Electromagnetic waves and Stokes parameters in the wake of a gravitational wave

Abstract

A theoretical description of electromagnetic waves in the background of a (weak) gravitational wave is presented. Explicit expressions are obtained for the Stokes parameters during the passage of a plane-fronted gravitational wave described by the Ehlers–Kundt metric. In particular, it is shown that the axis of the polarization ellipse oscillates, its ellipticity remaining constant.

Appendix

In this appendix, the EM field in the limit of flat space-time is worked out according to the formalism of the present paper. This permits to characterize the EM field before and during the passage of the gravitational wave and compare both cases. In flat space-time, the electric field is given by

$$\begin{aligned} {\mathbf E} =i(\omega {\mathbf a} - a_z {\mathbf k})e^{iS}, \end{aligned}$$

(5.1)

where $k^{\alpha } = ( \omega ,{\mathbf k})$ in Minkowski coordinates and $S =k_{\alpha } x^{\alpha }$. It is important to notice that the gauge used in this paper is such that $a_v =-a^u=0$, and therefore $a^t = a^z$. Thus the condition $k^{\alpha } a_{\alpha }=0$ implies

$$\begin{aligned} k_x a_x + k_y a_y = (\omega -k_z) a_z, \end{aligned}$$

(5.2)

and of course ${\mathbf E} \cdot {\mathbf k} =0$.

The unit vectors $\hat{\epsilon }_{\alpha }^{(i)}$ have purely space components:

$$\begin{aligned} \hat{ \epsilon }^{(1)}&= k_{\bot }^{-1} (-k_y,k_x,0) \nonumber \\ \hat{ \epsilon }^{(2)}&= \omega ^{-1} \left(\frac{k_z}{k_{\bot }} k_x, \frac{k_z}{k_{\bot }} k_y, -k_{\bot }\right) \end{aligned}$$

(5.3)

and thus

$$\begin{aligned} \hat{ \epsilon }^{(1)} \cdot {\mathbf E}&= i\omega k_{\bot }^{-1} (k_x a_y-k_y a_x ) e^{iS} \nonumber \\ \hat{ \epsilon }^{(2)} \cdot {\mathbf E}&= -i\omega k_{\bot }^{-1} (k_x a_x +k_y a_y) e^{iS}. \end{aligned}$$

(5.4)