Frequency shift and viewing direction variations in gravitational lensing

Abstract

In a gravitational lensing system, the relative transverse velocities of the lens, source, and observer induce a frequency shift in the observed radiation. While this shift is typically negligible in most astrophysical contexts, strategies for its detection have been proposed for both electromagnetic and gravitational waves. This paper provides a rigorous theoretical treatment of the effect, deriving general expressions for the frequency shift within a lensing system embedded in a cosmological spacetime. Our formulation remains valid for arbitrary distances and velocities - including highly relativistic regimes - under any Friedmann-Lemaître-Robertson-Walker metric. Expanding upon previous papers on moving lenses, we provide a detailed analysis of frequency effects induced by lenses moving at relativistic speeds. Furthermore, we extend standard lensing theory by deriving an exact formula for the variation in the source's viewing direction. This result is of interest for strongly anisotropic emitters, such as compact binary systems emitting gravitational waves. Finally, we quantify the apparent misalignment between the lens and the source's two images produced by time-delay effects in lens systems moving with high velocity.

Figures (8)

• Figure 1: The lensing setup in a FLRW spacetime: the fiducial light ray, or the line of sight (dotted line), parametrised backwards in time and passing through the observer, lens and source planes. The tangent vectors $\tilde{l}$ to the ray are denoted $\tilde{l}$, with the subscript denoting the point. We allow the 4-velocities of the observer $u_{\mathcal{O}}$, lens $u_{\mathcal{L}}$ and source $u_{\mathcal{S}}$ to be different from the comoving cosmological observers $U_\mathcal{O}$, $U_\mathcal{L}$ and $U_\mathcal{S}$ respectively.
• Figure 2: The lensed ray (dashed line) and the fiducial ray, or the line of sight (dotted line), as well as their tangent vectors at $\mathcal{O}$, $\mathcal{L}$ and $\mathcal{S}$. The point source is located and $\delta x_\mathcal{S}$. The lensed ray undergoes a momentary deflection at the lens plane, which leads to a discontinuity of the tangent vector, represented by the pair $\tilde{l}_\mathcal{L} + \Delta l_{\mathcal{L},\textrm{out}}$ and $\tilde{l}_\mathcal{L} + \Delta l_{\mathcal{L},\textrm{in}}$. The rays are parametrised backwards in time, from the observer to the source.
• Figure 3: Decomposition of the frequency shift effect into three distinct transformations as described by equation \ref{['eq:DZ_from_angles4']}. a)---b): Initial application of the frequency shift gradient, represented by the $\beta^{\bm A}$ term, across the unlensed source image. b)---c): The lensing map (modelled here as an isothermal ellipsoid) induces a nonlinear transformation, resulting in four partially overlapping images and a distorted frequency shift gradient. c)---d): Addition of a second frequency shift gradient on the lens plane, corresponding to the $\theta^{\bm A}$ term. The hues of red and blue indicate redshifts and blueshifts, respectively.
• Figure 4: Frequency shift as a function of the lensed position, decomposed according to equation \ref{['eq:DZ_from_thetaalpha_UO']} into the $\theta^{\bm A}_{U,\mathcal{O}}$ term in the form of a gradient (Left), the proper lensing term proportional to $\alpha_{U,\mathcal{O}}^{\bm A}$ (Centre), and the combined effect (Right). The gradient term is a long-range one, extending over the whole celestial sphere as a dipole, while the proper lensing term vanishes away from the lens. The lensing potential corresponds to a point mass, and the hues of red and blue indicate redshifts and blueshifts, respectively.
• Figure 5: Dependence of the kinematical prefactor $f$ on the direction of the lens's motion, given by the angle $\alpha$, for the lens velocities $v_\mathcal{L}$ of $10^{-5}$ (non-relativistic, dipolar dependence on $\alpha$), $0.6$ (mildly relativistic), $0.995$ and $0.99991$ (ultrarelativistic). Note that the scales on the $Y$ axes are different on each panel.