Electromagnetic Field Equation and Lorentz Gauge in Rindler Space-time

In this paper, we derived electromagnetic field transformations and electromagnetic field equations of Maxwell in Rindler space-time in the context of general theory of relativity. We then treat the Lorentz gauge transformation and the Lorentz gauge fixing condition in Rindler space-time and obtained the transformation of differential operation, the electromagnetic 4-vector potential and the field. In addition, charge density and the electric current density in Rindler spacetime are derived. To view the invariance of the gauge transformation, gauge theory is applied to Maxwell equations in Rindler space-time. In Appendix A, we show that the electromagnetic wave function cannot exist in Rindler space-time. An important point we assert in this article is the uniqueness of the accelerated frame. It is because, in the accelerated frame, one can treat electromagnetic field equations.


Introduction
In 2007, del Castillio and Sanchez [1] discovered Maxwell equations in vacuum in a uniformly accelerated frame and Maluf and Faria [2] derived electromagnetic field transformations in Rindler space-time in 2011 [3]. They used Maxwell equations for gravity field but we disagree with their approach because Maxwell equations for a uniformly accelerated frame have to be treated in flat Minkowski space-time and not in the curved space-time, which implies the presence of gravitational field. In this work, our aim is to find electromagnetic field equations in Rindler space-time, also in vacuum, but not in vacuum of the general relativity theory. In Sec. 2, after working out electromagnetic field equations in Rindler space-time, we derive the Lorentz gauge transformation and the Lorentz fixing condition, in addition to transformations for electromagnetic 4-vector potential in Rindler space-time. In Sec. 3, we define the electromagnetic field in Rindler space-time and we find the transformation of the electro-magnetic field. In Sec. 4, we obtain the electro-magnetic field equation in Rindler space-time and apply the gauge theory to Maxwell equations (worked out in earlier sections) in Rindler space-time for viewing the invariants of the gauge transformation.
We think it is important to know the electromagnetic wave function (radiation) in Rindler space-time but it is known that it does not _______________ * sangwha1@nate.com satisfy electro-magnetic wave equation mathematically (see Appendix A). Hence according to our arguments, many results published during the period 2007 -2011 (see Refs. [1] and [2]), especially the computation of electro-magnetic wave function, were incorrect. However, we do understand that electromagnetic wave function can exist in inertial frame as shown by Maxwell and Einstein.

Transformation of the Electro-magnetic 4-vector Potential, Lorentz Gauge Transformation and Lorentz Gauge Fixing Condition
The Rindler coordinate transformation is Now, the vector transformation is Therefore, the transformation of the electromagnetic 4-vector potential is given by the following equations: The transformation of differential coordinates is Lorentz gauge transformation in Rindler space-time is given by Where, Λ is a scalar function.
Here, Λ is a scalar function.
Therefore, the Lorentz gauge in Rindler spacetime can be written as Hence, Lorentz gauge transformation and Lorentz gauge in Rindler space-time are as follows: Electronic copy available at: https://ssrn.com/abstract=3508045 uniformly accelerated frame, which can be written as If we take the matrix of the transformation in the differential coordinate, then we have The inverse matrix of Eqn. (14) [12] and [14] can be obtained as Hence, we can obtain the matrix of transformation of differential operation as Electronic copy available at: https://ssrn.com/abstract=3508045 Therefore, the transformation of differential operation is (17) The above differential operation satisfies the following equations:

Space-time
Given the electromagnetic field ) , We need to perform computation in order to define the electro-magnetic field in Rindler space-time, which requires that we calculate electromagnetic field transformations in Rindler space-time. The computation is straightforward by using the electromagnetic 4-vector potential transformation, Eqn. (13) and the transformation of differential operation (Eqn. (17)). One therefore obtains The y-component of the electric field in the inertial frame is given as The z-component of the electric field in the inertial frame can be written as Now, for the magnetic field, the x, y and z components are given by Eqns. (23) to (25) as follows: Hence, we can define the electromagnetic field ) , ( in Rindler space-time. This is given as We then obtain the transformation of the electromagnetic field as Hence, we can find the matrix of the transformation of the electro-magnetic field. Electronic copy available at: https://ssrn.com/abstract=3508045 (see also Ref. [2]). Hence, the inversetransformation of the electromagnetic field is If we apply Lorentz gauge transformation, Eqn.
If we apply Lorentz gauge transformation, Eqn.
(10) to the transformation of the electromagnetic 4vector potential, and Eqn. (13), then we can obtain the following results.

Maxwell equation is
Electronic copy available at: https://ssrn.com/abstract=3508045 We shall now perform computation to derive Maxwell equations in Rindler space-time. For this purpose, we shall compute it by using the electromagnetic field transformation, Eqn. (27) and the transformation of differential operation as given in Eqn. (17).
Let us first deal with the transformation of 4vector, the charge density and the electrical current   The third law described by Maxwell equations in inertial frame is: The y-component is: We know that del Castillio and Sanchez [1] already discovered Maxwell equations in a uniformly accelerated frame in vacuum.
Hence, the transformation of 4-vector For instance, we know that the spherical charge density ξ ρ of a stationary accelerated frame in a charged huge sphere is Where, Λ is a scalar function. Electronic copy available at: https://ssrn.com/abstract=3508045 If we apply the Lorentz gauge transformation to Eqn. (52), we get 2 2  Electronic copy available at: https://ssrn.com/abstract=3508045

Conclusion
Since del Castillio and Sanchez already calculated Maxwell equations in uniformly accelerated frame in vacuum, and Maluf and Faria obtained electromagnetic field transformation in Rindler space-time [5] (see ArXiv preprint), we computed the electromagnetic field transformation and the electro-magnetic equation in a uniformly accelerated frame in a single theory. Generally, the coordinate transformation of accelerated frame (see Ref. [6]) is (I) ) sinh( ) ( If one uses Eqn. (59) to find Maxwell equations in Rindler space-time, one fails to do so. In Einstein's article (see [7]), he obtained Lorenz transformations for Maxwell equations in inertial frame and did not use Galilei transformations in inertial frame. In an accelerated frame, we think our choice of Rindler coordinate (I) is a better one can treat electromagnetic field equations in a manner similar to Einstein's choice.

Appendix A
In 2-dimensional Rindler space-time, if we use incorrect calculation, we think that the electromagnetic wave function will look like the expression given below: We conclude that it cannot exist as the electromagnetic wave function in Rindler spacetime.