1. Introduction

Maxwell’s equations are the foundation of electrodynamics, and they relate electric and magnetic fields to each other [1]. Historically, Lorentz used Maxwell’s equations to derive his space-time coordinate transformation rule [2]. Einstein in 1905 used Lorentz coordinate transformation rule to prove the validity of Maxwell’s equations in all inertial frames by using two postulates [3]; (i) The physics laws are invariant between two inertial frames. (ii) The speed of light is constant and independent of the direction of the motion of the emitting body in all inertial frames in free space. In electrodynamics, Maxwell’s equations and Lorentz force describe how the charge and current sources with densities and generate electric and magnetic fields ( and ) [1]:

(1)

(2)

(3)

Maxwell’s equations lead to several conservation laws [4], such as current continuity equation, conservation of electromagnetic energy, and momentum for which the electric and magnetic fields are Lorentz invariant between two inertial frames. Proof of Lorentz invariance of electromagnetic fields are interest over almost a century [5]-[8]. Einstein used the rates of momentum and of energy to transform electric and magnetic fields between two inertial frames [4]

(4a)

(4b)

where is Lorentz factor with velocity normalized to speed of light .

The electromagnetic field transformation between two inertial frames is also studied by using the electromagnetic field tensor, which is also called Faraday tensor [4]. In this formalism, one uses covariant and contravariant Faraday tensors and , and their duals and . To obtain the components of one uses in . Likewise, components of dual tensors and are obtained by using and in and , respectively. For boost along the x-axis, Faraday tensor in frame is obtained from the classical transformation , explicitly written as

(5)

Matching both sides of Eq. (5) one finds Eq. (4a) for Cartesian components of electric and magnetic fields in the inertial frame . Likewise, inverse transformation gives the Cartesian components of covariant tensor in frame , which then yields Eq. (4b).

The transformed field equations (4a) and (4b) state that: (i) the electric (magnetic) field are Lorentz invariant along the x-boost direction but not in the y and z directions. (ii) The scalar product of electric and magnetic fields is Lorentz invariant and (iii) the vector product of electric and magnetic fields is not Lorentz invariant , contradicting Lorentz transformation of vector quantity which must be invariant between two frames and .

In the classical Faraday tensor transformation, we observe that (i) Trace of the products of Faraday tensors and their duals is , which has no physical meaning since it yields zero electromagnetic field Lagrange density in free space, which is not realistic because electromagnetic waves transfer energy and momentum [4]. (ii) Furthermore, the trace of covariant and contravariant Faraday tensor and its dual is equal to , which suggests that Faraday tensor and its dual are orthogonal when scalar product of electric and magnetic fields is zero.

Following the work of Mignani and Recami [9], we recently proposed a 6-dimensional spacetime (3+3) frame [10,11] in which the transformed relativistic velocity is combined with energy conservation to successfully demonstrate the Lorentz invariance of electric and magnetic fields and Maxwell’s equations between two frames under rotation. In this work, we extend our recent study [10], [11] to study Lorentz invariance of relativistic quantities (e.g.position, velocity, momentum, force, electromagnetic fields, Poynting vector, Maxwell’s equations, and energy-momentum tensor in a four-dimensional spacetime in which both space and time coordinates are linearly inter-related. The outline of our presentation is as follows: In sections 2 and 3 we derive the metric equation and transform 4-velocity, 4-momentum and 4-force vector components between two frames. In sections 4 and 5 we use 4-vector velocity with the law of conservation of energy to prove the invariance of electric and magnetic fields under spatial rotation. In sections 6, 7, 8, and 9, we then combine the law of conservation of energy with Faraday tensor and its dual to study Lorentz invariance of electromagnetic fields, Maxwell equations, current continuity equation, and electromagnetic field energy-momentum tensor between two reference frames under a boost and spatial rotation. In section 10, we discuss the details and applications of the proposed theory in invariant relativistic electrodynamics and gravitational field theory.

2. Generalized Four-Dimensional Spacetime

We introduce a generalized four-dimensional spacetime wherein two “massive inertial frames” and , which coincide with a stationary inertial frame at time , and move relative to each other with an arbitrary velocity . The space and time coordinates are inter-related: in frame and and in frame . We require that Einstein’s two postulates are also valid in generalized 4-dimensional spacetime. We consider an event sending a light signal from the origin and second event of arrival at some points and in frames and . The square of the displacements are described by

(6a)

(6b)

where the differential space-time displacements are defined as

(7a)

(7b)

where in spherical coordinates with .

A pair of events with zero (null) separation connected by a signal at constant speed is described by

(8)

where is generalized Minkowski metric tensor. Equation (8) can be written as

(9)

where and . Matching both sides of Eq. (9) gives

(10)

as generalized Lorentz scaling factor, with and.

Figure 1 shows that are anisotropic and is uniform at any azimuthal angle .

Figure 1. Cartesian components of Lorentz factor as a function of azimuthal angle in spherical coordinates for polar angle and normalized speed (left) and (right)

The differential space and time coordinate displacements are then written in matrix form

(11a)

(11b)

where,, , in frame and , , , and in frame . Equations (11a) and (11b) is rewritten in a familiar matrix form as

(12)

The space and time differential displacement equations (7a) and (7b) are then replaced with

(13a)

(13b)

Using Eqs. (13a) and (13b) one then writes Cartesian components of velocity in frames and

(14a)

(14b)

(14c)

Combining Eqs. (13a) and (13b) reduces the four unknowns in Eqs. (14a), (14b) and (14c) to one

(15a)

(15b)

The Cartesian components of the velocity vector in frame are then written as

(16a)

(16b)

(16c)

Since and in Fig. 1, Eqs. (16a), (16b), and 16c) reduce to

(17a)

(17b)

which are identical to those in the 6-dimensional spacetime theory [10,11] and their use in finding relativistic mass, energy and Doppler effects is discussed at the end of the manuscript as appendix.

3. Vector transformation and Four Vectors

In this section we lay down the groundwork to study the invariance of relativistic vector quantities between two inertial frames. We momentarily set aside the relativity and focus on the three-dimensional vector transformation [12]. Since coordinate systems are used for convenience; we are free to use the following Figure 2 to define stationary 3-dimensional vectors and in frames and , both of which are initially coincide with a stationary universal inertial frame at .

Figure 2. The schematic diagrams of two stationary vectors and in terms of unit vectors in a counterclockwise rotation in frame through angle into frame plane (a) and in clockwise rotation in frame through angle in clockwise direction into frame (b) for

The stationary vectors and have same length from origin of two frames and , written as

(18)

which means that an ordinary vector transformation is identical as if a rotation causes no change of the magnitude of the vector quantity with respect to origin of any massive inertial frame. The unit vectors and in the massive inertial frames and are defined by using the classical vector transformation [12] and are related to each other according to following equations

(19a)

(19b)

Consequently, Cartesian components of and are then written in linear matrix form as [11]

(20a)

(20b)

It is noted that three-dimensional rotations can be around any of the three coordinate axes. The 4-dimensional analogue of 3-vector transformation equations given by equations (20a) and (20b) for rotation in counterclockwise and clockwise directions about the z-axis are written as

(21a)

(21b)

Here it is noted that, just like 3-dimensional rotations, 4-dimensional rotations can also be around any of 3-coordinate axes. The scalar product of vector with itself leads to , or , which states that the magnitude of a vector is Lorentz scalar between frames and .

The 4-vector velocities and are then defined as

(22a)

(22b)

where and , The scalar product of the 4-velocity vector with itself in frame is equal to the scalar product of in frame , which allows us to write . Since , the speed of light is Lorentz invariant

The scalar product of momentum 4-vectors and with themselves give the same length from the origins of two frames. The momentum 4-vector transformation is then identified as a rotation if it causes no change in their magnitudes.

4. Conservation of Energy Law and Invariance of Electromagnetic Fields

We will extend our recently proposed 6-dimensional space-time theory [10,11] to write the following linear matrix equations to describe the 4-force vectors and in terms of each other in frames and

(23a)

(23b)

where and . and are the Cartesian components of the 3-electromagnetic force vectors (Lorentz force) in the massive inertial frames and which, according to 3-dimensional vector transformation equations (21a) and (21b), are

(24a)

(24b)

Since in Eqs. (23a) and (23b), unlike the 4-velocity and 4-momentums, force vector is 3-dimensional as defined by Eqs. (24a) and (24b) in terms of unit vectors and in frames and . The scalar product of with itself leads to , or . The rates at which work is done on a particle by Lorentz force in frames and are

(25a)

(25b)

where and are Cartesian components of Lorentz force in frames and .

(26a)

(26b)

where and and are Cartesian components of and in Eqs. (22a) and (22b).

Considering the massive inertial frames and form a closed system in the stationary spacetime frame , the law of conservation of power (or energy) between them is written as

(27)

Using the transformation matrix equations (22a) and (22b) for and , Eq. (27) is written as

(28a)

(28b)

where . Using Eqs. (26a) and (26b) in Eq. (28a), we can write following equations

(29a)

(29b)

(29c)

Using Eq. (22a) for in Eqs. (29a), (29b), and (29c), we write following matrix equations

(30a)

(30b)

Using Eq. (22b) for in Eqs. (29a), (29b), and (29c) for inverse transformation, we write

(30c)

(30d)

which state that electric (or magnetic) field in frame is composed of electric (or magnetic) field in frame for spatial rotation of (x, y) plane about z-axis.

The scalar and vector products of electric and magnetic fields is essential in proving Lorentz invariance of electromagnetic energy, Poynting vector, current continuity equation, and densities of electromagnetic field energy and momentum. Using Eqs. (30a) -(30d) one writes

(31a)

(31b)

which suggests that and , so that and (Lorentz scalar invariants). The vector products of and in frame and of and frames and are

(32a)

(32b)

which states that is Lorentz invariant vector between the frames and .

5. Conservation of Energy and Faraday Tensor in Field Transformation

We will extend 3-dimensional Lorentz force to 4-dimensions as covariant and contravariant Lorentz tensors and , respectively. Here is the static electric charge, is the covariant contravariant antisymmetric second rank tensor, also known as Faraday tensor. and are covariant and contravariant 4-vector velocities. Considering the frames and form a closed system in the stationary frame , the law of conservation of power (energy) equation (27) can be written as

(33)

where and . Using and in Eq. (33) we write the following rule to transform in into for a counter-clockwise rotation about z-axis.

(34)

where is the transpose (inverse) of rotation matrix .

In tensor analysis, the covariant (contravariant) tensor is defined as vector product of any two vectors and is another vector and considered as second rank antisymmetric tensor [4], with . Covariant and contravariant 4-Lorentz forces as 4-tensors are defined as and in frame , written as

(35a)

(35b)

where is second rank antisymmetric covariant field tensor, also called Faraday tensors, with being any arbitrary 4-vector in frames and . are defined as

Similarly, one can define the contravariant Faraday tensor .

Dual Faraday tensor transformation between frames and is defined like Faraday tensor transformation equation (33). To do that we first introduce a “fictitious” magnetic charge [13] and define “fictitious” dual Lorentz force by using electromagnetic duality and in conventional Lorentz force . Here is the velocity of moving electrical charge. We then define 4- dual covariant and contravariant Lorentz tensors in frame as . Using and we can write the following rule to transform in into for a counter-clockwise rotation about z-axis.

(36)

where is the transpose (inverse) of rotation matrix . Covariant and contravariant fictitious 4-vector dual Lorentz forces and in frame are then written as

(37a)

(37b)

where is antisymmetric second rank covariant field tensor. Similarly, one can define contravariant dual tensor . Components of the covariant (contravariant) dual tensors are then obtained by using and in .

To eliminate the unnoticed error made in the classical use of Faraday tensor in field transformation [4] we use the generalized 4-dimensional Minkowski spacetime wherein 4-vector space coordinates are and in frames and .

Using the covariant 4-vector potentials and multiplying and with and the electric and magnetic fields in frames and are written as

(38)

We introduce covariant Faraday tensor with the following Cartesian components in frame

(39a)

(39b)

(39c)

Covariant and contravariant Faraday tensors and , and their duals and in frame are

(40a)

(40b)

(40c)

(40d)

where the components of the dual Faraday tensors and are constructed by using the so called the electromagnetic duality and in Faraday tensors and . Using equations (40a), (40b), (40c), and (40d) for the covariant and contravariant Faraday tensors ( and ) and their duals ( and ), one can write the following matrix expressions for the products and of their duals in frame

(41)

(42)

where , and are the x, y, and z components of the Poynting vector. Traces of and of are nonzero:

(43a)

(43b)

which is physically realistic because the electromagnetic waves transfer energy and momentum in free space. Traces of product of covariant and contravariant Faraday tensors with their duals are

(44a)

(44b)

Equation (44a) suggests that and are orthogonal only when is zero.

Equation. (44b) suggests that and are always orthogonal in frames and .

6. Faraday Tensor in Electromagnetic Field Transformation

The incremental displacement of a coordinate system relative to its initial position is known to be composed of a translation as well as a rotation. While the components of 4-vector quantities transform according to Eq. (21a) and (21b), the spatial translation has no effect on them. One can use Faraday tensors and their duals determine the components of electric and magnetic fields in frames and by using:

(i) Lorentz boost along the direction of motion, which is appropriate transformation for relative motion of two observers, and

(ii) Spatial rotation, which relates spacetime coordinates of two observers in two frames which are subject to planar rotation with respect to each other about a fixed axis.

6.1. Faraday Tensor and Field Transformation with Lorentz Boost

We use , and transformation rules to transform electromagnetic fields between frames and for Lorentz boost along x, y, and z-axes, which are written as

(45)

where with and . Here is called the boost parameter [4]. For a Lorentz boost along x-axis, the transformation allows us to obtain the Cartesian components of Faraday tensor in frame in terms of those in frame

(46)

which yields the covariant Faraday tensor in frame in terms of those in frame . Transformation and yield electric and magnetic field components in frames and

(47a)

(47b)

(47c)

(47d)

which are the same as Eqs. (4a) and (4b) and yield invariant scalar product and non-invariant vector product between two frames under Lorentz transformation.

In the next step, we introduce a circular boost, rather than hyperbolic one defined in Eq. (45), for motion along x, y, and z- axes in frame to find , which are

(48)

For a circular boost along x-axis, the components of covariant tensor in frame are obtained by using the transformation rule , which is explicitly written as

(49)

(50a)

(50b)

with inverse transformation rule one writes in frame and obtain

(50c)

(50d)

which yield Lorentz invariant scalar product of electric and magnetic fields between frames and . The vector products of electric and magnetic in frames and fields are

(51a)

(51b)

When unit vectors are equivalent in both frames Eqs. (51a) and (51b) yield and , contrary to the hyperbolic boosts in Eq. (45).

6.2. Faraday Tensor and Field Transformation with Spatial Rotations

It is possible to combine boosts with rotations to relate the electromagnetic fields to each other between the frames and , by using counterclockwise rotations of (x, y), (z, x), and (y, z) planes about z, y and x-axes, which are given by the following expressions

(52)

Replacing with one writes expressions for the clockwise rotation about x, y, and z-axes. As an example, the transformation allows us to determine the components of covariant Faraday tensor in frame in terms of the components of covariant in frame

(53)

with inverse transformation one writes similar equation for in frame . Matching both sides of Eq. (53) and of its inverse, one writes following matrix equations

(54a)

(54b)

(54c)

(54d)

which are identical to Eqs. (30a), (30b) and (30c), and (30d) and yield invariant scalar and vector products of electric and magnetic fields in frames and ,

(55a)

(55b)

which are both invariants between frames and .

7. Faraday Tensor and Invariance of Maxwell’s Equations

Using in and with for we write

(56a)

(56b)

(57a)

(57b)

Subtracting Faraday’s law of induction from Ampere-Maxwell’s equation and applying the divergence to the resultant equation we can write the following equation

(58)

Since and , equation (58) results in the conventional current continuity equation written as

(59)

where and are the total current and charge densities, respectively. In the following subsections we will prove that inhomogeneous and homogeneous Maxwell equations (56) and (57) are Lorentz invariant between two massive inertial frames. Recall the following expressions for the flux of a vector field through a spherical closed surface in frames and

(60)

Since any function (e.g., magnitude of electric and magnetic fields) is continuous at any point in space [16] in both frames and , we write the following chain rules of differentiation

(61)

(62)

7.1. Gauss Law of Electrostatics

Since electric field wave function is continuous at any point in space, taking and , and applying chain rule in Eq. (61) we write the charge density

(63)

where . Since and , matching both sides of Eq. (63) yields . where and are the charge densities in frames and , defined with respect to charge density in frame . We can write

(64)

which is Lorentz invariant Gauss law of electrostatics between the inertial frames and .

7.2. Gauss Law of Magnetostatics

Since magnetic field wave function is continuous at any point in space, taking and , and using chain rule in Eq. (61), we write

(65)

Matching both sides of Eq. (74), one finds for the space component of Lorentz scaling factor in Eq. (10), and covariant Eq. (65) is transformed into the invariant form

(66)

which is Lorentz invariant Gauss law of magnetostatics between the inertial frames and .

7.3. Faraday’s Law of Induction

Let us re-write the differential form of Faraday’s law of induction in Eq. (57b) in x, y, and z-directions of the Cartesian coordinates in frame

(67)

Applying the chain rule in Eqs. (61) and (62) to Faraday’s law of induction allows us to write

(68a)

(68b)

Side by side addition of Eqs. (68a) and (68b) allows us to write the covariant Faraday’s law

(69)

Matching both sides of Eq. (69) yield and for components of Lorentz scaling factor in Eq. (10). Covariant Eq. (69) is then transformed into invariant form

(70)

which is Lorentz invariant in the x, y, and z-directions between the inertial frames and .

7.4. Ampere-Maxwell Equation

Ampere-Maxwell’s equation (57b) are written as

(71a)

(71b)

(71c)

where and are the conductivities in frames and , defined relative to in the steady inertial frame . Applying the chain rule in Eq. (61) and (62) to differential form of Ampere-Faraday’s law in x, y, and z-directions and write

(72a)

(72b)

Combining Eqs. (72a), and (72b) side by side and adding and , we write

(73)

Matching both sides of Eq. (73), one finds and for space-time components of Lorentz scaling factor in Eq. (10). Then covariant Eq. (73) becomes invariant

(74)

which is Lorentz invariant Ampere-Maxwell equation between frames and .

7.5. Current Continuity Equation

Since Maxwell’s equations must satisfy the charge (current) continuity equation, using chain rules in Eqs. (61) and (62) we can write

(75a)

(75b)

where and are the total current and charge densities, respectively. Side by side additions of Eqs. (75a) and (75b) allows us to write the following covariant equation

(76)

Matching both sides of Eq. (76) we find and in Eq. (10), and

(77)

which is the Lorentz invariant current continuity equation between frames and .

8. Symmetric Electromagnetic Energy-Momentum Tensor and Conservation Laws

The classical symmetric energy-momentum tensor is described by a second rank tensor [4]

(78)

where is the contravariant field tensor and is mixed tensor, with. is energy density, is Cartesian components of Poynting vector, and , with Maxwell’s stress tensor defined as [4]

(79)

where is the Kronecker delta which is unity if and zero otherwise [4]. Mixed tensor in Eq. (78) has no explicit symmetry characteristics [4].

Consequently, we pursued a search for an alternative way to derive Lorentz invariant expression for a symmetric electromagnetic energy-momentum tensor. We discovered that the average sum of the product of covariant-contravariant Faraday tensor and transpose of the contravariant covariant dual Faraday tensors fits our requirement of the electromagnetic energy-momentum tensor. In the massive inertial frame they are written as

(80)

(81)

which have symmetric characteristics and consequently, algebraic sum of their traces is zero

(82)

Average sum of the product of covariant (contravariant) and transpose of contravariant covariant) Faraday tensors and of their duals, in frame is then written as

(83)

Cartesian components of the symmetric energy-momentum tensor in Eq. (82) are

(84a)

(84b)

(84c)

where is given by Eq. (79). Since is symmetric its trace must be zero:

(85)

9. Poynting Theorem and Conservation Laws

Using and Maxwell’s equations (56) and (57), divergence of Poynting vector can be written as

(86)

where, is the rate of work done by Lorentz force on moving charged particles and is the electromagnetic energy density. Divergence theorem yields

(87)

which is the well-known classical Poynting theorem [17]. The first and second terms on the left are the power flowing out of the volume and rate of stored energy density and the term on right is the power dissipated (or generated) in a closed electrical circuit.

We now demonstrate the use of symmetric energy-momentum tensor allows us to derive differential form of Poynting theorem and conservation of linear momentum with charge and current sources. Using the 4-vector current density in we write

(88)

Sum of the first-row elements on both sides of Eq. (88) yields the energy conservation law

(89)

which is the differential form of the Poynting theorem in free space given by Eq. (87). Further, adding the second, third and fourth row elements of Eq. (88) and combining them we write

(90)

which is the differential form of conservation of linear momentum with total current source.

Since for planar rotation about a fixed axis, Poynting vector is Lorentz invariant , letting (or ) and (or ), Eq. (89) can be decomposed as

(91a)

(91b)

(91c)

Side by side adding Eqs. (91a), (91b), and (91c) yields the following covariant energy continuity equation

(92)

which yield and in Eq. (10), which transforms covariant equation Eq. (92) into the following invariant form

(93)

Similarly, employing the differentiation chain rule to differential form of conservation of linear momentum equation with current source in Eq. (90) allows one to write the following equations

(94a)

(94b)

(94c)

Adding Eqs. (94a), (94b), and (94c) yields the covariant momentum continuity equation

(95)

Matching both sides of Eq. (95), one finds and in Eq. (10) for components of Lorentz factor and covariant Eq. (95) is transformed invariant form

(96)

which is Lorentz invariant linear momentum conservation equation between frames and .

10. Results and Discussions

In this work, we introduced so called generalized 4-dimensional massive inertial frames and , both coincide with stationary inertial frame at time , and move relative to each other with arbitrary velocity . The space and time coordinates are interrelated: , in and , in frame , where . We derived the generalized 4-dimensional spacetime metric equation (8), with Lorentz scaling factor in Eq. (10), which has anisotropic space and uniform time components. The role of this factor on the invariance of relativistic quantities such as energy, momentum, mass, time dilation and Doppler shift is discussed in appendix. In the following, we discuss Lorentz invariance of electromagnetic fields, Maxwell’s equations, and symmetric electromagnetic field energy-momentum tensor.

10.1. Comparison of Classical and New Way of Using Faraday Tensor in Field Transformation

By using the 4- velocity and 4-force vectors in the energy conservation law we demonstrated the Lorentz invariance of the scalar and vector products of electric and magnetic fields between two so called massive inertial frames for a spatial rotation about a fixed axis. As an example, consider point charge Q in frame . The produced electric and magnetic fields in frames and are

(97a)

(97b)

(97c)

(97d)

where . One can show that the scalar and vector products of electric and magnetic fields are Lorentz invariant and between two massive inertial frames.

To confirm the findings in section 4, we focused on the use of electromagnetic field strength tensor, also called Faraday tensor in sections 5 and 6 to derive expressions for the invariant electromagnetic fields. We demonstrated that the hyperbolic boost along x-axis classical use of Faraday tensor in field transformation given Eq. (6) and (45) for Lorentz boosts along the x, y and z-axes, lead to a non-invariant vector product of electric and magnetic fields, and in turn non-invariant Poynting vector, between two inertial frames. For a Lorentz boost along x-axis, one can use the transformation to find components of Faraday tensor in frame in terms of those in frame . Transformation and its inverse yield Cartesian components of the electric and magnetic fields in frames and in terms of each other, given by Eqs. (47a), (47b), (47c), and (47d), respectively. They are the same as Eqs. (4a) and (4b), with and . In zero velocity case, Eqs. (47a), (47b), (47c), and (47d) results in invariant vector product. However, this has no relativistic meaning since zero velocity case means that two frames coincide with each other at rest.

In the case of circular Lorentz boosts about the x, y, and z-axes given by Eq. (48), the result of Faraday tensor transformations surprisingly turns out to be quite different. For example, the circular Lorentz boost along x-axis with fixed z-axis, transformation yields components of covariant Faraday tensor in frame , which leads to Cartesian components of electromagnetic fields in frame as mixture of those in frame at any angle, according to Eqs. (50). However, as easily seen from Eqs. (51) and (52) for , contrary to the hyperbolic Lorentz boost in Eq. (47), the vector products of electric and magnetic fields is Lorentz invariant between frames and . This may be due to the difference between hyperbolic boosts in Eq. (45) and circular boost Eq. (48), which are derived from translational and rotational points of views for which and , respectively, in two dimensions.

In section 6.2 we demonstrated that if two frames and are related by spatial rotation of (x, y) plane with fixed z-axis, of (y, z) plane about x axis, and (z, x) plane about y axis, respectively, Cartesian components of covariant (contravariant) Faraday tensor in frame can be found according to the spatial rotation transformations given by Eq. (52). For a spatial rotation of (x,y) plane about the z-axis, covariant Faraday tensor in frame is obtained from , which yields Lorentz invariant electric and magnetic fields in frame in terms of those in frame according to Eqs. (54a) - (54d). Similar results are found under the spatial rotation of (y, z) plane about x-axis, and (z, x) plane about the y-axis, respectively.

10.2. Faraday Tensor and Lagrange Density of Electromagnetic Field

In section 6 we also proved that the new way of using Faraday transformation under the spatial rotations about fixed coordinate axes lead to Lorentz invariant properties (e.g., Poynting vector, inhomogeneous and homogeneous Maxwell’s equations, and equations of continuity) between two frames and , respectively. Considering a charged particles moving under the influence of an external electromagnetic field we use so called the action principle to derive its equation of motion [18]. Here L is the total Lagrange density of the charged particle which is equal to sum of the free space and external source contributions and is written as

(98)

According to Eqs. (43a) and (43b), the trace of the product of and of their duals are non-zero and yield a nonzero electromagnetic energy density in free space. Consequently, we write

(99)

which proves that contrary to the classical point of view [4], the electromagnetic Lagrange density is not zero in free space. This is physically realistic because electromagnetic waves transfer energy and momentum, which are the intrinsic properties of free space. Trace of product of covariant and contravariant Faraday tensors and of their duals in frames and are

(100a)

(100b)

which show that the product of and of their duals are Lorentz invariant between frames and . Meanwhile the trace of product of covariant (contravariant) Faraday tensors and of dual Faraday tensors are also invariant between the massive inertial frames and , written as

(101)

which is zero when , and , are invariant.

10.3. Symmetric Electromagnetic Energy-Momentum Tensor and Conservation Laws

In section 9 we used the symmetric electromagnetic field energy-momentum tensor to prove the invariance of the conservation of electromagnetic energy and linear momentum. We now extend this idea to cases under spatial rotations of planes about a fixed axis. For example, applying the 4-vector differentiation operator , we can extend Eq. (93) to

(102)

and using the chain rule in Eqs. (61) and (62 for the space and time differential operators, Eq. (102) can be written as covariant differential form of Poynting theorem with no current source

(103)

Matching both sides of Eq. (103) one finds and for the spatial rotation dependent space and time components of Lorentz scaling factor, which reduce to those in Eq. (10) without rotation, and transforms covariant Eq. (103) into invariant form

(104)

which is the Lorentz invariant classical differential form of Poynting theorem in free space.

Using the 4-vector operator and chain rule in Eqs. (61) and (62) we can extend Eq. (95) to the following equation for conservation of linear momentum

(105)

Matching both sides of Eq. (105) one finds and for space and time components of Lorentz scaling factor, which reduce to those in Eq. (10) without spatial rotation and transforms covariant Eq. (105) into invariant form. Following the similar steps to produce Eq. (104), we can write the conservation equations along the y- and z-components of linear momentum. Adding x, y and z components we write the following differential equation for 4-vector linear momentum conservation in the generalized 4-dimensional spacetime frame

(106)

which is the Lorentz invariant differential form of linear momentum conservation in free space.

10.4. Symmetric Electromagnetic Energy-Momentum Tensor for Field Transformation

It is also instructive to see if the use of the symmetric electromagnetic field energy-momentum tensor can be used to transform electromagnetic fields between two massive inertial frames and . We try this by using the counterclockwise rotation of (x, y) plane about the z-axis in Eq. (52) and the transformation yields the following symmetric tensor in frame

(107)

(108a)

(108b)

(108c)

where are the components of 3-D Maxwell stress tensor in frame related to those in frame . Equations (108a) and (108b) explicitly demonstrate that the electromagnetic field energy density, Poynting vector and momentum density are Lorentz invariant between frames and .

Furthermore, as a symmetric tensor, trace of energy-momentum tensor must be zero:

(109)

Likewise, the spatial clockwise rotation of (x, y) plane about fixed z-axis yields components of the inverse symmetric energy-momentum tensor in frame .

Since is derived from average sum of the tensor product of covariant and transpose of contravariant (vice versa) and of their duals, it is imperative to confirm that, and by back substitution we should be able to determine the Lorentz invariant electric and magnetic fields between frames and . Matching both sides of Eq. (107) and of its inverse, which is not written here to save space, for one obtains the following matrix equations for the Cartesian components of the electric and magnetic fields in frame those in frame ,

(110a)

(110b)

(110c)

(110d)

which are identical to Eqs. (54a), (54b), (54c), and (54d) for . We can then conclude that can reliably be used to find Lorentz invariant electric and magnetic field according to transformation for spatial rotation of (x, y) plane about the fixed z- axis, between two frames. This confirms the Lorentz invariance of symmetric electromagnetic field energy-momentum tensor between two massive inertial frames and .

Since Eq. (78) is representing the classical symmetric energy- momentum tensor and is described by using the 4-dimensional relativistic analogue of 3-dimensional Maxwell’s stress tensor [4], it is also imperative to see whether it also confirms the Lorentz invariance and trace characteristics described for the symmetric electromagnetic energy-momentum tensor in Eq. (107). This is expected since is also a symmetric tensor just like . In the framework of the classical Faraday field transformation, Lorentz invariance of the symmetric electromagnetic energy-momentum tensor in the massive inertial frame is written as

(111)

(112a)

(112b)

(112c)

Equations (112a) and (112b) yield invariant electromagnetic energy density, but Poynting vector and linear momentum density turn out to be non-invariant between the frames and . This contradicts Lorentz transformation which requires all components of relativistic vector quantities must be invariant. Since is a symmetric tensor, its trace is zero.

However, in the case of trigonometric circular Lorentz boost along the x-axis given by Eq. (48), applying in Eq. (78) one obtains symmetric tensor in frame

(113)

For symmetric energy-momentum tensor in Eq. (113) yields

(114a)

(114b)

(114c)

Equations (114a) and (114b) yield invariant energy density, Poynting vector, and momentum density between frames and . Furthermore, since is a symmetric tensor, Eq. (114c) predicts that trace of is zero in both frames, confirming that and are symmetric electromagnetic energy-momentum tensors. Component by component matching both sides of Eq. (113) and of its inverse for , one obtains the matrix equations (110a), (110b), (110c), and (110d) for Cartesian components of the electric and magnetic fields in frame in terms of those in frame . We conclude that using symmetric energy-momentum tensor is the most reliable way of finding invariant electric and magnetic fields for (i) the spatial rotation of planes about a fixed axis, and (ii) the circular Lorentz boost along the direction of motion and between two reference frames.

10.5. Symmetric Electromagnetic Energy-Momentum Tensor and Angular Momentum

Symmetrical electromagnetic field energy-momentum tensor is needed when we consider the conservation of angular momentum of the electromagnetic field [4]. We can write the angular momentum density of electromagnetic field in the following integral and tensor form [4]

(115)

As pointed out in section 8, classical construction of symmetric electromagnetic energy-momentum tensor in Eq. (78) is based on the mixed tensors which has no explicit symmetry characteristics [4]: it is neither symmetric nor asymmetric. Consequently, the symmetric angular momentum tensor must be constructed from the symmetric electromagnetic field energy-momentum tensor , which is defined according to equation (83), as the average sum of the product of covariant (contravariant) and transpose of contravariant covariant) Faraday tensors and of their duals, . Therefore, symmetric angular momentum density of electromagnetic field must be defined as

(116)

Conservation of total angular momentum of electromagnetic field is then defined as

(117)

Since is symmetric, then first and third terns n Eq. (117) can be eliminated and , which proves that conservation of angular momentum of electromagnetic field is symmetric.

10.6. Symmetric Electromagnetic Energy-Momentum Tensor and Einstein Field Equation

The proposed use of Faraday tensor, its dual, and symmetric electromagnetic energy-momentum tensor and its dual can have profound effect in the solution of Einstein equation [19]-[21], written as

(118)

where is the Ricci curvature tensor with R being the scalar curvature, and is the gravitational constant. In free space is the fundamental source of the electromagnetic and gravitational fields. Since is symmetric, then must be symmetric, so that we can write

(119)

Component by component matching the symmetric tensors in Eq. (119), one can write

(120)

Applying the 4-vector differentiation operator we can write

(121)

Using the chain rule in Eqs. (61) and (62) for space and time differential operators, Eq. (121) can be written as covariant differential form of Poynting theorem with no current source

(122)

Matching both sides of Eq. (120) one finds and for the spatial rotation dependent space and time components of Lorentz scaling factor, which reduce to those in Eq. (10) without rotation, and transforms covariant Eq. (122) into invariant form in which is the Lorentz invariant classical differential form of Poynting theorem in free space.

It is also important to point out that conservation of electromagnetic energy and linear momentum equations can also be written for the trigonometric circular Lorentz boost along the direction of motion following the steps to write Eq. (122). For a circular boost along the x-axis, we write the following matrix equation for the Poynting theorem under rotational boost

(123)

from which, similar to writing Eq. (123), one can write the covariant and then invariant form of Poynting theorem with no current source. Steps similar can be taken to write the equation for the conservation of linear momentum in free space.

It is also instructive to demonstrate the Lorentz invariance of the symmetric energy-momentum tensor in the massive inertial frame . Using Einstein’s gravitational field equation (118) we write

(124)

Since is Lorentz invariant, then must also be invariant between frames and . In other words, Einstein field equation must also be Lorentz invariant between frames and, written as

(125)

This demonstrates the ability of a simple and theoretically reliable derivation of the symmetric energy-momentum tensor and Lorentz invariance of relativistic quantities between two inertial frames in relativistic electrodynamics and in the classical field theory. Knowing the symmetric energy momentum tensor, one can solve Einstein’s field equation for the curvature of the universe.

11. Conclusions

We introduced a four-dimensional generalized Minkowski spacetime frame in which the space and time coordinates are linearly interrelated. After satisfying Lorentz invariance of metric equation between two massive inertial frames, we used classical vector transformation to derive general expressions for Cartesian components of the relativistic velocity, which is valid at any speed, including the speed of light. Considering two massive inertial frames form a closed and isolated system in four dimensional spacetime, we integrated the relativistic velocity components with conservation of energy and new way of using Faraday tensor and its dual in Minkowski space-time to prove that the electric and magnetic fields are Lorentz invariant under circular boost and spatial rotation of planes about a fixed axis. We demonstrate that the product of covariant and contravariant Faraday tensors and of their duals lead to non-zero electromagnetic field Lagrange density in free space, which is physically realistic since electromagnetic fields carry energy and momentum while they propagate free space. We derived analytical expressions for Lorentz invariant Maxwell’s equations, current continuity equation, and symmetric electromagnetic energy-momentum tensor between two inertial frames, with and without source. We further demonstrated that symmetric energy-momentum tensor can be used have a reliable derivation of Lorentz invariant electromagnetic fields between two inertial frames in the case of circular Lorentz boost and spatial rotation about a fixed coordinate axis. We believe that the proposed theory may have profound effect in creating new research areas in theoretical physics.

Data Availability Statement

This manuscript has no associated data, or the data will not be deposited.

APPENDIX I. Time Dilation, Doppler Shift, and Energy Dispersion Relation

Using Eq. (9) for the general form of four-covariant metric equation under Lorentz transformation, we write

(A1a)

(A1b)

where . Matching of Eqs. (A1a) and (A1b) yields

(A2)

For motion along x axis, and , Eq. (A2) reduces to

(A3)

Using , Eq. (A3) allows us to write the following expression for the relativistic invariance of Doppler shift between two massive inertial frames under Lorentz transformation

(A4)

For motion along the +x axis, and . Eq. (A4) reduces to for the relativistic Doppler shift in the classical four-dimensional spacetime theory.

Recall that Ives and Stilwell [22] who observed the wavelength of hydrogen atom emitted by canal rays with and against their motion by using a mirror and discovered the frequencies of displaced lines of incoming and outgoing light rays and their average are given by [23]

(A5)

Consider forward and inverse (incoming and outgoing) plane waves of frequencies and wave vectors in the massive inertial frames and , respectively, with wave functions

(A6)

where and in the massive inertial frames and , respectively. With Lorentz invariant phases of plane waves between and ( and ), we can write

(A7)

where is the incremental shift in the angular frequencies of plane waves in stationary inertial frame . and are the incremental shifts in and in frames and . Using the generalized time and velocity, equalities in Eq. (A7) lead to

(A8a)

(A8b)

where and , which can be proven by using and , and spacetime coordinate equations. Doppler shifts in the angular frequencies of forward and inverse plane waves and their averages in the massive inertial frames and are then written as

(A9a)

(A9b)

Expressions for angular frequencies in the massive inertial frames and are then written as

(A10)

where and are the background angular frequencies and and are the Doppler shifts with and . Multiplying and with and using and , one can write

(A11)

where and are the rest energies of a particle in and .

APPENDIX II. Relativistic Mass and Energy Dispersion Relation

Expressions for relativistic mass and energy are derived by considering the differential change in the energy of a particle moving under the influence of a force in frames and as [10], [11]

(A12a)

(A12b)

Using the change of variables, the integrals of expressions in Eq. (A12a) and (A12) are written as

(A13)

where with and in the massive inertial frame and with and in the massive inertial frame . The result integrals in Eq. (A13) give the relativistic masses in the massive inertial frames and

(A14)

where and are rest mass and initial velocities in both frames.

Since , the relativistic mass is Lorentz scalar in both frames.

The relativistic energy dispersion relations for a particle moving under the influence of a force in the and frames are found from the integrals of Eqs. (A12a) and (A12b) that are written as

(A15a)

(A15b)

where and , and , respectively.

Evaluating the first integrals in Eqs. (A15a) and A15b), then multiply both sides by and , and finally taking square root of the final results, one finds Einstein energy dispersion relation

(A16)

Furthermore, evaluating the second integrals in Eqs (A15a) and (A15b), one can also write

(A17)

Since by , and , relativistic energy is Lorentz scalar , which suggests that vector transformation does not affect the relativistic invariance of energy.

Dividing both sides of Eqs. (A16) and (A17) with and ,we and write

(A18a)

(A18b)

which suggest that the rest mass is linear function of frequency at any , and (i) at speed of light when nonstationary frame moves parallel to ±x axes of stationary frame and (ii) at v=0 as its limiting case at any angle The first result (i) proves that a relativistic particle has zero mass as it moves with the speed of light, frame independent. The energy dispersion relation (A16) becomes equal to

(A19)

The dynamic torsion balance experiment of Liu et al [24] yields an upper bound at f=7.41x10^-4Hz. Equations (A18a) and (A18b) predict = 4.34x10^-55kg suggesting that a particle at rest having a small magnitude but never zero. Comparison shows a good agreement with measurements [24] and astronomical observations [25].

The second result (ii) is compared with the prediction of Heisenberg uncertainty principle in rest frame, which yields an upper bound for the rest mass of photon

(A20)

where is the photon rest energy and . Notice that that the prediction of Eqs. (A18a) and (A18b) is times higher than that predicted by Eq. (A20] Equations (A18) and (A18n) give kg, compared with due to Eq. (A20) at , where year is the estimated age of the universe [26].