Electromagnetic radiation in a semi-compact space
Introduction
An accelerating charged particle emits electromagnetic radiation. If some of the spaces are compact and bounded by material walls, the radiation behaves differently. Some examples are microwaves propagating inside a compact waveguide, light propagating in an optical fiber, or black body radiation in a finite volume. A similar but slightly different situation appears in the string theory with higher dimensional spaces; spaces among which six-dimensional sub-spaces are compactified with periodic boundary conditions to describe our three-dimensional spaces. In string theory, we often consider D-branes, localized objects charged under the so called Ramond–Ramond (RR) fields (see [1], [2] for reviews). There are various types of D-branes: a Dp-brane is a p-dimensional object. In the brane world scenario, our universe is described by a D3-brane whose motion in the six-dimensional compact space is supposed to describe the early universe [3], [4], [5]. In such a situation, radiations of gravitational and RR fields in a compact space are important to be investigated [6], [7], [8].
In this short note, motivated by the studies of radiations from D-branes in motion, we study electromagnetic radiations from a revolving point charge in a compact space.1 When the size of the compact spaces is smaller than the typical wave length of radiation, one may expect that the radiation will be suppressed. The purpose of the note is to check whether it is correct or not. In the next section we introduce our setup and provide useful formulas to calculate the radiation. We especially evaluate the retarded Green's function in compact spaces. In section 3 we calculate energy flux of radiation which is defined at far infinity away from the revolving charge in the non-compact direction. The radiation has a discontinuous behavior as the size of the compact directions R is varied. It is also necessary to regularize divergences associated with resonances in the compact space, which also cause the discontinuities. We summarize the results in section 4. In Appendix, we list exact expressions of the electric and magnetic fields without using an approximation where d is the radius of the circular motion of a charged particle.
Section snippets
Setup and Green's function
The setup studied in this note is as follows. We consider four-dimensional space–time with time t and space coordinates , . The first and the second spacial directions are compactified by imposing the periodic boundary conditions with radius R, and the other direction z is extended to infinity. We introduce a point charge q revolving in the – plane with a constant angular frequency (see Fig. 1). The motion of the point charge is described by a vector
Energy flux of radiation
By using the retarded Green's function defined in the previous section, it is straightforward to calculate the electric and magnetic fields generated by the specific charge density (7) and the current (8). Suppose that is satisfied, we can expand the fields with respect to and neglect contributions of . The exact forms of the electric and magnetic fields are given in the Appendix A.
From the results listed in Appendix A, the energy flux of radiation defined in eq. (3) is
Summary
In this note, we have calculated the total flux of radiation from a charged particle circulating with an angular frequency in a compact space with periodicities . The radiation is measured at in the noncompact direction. The flux shows several interesting behaviors, which are characteristic to the radiation in a compact space. First, it changes discontinuously as a function of . As we increase ξ from small , a new mode (a higher oscillating mode on the plane)
Acknowledgements
This work is supported in part by Grant-in-Aid for Scientific Research 16H06490 from MEXT Japan. NK would like to thank KEK theory center for the kind hospitality.
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