Space-Time Geometry of Electromagnetic Field in the System of Photon

In the concept of general relativity gravity is the space-time geometry. Again, a relation between electromagnetic field and gravitational field is expected. In this paper, space-time geometry of electromagnetic field in the system of photon has been introduced to unify electromagnetic field and gravitational field in flat and curvature space-time.


INTRODUCTION
In physics, a unified field theory is a type that allows all fundamental forces and elementary particles to be written in terms of a single field.
The term was proposed by Einstein, who attempted to unify the general theory of relativity with electromagnetism. According to Einstein"s general relativity [2,3], gravity is the space-time geometry. Also, he suggested [4] the field equation for the gravity of an electromagnetic wave as where, G  is the Einstein tensor, and K is the coupling constant. But, the problem of the unification of fundamental fields into a single theory has not been solved until now in a satisfactory manner, although, in different time, a lot of papers have been published which attempt to unify the fundamental fields.
Recently, in [1], a relation between electromagnetic field and gravitational field has been introduced by considering a super system in photon. In this paper a trial has been made to introduce a geometrical relation between electromagnetic field and gravitational field.

SPACE-TIME GEOMETRY OF SYSTEMS
In [1], to clarify two simultaneous superimposed motion (either linear or rotational), three types of system has been assumed which are L-L system, S-S system and S-L system; depending upon the S-L system SSP picture of photon has been considered; also, using this picture (SSP) a connection between electro-magnetic field ( ( , ) rt   ) and gravitational field ( ( , ) G r t     ) has been introduced by the relation where, ij Z are transformation matrix in the picture of SSP. It is also pointed out that to clarify L-L or S-S or S-L system, four reference frames (S, S 1 , S 2 , S 3 ) has been considered in a simultaneous superimposed form.
Relation for co-ordinate transformation from S 3 to S in S-L system [1] is where, ij Z is co-ordinate transformation matrix and the co-ordinates of an event in S 3 be ( , , , ) with respectively in S. Now, following the space-time geometry as in [5], one can introduced the Space-time geometry of the said system as stated below From (2) where, Using (4) and (7) we obtain the space-time geometry in S-S system as in (5)

SPACE-TIME GEOMETRY OF ELECTROMAGNETIC FIELD IN PHOTON
Since picture of SSP depends upon the S-L system so, following (1) and using where, superscript "g" represents the gravitational system and superscript em represents electromagnetic system. Following the convention as in (1), one may assume a relation between electromagnetic field and gravitational field as where, 1  is a constant and ij S is transformation matrix in S-S system.
This means that, in S-S system, gravitational field of frame S 3 would be electromagnetic field with respect to frame S. For this system space-time geometry of electromagnetic field would also be as in (8) where, transformation matrix would be ij S However, equation (8) would be the space-time geometry of electromagnetic field connecting gravitational field and electromagnetic field in the system of photon.

CONCLUSION
Equation (8) represents a picture of space-time geometry of the electromagnetic field in the system of photon. This implies that a geometrical relation is existed in between electromagnetic field and gravitational field.