Electromagnetic field near a cosmic string

The retarded Green function of the electromagnetic field in spacetime of a straight thin cosmic string is found. It splits into a geodesic part (corresponding to the propagation along null rays) and to the field scattered on the string. With help of the Green function the electric and magnetic fields of simple sources are constructed. It is shown that these sources are influenced by the cosmic string through a self-interaction with their field. The distant field of static sources is studied and it is found that it has a different multipole structure than in Minkowski spacetime. On the other hand, the string suppresses the electric and magnetic field of distant sources—the field is expelled from regions near the string.

Figure 1

Three-dimensional spacetime diagram of the plane orthogonal to the cosmic string (i.e., the spacetime diagram with coordinate $z$ is suppressed). A rescaled coordinate $\tilde{\phi }=\nu \phi$ (periodic with the period $2\pi$) is used instead of geometrical angular coordinate $\phi \in (-\pi /\nu ,\pi /\nu )$. Two spacetime points $x$ and $x^{\prime }$ can be joined by more than one geodesic as indicated in the diagram. The geodesics bend around the string due to the curvature localized on the string. Of course, the bending of the geodesics is only apparent—it arises because the rescaled coordinate $\tilde{\phi }$ is employed. The geodesics are straight lines in locally flat geometry. However, the cosmic string causes the angle deficit and the intersection of geodesics behind the string is a real effect. Spatial projection of the geodesics into hypersurface $t=\mathrm{const}$ is also shown. These are spatial geodesics with length $r_k$. Also, cf. Fig. 5.

Figure 2

The support of the geodesic part ${\mathit{G}}^{\mathrm{gd}}(x|x^{\prime })$ of the Green function is localized on the future light cone of the source point $x^{\prime }$, i.e., on the null hypersurface generated by future null geodesics from $x^{\prime }$. Near vertex $x^{\prime }$ the hypersurface has the standard structure of the light cone in Minkowski spacetime. However, at the cosmic string the light cone deforms—it intersects itself and becomes a hypersurface with a boundary (given by null rays propagating near the string).

Figure 3

The support of the scattered part ${\mathit{G}}^{\mathrm{sc}}(x|x^{\prime })$ of the Green function is localized inside light cones with vertices $y$ on the cosmic string which are connected with the source by null geodesics. In the figure only a spacetime diagram of the plane going through the charge orthogonally to the string is shown. The electromagnetic field given by ${\mathit{G}}^{\mathrm{sc}}(x|x^{\prime })$ can be interpreted as the electromagnetic field scattered on the singular curvature localized on the string.

Figure 4

The graph of the scalar potential of a static charge near a cosmic string. Horizontal plane corresponds to the plane of the charge orthogonal to the cosmic string. The values of the scalar potential are drawn in the vertical direction. The full potential (3.1) [diagram (c)] is split into the geodesic part (a) and the scattered part (b). The whole potential is smooth except at the locations of the charge and at the string. Both geodesic and scattered parts are discontinuous at surfaces $\mathcal{B}$ where a number of spatial geodesics from the charge changes (cf. Fig. 5), however, the discontinuity of these two parts of the Green function cancels each other out. The scattered part of the field is finite and smooth near the charge; this part is responsible for self-interaction of the charge.

Figure 5

The spatial diagram of the hypersurface $t=\mathrm{const}$ (cf. spacetime diagram in Fig. 1). Points $\overline{x}$ and ${\overline{x}}_o$ are spatial projections of spacetime point $x$ and of the worldline of the static charge. Rescaled coordinate $\tilde{\phi }=\nu \phi$ is used. geodesic part (3.6) of the electric field at $\overline{x}$ of the charge at ${\overline{x}}_o$ is given by standard flat-space Coulomb contributions for each spatial geodesic joining $\overline{x}$ and ${\overline{x}}_o$. Surface $\mathcal{B}$ (indicated by dashed line) divides the space into domains, points of which are connected with the charge by one geodesic or by two geodesics, respectively. For sufficiently large angle deficit there exist also points connected with the charge by more geodesics.

Figure 6

Magnetic lines around the electric current along the line through ${\overline{x}}_o$ parallel to the string. Planes $\mathcal{B}$ (indicated by dashed lines) separate the space into domains in which the points are connected with the source by one, respectively, two geodesics. A rather large value $\nu =1.8$ of the inverse conicity parameter was chosen to emphasize the deformation of magnetic lines near the string. As discussed in Sec. 6, for physically relevant values, $\nu \approx 1$, the domain near the string in which the magnetic lines are deformed is too small to be visualized (cf. also Fig. 8).

Figure 7

Scalar potential of the static charge near the string with a large deficit angle (namely, $\nu =1.8$). We can see that it has a small plateau near the string. The electric field $\mathit{E}$ is suppressed here. Such a plateau exists for any value $\nu \ne 1$, however, it is very small for $\nu$ close to 1, cf. Figs. 4, 8.

Figure 8

The diagram illustrates the dependence of a typical size of the domain near the string, where the electric field $\mathit{E}$ of the static charge is suppressed, on the parameter $\nu$. To estimate a size of the domain, a value of the field is evaluated along a line going through the charge orthogonally to the string. The horizontal direction in the figure corresponds to the distance from the string. The vertical direction corresponds to different values of the parameter $\nu$. The contours of constant values of the field are drawn, with shading indicating the strength of the field (the shading is drawn only in discrete steps given by contours). For $\nu >1.5$, the interval of small field (light shading) is large; for $\nu \to 1$ this intervals shrinks very rapidly and it disappears for $\nu =1$. See also graphs of scalar potential for $\nu =1.2$ in Fig. 4c and for $\nu =1.8$ in Fig. 7. Only in the latter one a flat domain near the string can be easily identified. A similar analysis applies also for magnetic field $\mathit{B}$ discussed in Sec. 4. A domain of the expulsion of magnetic lines near the string can be identified, e.g., in Fig. 6.