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The Geometrization of the Electromagnetic Field

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*the unity of objective reality as the foundation of the unity of science*. Einstein’s engagement along his unification program was burdened with a number of difficulties and lastly in vain. Nevertheless, a successful geometrization of the gravitational and the electromagnetic fields within the framework of the general theory of relativity is possible. Thus far, it is a purpose of the present contribution to geometrize the electromagnetic field within the framework of the general theory of relativity.

KEYWORDS

1. Introduction

It is very easy to get lost in the many [1] and conceptually somewhat very different attempts at the unified field theories. Lastly, the progress [2] at unification has been very slow. Therefore, in this paper in order to “geometrize” the electromagnetic field I will follow neither the scalar gravitational theory of electromagnetism and its introduction of an additional (four spatial and one time dimension) space dimension (Nordström [3] , Kaluza [4] ), nor Weyl’s trial for generalising Riemannian geometry and his concept of “gauging” (Weyl [5] ), nor will I use an asymmetric Ricci tensor (Eddington [6] ), nor will I try to add an antisymmetric tensor to the metric (Bach [7] , Einstein [8] ), nor will I use the framework of quantum field theory et cetera as the point of departure to “geometrize” the electromagnetic field. Theoretically, it seems to be possible to approach unification in the framework of quantum field theory. Still, a satisfactory inclusion of gravitation into the scheme of quantum field theory is not in sight. From this point of view, Finsler [9] geometry introduced by Randers [10] , as a kind of a generalization of Riemann geometry, is another and alternative approach to the geometrization of electromagnetism and gravitation. Taken all together, the point of departure for including the electromagnetic field into a geometric setting will be general relativity. In this context, at least one point has to be considered.

Taken Einstein for granted, we must give up general relativity theory. Einstein himself in his hunt for progress at the unification went so far to force us to give up his own general theory of relativity and the successful geometrization of the gravitational field. According to Einstein, a generalization of the theory of the gravitational field is necessary with the consequence that we must go beyond the general theory of relativity. In this context, Einstein’s position (see Figure 1) concerning the unified field theory is very clear and strict.

Anyhow, if we follow Einstein’s proposal at this point to account for a classical unified field theory of the gravitational and electromagnetic fields with the conceptual unification of the gravitational and electromagnetic field into one single and unique hyper-field [12] , it appears to be necessary and justified on a foundational level to concentrate at the heart of general relativity, the crucial mathematical concept of the metric tensor field g_{µv}.

The following paper can be characterized as follows. The attempt to develop some new, basic and fundamental insights is grounded on a deductive-hypothetical methodological approach. In the section material and methods the basic mathematical objects and tensor calculus rules needed to achieve the “geometrization” of the electromagnetic field will be defined and described.

In this context, physicists should be able to follow the technical aspects of this paper without any problems, while reader without prior knowledge of general relativity or of the mathematics of tensor calculus might gain an insight into the new methods and the scientific background involved. In general, it is necessary to decrease the amount of notation needed. Thus far, I will restrict myself as much as possible to covariant second rank tensors. I apologize for the shortcoming.

Especially, to enable the fusion of quantum theory and relativity theory into a new and single conceptual formalism the starting point of all theorems in the section results is axiom I or +1 = +1 (lex identitatis). The same axiom I possess the strategic capacity to

Figure 1. Einstein and the problem of the unified field theory.

serve as a common ground for relativity and quantum theory with regard to unified field theory. The section discussion examines some the consequences of the theorems proved. This paper does not provide any proof, whether Einstein’s general theory of relativity is correct or not, this publication assumes only that Einstein’s general theory of relativity is correct.

In this context, from the conceptual point of view of the unified field theory, it is the purpose of this publication to in find a convincing formulation of a geometrization of the electromagnetic fields under conditions of the validity of the general theory of relativity.

2. Material and Methods

2.1. Definitions

Einstein’s General Theory of Relativity

Definition: Einstein’s field equations

Einstein field equations (EFE), originally [13] published [14] without the extra “cosmological” term L ´ g_{µv} may be written in the form

(1)

where G_{µv} is the Einsteinian tensor, T_{µv} is the stress-energy tensor of matter (still a field devoid of any geometrical significance), R_{µv} denotes the Ricci tensor (the curvature of space), R denotes the Ricci scalar (the trace of the Ricci tensor), L denotes the cosmological “constant” and g_{µv} denotes the metric tensor (a 4 × 4 matrix) and where p is Archimedes’ constant (p = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 9∙∙∙), g is Newton’s gravitational “constant” and the speed of light in vacuum is c = 299 792 458 [m/s] in S. I. units.

Scholium.

The stress-energy tensor T_{µv}, still a tensor devoid of any geometrical significance, contains all forms of energy and momentum which includes all matter present but of course any electromagnetic radiation too. Originally, Einstein’s universe was spatially closed and finite. In 1917, Albert Einstein modified his own field equations and inserted the cosmological constant L (denoted by the Greek capital letter lambda) into his theory of general relativity in order to force his field equations to predict a stationary universe.

“Ich komme nämlich zu der Meinung, daß die von mir bisher vertretenen Feldglei- chungen der Gravitation noch einer kleinen Modifikation bedürfen…” [15] .

By the time, it became clear that the universe was expanding instead of being static and Einstein abandoned the cosmological constant L. “Historically the term containing the “cosmological constant” λ was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ” [16] . But lately, Einstein’s cosmological constant is revived by scientists to explain a mysterious force counteracting gravity called dark energy. In this context it is important to note that neither Newton’s gravitational “constant” big G [17] [18] nor Einstein’s cosmological constant L [19] is a constant.

Definition: General tensors

Independently of the tensors of the theory of general relativity, we introduce by definition the following covariant second rank tensors of preliminary unknown structure whose properties we leave undetermined as well. We define the following covariant second rank tensors of yet unknown structure as

(2)

while the tensors A_{µv}, B_{µv}, C_{µv}, D_{µv} may equally denote something like the four basic fields of nature. Especially, the Ricci tensor R_{µv} itself can be decomposed in many different ways. In the following of this publication we define the following relationships. We decompose the Ricci tensor R_{µv} by definition as

(3)

to assure that both gravitation and electromagnetism is geometrized simultaneously. Since everything is expressed in terms of curvature tensor, the electromagnetic field itself is completely geometrized from the beginning. By the following definition, the electromagnetic stress energy tensor, denoted as B_{µv}, appears as part of Einstein’s stress- energy tensor, while the tensor A_{µv}, also part of curvature, denotes the stress energy tensor of “ordinary” matter. Thus far, we obtain

(4)

Scholium.

By this definition, we are following Vranceanu in his claim that the energy tensor T_{kl} can be treated as the sum of two tensors one of which is due to the electromagnetic field.

“On peut aussi supposer que le tenseur d’énergie T_{kl} soit la somme de deux tenseurs dont un dû au champ électromagnétique…” [20] .

In English:

“One can also assume that the energy tensor T_{kl} be the sum of two tensors one of which is due to the electromagnetic field”.

In other words, the stress-energy tensor of the electromagnetic field B_{µv} is equivalent to the portion of the stress-energy tensor of energy which is determined by the stress energy tensor of the electromagnetic field B_{µv} itself. The stress-energy tensor T_{µv} has the unit of energy density [J/m^{3}] or pressure [N/m^{2}] which are actually the same unit. In the International System of Units the joule is a derived unit of energy and is defined as 1[J] = 1[kg×m^{2}/s^{2}] = [N×m] while 1[N] = 1[kg×m/s^{2}] = 1[J/m] denotes the unit of force. Figure 2 may illustrate the relationship above in some more detail.

Einstein himself was demanding something similar.

“Wir unterscheiden im folgenden zwischen “Gravitationsfeld” und “Materie” in dem Sinne, daß alles außer dem Gravitationsfeld als “Materie” bezeichnet wird, also nicht nur die “Materie” im üblichen Sinne, sondern auch das elektromagnetische Feld.” [14]

We define an anti tensor [2] of Einstein’s stress energy tensor T_{µv}, as

(5)

while Einstein’s tensor G_{µv} is defined by

(6)

where A_{µv} is the known the stress energy tensor of “ordinary” matter.

Scholium.

One consequence of the definition before is that the tensor of ordinary matter A_{µv} becomes a joint tensor since the same tensor is a determining part of the Einstein’s stress energy tensor and equally a determining part of Einsteinian tensor G_{µv}. In probability theory, such a tensor would represent a joint distribution function. The Ricci scalar curvature R[1/m^{2}] is a contraction of the Ricci tensor R_{µv}[1/m^{2}]. The Ricci tensor itself is a contraction of the Riemann tensor while a contraction as such doesn’t change the units.

Finally, we define an anti-tensor G_{µv} of Einsteinian tensor G_{µv}, as

(7)

Scholium.

The following 2 × 2 table (Table 1) may illustrate the basic relationships above

Figure 2. The relationship between ordinary matter A_{µv} and the electromagnetic field B_{µv}.

Table 1. The decomposition of the Ricci tensor R_{µv} in general.

Definition: The stress energy tensor of the electro-magnetic field B_{µv}

We define the second rank covariant stress-energy tensor of the electromagnetic field B_{µv}, an anti tensor [2] of the tensor A_{µv}, as

(8)

Under conditions of general relativity, where A_{µv} denotes the stress energy tensor of ordinary energy/matter, the stress-energy tensor of the electromagnetic field B_{µv} is an anti tensor [2] of ordinary energy/matter A_{µv}. Under conditions of general relativity, the second rank covariant stress-energy tensor of the electromagnetic field B_{µv} is determined by an anti-symmetric second-order tensor known as the electromagnetic field (Faraday) tensor F. In general, under conditions of general relativity, the second rank covariant stress-energy tensor of the electromagnetic field B_{µv} in the absence of “ordinary” matter, which itself is different from the electromagnetic field tensor F, can be derived many different ways. One form of this tensor is

(9)

where F is the electromagnetic field tensor and g_{µv} is the metric tensor of general relativity.

Scholium.

The probability tensor [2] of the second rank covariant stress-energy tensor of the electromagnetic field B_{µv} is defined as

(10)

One possible theoretical geometric formulation of the stress-energy tensor of the electromagnetic field [2] follows as

(11)

Definition of Tensors with relation to the Ricci scalar R

In general, we define the second rank covariant metric tensor _{X}g_{µv} of the tensor X_{µv} under conditions of a Ricci scalar as

(12)

where R denotes the Ricci scalar and X_{µv} denotes a second rank (even a metric) tensor. Thus far, even a metric tensor can possess a metric. A straightforward consequence of this definition is the relationship between the metric tensor _{X}g_{µv} and the probability tensor p(X_{µv}) as p(X_{µv}) Ç _{Ric}g_{µv} = _{X}g_{µv}. Further, we define n(X_{µv}) as

(13)

where R denotes the Ricci scalar and X_{µv} denotes a second rank (even a metric) tensor. Further, we define as

(14)

where R denotes the Ricci scalar and X_{µv} denotes acovariant second rank (even a metric) tensor.

Definition: The tensor n(A_{µv})

We define the second rank tensor n(A_{µv}) as

(15)

where R denotes the Ricci scalar.

Definition: The metric tensor _{A}g_{µv}

We define the second rank metric tensor _{A}g_{µv} as

(16)

where R denotes the Ricci scalar.

Definition: The tensor n(B_{µv})

We define the second rank tensor n(B_{µv}) as

(17)

where R denotes the Ricci scalar.

Definition: The metric tensor _{B}g_{µv}

We define the second rank metric tensor _{B}g_{µv} as

(18)

where R denotes the Ricci scalar.

Definition: The Tensor n

We define the second rank tensor as

(19)

where R denotes the Ricci scalar. Due to our definition before it is n(A_{µv}) = A_{µv}/R and n(B_{µv}) = B_{µv}/R while at the same time it holds true that, the stress energy tensor is decomposed into the tensors A_{µv} and B_{µv}.

Definition: The metric tensor _{E}g_{µv}

We define the second rank metric tensor _{E}g_{µv} as

(20)

where R denotes the Ricci scalar.

Definition: The tensor n(C_{µv})

We define the second rank tensor n(C_{µv}) as

(21)

where R denotes the Ricci scalar.

Definition: The metric tensor _{C}g_{µv}

We define the second rank metric tensor _{C}g_{µv} as

(22)

where R denotes the Ricci scalar.

Definition: The tensor n(D_{µv})

We define the second rank tensor n(D_{µv}) as

(23)

where R denotes the Ricci scalar.

Definition: The metric tensor _{D}g_{µv}

We define the second rank metric tensor _{D}g_{µv} as

(24)

where R denotes the Ricci scalar.

Definition: The tensor

We define the second rank tensor as

(25)

where R denotes the Ricci scalar.

Definition: The tensor _{Ø}_{E}g_{µv}

We define the second rank metric tensor _{Ø}_{E}g_{µv} as

(26)

where R denotes the Ricci scalar.

Definition: The tensor n(G_{µv})

We define the second rank tensor n(G_{µv}) as

(27)

where R denotes the Ricci scalar and G_{µv} is the Einsteinian tensor.

Definition: The metric tensor _{G}g_{µv}

We define the second rank metric tensor _{G}g_{µv} as

(28)

where R denotes the Ricci scalar and G_{µv} is the Einsteinian tensor.

Definition: The tensor n(G_{µv})

We define the second rank tensor n(G_{µv}) as

(29)

where R denotes the Ricci scalar and G_{µv} is the anti Einsteinian tensor.

Definition: The metric tensor _{Ø}_{G}g_{µv}

We define the second rank metric tensor _{Ø}_{G}g_{µv} as

(30)

where R denotes the Ricci scalar and G_{µv} is the anti Einsteinian tensor.

Definition: The metric tensor _{gr}g_{µv} of the metric tensor g_{µv}

We define the second rank metric tensor _{gr}g_{µv} as

(31)

where R denotes the Ricci scalar where g_{µv} denotes the metric tensor of general relativity theory.

Definition: The tensor n(R_{µv})

We define the second rank tensor n(R_{µv}) as

(32)

where R denotes the Ricci scalar and R_{µv} denotes the Ricci tensor.

Definition: The metric tensor _{Ric}g_{µv}

We define the second rank metric tensor _{Ric}g_{µv} as

(33)

where R denotes the Ricci scalar and R_{µv} denotes the Ricci tensor.

Scholium.

The following 2 × 2 table (Table 2) may illustrate the basic relationships above

The 2 × 2 table (Table 3) illustrates the basic relationships between the metric tensors.

Definition: The tensor g_{µv}

The mathematics of general relativity are more or less complex. As a result, the curvature of space (represented by the Einstein tensor G_{µv}) is caused by the presence of matter and energy (represented by the stress-energy tensor T_{µv}) and vice versa. The cur- vature of space is the cause or determines how matter/energy has to move. The Riemannian metric tensor for a curved space-time of general relativity theory, a kind of generalization of the gravitational potential of Newtonian gravitation, is denoted as

(34)

In the following, let us define the following. Let

Table 2. The decomposition of the Ricci tensor R_{µv}.

Table 3. The decomposition of the Ricci tensor R_{µv} in terms of metric tensors.

(35)

and

(36)

The (unitless) metric tensor g_{µv} is a central object in general relativity and describes more or less the local geometry of space-time while representing the gravitational potential. The metric tensor determines the invariant square of an infinitesimal line element, denoted as ds and often referred to as an interval. In general, the generalization of the standard measure of distance ds between two points in Euclidian space due to the Pythagorean theorem is defined as

(37)

or

(38)

In general, a coordinate system can be changed from the Euclidean Y’s to some coordinate system of X’s then

(39)

and

(40)

The Pythagorean theorem is known to be defined as

(41)

where δ_{mn} denotes the known Kronecker delta. Using Einstein’s summation convention, the metric tensor g_{µv} is

(42)

and a curved space compatible formulation of the Pythagorean theorem follows as usual as

(43)

In other words, it is

(44)

Under conditions, where

(45)

it is

(46)

Dividing the equation before by the speed of the light squared, c^{2}, it is

(47)

where _{t}g_{µv} = (1/c^{2}) ´ g_{µv}. The term ds^{2}/c^{2} yields the time squared or ds^{2}/c^{2} = d_{R}t^{2} as do the other terms. The equation before can be rearranged as

(48)

Rearranging equation, we obtain

(49)

or while using Einstein’s summation convention,

(50)

or

(51)

Multiplying by ((R/2) − Λ), it is

(52)

Definition: The metric tensor of the electromagnetic field _{EM}g_{µv}

We define the second rank metric tensor of the electro-magnetic field _{EM}g_{µv} of preliminary unknown structure as

(53)

where Y denotes an unknown (i.e. scalar) parameter. Due to this definition, it is B_{µv} = Y ´ _{EM}g_{µv}.

Definition: The anti metrictensor of the electromagnetic field _{0}g_{µv} or _{W}g_{µv}

We define the second rank anti metric tensor of the electro-magnetic field _{W}g_{µv} of preliminary unknown structure as

(54)

where Y denotes an unknown (i.e. scalar) parameter. Due to this definition, it is D_{µv} = Y ´ _{W}g_{µv}.

Definition: The relationships between the metric tensors

In general, the metric tensor for a curved space-time of general relativity theory is equally determined as

(55)

where _{EM}g_{µv} = g_{µv} ? _{W}g_{µv} and _{EM}g_{µv} denotes the second rank metric tensor of the electro- magnetic field while _{W}g_{µv} is the second rank anti metric tensor of the electro-magnetic field. Both tensors are of still unknown structure. From this definition, it follows that

(56)

Scholium.

The true meaning of the metric tensor _{W}g_{µv} is not clear at this moment. One is for sure, the same tensor is an anti-tensor of the metric tensor of the electromagnetic field _{EM}g_{µv}. There is some theoretical possibility that the tensor _{W}g_{µv} is related to something like the metric tensor of the gravitational waves, therefore the abbreviation _{W}g_{µv}.

Definition: The tensor of energy of the unified field theory _{R}E_{µv}

In order to assure compatibility between general theory of relativity and the unified field theory, we define the following relationship between the stress energy tensor of general relativity and the tensor of energy [2] of unified field theory _{R}E_{µv} as

(57)

while the value of X is undermined at this moment. The value of X can be X = 1.

Definition: The tensor of time of the unified field theory _{R}t_{µv}

In order to assure compatibility between the second rank tensor of general theory of relativity and tensor of time [2] _{R}t_{µv} of the unified field theory, we define the following relationship.

(58)

while the value of X is undermined at this moment. The value of X can be X = 1.

Definition: The tensor of space of the unified field theory _{R}S_{µv}

In order to assure compatibility between the second rank Ricci tensor R_{µv} of general theory of relativity and tensor of space [2] _{R}S_{µv} of the unified field theory, we define the following relationship.

(59)

while the value of X is undermined at this moment. The definition before does not exclude the case that X = 1.

2.2. Axioms

2.2.1. Axiom I. (Lex Identitatis. Principium Identitatis. The Identity Law)

The foundation of all what may follow is the following axiom:

(60)

2.2.2. Axiom II

(61)

2.2.3. Axiom III

(62)

Remark.

In order to avoid any kind of an inconsistency and to obtain a theory that also prove to be universally valid, the axioms and rules of this publication are chosen very carefully. Especially to face singularity problems of general relativity, the axioms II and III can be of use. However, today it is far from clear whether axiom II and axiom III can be treated as generally valid. Usually the chance of proving the consistency of axioms is sometimes temporary limited. By time objective reality or scientific experiments or human practice as such is the main proof of any axiomatic theory. More precisely, special experimental settings satisfying some certain conditions can disproof any formal deductive systems or axiomatic theory. There is already some evidence, that axiom III [21] is correct.

3. Results

3.1. Theorem. The Unification of Gravity and Electromagnetism

Claim. (Theorem. Proposition. Statement.)

In general, the gravitational and the electromagnetic field can be joined into one single hyperfield which itself is completely determined by the geometrical structure of the space-time. We obtain

(63)

Direct proof.

In general, using axiom I is it

(64)

Multiplying this equation by the stress-energy tensor of general relativity, it is

(65)

where γ is Newton’s gravitational “constant” [17] [18] , c is the speed of light in vacuum and, sometimes referred to as “Archimedes” constant’, is the ratio of a circle’s circumference to its diameter. Due to Einstein’s general relativity, the equation before is equivalent with

(66)

where R_{µv} is the Ricci curvature tensor (the trace of Rimanian curvature tensor), R is the scalar curvature, g_{µv} is the metric tensor, Λ is the cosmological constant and T_{µv} is the stress-energy tensor. By defining the Einstein tensor as, it is possible to write the Einstein field equations in a more compact as

(67)

Due to our definition above it is G_{µv} = A_{µv} + C_{µv}. Substituting this relationship into the equation before, we obtain

(68)

Under these conditions, we recall our definition before where . Substituting this relationship into the equation before, we obtain

(69)

where A_{µv} denotes the stress energy tensor of ordinary matter and B_{µv} denotes the stress energy tensor of the electromagnetic field. Simplifying equation, it is

(70)

where C_{µv} denotes the gravitational field due to the stress energy tensor of ordinary matter A_{µv}. We add the tensor C_{µv} on both sides of the equation before. The unification of gravity and electromagnetisms under conditions of general relativity theory follows as

(71)

or in general as

(72)

Quod erat demonstrandum.

Scholium.

The theorem of the unification of gravity and electromagnetism does not contain any additional fields and is formulated in four-dimensional space-time. Thus far, within the sum of the tensors B_{µv} + C_{µv}, both electromagnetism and gravity are successfully unified and linked to each other. Up to now, there is neither theoretical nor experimental evidence that there might be unobserved additional fields or extra dimensions necessary for the unification of gravity and electromagnetism. Therefore, for geometry underlying the theorem of the unification of gravity and electromagnetism we choose Riemannian geometry which is known to be suitable for gravitational interaction. Accordingly, until today all attempts known to geometrize electromagnetism or unify electromagnetism with gravitation in the framework of Riemannian geometry were in vain. Still, the theorem of the unification of gravity and electromagnetism demonstrates equally that Riemannian geometry is appropriate for unification of gravitation and electromagnetism. The most important and interesting thing is a prediction of the theorem of the unification of gravity and electromagnetism that the stress energy tensor of the electromagnetic field B_{µv} is a source for ordinary gravitational field C_{µv}. From physical point of view, this prediction can be confirmed by experiments in strong electromagnetic field very precisely.

Lastly, the electromagnetic field B_{µv} is a source for the ordinary gravitational field C_{µv} since both fields are related by the equation and the ordinary gravitational field C_{µv} is itself is a determining part of Einstein’s tensor G_{µv}. Furthermore, the ordinary gravitational field C_{µv} when the stress energy tensor of the electromagnetic field B_{µv} is equal to zero (B_{µv} = 0) is still determined by the equation. Thus far, under these circumstances it is and the ordinary gravitational field C_{µv} is given exactly by the equation C_{µv} = −Λ ´ g_{µv}.

In conclusion, we note that when the ordinary gravitational field C_{µv} is equal to zero or C_{µv} = 0 the stress energy tensor of the electromagnetic field B_{µv} derived from the equation is determined in this context by the equation . In other words, we obtain Λ ´ g_{µv} = B_{µv}. Under these conditions, the metric tensor _{EM}g_{µv} of the stress-energy tensor of the electromagnetic field B_{µv} follows in general from the equation as .

3.2. Theorem. The Anti Einsteinian Tensor G_{µv}

Claim. (Theorem. Proposition. Statement.)

The anti Einsteinian tensor G_{µv} is determined as

(73)

Direct proof.

In general, using axiom I is it

(74)

Multiplying this equation by the Ricci Tensor R_{µv}, it is

(75)

Due to our definition above, it is. Substituting this relationship into the equation before, we obtain

(76)

The sum of the tensor G_{µv} = B_{µv} + D_{µv} can be obtained as

(77)

which can be simplified as

(78)

Due to our definition, Einsteinian tensor G_{µv} is defined as G_{µv} = A_{µv} + C_{µv}. Rearranging equation above, it is

(79)

Einstein’s tensor G_{µv} is defined as. Substituting this relationship into the equation before, we obtain

(80)

Rearranging equation, we obtain

(81)

At the end, we obtain

(82)

Quod erat demonstrandum.

3.3. Theorem. The Determination of the Unknown Parameter Y

Claim. (Theorem. Proposition. Statement.)

The unknown parameter Y is determined as

(83)

where R denotes the Ricci scalar.

Direct proof.

In general, using axiom I is it

(84)

Multiplying this equation by the anti Einsteinian tensor G_{µv}, it is

(85)

The same tensor was determined by the theorem before as B_{µv} + D_{µv} = G_{µv}. Substituting this relationship into the equation before, we obtain

(86)

Due to our definition above, it is as B_{µv} = Y ´ _{EM}g_{µv} and D_{µv} = Y ´ _{W}g_{µv}. Substituting this relationship into the equation before, we obtain

(87)

Due to our definition g_{µv} = _{EM}g_{µv} + _{W}g_{µv}, the equation before can be rearranged as

(88)

In other words, it is

(89)

A further manipulation of the equation before yields the result that

(90)

Quod erat demonstrandum.

Scholium.

Such a result is logical too. Due to our definition it is

(91)

where _{EM}g_{µv} = g_{µv} − _{W}g_{µv} and _{EM}g_{µv} denotes the second rank metric tensor of the electro- magnetic field while _{W}g_{µv} is the second rank anti metric tensor of the electro-magnetic field. Multiplying equation above by the term (R/2), we obtain

(92)

which is exactly the result as obtained above.

3.4. Theorem. The Geometrization of the Stress-Energy Tensor of the Electromagnetic Field

Claim. (Theorem. Proposition. Statement.)

The geometrization of the stress-energy tensor of the electromagnetic field under conditions of general relativity follows as

(93)

where R denotes the Ricci scalar and _{EM}g_{µv} denotes the metric tensor of the electromagnetic field.

Direct proof.

In general, using axiom I is it

(94)

Multiplying this equation by the stress-energy tensor of the electromagnetic field, denoted as B_{µv}, it is

(95)

Due to our definition above, the stress-energy tensor of the electromagnetic field was determined by the relationship B_{µv} = Y ´ _{EM}g_{µv}. Rearranging the equation before we obtain

(96)

According to the theorem before, the unknown parameter Y is determined as Y = R/2. The geometrization of the electromagnetic field under conditions of general relativity follows as

(97)

Quod erat demonstrandum.

Scholium.

In a more far reaching development, at least since general relativity theory brought the geometry to the scenario of physics, many attempts were made to extend general relativity’s geometrization of gravitation to non-gravitational fields. In particular, the geometrization of the electromagnetic field became a principal focus and a cornerstone of physical interest and inquiry. The many geometric theories of electromagnetism as published meanwhile are still not consistent with the framework of the quantum theory or self-contradictory, despite the fact that the electromagnetic theory was consolidated in the 19th century. The present theorem before describes the stress-energy tensor of the electro-magnetic field as directly related or determined by the space-time geometry or the metric tensor _{EM}g_{µv}. A unified field theory, in the sense of a completely geometrical field theory of all fundamental interactions, is no longer only a theoretical desire.

3.5. Theorem. The Determination of the Metric Tensor of the Electromagnetic Field _{EM}g_{µv}

Claim. (Theorem. Proposition. Statement.)

The metric tensor of the electromagnetic field _{EM}g_{µv} under conditions of general relativity is determined as

(98)

where R denotes the Ricci scalar and _{EM}g_{µv} denotes the metric tensor of the electromagnetic field.

Direct proof.

In general, using axiom I is it

(99)

Multiplying this equation by the stress-energy tensor of the electromagnetic field, abbreviated as B_{µv}, it is

(100)

Due to our theorem before, the geometrization of the electromagnetic field under conditions of general relativity is determined as

(101)

The stress-energy tensor of the electromagnetic field B_{µv} is determined in detail i.e. by the relationship

(102)

where F is the electromagnetic field tensor and g_{µv} is the metric tensor. The equation before changes to

(103)

The metric tensor of the electromagnetic field _{EM}g_{µv} under conditions of general relativity is determined as

(104)

Quod erat demonstrandum.

3.6. Theorem. The Tensor C_{µv}

Claim. (Theorem. Proposition. Statement.)

The tensor C_{µv} is determined as

(105)

Direct proof.

In general, using axiom I is it

(106)

Multiplying this equation by the stress-energy tensor of the electromagnetic field, denoted as B_{µv}, it is

(107)

Due to our theorem before, the metric tensor of the electromagnetic field under conditions of general relativity is determined as

(108)

where R denotes the Ricci scalar and _{EM}g_{µv} denotes the metric tensor of the electromagnetic field. Due to the another theorem above, it is

(109)

The tensor C_{µv} is determined by the stress energy tensor of the electromagnetic field B_{µv} as

(110)

Quod erat demonstrandum.

Scholium.

Lastly, the stress-energy tensor of the electromagnetic field B_{µv} is a source or a determining part for the ordinary gravitational field C_{µv}. From physical point of view, this theorem can be confirmed by experiments in strong electromagnetic fields.

3.7. Theorem. The Determination of Tensor of the Hyperfield of Gravitation and Electromagnetism

Claim. (Theorem. Proposition. Statement.)

The geometrized form of the hyper-tensor of unification of gravitation and electromagnetism is determined as

Direct proof.

In general, using axiom I is it

(111)

Multiplying this equation by the stress-energy tensor of the electromagnetic field, denoted as B_{µv}, it is

(112)

Due to our theorem before, the stress energy tensor of the electromagnetic field under conditions of general relativity is determined as

(113)

where R denotes the Ricci scalar and _{EM}g_{µv} denotes the metric tensor of the electromagnetic field. Due to a theorem before, it is

(114)

The tensor C_{µv} is determined by the stress energy tensor of the electromagnetic field B_{µv} as

(115)

Adding the stress-energy tensor of the electromagnetic field B_{µv} = (R/2) ´ _{EM}g_{µv} to the equation before, we obtain the geometrized form of the hyper-tensor of C_{µv} puls B_{µv} (i.e. the unity of gravitation and electromagnetism) as

(116)

Quod erat demonstrandum.

3.8. Theorem. The Tensor D_{µv}

Claim. (Theorem. Proposition. Statement.)

The tensor D_{µv} is determined as

(117)

Direct proof.

In general, using axiom I is it

(118)

_{µv}, it is

(119)

Due to our theorem before, the stress energy tensor of the electromagnetic field under conditions of general relativity is determined as

(120)

where R denotes the Ricci scalar and _{EM}g_{µv} denotes the metric tensor of the electromagnetic field. Adding the tensor D_{µv}, we obtain

(121)

According to a theorem before, this relationship is equivalent with

(122)

Rearranging the equation before, the tensor D_{µv} is determined as

(123)

Quod erat demonstrandum.

3.9. Theorem. The Geometrization of the Stress Energy Tensor of “Ordinary” Matter A_{µv}

Claim. (Theorem. Proposition. Statement.)

The geometrization of the stress-energy tensor of “ordinary” matter A_{µv} can be obtained as

(124)

Direct proof.

In general, using axiom I is it

(125)

Multiplying this equation by the Ricci Tensor R_{µv}, it is

(126)

Due to our definition above, it is. Substituting this relationship into the equation before, we obtain

(127)

The tensor of ordinary matter A_{µv} follows as

(128)

The tensor B_{µv} itself was determined as. The addition of the tensors D_{µv} plus C_{µv} is determined as. The equation before changes to

(129)

Rearranging equation before, we obtain

(130)

The tensor (R/2) ´ g_{µv} is determined as. The equation can be simplified as

(131)

The geometrization of “ordinary” matter follows in general as

(132)

Quod erat demonstrandum.

3.10. Theorem. The Probability Tensor of the Electromagnetic Field p(B_{µv})

Claim. (Theorem. Proposition. Statement.)

The probability tensor p(B_{µv}) of the stress-energy tensor of the electromagnetic field is determined as

(133)

Direct proof.

In general, using axiom I is it

(134)

_{µv}, it is

(135)

Due to our theorem before, the geometrization of the electromagnetic field under conditions of general relativity is determined as

(136)

The stress-energy tensor of the electromagnetic field B_{µv} was determined i.e. by the relationship

(137)

where F is the electromagnetic field tensor and g_{µv} is the metric tensor. The equation before changes to

(138)

The Ricci scalar R is defined as the contraction of the Ricci tensor R_{µv} or it is R = g^{µv}R_{µv}. The equation before changes to

(139)

A commutative division [2] by the Ricci tensor R_{µv} leads to the relationship

(140)

This equation is identical with the probability tensor p(B_{µv}) of the stress-energy tensor of the electromagnetic field. In general it is

(141)

Quod erat demonstrandum.

3.11. Theorem. The Probability Tensor Is Determined by the Metric Tensor

Claim. (Theorem. Proposition. Statement.)

In general let n(X_{µv}) = X_{µv}/R where X_{µv} denotes a second rank co-variant tensor and R denotes the Ricci scalar, the contraction of the Ricci tensor as R = g^{µv}R_{µv}. Further, p(X_{µv}) = X_{µv}/R_{µv} denotes the probability tensor [2] of the tensor X_{µv}. The probability tensor p(X_{µv}) of a tensor X_{µv} is determined by the metric tensor as

(142)

Direct proof.

In general, using axiom I is it

(143)

Multiplying this equation by the tensor X_{µv}, it is

(144)

A commutative [2] division of the tensor X_{µv} by the Riccitensor R_{µv} yields

(145)

which is equivalent with

(146)

or with

(147)

Rearranging equation it is

(148)

Changing equation, we obtain

(149)

Due to the relationship R = g^{µv}R_{µv}, the equation before simplify as

(150)

or as

(151)

In general, it is n(X_{µv}) = X_{µv}/R. The probability tensor of a tensor X_{µv} is determined by the (inverse or) conjugate metric tensor as

(152)

Quod erat demonstrandum.

Scholium.

Quantum physics (quantization) focuses on the probability (amplitudes) while general relativity theory relies on geometry (tempo-spatial points). The definition of a probability tensor p(X_{µv}) of a tensor X_{µv} marks a remarkable degree of interaction between probability theory and the highly dimensional theory of general relativity and is a key step to the unification of quantum physics and general relativity by probabilitizing general relativity’s geometric background. In principle, a contradiction free transformation of a geometrical mathematical framework into a probabilistic mathematical framework and vice versa is possible. A geometrization of probability theory appears to be necessary too.

3.12. Theorem. The Normalization of the Relationship between the Tensors of General Theory of Relativity

Claim. (Theorem. Proposition. Statement.)

The relationship between the tensors of general theory of relativity can be normalized as

(153)

Direct proof.

In general, using axiom I is it

(154)

Multiplying this equation by the stress-energy tensor of general relativity, it is

(155)

where γ is Newton’s gravitational “constant” [17] [18] , c is the speed of light in vacuum and π, sometimes referred to as “Archimedes” constant’, is the ratio of a circle’s circumference to its diameter. Due to Einstein’s general relativity, Einstein’s field equations are determined as

(156)

where R_{µv} is the Ricci curvature tensor (the trace of Rimanian curvature tensor), R is the scalar curvature, g_{µv} is the metric tensor, Λ is the cosmological constant and T_{µv} is the stress-energy tensor. Rearranging equation we obtain

(157)

A commutative [2] division of tensors simplifies the equation as

(158)

where 1_{µv} denotes the tensor of unified field [2] . In general, a normalization of some tensors of general relativity follows as

(159)

Quod erat demonstrandum.

3.13. Theorem. The Normalization of the Relationship between the Tensors of the Unified Field Theory

Claim. (Theorem. Proposition. Statement.)

The relationship between the tensors of the unified field theory normalized as

(160)

Direct proof.

In general, using axiom I is it

(161)

Multiplying this equation by the energy tensor of general relativity _{R}T_{µv}, it is

(162)

The field equations of the unified field theory [2] are determined as

(163)

where _{R}S_{µv} is the tensor of space [2] of the unified field theory and _{R}t_{µv} is the tensor of time [2] of the unified field theory. Rearranging equation it is

(164)

A commutative [2] division of tensors simplifies the equation as

(165)

where 1_{µv} denotes the tensor of unified field [2] . In general, a normalization of some tensors of the unified field theory follows as

(166)

Quod erat demonstrandum.

3.14. Theorem. The Determination of X

Claim. (Theorem. Proposition. Statement.)

The unknown parameter X, which should equal 1, can be determined as

(167)

Direct proof.

In general, using axiom I is it

(168)

Multiplying this equation by the tensor of the unified field 1_{µv}, of the unified field, it is

(169)

or

(170)

Due to the theorem about the normalization of some tensors of the unified field theory, this equation rearranges to

(171)

According to the theorem about the normalization of some tensors of the general theory relativity, the equation before rearranges to

(172)

A commutative multiplication and division [2] of tensors changes the equation before to

(173)

Due to our definition, it is and equally. Substituting these relationships into the equation before we obtain

(174)

Due to commutative operations [2] , this equation can be simplified as

(175)

which itself can be simplified as

(176)

or as

(177)

or as

(178)

The determination of the value of X follows as

(179)

Quod erat demonstrandum.

Scholium.

The straightforward question is, must we accept that (_{R}S_{µv}/R_{µv}) = 1_{µv} or (_{R}S_{µv}/R_{µv}) = c^{2} or (_{R}S_{µv}/R_{µv}) = 1/c^{2}? The probability tensor of the stress-energy tensor of the theory of general relativity is defined as

.

The energy tensor of the unified field theory is defined as

.

The value of X is determined as (_{R}S_{µv}/R_{µv}). The equation before changes to as

.

In general, the probability tensor is of use to express the energy tensor as

.

The probability tensor simplifies this equation simplifies in other word to

.

The probability tensor of the tensor is defined in general something as

.

The value of X is determined equally as (_{R}S_{µv}/R_{µv}). The tensor of time

follows as

or as

or as

.

In last consequence, the relationship between the field equations of unified field theory and Einstein’s theory of general relativity is determined by the equation

(180)

or by the equation

(181)

Einstein’s field equation yields the result

(182)

What is the physical meaning of Einstein’s field equation, if we multiply the same by the term (1/c^{2}). In this case we obtain

(183)

In the International System of Units the joule, the unit of energy, is defined as 1[J] = 1[kg∙m^{2}/s^{2}] = 1[N ´ m] while 1[N] = 1[kg∙m/s^{2}] = 1[J/m] denotes the unit of force. The stress-energy tensor T_{µv} without the mathematical term has the unit of energy density [J/m^{3}], it is 1[J/m^{3}] = 1[(kg∙m^{2})/(s^{2}∙m^{3})]. Let us multiply the stress-energy tensor T_{µv} by (1/c^{2}), we obtain which is equivalent with. Conse- quently, the term s^{2} and m^{2} cancels out and we obtain the unit . In other words, the stress-energy tensor T_{µv} changes to the stress tensor of matter. Thus far, under these conditions there is some evidence that it makes sense to assume that [2] . This assumed as correct, the tensor _{R}g_{µv} [2] is determined as where the term (1/c^{2}) ´ g_{µv} can denote something like the metric tensor of time _{t}g_{µv} = (1/c^{2}) ´ g_{µv}.

3.15. Theorem. The Geometrization of the Stress-Energy-Tensor T_{µv}

In general theory of relativity the gravitational field is completely geometrized. Still, Einstein failed to geometrize the stress-energy-momentum tensor T_{µv} too. Einstein was convinced that the main problem in the unified field theory was the geometrization of the stress-energy-momentum tensor of matter on the right-hand side of his field equations known to be determined as. The geometrization of the stress-energy-momentum tensor of the matter T_{µv} should result in the geometrization of the quantum i.e. matter fields.

Claim. (Theorem. Proposition. Statement.)

The total geometrization of all fields or Einstein’s field equations with geometrized stress-energy-momentum tensor of the energy (T_{µv}) are determined as

(184)

Direct proof.

In general, using axiom I is it

(185)

Multiplying this equation by the tensor of the stress-energy tensor , we do obtain

(186)

Due to Einstein’s general theory relativity, this equation can be rearranged as

(187)

Multiplying Einstein’s field equation by the term (2/R), we obtain

(188)

Due to our definition, this equation is equivalent with

(189)

Multiplying the equation before by the term (R/2), Einstein’s field equation completely geometrized follows as

(190)

Quod erat demonstrandum.

Scholium.

In view of Einstein’s geometrization of gravity, the stress-energy tensor T_{µv} is the source-term of Einstein’s field equations. From the geometrical point of view, the stress-energy tensor T_{µv} is still a field without any geometrical significance. In particular, the main goal of the very transparent and also highly general theorem above is to describe matter as an inherent geometrical structure and to incorporate both, the principles of general relativity and quantum theory, in one mathematical formula. Such a theorem is expected to be able to provide a satisfactory (geometrical) description of the microstructure of spacetime, to geometrize the quantum. We rearrange the equation before as

(191)

or as

(192)

Einstein’s vacuum field equations can be obtained when the known the stress-energy- momentum tensor is to be determined as (R/2) ´ _{E}g_{µv} = 0. Under these conditions (T_{µv} = 0), the Einstein vacuum equations are determined by the fact that

(193)

One focus of this paper is the attempt to build a bridge between quantum theory and classical geometry. The equation before can be changed as

(194)

In this context we multiply the equation before by the wave-function Ψ. We obtain the Schrödinger’s equation as

(195)

or a kind of a “normalized” Schrödinger’s equation where R = () ~ h/π as

(196)

an equation where quantum meets geometry and vice versa, where the metric (i.e. geometry) is a determining part of this equation, but the wave-function (i.e. quantum) too. In last consequence, the gravitational field itself can be quantized. A profound methodological challenge for the physicist was the geometrization of the stress-energy tensor T_{µv}. This problem is solved by this theorem. The mathematical term (R/2) ´ _{E}g_{µv} denotes the geometrical description of the stress-energy tensor T_{µv} of general theory of relativity. Thus far, in general it is, where _{E}g_{µv} denotes the metric tensor of the stress energy tensor T_{µv}.

4. Discussion

A new approach to quantum gravity and the unified field theory developed by the author is already published [2] . Besides of the misprint in Equation (76) in [2] , in this paper Equation (197)

(197)

which should be changed to

(198)

one way how to geometrize the electromagnetic field is already provided. In this paper the geometrization of the electromagnetic field under conditions of general relativity theory is developed in more (technical) detail. This paper has answered the question about the geometrization of the electromagnetic field under conditions of general theory of relativity.

Thus far, this publication has not answered the question whether does geometrization excludes quantization and vice versa. In other words, is there a dualism between geometrization and quantization in the sense either geometrization or quantization. This geometry-quantum dilemma leads straight forward to the question which came first, the geometry or the quantum? In general, are the rules of quantization more fundamental than the rules of (classical) geometry or vice versa? The question about the very first geometry or the very first quantum also evokes the question of how the development of this universe in general began. Thus far, quantizing geometry is not only a major undertaking but a theoretical necessity and vice versa. Geometrizing the quantum should be provided by a self-consistent deterministic formulation of a unified field theory of nature. In this context, the geometric entity “line” (in the framework of string-theory: the string) is determined by points. But what is a point, how does geometry defines a point? A point appears to be something quantized. In other words, within geometry (a line, a string), the quantum (a point) can be found and surely vice versa. Within the quantum (a point) the geometry (a line) can be found. The one cannot without its own other and vice versa. Today, a unified description of all physical phenomena is endangered especially by the incompatibility between the deterministic geometrical formulation of general relativity and the claimed indeterministic nature of quantum mechanics. Still, the problems of quantizing geometry or geometrizing the quantum are solveable. The answer to such and similar questions may be considered for future work. In this paper, the tensor D_{µv} was derived as

(199)

In particular, the physical content of the tensor D_{µv} is not clear at this moment. Still, a further lack of clarity may not stem from this fact. In order to express the physical content of the tensor D_{µv} it is necessary to distinguish clearly between the tensor D_{µv} and the tensor B_{µv}. To within acceptable margin of error, the information carried by the tensor D_{µv} is very different from the information as carried by the stress-energy tensor of the electro-magnetic field B_{µv}. In this context, the tensor D_{µv} is an anti-tensor of the stress-energy tensor of the electro-magnetic field B_{µv}. But, as noted above, there are some aspects connected with the tensor D_{µv}. In fact, the tensor D_{µv} is a sub-tensor of the metric tensor g_{µv} of Einstein’s gravitational theory of curved spacetime. Thus far, the metric tensor _{W}g_{µv} has to do something with the gravitational field. Moreover, it is possible and highly desirable that the metric tensor _{W}g_{µv} is determined by fuctuations of gravitational fields and that the same tensor represents something like “ripples” in spacetime. The interaction between electromagnetic and gravitational waves and the transformation of one wave into another became already a principal focus of theoretical interest and inquiry and has been discussed [22] in literature. Under these assumptions, the tensor D_{µv} could be determined by the metric tensor of the gravitational waves. More precisely stated, it may be rather difficult to understand the significance that has to be accorded the tensor D_{µv} but the assumption that the tensor D_{µv} represents a fourth and until today unknown “force” does not make any sense so far. Despite Einstein’s intent to realize something like a unified field theory, there is considerable disagreement about the extent to which, if at all, such a theory is possible. And yet, from the epistemological standpoint, despite the long history of trials about a completely geometrical field theory of all fundamental interactions under conditions of Einstein’s gravitational theory of curved space-time, it is possible to go beyond general relativities’ definite’ advance in physical knowledge. Furthermore, besides of the influence of Einstein’s reduction of physics to geometry, geometry is nothing absolute but something relative. In fact, striving towards an extension of Einstein’s gravitational theory, we may append an unknown tensor X_{µv} to Einstein’s filed equations for this purpose. To be sure, the Einstein field equations (EFE) with the extra term X_{µv} may be written in the form

(200)

and it follows that

(201)

More precisely, one possible extension of general relativity is viewed within Table 4.

5. Conclusion

In the 1940s, the theoretical framework of quantum electrodynamics consolidated electromagnetism with quantum physics. It has also to be noted that the trial to geometrize the stress energy tensor or the electromagnetic field within the theoretical framework of general relativity has still not met with much success. In this publication, the stress- energy tensor and the electromagnetic field have been geometrized under conditions of general theory of relativity.

Table 4. A possible extention of the theory of general realtivity.

Acknowledgements

None.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Journal of Applied Mathematics and Physics*,

**4**, 2135-2171. doi: 10.4236/jamp.2016.412211.

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