Newtonian Origin of General Relativity and Impossibility to Unify the Fundamental Forces

by Salim Georges Khoury

This paper shows that there exists in "Classical Newtonian Mechanics" a relationship between the curvature of space and the density of energy identical to that of "General Relativity". It is as if the General Theory of Relativity has replaced, in this relationship of Newtonian Mechanics the Scalar quantities (space curvature and Energy density) by their equivalent Tensors. From this point of view, the Theory of General Relativity can be considered an extension of Newton's classical mechanics, an extension from Euclidian flat space-time to curved space-time of Riemann. This paper also demonstrates the impossibility to having for the other fundamental forces a Theory equivalent to that of General Relativity.

INTRODUCTION

In General Theory of Relativity [1], [2], [3], the Einstein's equations relate the geometry of spactime curvature (expressed by the Eistein tensor) with the distribution of matter within it (expressed by the stress-energy tensor).

where

     is the Eistein tensor, it depends on the coordinates,

     is the stress-energy tensor, it depends on the mass-energy distribution,

    G is the Newtonian constant of gravitation and c is the velocity of light in vacuum.

The coeeficient is called The Einstein gravitational constant. 

The tensors in Einstein's equations are symmetric of dimension 4x4. so we have ten different equations to solve. The term of the tensor are the entries of the problem. the goal is to find the relation between the coordinates which describes the system and which could be found in each term of the tensor . The search for exact solutions to the equation is therefore an extremely complicated exercice.

The solving Einstein's equations is only possible in very simple cases like:

- A spherical distribution of static mass-energy : this is the Schwarzschild metric that results in the calculation of the horizon of the black hole or Scharzschild radius [4], [5],

- Weak field and low velocity approximation, the ten components of Einstein's equation reduce to a single equation which is the Poisson equaion of Newton gravity [6].

Despite the resolution of Einstein's equation for very particular cases, the General Relativity remains a very difficult theory to assimilate and Newton's mechanics is considered as an approximation to this theory.

The objective of this paper is not to present a new theory, nor to find new solutions to Einstein's equations but to show that there exists in Newton's mechanics a relation between the scalar curvature in Euclidean flat space-time and the density of energy similar to that of the General Relativity with the equivalent tensors in the curved space-time of Riemann.

Einstein was probably inspired by this relation in Classical Mechanics to establish the equation of his theory, this is important for the physicists to know this and then to understand that the theory of General Relativity can be considered as an extension of Newton's Classical Mechanic: an extension from Euclidean flat space-time to curved space-time of Riemann.

This paper also explains why the four fundamental forces cannot be unified within the frame of the theory of the General Relativity.

DESCRIPTION

Consider a point mass which is not subject to any external force. According to the Newton's first law, its motion is uniform rectilinear and its velocity is constant equal to

Figure 1: uniform rectilnear motion

If this point mass passes near a mass with , it starts to turn around under the effect of gravitational force (Newton's law attraction). Its trajectory is quasi-circular with radius .

Figure 2: uniform circular motion

First Remark:
The mass seems to curve the linear space of the mass  but not in the meaning of space-time curvature of General Relativity.

The force exerted by the mass on is, according to Newton's law of attraction:

                (1)

According to the Newton's second law of motion, we can write

                     (2)    

Therefore

                   (3)   

which is equal at the same time to the centripetal acceleration of the uniform circular motion

                        (4)   

hence

                     (5)     

                      (6)     

Multiplying and dividing the second term by

               (7)           

          = curvature of the circular trajectory of radius  (according to the definition of curvature).
     = fictitious areal density of M. It is as if this mass M is homogeneously distributed (by its effect or its gravitational field) over the surface of the sphere of radius where the mass moves.

Equation (7) is a good example to understand this relationship of proportionality between curvature of space and density of mass (or density of energy, see below equation 18).

Indeed, when the radius   decreases(*) both the curvature   of a smaller circle and the area density of mass   on a smaller sphere increase and in the same ratio. 

(*) according to (5) the radius 

                        (8)      

decreases when the velocity of rotation of the mass increases

                     (9)         

We will demonstrate now that the relation (7) between the curvature of space and the density of mass is independant of the variation of the velocity ,

Demonstration

Multiply the velocity  of   around the the same mass   by the factor

 

 which corresponds, according to equation (9), to a radius divided by

 

the equation (7) becomes

 

 

The curvature   and the density of mass   have been multiplied by, the proportionality factor   remained the same as before multiplying the velocity by .

Second Remark:

When the velocity is multiplied by a factor , the curvature and the mass density will be multiplied by , the proportionality factor   between these two quantities remains the same, it is independant of the variation  of the velocity . We can therefore choose a reference velocity .

Which value should we choose for ?
It makes more sens to choose as a reference velocity  , for a mass in uniform circular motion, the maximum velocity it can on its orbit.

The refernce velocity is given by the equation (9)

             (10)

Its maximum value corresponds to the maximum escape velocity   [7]. is calculated according to the principle of energy conservation : 

kinetic energy + potential energy = 0

                    (11)

 

 taking into account equation (10)

                   (12)

therefore

        (13)

we replace by its value in (7) we obtain :

         (14)

What becomes of this equation (14) with the Special Relativity?
According to Special Relativity [8] and [9]

                  (15)

                           (16)

Equation (16) allows us to write

                              (17)

where represents the celerity of light, and   the energy associated with the the mass .

By replacing  and  by their value in the equation (14), we obtain :

                   (18)

where

           = curvature of the cicular trajectory of radius

     = areal energy density of M. This energy is distributed over the surface of the sphere of radius where the mass   moves.

 

Third Remark:

Equation (18) shows a relationship between curvature and energy density similar to the equation of General Relativity (19)

                  (19)

where
      = Einstein tensor, it represents the curvature of space-time
     = Energy-impulse tensor, it represents the total energy density (mass and energy)

Let's continue our analysis to find the meaning of . From (11) and (15), we deduce

         (20)

is the minimum radius of the orbit that corresponds to the maximum escape velocity  . Below this value the escape velocity becomes greater than , and since it is impossible, the mass can never escape and the mass becomes a black hole. is the Schwarzschild radius in General Relativity [10]. For the sun, = 2.948km. This means that if we concentrate the mass of the sun is a sphere with a radius less than , it becomes a black hole for everything that passes at a distance less than from it.

Fourth Remark:

The value of the reference velocity chosen in the equation of General Relativity corresponds to the maximum escape velocity or to the orbit whose radius ia equal to Schwaezchild radius.

 

Fifth Remark:

If the inertial mass in of equation (2) were not equal to the gravitational mass in of equation (1), we would not have had equation (3) to compare it to equation (4) and then establish equation (7) or (18). This explains why Einstein sait that Newton's principle of equivalence between inertial mass and gravitational mass was the starting point of the famous theory of General Relativity.

Sixth Remark:

The principle of equivalence cannot be applied to electric force, because we have a mass on one side and a charge on the other side

            (21)

where "q" is the charge of mass and "Q" is the charge of mass , therefore, we cannot simplify to an equation similar to equation (3), i.e.

                        (22)

to then compare it to the equation (4) and establish an equation for the electric charge equivalent to equation (18), that is to say :

                (23)

The conclusion is that we cannot have a relation between a "charge density tensor" and "space curvature  tensor" equivalent to the relation of Einstein's General Relativity. Therefore, it is impossible to unify the four fundamental forces, especially the gravitational force and the electromagnetic force, into one theory ithin the frame of General Relativity [11]. 

CONCLUSION

This paper is a new approach in the understanding of the General Theory of Relativity. It shows the origin of its equations in Newton's mechanics and it enables a better understanding of this relationship between the curvature of space and the density of energy without going through the non-intuitive tensorial equation of this theory. The paper also shows the importance of the principle of equivalence "gravitational mass and inertial mass" in the establishment of the equation of this theory. It demonstrates the impossibility to having an euivalent theory for the other fundamental forces like the electric force. In other words, the unification of the fudamental forces is impossible within the frame of the teheory of General Relativity.

REFERENCES

[1]    A. Einstein, Die Feldgleichungen der Gravitation, Sitzung der physikalish-mathematicschen, 25 November 1915, pp. 844-847

[2]    Henk W.J. Blöte, Introduction to General Relativity, Lorentz Institute for Theoretical Physics, March 2016

[3]    D.F. Lawden, introduction to tensor calculus, Relativity and  Cosmology, Dover Publications, 2003, pp. 6-10

[4]    K. Schwarzschild, Uber das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie, Sitzung de phys.math, 23 März 1916, pp. 424-434

[5]    The Schwarzschild Metric and Application, http://physics.gmu.edu/~joe/PHYS428/Topic10.pdf

[6]    E. Bertschinger, Gravitation in the Weak-Field Limit, Massachusetts Institute of Technology, spring 2000

[7]     Adrian G. Cornejo, The equivalent expressions between escape velocity and orbital velocity, Lat. Am. J. Phys. Educ. Vol. 4, N0 3, Sept. 2010, pp. 575-579

[8]     A. Einstein, 30 June1905, Zur Elektrodynamik bewegter Körper, Annalen der Physik, 17, 891-921

[9]    A. Einstein, 27 September 1905, Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig? Annalen der Physik, 18, 639-641

[10]    Hall, Michael J.W., General Relativity, An introduction to Black Hole, Gravitational Waves, and Cosmology, Morgan  Claypool Publishers, 2018, Chapter 8.

[11]    Wang LJ. Classical unification of gravitational and electromagnetic forces (2020) Edelweiss Appli Sci Tech 4:9-18.

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