Laboratory Frame Representation for General High-Frequency Gravitational Waveforms

TL;DR

This paper develops a general framework to transform high-frequency gravitational-wave signals from TT coordinates into the proper detector (PD) frame, essential when GW wavelengths approach detector size ($ω_g L_{\text{det}} \gtrsim 1$). It derives exact PD-frame metric components in terms of the TT metric, its integral, and derivatives, and provides efficient approximations based on retarded-time Taylor expansions and monochromatic limits for chirp-like waveforms from compact binaries. The method is applied to quasi-circular inspirals, with analytic expressions for the TT strain and a treatment of ISCO effects, enabling accurate predictions of detector signals in electromagnetic detectors via the effective current $J_{\text{eff}}^{\mu}$. By comparing exact and approximate transformations, the work delineates the validity of monochromatic and chirp-based approximations and demonstrates how these transformations inform the spectral response and coupling of detectors across the MHz–GHz range, supporting the design and interpretation of next-generation high-frequency GW experiments.

Abstract

Next-generation gravitational wave (GW) experiments will explore higher frequency ranges, where GW wavelengths approach the size of the detector itself. In this regime, GWs may be detected not just through the well-known mechanical deformation by tidal forces but also via induced effective currents in electromagnetic background fields. However, the calculation of this signal requires the GW metric in laboratory coordinates of the detector, and an accurate transformation to all orders into this frame is necessary. In this work, we derive a closed-form expression for the metric transformation of general chirp-like waveforms expressed in terms of the transverse-traceless GW metric, its integral, and its derivative. For more complex signals, where analytical integration is impractical, we provide an efficient approximation based on Taylor expansions of the retarded time to coalescence. Finally, we demonstrate how these results can be applied to calculate the signal response of a large class of detectors. Our approach provides essential tools for designing and interpreting high-frequency GW experiments that search for compact object mergers at MHz to GHz frequencies beyond the long-wavelength limit.

Figures (3)

• Figure 1: Effective current of a chirp signal. For a GW traveling along the $z$-axis with $M_c\approx10^{-6}M_\odot$, $\iota=\frac{\pi}{2}$, $\dot{\phi}(t)=\text{GHz}$ the effective current is calculated before coalescence in a cylindrical volume within a static magnetic field along the $z$-axis. The solid curve corresponds to $J_\text{eff}$ as obtained from equation \ref{['eq:exact_PD_components_from_TT']}, the dashed curve to equation \ref{['eq:PD_chirp_strain']}, the dotted curve to the monochromatic approximation \ref{['eq:monochromatic_approximation']} and the dash-dotted curve to $J_\text{eff}^x$ evaluated in TT coordinates without approximation.
• Figure 2: Numerical analysis of the relative deviation of the effective current for the approximate metric components. (a) The relative deviation of the effective current obtained for a gaussian burst and quadratic phase modulation for the monochromatic approximation \ref{['eq:monochromatic_approximation']}. The GW is traveling along the same axis as a static magnetic field, for a length of $1\,\text{m}$ with $\omega_0=2\pi\,\text{GHz}$. (b) The relative deviation of the effective current obtained for a compact binary as described in section \ref{['sec:compact_binaries']}. The GW is traveling along the same axis as a static magnetic field, for a length of $1\,\text{m}$ and at different times to coalescence where $\dot{\phi}(t)=2\pi f_g$. The darker color corresponds to using equations \ref{['eq:PD_chirp_strain']} and the lighter color to a monochromatic approximation \ref{['eq:monochromatic_approximation']} for different frequencies $f_g$. The line is drawn dashed for unphysical combinations where $f_g>f_\text{ISCO}$.
• Figure 3: Coupling coefficient of chirp signals to electromagnetic detectors. We plot the coefficient $\kappa$ from equation \ref{['eq:coupling_coefficient']} for a chirp signal with $M_c=10^{-6}M_{\odot}$ as shown in figure \ref{['fig:Jeff']}, traveling parallel to a static magnetic field aligned with the symmetry axis of a cylinder with length and radius of $1\,\text{m}$ with the resonant TE$_{212}$ mode. (a) We show the envelope of the time evolution of $|\kappa(t)|$ and the accuracy of the approximations according to equation \ref{['eq:Delta_kappa']} close to the merger at $t_0$ in the inset. (b) The absolute value of the Fourier transform of $|\kappa(t)|$ in a stationary-phase approximation is shown.