Electromagnetic Wave Equation on Differential Form Representation

DOI : One of the indispensable part of the theoretical physics interest is geometry differential. This one interest of physical area has been developed such as in electromagnetism. Maxwell's equations have been generalized in two covariant forms in differential form representation. A beautiful calculus vector in this representation, such as exterior derivative and Hodge star operator, lead this study. Electromagnetic wave equation has been expressed in differential form representation using Laplace-de Rham operator. Explicitly, wave equation shows the same form in Minkowski space-time like vector representation. This study is able to introduce us to learn application of differential form in physics.


Introduction
Electromagnetism as a basic science has a role to develop mathematics models.Earlier, In 1865 Maxwell proposed four equations called Maxwell's equations.These equations are the core or the summary of natural phenomenon that is related to electromagnetic fields.The form of the Maxwell equation in covariant form has been studied in many books (Griffith, 1999 andJackson, 1999).Correspondingly, the study of physical phenomena in any coordinate system developed quite rapidly through the theory of mathematical geometry in a more general form.We start with the formulation of electromagnetic equations in differential form.
Electromagnetic studies in differential form representations have been carried out in various studies, which are written in various papers and texts such as [3][4][5][6].In 1981 the relationship of external derivatives to electromagnetic equations in 4-dimensional space (space-time) was studied [3].The relationship diagram of the electromagnetic equation is made in such a way that the exterior derivative successfully satisfies the electromagnetic equation.Electromagnetic studies in differential form representations focusing on visual advantages obtained from differential form representations have also been described [4].The form order for electromagnetic quantities are presented in visual form for the more easily understood of electromagnetism [4].The steps to reduce Maxwell's equations in differential form representations are well explained by Owere [5] and Hossine and Ali [6].
The study of electromagnetic equations in differential form representations which develop very rapidly allows electromagnetics to be studied in free coordinate space and other advantages.This encourages all matters related to electromagnetics to be assessed in the differential form representation.Actually, vector analysis is a real form of differential form, so differential form does not replace vectors.In particular cases, differential form and vector replace each other [4].Therefore, in this paper, the explicitly component uses alternating differential forms and vectors.

Metric and Minkowski Space Field Tensor
The study of geometry and topology theory in physics has been written in many books.To understand many things about the application of geometrical concepts, especially the differential form can be seen in Nakahara [7].However for more details, we start from the concept of tensor metric.A metric tensor introduces the length of a vector and an angle between every two vectors.The components of the metric are defined by the values of the scalar products of the basis vectors.
In elementary geometry, the inner product between two vectors U and V is defined by , where Ui and Vi are the components of the vectors in R m .On a manifold, an inner product is defined at each tangent space TpM.Let M be a differentiable manifold.A Riemannian metric g on M is a type (0, 2) tensor field on M which satisfies the following axioms at each point p ∈ M [7]: where the equality holds only when U = 0.
A tensor field g of type (0, 2) is a pseudo-Riemannian metric if it satisfies (i) and If g is Riemannian, all the eigenvalues are strictly positive and if g is pseudo-Riemannian, some of them may be negative.If there are i positive and j negative eigenvalues, the pair (i,j) is called the index of the metric.If j = 1, the metric is called a Lorentz metric.Once a metric is diagonalized by an appropriate orthogonal matrix, it is easy to reduce all the diagonal elements to ±1 by a suitable scaling of the basis vectors with positive numbers.If we start with a Riemannian metric we end up with the Euclidean metric δ = diag(1, . . ., 1) and if we start with a Lorentz metric, the Minkowski metric η = diag(−1, 1, . . ., 1) [7].
Minkowski metric is the Lorentz metric on R 4 that is written in terms of coordinates (ct; x; y; z) as where x, y and z are spatial dimensions, t is time dimension and c is speed of light.We use indices for space-time coordinates as follows: which can be written as where   is the matrix as we say before.This metric can be used to raising or lowering tensor index [7].

Maxwell Equation
Electromagnetism is lead by Maxwell's equations, which gives a calculation of how the nonpermanent electric field can generate an impermanent magnetic field and vice versa.The four Maxwell equations are as follows.
(5) ( In the equation above, E is an electric field, B is a magnetic flux density, D is a electric flux density, H is a magnetic field, ρ is a charge density (charge per unit volume), and J is the total current density.In addition, there is also a constant μ which is the medium permeability and ε is the permivisity of the medium.For vacuum space, then the electromagnetic entity has a reference value of μ0 and ε0.This value is a universal constant through a relationship that is c = 1= 1/√ , where c is the speed of light in a vacuum.The charge density and current density have a relationship as follow (9) called continuity equation.
Using vector calculus of divergence, from (6) we can obtain hat B must be a curl of a vector function, namely the vector potential A, can be write as (10) Substituting (10) into equation ( 7), we obtain ./ P-ISSN : 2615-1278, E-ISSN : 2614-7904 which means that the term in the bracket can be written as the gradient of a scalar function, namely the scalar potential ϕ (11)

Manifold and Differential Form
Let is manifolds.Suppose that a tangent space on open subset as a vector space which is written by .A map is a linier mapping from to .Vector space consisting of all linear mapping is called the dual tangent space for at , denoted by .
Suppose there is a mapping which is a multi linier mapping from the tangent space to .The vector space consisting of all multi linier mapping is referred to as space -forms, denoted as ( ).In addition, there is a multi linier mapping which is a multi linier mapping from dual tangent space to .The set of all mappings from the dual tangent space is denoted by ( ).
Suppose which is multi linier mapping from the dual tangent spaces and tangent spaces to .This multi linier map called as tensor type ( ).Vector field can be defined as "a way of embedding" vector at a point.At ∈ , there are various ways of embedding that produces the tangent vector at the point .Vector field on can be written as a mapping Tensor fields have the same as the definition of a vector field.Tensor field of type ( ) is a way of embedding the point that produces a tensor of type ( ).Tensor field on is differentiable cross section which can be written as The same type of tensor fields on form a vector space over and form a module over ring.The tangent bundle on subset is a collection of all the tangent space at , i.e.
⋃ ∈ Also, it can be defined dual tangent bundle, i.e.

⋃ ∈
Manifolds as a place of tangent bundle is defined as the basic space.More generally, defined outer fiber bundle on , which is as follows.

( ) *( ) ∈ ∈ ( )+
This outer bundle is a bundle which its fiber is outer algebra of dual tangent space ∈ Outer fiber bundle on base define as ) In addition, there is also a Hodge operator, which is mapping from which is working on basis vector which defined as follows.
where є is the totally anti-symmetric Levi-Civita permutation symbol.How this operator work to the basis form have been written in Hossine [6].

Conclusion
In this paper, we studied the electromagnetic theory in 4 spacetime.By using differential forms, we formulate electromagnetic wave on differential form representation.This studied show that when expressed in terms of component, differential form can back again like equation in vector form.But, we know that differential form more general than vector as we desire.
forms on differentiable manifold ( ) can be defined in two points of view, namely from the viewpoint of algebra and geometry standpoint.