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Re Birth of Einstein’s Relativity via Unified Transformation Law

N. Hussain, H.A. Hussain, A. Tariq, M. F. Yaseen, J. Amjad, A. Qayyum*

Department of Mathematics, Institute of Southern Punjab Multan Pakistan

Abstract: Einstein principle of relativity (EPR) demands that all the spacetime laws of physics must remain same in every frame of reference. Einstein provided us powerful techniques to express the laws of physics in terms of 4-vectors, tensors and their unification. Maxwell’s equations in tensor form as divergence of electromagnetic field tensor is equal to 4-current density that is a physical relation between tensor and 4-vector. The first problem arises when we transform divergence of electromagnetic field tensor and 4-current density under Lorentz transformation matrix (LTM) by different transformation laws then the results of both sides do not agree. This situation led to discover a single transformation law (STL) for 4-vectors and tensors and universal Lorentz transformation matrix (ULTM) such that their results after transformation remain same. The second problem lies with the present form of LTM as 4-vectors and tensors do not remain same under LTM except inner product of 4-vectors and tensors. Special relativity (SR) is based on inertial frame while general relativity (GR) requires noninertial frame. ULTM is valid for SR and GR equally to fulfill the requirement of EPR. In this paper, we will mainly focus on transformation of electromagnetic theory consisting of 4-current density, electromagnetic field, Maxwell’s equations and conservation law in tensor notation. Matrix method is employed for all calculations. As a consequence of STL, new symmetry terms appear in electromagnetic field, Maxwell’s equations and conservation law. These terms not only obey EPR but also Noether’s symmetry principle. The significance of discovery of STL and ULTM is that all the spacetime laws of physics namely 4-vectors, tensors and their inner product remain same independent of inertial or noninertial frame. Constancy of speed of light is a special case of 4-velocity. This is the reason that it is named as rebirth of Einstein’s theory of relativity.

Keywords: unification of SR and GR via ULTM, single transformation law for 4-Vectors and tensors, emergence of new symmetry terms, form invariance of all the laws of physics, constancy of speed of light and 4-velocity

Received November 12, 2022; Revised December 19, 2022; Accepted December 27, 2022

Citation: N. Hussain, H.A. Hussain, A. Tariq, M. F. Yaseen, J. Amjad, A. Qayyum, Re Birth of Einstein’s Relativity via Unified Transformation Law. International Journal of Physics, 11 (1). 6-16. January 2023. doi:10.12691/ijp-11-1-2

1. Introduction

What is the importance of principle of relativity? Why do we need same laws of physics everywhere? Do we have a comprehensive and consistent framework of same laws of physics at the moment based on EPR? The principle of relativity sets up a standard for the validity of any physical theory. Every physical theory must obey EPR. The answer to the second question is that same laws of physics are needed to explore other regions of our universe for our safe survival. Thirdly, we do not have a framework of same laws of physics yet based on EPR. In the subsequent development, these issues are taken up by pointing out the existing gaps in the achievement of same laws of physics.

The elegance of Einstein’s theory of relativity teaches us how to unify laws of physics in compact form in terms of 4-vectors and tensors including mathematical operators. Electromagnetic theory is one of the physical example that fits naturally in relativistic dress. My earlier research was to develop general covariance of Maxwell’s equations in curved and noninertial spacetime [1]. In order to see how do the laws of physics remain same and the mathematical tools for them then it was found that laws of physics are transformed under LTM. Tensors like electromagnetic field was transformed by similarity transformation technique and 4-vectors by usual transformation under LTM. In both cases, laws of physics in their final form were not independent of effect of frame of reference contradicting with EPR. In other words, laws of physics were not found same after transformation under LTM. Only inner product of 4-vectors and tensors was shown same that was based on non-invariant assumption.

Near about twelve decades have passed throughout the literature or standard textbooks on relativity and relativistic electrodynamics but no one [2-11] has ever noticed that 4-vectors and tensors do not remain same under LTM and even spacetime itself. The need of arbitrary frame of reference is discussed in the papers by Nicholie [12] used monad Lie derivative having no simple solution and Scorgie [13] employs Frenet equations to introduce intrinsic coordinates. But but what required is that the frame should be constructed employing such coordinates. both models do not provide a simple solution, EPR is the fundamental postulate of relativity theory. If any one or more examples violate EPR then we must think over the problem seriously. Where does the problem lie? The problem lies with LTM. How this deficiency was noticed? While working on Maxwell’s equations in tensor form whose left side is divergence of tensor and right side is a 4-vector. What happens to their results when we transform both sides under LTM? Since in the contemporary world tensors are transformed using similarity transformation where LTM is applied twice while in the transformation of 4-vector LTM is used once only. There results after transformation do not agree. In principle, their results after transformation must remain same. This situation led to discover a single transformation law STL valid for both sides of equation. The problem of LTM is replaced by ULTM. Why LTM is unable to provide form invariance of laws of physics? It doesn’t behave as a physical identity matrix because only physical identity matrix can provide same laws of physics. ULTM is nothing but physical identity matrix so all the laws of physics remain same under this matrix. LTM and ULTM both have the same geometrical origin i. e. hyperbolic geometry. LTM is a rotation matrix in hyperbolic space possesses discontinues symmetry while ULTM is rotation on a unit hyperbolic space possesses continues symmetry. ULTM is developed in 2-dimesions as well as in 4-dimensions which are entirely new results. Their inverses are very strange in the sense that they both behave as a physical identity matrices.

It is well-known that the speed of light is constant independent of the motion of the source or the observer. This theoretical and empirical fact indicates there must exist a framework for all the laws of physics. ULTM has provided the solution. Speed of light is a temporal component of 4-velocity (C + V) that remains same independent of the motion of the source or the observer under ULTM so faster than speed of light objects are included in this model. ULTM like rotation matrix is also developed that is associated with unit circle in terms of trigonometric functions. Special relativity theory has not cared about it but it is an integral part which has never been pointed out. Mathematically, it will be discussed in the subsequent development.

Since standard electromagnetic theory is the most empirically established and well-known physical theory consisting of 4-vector and tensor viz. 4-current density, electromagnetic field tensor, Maxwell’s equations and conservation law in tensor form. When applied STL and ULTM, then new symmetry terms appeared along the diagonal of the tensors written in components. These terms contribute in such a way that conservation law and form invariance of laws is not altered that satisfies EPR and symmetry principle. It is to be noted that time component alone doesn’t remain same and similarly space components but their combination always remains same. Einstein gravitation tensor is symmetric tensor. Its transformation under ULTM doesn’t alter its symmetry so GR is naturally unified with SR. This includes all types of motion constant velocity, variable velocity, rotation etc.

Matrix method is valid throughout all transformations. The existing confusion in relativity theory that the following quantities do not remain same: e.g. 4D displacement, linear spacetime operator () and () do not remain same only their inner product () remains same. Here these all remain same. Similarly, 4-vectors and tensors including their inner product remain same. Acceleration, rotation and their combination remain same. All the final results of each transformation remain independent of effects of frame of reference. Einstein’s theory of relativity has taken a new birth and became comprehensive for a common student of science.

In sec-1, problem with present form of LTM showing 4-current density and electromagnetic field tensor, do not remain same, inner product is given for comparison; inconsistency in transformation of Maxwell’s equations in tensor form is pointed out and its consistency is provided as a unified transformation law; ULTM in 2D as well as in 4D with inverses are developed. In sec-2, Electromagnetic theory under 4D ULTM is developed containing transformation of 4- current density, Electromagnetic field, Maxwell’s equations, conservation law and inner product of Electromagnetic field tensor; general relativistic law, noninertial field tensor transformation. In Sec-3 2D ULTM is applied to 4-velocity, inner product of 4-current density, wave operator; trigonometric rotation matrix is used to transform 4D coordinates. Sec.4 is based on discussion and comparison with the contemporary work; Sec5-consists of conclusion and final section contains important references.

Sec-1: Mathematical Development of The Model

Notations in this model are adopted according to modern approach of relativity. Greek alphabets …runs from 0 to 3 and Latin letters i, j, k, .. from 1 to 3. Comma (,) denote partial differentiation e.g. Partial derivative of electric field w. r. t. time, Partial derivative of electric field w. r. t. x-axis, Partial derivative of electric field w. r. t. y-axis, Partial derivative of electric field w. r. t. z-axis, Fµν,ν means 4-dimensional or spacetime partial derivative of EMF tensor.

4 dimensional Coordinates xμ = (x⁰, x¹, x², x³) = (ct, x, y, z) =(ct, xi) with x⁰ = ct and xi = (x, y, z). Time component ct is scalar while space components xi is vector such that xμ is the unification of time and space. The dimensions of all components are that of length.

In this model, 4-vectors and electromagnetic theory in tensor form is considered. We transform these objects under ULTM in 4D and 3D notations like electromagnetic field, Maxwell’s equations and conservation law.

1.1. Problem with Contemporary LTM

[picture] | (1)

Det

In mathematics 4 by 4 identity matrix is

Det = 1.

Einstein and his contemporaries perhaps thought that, since the determinant of LTM and mathematical identity matrix is 1 so it will give us same laws of physics under LTM. Unfortunately, the idea was not consistent. A physical matrix having determinant 1 doesn’t mean will provide same results after transformation.

For a physical identity matrix, it is required that each row and column must contain a physical identity equal to 1.

1.2. Examples that do not Show Form Invariance under LTM

We are considering only two examples one for 4-vector and one for tensor that do not remain same under LTM otherwise it is true for all 4-vectors and tensors

Example-1: Transformation of Contra-variant 4-current Density Under LTM

[picture] | (2)

4-current density does not remain same under LTM

The transformation is also true for covariant 4-current density

4-current density doesn’t remain same under LTM

Example 2: Transformation of EMF Tensor in 2D LTM

[picture] | (3)

Similarity transformation is used by the contemporary physicists. The above matrix retains its anti-symmetry but its constituent terms are not independent of reference frame. Furthermore, the original tensor is not obtained or its form invariance is gone.

1.3. Inner Product of Electromagnetic Field Tensor under LTM

Only inner product of 4-vectors and tensors remain same under LTM. It is to be noted that similarity transformation is applied for tensor as in equation ()

The positive components of electric field are from the above expression

[picture] | (4)

The positive components of magnetic field are from the above expression

Squaring all the components and Subtracting square of B components from square of E

[picture] | (5)

Gives non-invariant form of Maxwell’s equations under similarity transformation based on LTM

1.4. Adventure of (ULTM) in 2D and 4D

The presence of number 1 in third and fourth row of matrix (1) provided a clue that first and second row must consist of such physical terms whose sum can give 1 in both rows. Fortunately, these terms are well known in unit hyperbolic geometry in the form of identity

By substituting in first column and in the second column of first row and reversing their order in second row gives us universal Lorentz matrix.

[picture] | (6)

The inverse of matrix () is calculated as

[picture] | (7)

It is very strange to note that it also acts as a ULTM

1.5. Universal Rotation Transformation Matrix in Trigonometry

[picture] | (8)

It possesses continues symmetry at each angle. All the spacetime laws of physics remain same under transformation of this matrix.

Matrices (6) and (8) both give the same results that are also entirely new findings.

1.6. 4 Dimensional ULTM

What to talk about ULTM in 4D, there doesn’t exist ULTM in 2D. It is entirely new result having no counter in present context of relativity theory.

The generalization of Lorentz matrix () in 4 dimensions is

[picture] | (9)

Its inverse is

[picture] | (10)

The above matrix is also a 4 by 4 physical identity matrix where faster than speed of light objects possible to accommodate to verify the validity of Eq. (9) and (10), we multiply them to get identity matrix

1.7. Incompatibility of Maxwell’s Equations in Tensor Form

Maxwell’s Equations in tensor form are well known

[picture] | (11)

The transformation of both sides in the literature on spacetime physics is carried out as

[picture] | (12)

Both sides of equation are transformed under different transformation laws so the results of both sides are not compatible.

Consistency lies in the following transformation laws

[picture] | (13)

Both sides of equation are transformed under the same transformation law and the results are in complete agreement.

Eq. (13) can be written in a single unified transformation law as

[picture] | (14)

This also implies the validity of transformation of electromagnetic field and conservation law

[picture] | (15)

[picture] | (16)

The relations (13), (14), (15) and (16) are the unification of 4-vectors and tensors through a single transformation law leading to new perspective of spacetime physics. These relations along with ULTM decoding the physical significance of electromagnetic singularities as new symmetry terms shown in the subsequent development

2. Application of 4D ULTM and STL on Electromagnetic Theory

2.1. Transformation of Contra-variant 4-Current Density Jμ

[picture] | (17)

Adding all the components on left and right hand side

Re arranging the terms

Result

[picture] | (17a)

4- Current density remains same after transformation.

The transformation is also true for covariant 4-current density

[picture] | (17b)

It is to be noted that all 4-vectors like 4D coordinates , 4D velocity etc. can be transformed by following the above procedure

2.2. Transformation of Electromagnetic Field Fμν

[picture] | (18)

Electromagnetic field tensor is related to its components electric field E and magnetic field B as follows

Electromagnetic field tensor in component form is needed to get new terms along the diagonal of Electromagnetic field. It is represented as 4 by 4 antisymmetric matrix

Now applying transformation

Note that electric field is transformed into a mixture of electric and magnetic field. Similarly, magnetic field is a mixture of electric and magnetic field. After simplification and by putting the values of identities, we get the original form of electromagnetic field. The above transformed matrix apparently seems to alter the symmetry of electromagnetic field but surprisingly, the final result will remain same i. e. antisymmetric electromagnetic field remains antisymmetric in its original form. The new terms appearing along the diagonal have physical values. These terms are the missing links that are uncovered by new transformation law (13). If we do not take these terms into account then the form invariance, conservation and symmetry of electromagnetic field all are lost.

Hidden Symmetry or new symmetry terms

The term (18a) contribute in electric field and (18b) to (18d) in magnetic field.

in usual electromagnetic theory

Substituting the value in above, we get

RESULT:

[picture] | (18e)

This is original form of Electromagnetic field tensor. All the terms on RHS cancel out due to anti-symmetry

Electromagnetic field tensor remains same in its original form with its antisymmetric property

2.3. Transformation of Maxwell’s Equations Fμν,ν

Maxwell’s equations in tensor form is the sum of Gauss’s law and Ampere’s law. First row of the matrix is Gauss’s law and remaining three rows constitute Ampere’s law

[picture] | (19)

Maxwell’s equations in terms of tensor in matrix form are

Hidden Symmetry or new symmetry terms

(19a) contribute to complete the structure of time varying electric field while (19b) to (19d) to complete Gauss’s law and Ampere’s law. These terms retain the symmetry of Maxwell’s equations original form in the final result

Substituting the value of in above, we get

[picture] | (19e)

where

Sum of Gauss’s law and Ampere’s law

RESULT:

Maxwell’s equations remain same in its original form whereas the contemporary frameworks are devoid of it

2.4. Transformation of Conservation Law Fμν,νμ

Conservation law can be written in tensor form using matrix is very simple but no one has ever noticed this simplicity. First row tells us about conservation of Gauss’s law and remaining three give conservation of Ampere’s law and their superposition results in conservation of both electromagnetic laws

[picture] | (20)

Hidden Symmetry or new symmetry terms

Substituting the value of in above we get

Sum of conservation of Gauss’s law and Ampere’s law

Result:

[picture] | (20e)

Electromagnetic conservation law remains same in its original form.

3. Application of 2D ULTM

Electrodynamics in 2 Dimensional ULTM

Since electrodynamics in 2D is a special case of 4D and both formulations have the same final results so we leave it for the reader. Some results like inner product of 4-vectors and tensors are done in 2D.

3.1. Transformation of Inner product of Tensors FμνFμν=trace2[E²-B²]

The inner product of cotravariant and covariant electromagnetic field tensor whose trace gives us scalar quantity called invariant of electromagnetic field as

[picture] | (21)

The product is decomposed in to 4 components that directly gives the trace of product

the transformation is now become very simple

After multiplying, adding all the terms on left and right hand sides, we have

Now, putting the values of the terms in small brackets and of in above equation

Result:

[picture] | (21a)

Inner product of Contra and covariant electromagnetic field tensor remains same

Inner product of cotravariant EMF and dual of its covariant tensor:

[picture] | (22)

The above product is also decomposed in to 4 components that only represent the trace of the product

Trace

After multiplication, adding all the terms on left and right hand side

Result:

[picture] | (22a)

inner product of electromagnetic tensor with its duel remains same.

3.2. General Relativistic Transformations

Einstein gravitation tensor, metric tensor, Ricci tensor are the objects of general relativity. These tensors are symmetric in nature. Under ULTM, all types of tensors remain same independent of symmetric or asymmetric

Einstein Gravitation Tensor Transformation

[picture] | (23)

Similarly, metric tensor, stress energy tensor, Ricci tensor remain same under ULTM in their original form.

3.3. Noninertial Frame Transformation

Noninertial field tensor Ωµν or [14] is second rank anti-symmetric tensor. Translational acceleration a and angular velocity are its components.

Ωµν is related to its components as follows

[picture] | (24)

Electromagnetic field tensor in component form is needed to get new terms along the diagonal of electromagnetic field that is represented as 4 by 4 antisymmetric matrix

[picture] | (25)

The transformation of this tensor gives the same results as that of electromagnetic field tensor

[picture] | (26)

3.4. Transformation of 4-Velocity νμ

[picture] | (27)

After simplification we get

3.5. Transformation of Inner Product of 4-current Density Jμ′Jμ

[picture] | (28)

After multiplication, adding all the terms on left and right hand sides

Result:

[picture] | (28a)

Inner product of 4-currenr density remains same in single step transformation

3.6. Transformation of 4D-wave Operator □²= ∂²/∂t²- ▽²

[picture] | (29)

Adding and rearranging all terms on left and right hand side, we get

4-D wave operator or de Alembertian operator remains same in its original form

3.7. Transformation of Contra-variant 4D Coordinates xμ=(x⁰,x) in 2D ULTM

Only cotravariant 4-vectors are transformed for simplicity otherwise transformation law is valid for covariant 4-vectors

[picture] | (30)

The transformation is also true for covariant 4-position

4. Discussion and Comparison

Present day spacetime laws of physics do not satisfy the fundamental principle of relativity as 4-vectors, tensors and spacetime itself suffers from non-invariance under LTM viz. equations (3) and (4). The inner product of electromagnetic field tensor via equation (5) is also based on non-invariant quantities. Only constancy of speed of light is independent of motion of the source or the observer. The remaining laws of physics remained unattended on the same footing. The unavailability of ULTM and STL for 4-vector and tensor caused a lot of confusion among the relativistic context e.g. Bowler [15] says that relativity theory is difficult to understand. Another physicist [16] writes space and time are incomplete and relies on Galilean relativity.

The discovery of ULTM and STL for 4-vetors and tensors based on matrix method has rectified the above mentioned via equations (6)-(16). These results are entirely new having no counter example in the contemporary world. Complete framework of electromagnetic theory is transformed under ULTM in 4D in 2D 4-current density, electromagnetic field, Maxwell’s equations, conservation law in equations from (17) to (20). Since ULTM in 4D and in 2D give the same final results so ULTM in 2D is employed in the transformation of inner product of electromagnetic field tensor to get the same form of Lorentz invariants E.B and by equations (21) and (22). Other relations gravitational tensor, noninertial tensor field, 4-velocity, inner product of 4-current density, 4D wave operator are transformed from Eq.(21) to (29) respectively. Universal trigonometric rotation matrix is applied only to show the form invariance of 4D coordinates Eq. (30), otherwise it is applicable to whole physics.

One can analyze from equation (12) through the calculations via equations (2) and (5) that both sides of maxwell’s equations in tensor form give different results due to different transformation law under LTM.

Relation (13) is the most important discovery that has unified 4-vector and tensor. Re-writing the results for clarity

[picture] | (13)

By eq. (17a) and (19e), their equality is given below

[picture] | (17a)

[picture] | (19e)

Comparing (17a0 and (19e), we observe that

where

and

In relativistic electrodynamics, Gauss’s law and Ampere’s law individually do not remain same but their combination remains same as clear from above equations. Similarly, charge density and current density do not remain same but their combination remains same.

Relation (13) or unified STL is written in more compact form by equation (14) i.e.

[picture] | (14)

Since the model is entirely new having no counter in the present context of spacetime physics so the comparison with the contemporary models is given below:

There exists only one model of spacetime electromagnetism by H. A. Atwater [17], page129, who pointed out new symmetry terms in electromagnetic field only. On reproduction of his work with newer scheme, it is observed that the diagonal terms do not cancel out whereas in scheme they do cancel. Since his model doesn’t provide form invariance of electromagnetic field in its original form so cannot be persuaded further. Anyhow, his model provided a reasonable guideline.

All concepts of Electromagnetic theory can be expressed in terms tensors in matrix form as worked out above. In the contemporary relativistic electrodynamics, only electromagnetic field is used in matrix form but do not extend it to Maxwell’s equations and conservation law. Atwater [17] has worked out Maxwell’s equations in matrix form but using Galilean metric in similarity transformation which also remained non-invariant. Why matrix method is important? Einstein’s summation convention method doesn’t work when derivatives of 4-vetors or tensors are involved. Matrix method works well throughout the development.

The role of symmetry principle is necessary to consider over here as the new symmetry terms have emerged in this model. It states that “Any principle of relativity prescribes a in natural law: that is, the laws must look the same to one observer as they do to another. According to a theoretical result called , any such symmetry will also imply a alongside so symmetry, conservation law and form invariance of laws of physics are deeply related to each other [18 ]. The new symmetry that appeared only in tensor approach has drastically simplified the problem of conservation law and form invariance of all the laws of physics in all frames of reference.

5. Conclusion

Einstein’s relativity theory is fundamentally based on EPR but the contemporary frame work of spacetime physics doesn’t satisfy the above requirement. What was missing or un-noticed gap to meet the standard of the theory? By developing ULTM in 2D, 4D and a unified STL for 4-vectors and tensors has filled the gap with calculations step by step.

The significance of this work is that all the spacetime laws of physics remain same after transformation in their original form independent of inertial or noninertial frame. SR and GR are unified.

Speed of light is already time component of 4-velocity so the transformation of 4-velocity that remains same under ULTM via equation (27) doesn’t put any restriction on the constancy of speed of light. It provides us a room for the objects moving faster than speed of light without violating EPR.

Electromagnetic theory has become completely relativistic, an elegant example of a physical theory due to which we named it as a Rebirth of Einstein’s theory.

Since most of the systems in the universe are noninertial like our planet earth, particle accelerators, Large Hadron Collider, stars, galaxies, supernova explosion etc. It is now empirically verified that expansion of our universe is accelerating [19] so physics in noninertial frame will be real time physics. We need to formulate spacetime laws of physics relative to noninertial frame of reference. It will include ULTM for noninertial frame in terms of acceleration and rotation. For the interest of the readers, formula of energy of an object that is accelerating and rotating is

[picture] | (31)

Where m is the mass, a is acceleration and is angular velocity [unpublished data].

List of Abbreviations Used

EPR: Einstein Principle of Relativity

LTM: Lorentz Transformation Matrix

ULTM: Universal Lorentz Transformation Matrix

STL: Single Transformation Laws

References

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