Nonlinear Dynamics in General Relativity

Abstract

Black holes and gravitational waves are consequences of the nonlinear character of the Einstein equations. Yet, the remarkable properties of General Relativity point to the existence of other effects. Here we uncover new nonlinear facets of gravity. We establish higher harmonic generation, spectral broadening and focusing in the Einstein Klein-Gordon system. In vacuum, we show that scattering of monochromatic waves at quadratic order is weakly sensitive to frequency, at large wavelengths. These aspects can both explain the seemingly smooth behavior of mergers, but also caution us against too simplistic an interpretation of waveforms.

Figures (6)

• Figure 1: Left: Profile of the linear $r\phi_1$ and first nonlinear correction $r\phi_3$ at different times (see colors), for quasi-monochromatic initial data $\Omega_1=\Omega_2=\Omega_3=\Omega_d=1$ with $r_0=200,\sigma=30$. The bulk of the nonlinear field is generated in the focusing region $r\sim 0$, and spatial focusing is apparent. Center: Fourier transform of the field of the left panel observed at $r=200$ at early times (blue), and late times (red), demonstrating the excitation of the third harmonic. Right: Spectrogram (top), signaling the excitation of the third harmonic, and spectral broadening, and the reconstructed spatial profile of the different harmonic components (bottom), demonstrating clear focusing.
• Figure 2: Decay properties of each of the higher harmonics generated nonlinearly, with frequencies $\omega_i$, for three-harmonic initial data with $(\Omega_1,\Omega_2,\Omega_3)=(1,2,3)$.
• Figure 3: Susceptibility \ref{['eq:q_def']} of the excitation of the higher harmonic as a function of the dimensionless driving frequency $\Omega_d M$ of the axial quadrupolar with $\ell=2,3,4$ in red, blue, and green, respectively. At low frequencies the susceptibility vanishes, in agreement with the flat space limit \ref{['eq:flat_space_q']}. The susceptibility peaks close to the resonance condition $2\Omega_d=\Re\omega_{\ell 00}$, indicated with dashed lines. We have cut the $\ell=2$ line since the integral exhibits poor convergence at high frequencies.
• Figure 4: Fourier transform of the quadrupole $r\Psi_4$ extracted at three different observing radii $r_{\rm obs}/M=4,15,20$. The dashed lines correspond to $k\Re[\omega_{220}]$ for $k=1,2,3$, with $\omega_{220}$ the fundamental mode of the remnant BH. The inset shows the time-domain response, where the waveforms are aligned to the peak of $r_{\rm obs}=4M$. The Fourier transform is only applied in the time window shown in the inset, corresponding to the merger-ringdown phase.
• Figure 5: Residuals in the envelope of the nonlinear correction to the scalar field $r\phi_3$ observed at $r_{\rm obs}=300\Omega_1$, between coarse ($\Delta r=0.1$) and medium ($\Delta r=0.05$) resolutions, in black, and between medium and fine ($\Delta r = 0.025$) resolutions, rescaled by the expected fourth order convergence factor, in red. The two lines overlap, signaling fourth order convergence of our results.