. For example, despite the claimed advantage of conserving wave equations, Lorentz transformation astoundingly fails to conserve the more fundamental second law and the law of gravitation. Considering the way Lorentz transformation was obtained, it is not surprising that these two major laws of mechanics are not Lorentz invariant. This means that Lorentz transformation is not a relativistic transformation as it violates the principle of relativity.

We can further show that the claimed invariance of wave equations under Lorentz transformation is false. To make the argument more articulate, let us discuss the issue under a general setting.

Similarly

Similarly

This is valid only under the condition $v=c=\omega.$ The second equality comes from the fact that $\omega$ is the wave speed. The first equation implies that the frame speed is $c$ which is not possible in the special theory of relativity. This means that Einstein’s claim that the electromagnetic wave equation is invariant under the Lorentz transformation is invalid. It is a well understood fact that there is no reference frame for light at the pain of contradiction. If $v=\omega$ then the gamma factor becomes undefined and there is no Lorentz transformation for such a frame.

All of this was well expected logically. Lorentz transformation is defined in terms of the constant $c$, which is the speed of electromagnetic waves in vacuum. So, there is no convincing reason why this transformation will conserve wave equations, which are not electromagnetic wave equations of Maxwell.

There are two criticisms of the way how this historic experiment was treated in theoretical physics. This experiment was interpreted under the assumption that light is an electromagnetic wave, in accordance with Hertz’s proposal. Even to this day though, we are not sure what light is. We were told to accept the view of Hertz without a substantial proof thereof.

Moreover, electromagnetic waves are not physical reality as the concept of electromagnetic field is not a physical reality. This concept is what logicians and philosophers call “modality”, “counter factual modality” to be precise. The spatial distribution of electromagnetic force per a unit charge is not reality. Such distribution should appear in reality only when we place unit charge everywhere in the space. But, if we place a unit charge at every point in the electromagnetic field, the source which created such a field will be affected and the electromagnetic field will not be maintained. Moreover, the placed unit charges will react with each other making it impossible to sustain such a configuration. Putting it more mathematically, there is not enough charges to fill all points in a space. Pure mathematics will put this as follows: There are uncountably many points in a geometric space and there are only countably many charges in the universe. This implies that the concept of electromagnetic field has nothing to do with reality. To be precise, the so-called electromagnetic wave is an “action at a distance transmission” of the change in electromagnetism at the source to a charge placed at a certain location in the space. There is no wave. This is precisely why without wave medium the so-called electromagnetic waves “travel” with speed $c$. So, there is no need for the “aether”. There is no physical realism that supports the counter-factual modality. All of this calls to question the treatment of light as an electromagnetic wave by the Michelson-Morley experiment.

Galilean theory of relativity did not consider reference frames that are under acceleration relative to each other. This was because acceleration, through the second law, violates the principle of relativity. This only restriction imposed by kinematics on relativity theory was too limiting for Einstein. Considering that way before this setback, already at the most basic level of Galilean theory of relativity the concept of relativity is insurrectionist, Einstein should have abandoned the idea of relativity. It is unfortunately not what happened.

The ultimate product of this highly questionable wave-particle duality manifested itself most dramatically in the quantum electrodynamics (QED) of Dirac. Quantum electrodynamics seems to be the ultimate end-product of Einstein’s special theory of relativity, which is inconsistent. Here is a summary of the complications we have regarding this issue.

1. 1.

   The reflection of light on the mirror is a complex problem of statistical particle physics and only material science can bring us closer to the truth. The problem we have here is that we have not such material science. Also the material science which digs into this kind of problems must come from a satisfactory quantum mechanics, which we have not, as our quantum mechanics is based upon the special theory of relativity, which came from the Michelson-Morley experiment. To make the matter even worse, now Wheeler’s result seems to support the concern that Michelson-Morley’s experiment could have been interpreted wrongly.

2. 2.

   Even if the Michelson-Morley experiment is limited to electromagnetic waves only, these waves are not physical reality. They are counter-factual modality. Moreover, when we operate upon the assumption of wave-particle duality, the uncertainty principle creeps in and this makes the constancy of the speed of light claim “statistical”. When a most fundamental assumption of our theory is of statistical nature, we do have some serious concern. True, the original CSL argument is not statistical. It was based upon abstract (not physical) wave theory of counter-factual modal waves. But the recent argument by Wheeler again is statistical.

Early quantum mechanics in principle avoided getting into this kind of problems when the problem of particles enclosed by walls etc. was considered. Walls are represented not as a complex material, but as potential barriers, which is nothing but a mathematical entity. Here, one is trying to study the reflection of light at the mirror. Quantum mechanics cannot really handle the reflection of light on a mirror as we do not understand what mirrors are sufficiently enough to discuss these issues. To understand it requires perfect quantum mechanics, which we have not.

We use macro materials, which are microscopically incredibly complex, to do our experiments. It is not a problem when we do macro-level physics. But when we deal with micro-level physics, we have no solution. Our experimental instruments belong to the macro-level physics. The cosmology shares the same problem. We can never conduct experiment or measurement at this incredibly large scale level. It is not promising to do physics of the cosmos through looking at billions of stars light years away from us. The only tool we have available here is the relativistic Doppler effect and we do not even know what light is.

What is striking is that regardless of the status of quantum mechanics, Wheeler’s experiment shows that while passing the second splitter, the light “knows” which part it came out from the first splitter. In the second splitter we get the Hadamard-Walsh gate, which is an “operator” in quantum computing. This is sufficient for engineering. But it is unfortunate that our theory will not make us understand all of this.

Philosophically, what we are facing is a crisis where the most basic laws of physics are no longer macroscopic laws. To make it macroscopic we went through a statistical argument, which makes us wonder what laws of physics are.

Moreover, all of these problems by which we become overwhelmed make us wonder what do we really mean by experimental observation, which is the essence of empiricism. To make experiments at the micro-level, we need a functional theory of the micro-level physics. But such a theory must come from an acceptable micro-level experiment. We are going around a big circle. As we said above, cosmology is facing the same difficulty at the other end of the spectrum. In the case of micro-level physics, we have a distinguished difficulty of uncertainty issue which makes the theory probabilistic. All of these issues have to be dealt with even after we manage to free quantum physics from the inconsistency of relativity theory. A long way to go lies ahead of us.

So, it is no longer just an isolated problem of theoretical physics. The dynamic linkage between theory and empiricism has to be re-examined and new working link must be established.

Einstein assumed that if an accelerating reference frame can be reduced to an inertial frame in which acceleration induces “gravitational field”, then it is possible to treat accelerating frames as inertial frames within the theory of relativity, which rejects reference frames under acceleration for a legitimate reason. He called this the principle of equivalence. To that end he proceed as follows:

Assume a spaceship is in inertial motion in our reference frame. Moreover, a force accelerates this spaceship with a rate $\mathbf{\alpha}.$ A body $m$ in the spaceship experiences a force $\mathbf{f}$, which is due to the acceleration of the spaceship that makes the body $m$ move with an acceleration of rate $\mathbf{a}$ in the frame of the spaceship. Putting aside what the force $\mathbf{f}$ is, this means $\mathbf{f}=m\mathbf{a}.$ Then from our perspective, $m$ in the spaceship experiences the acceleration with rate $\mathbf{\alpha}+\mathbf{a}.$ So, $m$ will experience $\mathbf{f}=m(\mathbf{\alpha}+\mathbf{a}).$ Therefore,

This means that, from our perspective, the acceleration $\mathbf{\alpha}$ on the spaceship induces an “additional” force $-m\mathbf{\alpha}$ on $m,$ which Einstein called “inertial force” upon the mass $m$ and the equation (IF) yields the force $m$ experiences in the accelerating spaceship. This Einstein called the second law in the accelerating frame of the spaceship. According to him, upon the modification of $\mathbf{f}$ to $\mathbf{f}-m\mathbf{\alpha}$, the second law is conserved under the choice of accelerating reference frames.

There are several issues to be considered with respect to that.

1. 1.

   Putting aside the invalidity of the special theory of relativity (as we discussed in the foregoing), some issues have been overlooked here. Namely, according to the special theory of relativity, even addition of speeds is not classical addition. One has to use the so-called relativistic addition of speeds $v\oplus v^{\prime}.$ So, how can the addition of accelerations be the same as classical addition, hoping that in the words of Einstein, acceleration is still the time derivative of speed. This confusion is closely related to the issue of the disjointedness between the acceleration and relativistic reference frames. For that reason the special theory of relativity does not consider acceleration, as we have been told.

2. 2.

   As an important example, this inertial force is also closely related to the issue of “fictitious force” on a mass inside an orbiting object. Fictitious force is a “force in fiction”, not reality. The reason why we have a problem with the fictitious force for an orbiting spaceship is because orbiting spaceships are under centripetal acceleration. Therefore, it is not an inertial frame. It is wrong to claim that just because inside the orbiting spaceship a fictitious force called centrifugal force is “created?”, the orbiting spaceship becomes an inertial reference frame.

3. 3.

   This fictitious force is the creation of the relativistic interpretation of the second law of Newton. This law can be interpreted in two ways. First, when force $\mathbf{f}$ is applied to a mass $m$ , it accelerates the mass with the rate $\mathbf{a}=\mathbf{f}/m.$ Second, when $m$ is accelerating with a rate $\mathbf{a}$, a “fictitious force” $\mathbf{f}$ appears. Ontologically the second interpretation is invalid. It is always the case that a force applied to a mass causes an acceleration of the mass. It becomes clear when we consider the case of an accelerating train. Assume a stationary train accelerates. Then, a passenger in the train will feel that he/she is pushed back (against the direction of the train’s acceleration). This is simply because the observer tries to stay where he/she is in the train due to the first law of Newton. If the observer can see the outside, which is not moving, he/she will see that it is not him/her but the train that is moving under the force.

4. 4.

   Now it is clear that the problem of inertial force (fictitious force) is caused by the faulty relativistic interpretation of the second law of Newton. This is to say that the second law is not “relativistic”, further confirming that the relativism as per Galileo and Einstein is untenable. With this, we have that the relativism violates not only the third law of Newton but also the second law.

5. 5.

   It also is important to note that the fictitious force violates the third law of Newton.

An even more fundamental issue here is considering the spaceship (or train). In the theory of dynamics, as Newton made it clear, theoretically there is no such thing as a spaceship. All physical bodies must be point masses, and rightly so. For this reason, there is no such thing as a mass $m$ inside a spaceship. In dynamics, there is no such thing as a spaceship to begin with, for very good mathematical reasons, as incisively explained by Newton.

Moreover, a body attached firmly to the wall of a spaceship will not experience any special force. This is because when we consider the dynamics of the spaceship it is just a point mass and there is no “inside” of it to which this extra mass is to be attached. Even if we reluctantly allow spaceships, according to the law of inertia without force exerted, a mass will continue constant speed motion in a frame. So, what force is supposed to be exerted upon this body “inside” the spaceship”? Does this firmly attached mass move inside the spaceship?

The usual response is that we experience such force even if we are firmly attached to the inside wall.

However, our body is not just a solid. Our body is beyond the category of physical objects. Our body has incredibly complex internal system for perception. This is why we feel such a pressure.

The most fundamental reason why Newton correctly reduced all moving masses to point masses is simple. It is purely mathematical and conceptual. Newton correctly observed that the best we can do is to consider a physical body as a point object with size of a geometric point. Without this assumption, how can one define motion mathematically? With this assumption Newton found a solid mathematical representation of motion in a space as a function from time to space. With this he obtained the legitimate concept of speed as the first order derivative of the motion function and the legitimate concept of acceleration as the first order derivative of the speed function. Without all of this basics, we have no theory of motion upon which to build dynamics. Mathematics is an essential part of physics and not just a language for physics.

Moreover, for dynamics, we have yet another important reason to reduce a mass to a point mass. It is because force is a vector, a pointed entity. So, the only entity to which we can exert a force is a point object (mass).

There are some more issues to be discussed regarding the “gravitational field” that Einstein introduced to a space under acceleration. As discussed in the foregoing, the concept of any force field, in general, violates the action-reaction law, which in turn violates the principle of relativity. Moreover, the gravitational field that Einstein introduced to an accelerating space is a force field which has no source for the gravitational forces spreading all over the space. This is yet another violation of the third law in a different sense. The “uniform gravitational field” near the surface of our planet is a gravitational field in “approximation”.

More importantly, the “force field”, whatever it may be, that Einstein introduced to frames under acceleration is not gravitational at all. Gravitational fields are to be the modal representation of the effect of gravitational force created by Newton’s law of gravity. So, it is not a uniform field. A possible argument to counter that would be that near the surface of the Earth the gravitational field is “almost” uniform. This is however not acceptable in precise science such as theoretical physics. Almost uniform is not uniform. The Universe is not approximate.

It appears that the idea of associating “acceleration” with “gravitational force” comes from the old concept of “aether” by Descartes. Descartes wrongly considered a spaceship that contains an object which is under acceleration and identified the fictitious (inertial) force with gravitational force. As the name “fictitious force” clearly indicates, such a force is just an imaginary force that in fact does not exist. It is not that the acceleration exerts such a force but as the object in the accelerating frame is not a part of the frame (spaceship) it appears that everything in the spaceship moves with acceleration relative to the object. Hence, it is not a real force. It is a fictitious force.

Einstein assumed a laboratory that is free falling under the gravitational force. Assume we emit a light beam upward from the floor to the ceiling. Due to the acceleration, by the time the light reaches the ceiling, the ceiling is moving faster than the source on the floor was when the light beam left it. In other words, the receiver at the ceiling is approaching the source (where it was when the light left). Therefore the observer in the lab will notice the blue shift due to the Doppler effect. This will make this observer notice the downward acceleration. This contradicts the equivalence principle which states that a free falling body will not notice its free falling. So, Einstein postulated that there must be a red shift due to the light moving upwards against the gravitational force to offset this blue shift. Unfortunately, it is not the observer in the lab who sees that the ceiling is moving towards the floor. It is an observer outside of the falling lab who will see that the ceiling is falling towards where the floor used to be. So, the observer inside the lab will not observe the blue shift. This is how Einstein obtained the red shift effect.

The situation is rather complex and we have to consider many elements involved in this apparently simple thought experiment. This is an instance of the confusion coming from the ambiguity that a rest point in a frame F1 is also a moving point in a reference frame F2 that is moving relative to F1. The problem which haunted the special theory of relativity, which we presented as the “power pole-power line paradox” in the context of time dilation and length contraction, came back to haunt the general theory of relativity, in the context of accelerating frames this time. As inertial frames and accelerating frames are both moving frames, the same problem affects both of them. So the equivalence principle which is to reduce an accelerating frame to an “inertial frame” with induced gravitational field has no capability to avoid this fatal contradiction.

Moreover, the connection between relativistic dynamics and the general theory of relativity is not clear at all. Are they equivalent? Both of them are supposed to be the relativity theory of dynamics dealing with more than relativistic kinematics, which is known as the special theory of relativity. We know the former is inconsistent as it violates the limitation to non-accelerating frames, which is badly needed to avoid the contradiction against the third law of dynamics, an absolutely essential assumption for any dynamics.

Putting aside the problem associated with the inconsistency of the relativistic dynamics, there are some more lessons to be learned from using inconsistent theories such as the special theory of relativity, relativistic dynamics and the general theory of relativity. Neither the special theory of relativity nor relativistic dynamics can handle energy. What about the general theory of relativity, which is based upon the equivalence principle? Unfortunately the answer is negative as well. As we have discussed at the beginning of this section, the special theory of relativity claims that the addition of speeds should not be $v+v^{\prime}$ but $v\oplus v^{\prime}.$ This contradicts the second law of dynamics.

All of this clearly suggests that Newton’s dynamics is a theory of dynamics in the absolute frame only. Relativity is introduced not in the way Galileo and Einstein did in their relativity theory. Relative motions are defined as the difference of two absolute motions. All attempts to relativize Newton’s dynamics thus far have failed rather expectedly.

As discussed above, when we represent a constant speed motion in the 3D Euclidean space as its graph, it becomes a straight line in the 4D space-time. If the motion is an accelerating one, then the graph becomes a curved line in the 4D space-time. Einstein represented gravitational force field, which will cause accelerating motions, as the 4D space-time manifold so that all motions caused by the gravitational field will appear as straight lines (geodesics) in the manifold. Applying this idea to gravitational fields created by a system of masses, Einstein obtained the so-called gravitational field theory. In this theory the spatial distribution of masses determines the 4D space-time manifold representing all possible “motions”.

According to the classical dynamics, when we have a system of masses in a space, the mutual gravitational pull makes them converge to a single point called the centre of gravity. It is unclear how this process of convergence is dealt with in the gravitational field theory of Einstein. This question relates to the capacity of this theory to express dynamic processes.

Einstein studied the rest mass $0$ photon under constant acceleration. He considered a photon moving with speed $c$ in the $x$-direction while it is in a frame that is under acceleration $a$ in the $y$-direction. So, we have

If $\theta$ is the angle made by a tangent of the light ray to the $x$-axis, we have

and we can assume that $\theta$ is very small. So we have

But the GTR predicts otherwise, i.e.

Note that Eddington proved experimentally that the GTR’s prediction $(3)$ is correct.

It is clear that all of this uses nothing but the kinematic concept of acceleration and in kinematics there is no concept of mass. Putting aside the issue of inconsistency coming from the assumption of the mass $0$ point mass as discussed in section 3.1.3, the concept of photon belongs to dynamics. The photon is a point mass of mass $0$. Having no mass and having mass $0$ are entirely different categories. The problem with considering a point object whose mass is $0$ is that the second law fails for the mass $0$ point object. So, it makes no sense to say that the light (trajectory of photon) bends due to the gravitational force of the Sun. Thus, Einstein’s argument here diverted this difficulty by replacing the “gravitational force upon mass $0$ photon” with the “reference frame of photon accelerated by the Sun’s gravitation”. It is truly astounding that this even worse confusion has never been detected till now. The reference frame of this photon cannot be accelerated unless there is some mass stationary in it. The second law was never meant to be applied to mass zero objects.

So, the only apparently appropriate thing to say here is that the claim that light bends due to the acceleration of the reference frame caused by the gravitation force is also false. In the end, we do not know what is really happening here. If the curved 4D spacetime GTR predicts the equation (3), then clearly something that went wrong in the development of 4D spacetime GTR. Unfortunately, Einstein’s rest mass $0$ particle that moves with speed $c$ was brought into physics with serious consequences.

As discussed in the foregoing, the concept of any force field in general violates the action-reaction law, and in turn violates the principle of relativity. Moreover, the gravitational field that Einstein introduced to an accelerating space is a force field that has no “external source” creating the “gravitational forces per unit mass” spreading all over the space. This is yet another violation of the third law in a different sense. The “uniform gravitational field” $g$ near the surface of our planet is a gravitational field in “approximation”. It is “not” a gravitational field. So, it is wrong to call Einstein’s “induced” force field a gravitational force field.

To simplify this already confusing situation, let us put it this way. In classical post-Newtonian dynamics, a field of gravitational force applied to each location on a unit mass placed there was considered. This concept itself fundamentally violates the action-reaction law and so must be abandoned. The situation here with Einstein’s gravitational field created by the empty reference frame under acceleration makes things even more problematic. The former ignored the source mass that created the force field. Here such a source mass to be ignored does not exist at all.

We must stop identifying entirely different things in approximation as it was done in quantum field theory.

Moreover, the force field that Einstein introduced to the frame of the spaceship is not gravitational at all. Gravitational fields are to be the field representation of the effect of gravitational force created by Newton’s law of gravity. So, it is not a uniform field.

It was Minkowski’s 4D spacetime theory of relativity which taught Einstein that the motion line (geodesic) in the 4D spacetime bends in the presence of acceleration. Combining this with the second law of dynamics, which connects acceleration and force through mass, and the law of gravity, Einstein concluded that through gravitational force masses distributed in the 4D spacetime bend the 4D spacetime. Now the 4D space itself is curved and the motion line is a straight line (geodesic) in this curved 4D spacetime. So, there is no need for TD, LC and Lorentz transformation. But then what about Einstein’s claim that Minkowski spacetime is a local tangential space-time in the curved spacetime?

Judging from the argument to claim the equivalence principle, it appears that Einstein was considering only local situations, such as a spaceship, as an accelerating frame. In this local setting it appears possible to treat all accelerating frames as inertial frames. Without knowing that the equivalence principle is false, which we explain above, Einstein made a move to represent all accelerating reference frames as “local inertial frames” in which acceleration is replaced by the induced gravitational field. This ill-fated idea lead Einstein to consider the general coordinate system upon which all accelerating frames are treated as local inertial frames with induced gravitational field. Hence, Einstein moved on to develop the concept of general absolute reference frames to which we will turn in what follows.

Einstein assumed that the whole cosmos is occupied by a fluid whose molecules are “clocks” of any variety. This fluid can flow in any manner except that it will be assumed that there is no turbulence, so that neighbouring molecules always have almost equal “speed” and the velocity of the flow is a continuous function.

Each clock is allocated three coordinates $(x_{1},x_{2},x_{3})$ in such a manner that:

1. 1.

   No two clocks will have the same coordinates, and

2. 2.

   Neighbouring clocks have neighbouring coordinates. Therefore, coordinates are also continuous with respect to spatial displacement.

3. 3.

   It is understood that the coordinate of each clock remains the same through time. As time elapses at each clock, its readings are assumed to increase but the rate of increase is not necessarily uniform as compared with a local standard clock. No attempt is made to synchronize distant clocks. Neighbouring clocks are assumed to be “sufficiently synchronized” so that the clock readings are continuous with respect to spatial displacement.

4. 4.

   The reading of a clock will be denoted by $x_{0}.$

It is unfortunate that this paradigm is not possible for the following reasons:

1. a)

   No clock has any specific coordinate as it is not a point object.

2. b)

   In the continuum, there is no such thing as a point next to another point. So, the concept of a “neighbouring point” to a point is invalid. There is no such thing as a rational number next to a given rational number. This is because in between any two rational numbers, we can always find a rational number. This property is called the density of the set of rational numbers.

3. c)

   As pointed out in b), there is no such thing as the coordinate of a clock in this setting. Clocks are made by continuumly many points. They are by themselves very complex physical structures. In a) Einstein assumes the fluid of clocks and yet in c) he says that the coordinate of each clock remains the same.

More generally, the following further questions remain to be answered.

1. (1)

   Upon what time and space the mechanics of such molecule clocks is defined? Each clock is a physical system and so it is operating in a spacetime that is not the same as the spacetime defined by the clock. This is to say that the spacetime $(x_{0},x_{1},x_{2},x_{3})$ does not define the inside dynamics of the clock at $(x_{0},x_{1},x_{2},x_{3})$. Moreover, where is the clock which governs the spacetime in which this clock operates? According to the general theory of relativity, the time of this spacetime $(x_{0},x_{1},x_{2},x_{3})$ and that of the spacetime in which this clock operates are not the same and how much they are synchronized depends upon the location of the clock that defines the spacetime that defines the clock. This problem is related to a more general problem associated with the instrumentalist view of time as clocks. This view falls into the following vicious circle: The clock, which is supposed to define time, must operate as a dynamical system upon some time and space. Then how this time and space are supposed to be defined? 

2. (2)

   It is a common understanding among researchers in “dynamical system theory” that time has a special status and is different from all other coordinates of the system. This is in agreement with the idea of Newton in his classical dynamics. Newton said that time, unlike other coordinates, has a natural flow that “moves” forward only. This makes it impossible to consider time as reading of clocks. Time is an entity that transcends empiricism and operationalism.

   Once we violate the most fundamental assumption on time, anything can happen and relativity apparently made it happen.

3. (3)

   Clocks are physical entities. There are at most countably many clocks in this universe. No mater how closely we put clocks together, we cannot form a continuum of clocks. No matter how one puts countably infinite particles together, one will not create a mathematical continuum. This is mathematically the same problem as the problem of photons, which are supposed to exist for each frequency: as the frequency has a continuous spectrum, there must exist uncountably infinite particles called photons. Countably infinite points will never form real continuum. We need continuumly many points to form mathematical continuum.

4. (4)

   What does it mean to be sufficiently synchronized? The concept of synchronization presupposes external absolute time which contradicts the concept of relativism. Here, we have to check time of each clock at precisely the same moment in absolute time.

Suppose that at a point P in a “gravitational field”, which is sea of infinitely many clocks, a freely falling non-rotational (relative to distant stars) local inertial frame is “constructed”. We further assume that the axioms of the special theory of relativity are valid within this frame as it is supposed to be an inertial reference frame. So, we can set up a Cartesian coordinate system $(\emph{Px,Py,Pz)}$ at this point P. Furthermore, we can distribute clocks over the frame, all of them synchronized to the clock at $P$.

As we assumed that the universe is a sea of clocks that are not overall synchronized, this implies that such a coordinate system $(Px,Py,Pz)$ is not universal. It is a local coordinate system around $P$.

Using this frame and clocks, events that occur in the vicinity of P over a suitably restricted time period, can be allocated space-time coordinates $(t,x,y,z)$.

It is not quite clear why the time period must be restricted.

Now suppose that in this local inertial frame a pair of neighboring events have space-time coordinates $(t,x,y,z)$ and $(t+dt,x+dx,y+dy,z+dz).$ Then, if $d(\tau)$ such that

is Lorentz invariant, then it also is called the Minkowski distance. This serves as the correct metric on the 4D Minkowskian spacetime.

There are some mathematical problems regarding this metric on the “local” 4D spacetime. (1): It is not a topological metric that is used in topology. This is to say that Minkowski 4D space time is not a metric space. (2): The Minkowski distance between two events that happen at the same time is zero regardless of the 3D geometric distance between these two events. (3): Here, Einstein is assuming that in this “freely falling” local inertial frame, in which a gravitational field is induced by the equivalence principle, all clocks are synchronized. The local inertial frame must be accompanied by a gravitational field. So, all of these clocks are under gravitational acceleration. How is it possible then that all of these clocks are synchronized? Einstein also claims that all clocks under acceleration slow down. Do they slow down uniformly? As the acceleration is inertial “locally”, and time dilation is relative to the inertial speed, this slowdown is not uniform at all. 

After all, as we have shown in the foregoing, the Minkowski distance has no relevance to theoretical physics. It is mathematically irrelevant too as it is not a topological metric. This clearly shows where relativity theory should be placed in science. It is neither physics nor mathematics. It appears that it belongs to its own category.

The local inertial frame is suitable only for the description of very limited situations. For a larger scale (temporal, as well as spatial) issues, it is necessary to use one of the general reference frames. If $x_{i}$ are space-time coordinates relative to such a general frame, transformations of the form $x_{i}\prime=\pi(x_{0},x_{1},x_{2},x_{3})$ must exist relating $(x_{0},x_{1},x_{2},x_{3})$ to $(t,x,y,z)$ [the local inertial frame]  such that

Then, if $x_{i}$ are subjected to increments $dx_{i}$, the corresponding increments in $t,x,y,z$ will be given by

etc. and substitution in equation of proper time interval

will result in an expression $d\tau{{}^{2}}$ that is quadratic in the increments $dx$, i.e. whose terms will either involve squares of the $dx_{i}$ or the product of two different $dx_{i}$. Thus

where the coefficients $(g_{ij})$ will be the functions of $x_{i}$.

Now, a continuum in which the interval between neighbouring points is given by a quadratic form like $(R)$ is called a “Riemannian space” and the quadratic form like $(R)$ is called its metric. Thus, the space-time continuum is a four-dimensional Riemannian space whose interval is everywhere identified with the proper time interval between neighbouring events in a local inertial frame.

Here are some issues to be discussed:

1. 1.

   What is important for physics is not that we use 4D spacetime manifold of Riemann. What has been questioned here is the relevance of such a mathematical structure to physics. This structure $(R)$ is obtained by substitution

   $$dx=(\partial(\theta)/\partial(x_{0}))dx_{0}+(\partial(\pi)/\partial(x_{1}))dx_{1}+(\partial(\psi)/\partial(x_{2}))dx_{2}+(\partial(\gamma)/\partial(x_{3}))dx_{3}$$

   etc. in the equation of proper time interval

   $$(d(\tau)){{}^{2}}=(dt){{}^{2}}-(1/c)((dx){{}^{2}}+(dy){{}^{2}}+(dz){{}^{2}}).$$

   As we have shown, this whole mathematical argument makes little physical sense. A most serious flaw in all of this is that the theory of general relativity was obtained from the special theory of relativity which is both mathematically and ontologically inconsistent. So, the general theory is false too. As discussed above, the special theory of relativity yields the relativistic addition of speed which contradicts the addition of acceleration of a mass which is governed by the second law of Newton. The equivalence principle, which is the most fundamental assumption of the general theory of relativity, is based upon the second law of dynamics. Einstein explains the curved space in the general theory of relativity using a rotating disc; the radius of the disk does not contract as the motion of the disk is not along this direction. It comes under the influence of the length contraction around the perimeter of the disk as the motion is in the direction of the tangential speed of the point on the perimeter of the disk. There are two errors in this argument. Firstly, the tangential speed of a circular motion is not inertial, it is under acceleration and so, the length contraction should not apply. Secondly, length contraction is false.

2. 2.

   More fundamentally, Einstein was clearly not aware of the difference between a countably infinite and a continuum. Cantor’s diagonal argument clearly shows that there are more points in the geometric continuum than the discrete collection of points. The Lebesgue integral of the Weierstrass function over $[0,1]$ shows that the geometric continuum has unimaginably more points than the “space” of countably many points has. For example, on the real number line almost all points are irrational numbers. So, one cannot cover the entire global space with clocks as there are only finitely many clocks. This makes the most fundamental assumptions of Einstein’s general theory of relativity untenable. There is no such thing as the “global spacetime” prescribed by Einstein. define neither differentiation nor integration.

3. 3.

   Also, Einstein’s description of the clocks used to define the global spacetime is off. To begin with, it must be required that the neighbouring clocks are of infinitesimal distance and the time difference between each pair of neighbouring clocks must be infinitesimally small. Otherwise we cannot use calculus to calculate on such a structure. Physically, it is impossible to make enough clocks to do this and place them in the way expected, as we stated above. As an infinitesimal means a number that comes in between $0$ and any positive number, this method is ontologically untenable. Clearly Einstein was unaware of what infinitesimals were as this concept was articulated only later in 1960’s by Abraham Robinson. Even in pure mathematics, the work of Robinson is understood only by a small number of mathematical logicians. The general theory of relativity was developed before the development of Robinson’s infinitesimal calculus and so it is understandable that the Hilbert school of mathematics and physics did not have the concept of infinitesimals. By the time Robinson’s work came out, the separation between pure mathematics and theoretical physics became material and communication between these two communities became almost non-existent.

4. 4.

   In short, contrary to what Einstein proposed, the universe cannot be a sea of clocks.

   In addition to this topological problem the general theory of relativity suffers, there is an even more fundamental issue of logical deficiency in the idea of the general reference frame which is the sea of clocks. Clocks are physical entities and it requires physics to make them. One cannot use clocks to define clocks at the pain of vicious circle. So, there is no such thing as metaphysical clocks though time is certainly a metaphysical entity, as Newton thought. It was relativity theory, the special and the general, which tried to use empirical clocks that lead the world of physics to the current confusion about time.

   Logically speaking, modern physics started with the wrong idea of what is time. Contrary to the special theory of relativity, time cannot be defined in terms of speed as speed is defined in terms of time. And as we have discussed here, the universe is not a sea of clocks contrary to the general theory of relativity. From the combination of these wrong assumptions, it is expected that we ended up with questioning what time is.

   Our understanding of time as in relativity theory is completely wrong.

We have shown that the general theory of relativity came from the special theory of relativity, which is false. Therefore, the general theory of relativity is also false. Any theory which contains an inconsistent theory is inconsistent.

When we express a linear function of one variable on a 2D space, then the function’s graph becomes a straight line. The coefficient of the first order variable is the slope of the line. This idea was extensively exploited by train companies to visualize train operation on a 2D space where one coordinate is the time coordinate and the other coordinate is the location coordinate expressed at the distance from the origin station. It is called “operational diagram”, which pure mathematicians came to treat as a 4D spacetime. In the 4D space time all constant speed 3D motions should be just straight lines and the slope of the line is the constant speed 3D motion.

So, in 4D spacetime geometry of 3D motions, the “Euclidean geometric distance” between two points $P_{1}(t_{1},x_{1},y_{1},z_{1})$ and $P_{2}(t_{2},x_{2},y_{2},z_{2})$ in the 4D spacetime is:

The slope of the line segment $P_{1}P_{2}$ is given as

In general theory of relativity, this changes: the distance, as par Minkowski, between $P_{1}$ and $P_{2}$ is

It can be shown that this distance is smaller than any other “Euclidean distance” along any curve connecting $P_{1}$ and $P_{2}$ in the 4D space. It goes as follows. Clearly,

But for any path $l_{P_{1},P_{2}}$ between $P_{1}$ and $P_{2}$,

where $\overline{l_{P_{1},P_{2}}}$ is the Euclidean length of $l_{P_{1},P_{2}}.$

From this, relativity theory concludes that the 4D spacetime with Minkowski metric is not a Euclidean space but a Riemannian space that is curved. There are some unclear issues to be addressed here.

1. 1.

   In Euclidean geometry, we define the distance between two points P and Q using the Pythagorean formula and we show that the length of any path connecting the two points P and Q is longer or equal to the distance between them. We have not seen how to define the length of a curved path between P and Q in Minkowski 4D spacetime.

2. 2.

   Putting this purely mathematical issue aside, the only reasoning for admitting $\widetilde{P_{1}P_{2}}$ , the Minkowski distance between $P_{1}$ and $P_{2}$, is because this distance is invariant under Lorentz transformation. There are two questions to be answered here however. (1) Why Lorentz transformation is the issue in the general theory of relativity? This transformation belongs to the special theory of relativity. (2) This transformation is said to be essential for the special theory of relativity as it is said to preserve all wave equations and axioms of Maxwell’s electromagnetic field theory. (3) We have shown that this claim is false in our paper “Reference Frame Transformations and Quantization”.

Nonetheless ,Einstein continued to develop his theory as follows: Under the equivalence principle,

From this, Einstein concludes that the free fall in the gravitational field is a geodesics [in the 4D spacetime]. This is called the “geodesic principle”. Under this principle we obtain equations of motion for bodies falling freely in a gravitational field.

Using Riemannian geometry for the “general metric”

we can show the following “general equation of a geodesic”

For the metric

these equations reduce to

This is equivalent to

where $a,b,c$ are constants.

The discussion above shows that $g_{ij}$ determines the motion (geodesic) in general relativity theory. Given the energy-momentum tensor $T_{ij}$ which describes the distribution of mass, energy and momentum of the system, Einstein’s equations of gravitation yield corresponding $g_{ij}$ enabling one to calculate the geodesics of the system. It was Schwarzschild who first obtained an exact solution of Einstein’s equations for a spherically symmetric field. He used the solutions to calculate the motion of a planet in the field of its sun.

Putting aside the problems of the general theory of relativity as discussed above, this result of Einstein and Schwarzschild poses some issues to consider. The question is what this momentum-energy tensor is about. There are several issues to be cleared.

1. 1.

   The momentum-energy relation is a problem. The former is a predicative concept but the latter, as the potential to do work, is a modal concept, and so, connecting them in the same category is not the right thing to do.

2. 2.

   Logically speaking waves in wave mechanics have no momentum. This is because momentum is the product of a mass and its speed. In wave mechanics, no mass moves in the direction of the wave. What is moving towards the direction of the wave is the local vibration of the medium. So, waves have no momentum. When it comes to energy, as the work needed to accelerate from $m0$  to $mv$ is not necessarily $(1/2)mv^{2},$ the concept of kinetic energy is false. The work needed for this acceleration depends upon how we accelerate from $m0$ to $mv.$

3. 3.

   So, the energy-momentum relation does not represent the state of a physical system properly. Then how can the energy-momentum tensor describe the physical system properly?

4. 4.

   Is the energy-momentum relation as energy-momentum tensor used here classical or relativistic? If it is relativistic, the entire argument by Einstein becomes viciously circular. If not, then how does it relate to the claim that the classical physics is invalid. This is a contradiction, is it not? Classically we have some tension between momentum and energy. Moreover, the relativistic energy-momentum relation is also false. It is because $e=mc^{2}$ is false as we discussed earlier in this paper. This equation came from the false assumption that the $v$ in the gamma factor in $m$ is time dependent, which is not allowed in relativity theory. Considering that the concept of energy is not a physical reality but a modality and understanding that, as philosophy asserts, the modality and reality are of different category, it is astounding that the momentum-energy tensor plays the most fundamental role in general relativity theory. Mathematics is not just a language for physics. It is the only way to understand physical nature around us. It is the most articulate way of thinking correctly.

5. 5.

   What is the most fundamental issue here is that the practice of bootstrapping classical mechanics to relativistic mechanics is not a legitimate thing to do.