Gertsenshtein effect on the spacetime curved by background magnetic field with geometric optics

TL;DR

This work addresses how spacetime curvature sourced by a background magnetic field affects the inverse Gertsenshtein effect. By solving the linearized Einstein equations for a static, uniform B field with cylindrical symmetry, the authors obtain a background metric accurate to ${\cal O}(B^2)$ and apply a geometric-optics framework to derive coupled propagation equations for EMWs and GWs on this curved background. They provide explicit solutions for plane- and spherical-wave beams, showing that curvature-induced focusing and photon–graviton conversion compete: plane waves exhibit a precise cancellation leaving the EMW amplitude unchanged, while spherical waves experience a net attenuation due to dominant conversion. The results yield gauge-invariant insights into curvature corrections at order ${\cal O}(B^2)$ and have implications for interpreting magnetic-field constraints using the Gertsenshtein effect in realistic curved spacetimes.

Abstract

When electromagnetic (or gravitational) waves propagate in the presence of a background magnetic field, a portion of the waves converts into gravitational (or electromagnetic) waves. This phenomenon, known as the (inverse) Gertsenshtein effect, is typically analyzed in Minkowski spacetime, neglecting the spacetime curvature induced by the magnetic field itself. This paper investigates, for the first time, the influence of spacetime curvature on the (inverse) Gertsenshtein effect. To this end, we first determine the metric perturbation from Minkowski spacetime up to second order in the magnetic field strength, assuming cylindrical symmetry. We also discuss the ambiguities in the form of the metric perturbation arising from gauge freedom and boundary conditions. Using the geometric optics approximation, we then derive a set of coupled equations governing the propagation of electromagnetic and gravitational waves in the resulting curved spacetime. These equations are solved for two specific scenarios: a plane wave and a spherical wave. From the solutions, we compute the evolution of the wave amplitudes and the associated energy fluxes. Our analysis reveals that two competing effects govern the amplitude evolution: magnification due to the focusing of waves by spacetime curvature, and attenuation due to wave conversion via the Gertsenshtein effect. In the plane wave case, these effects precisely cancel, resulting in no net change in amplitude. In contrast, for the spherical wave, the Gertsenshtein effect dominates over focusing, leading to an overall reduction in amplitude.

Figures (5)

• Figure 1: The region $V$ containing a uniform magnetic field along the $+z$ axis is depicted in purple.
• Figure 2: A narrow beam emanating from the origin moves in the $x$--$z$ plane with an initial angle $\theta$ from the $+z$--axis. The three-dimensional vector ${\bm k}$ which is the spatial component of $P^\mu$ represents the direction of the beam.
• Figure 3: (a) Congruence of null geodesics (dark cylinder) corresponding to a plane wave. The two plates perpendicular to the cylinder represent $\Sigma$ with different values of the phase. (b) Congruence of null geodesics (dark thin cone) corresponding to a spherical wave. The sphere perpendicular to the cone represents the surface with a fixed value of the phase.
• Figure 4: This figure illustrates the evolution of the cross-sectional area during the propagation of a null congruence constructed from geodesics parameterized by $(s, t)$ satisfying $s^2 + t^2 < 1$ on the Riemann normal coordinate system (see also Appendix \ref{['appendix:tetrad']}). Each panel (left to right) corresponds to different values of the free parameter $\alpha$ in $g^{(B)}_{\mu\nu}$. The chosen values $\alpha = -2, 0, 2$ represent the behavior of the cross-sections in the regions $\alpha < -1$, $-1 < \alpha < 1$, and $\alpha > 1$, respectively. Geodesics through the origin in directions $\theta = 0, \pi/4, \pi/2$ are shown as black straight lines on the $\bar{x}$-$\bar{z}$ plane (i.e., the $\vec{k}$--$\vec{B}$ plane). Along each geodesic, the cross-sections are illustrated by filled-in ellipses centered at $\overline{x}^\mu(\lambda)$ with $80 \lambda = 12,29,46,63,80$ (the cross-sections at $\lambda=0$ are omitted to avoid confusing overlapping regions). The shape of each cross section is to be judged against a circle of radius 1 in the plane of the cross section (the white disk accompanying each filled-in ellipse).
• Figure 5: As in Fig. \ref{['CSplane']}, but for spherical waves emanating from the origin and the radii of the reference circles rescaled by $\lambda$ to better follow the spherical expansion of the waves.