Efficient pseudospectral method for the computation of the self-force on a charged particle: Circular geodesics around a Schwarzschild black hole

The description of the inspiral of a stellar-mass compact object into a massive black hole sitting at a galactic center is a problem of major relevance for the future space-based gravitational-wave observatory Laser Interferometer Space Antenna (LISA), as the signals from these systems will be buried in the data stream and accurate gravitational-wave templates will be needed to extract them. The main difficulty in describing these systems lies in the estimation of the gravitational effects of the stellar-mass compact object on his own trajectory around the massive black hole, which can be modeled as the action of a local force, the self-force. In this paper, we present a new time-domain numerical method for the computation of the self-force in a simplified model consisting of a charged scalar particle orbiting a nonrotating black hole. We use a multidomain framework in such a way that the particle is located at the interface between two domains so that the presence of the particle and its physical effects appear only through appropriate boundary conditions. In this way we eliminate completely the presence of a small length scale associated with the need of resolving the particle. This technique also avoids the problems associated with the impact of a low differentiability of the solution in the accuracy of the numerical computations. The spatial discretization of the field equations is done by using the pseudospectral collocation method and the time evolution, based on the method of lines, uses a Runge-Kutta solver. We show how this special framework can provide very efficient and accurate computations in the time domain, which makes the technique amenable for the intensive computations required in the astrophysically relevant scenarios for LISA.

Figure 1

The figure shows the structure of the one-dimensional spatial grid, the division in subdomains, and the location of the particle at the interface between two of them. The boundaries of the domain have coordinates $r_{\mathrm{H}}^{*}$ (this coordinate will have a finite value and is meant to approach the BH horizon at $r^{*}=-\infty$) and $r_{\mathrm{I}}^{*}$ (this coordinate will also have a finite value and is meant to approach spatial infinity at $r^{*}=+\infty$). The particle is at the boundaries of two different boundaries (which are identified), which have coordinates $r_{a,R}^{*}$ and $r_{a+1,L}^{*}$ that satisfy the relation (47).

Figure 2

This figure shows the dependence of the truncation error (estimated by the logarithm of the absolute value of the last spectral coefficient, ${log}_{10}|a_N|$, associated with the field $\psi$) with respect to the number of collocation points. This error corresponds to a snapshot of the evolution of the classical wave equation, for an initial condition given by a moving Gaussian packet. The data indicates the exponential convergence of the numerical method.

Figure 3

We show snapshots of the evolution of the scalar charged particle in circular motion at the LSO for the mode $\ell =m=2$. These simulations used 12 subdomains and 50 collocation points per subdomain. They show the evolution of the variables ${\psi }_{\ell }^m$ (left), ${\varphi }_{\ell }^m$ (center), and ${\phi }_{\ell }^m$ (right), after a substantial time has passed and a number of wave cycles have been generated. In particular they show how the jump in the radial derivative of the field ${\Phi }_{\ell }^m$ is resolved (this information is encoded in the variable ${\phi }_{\ell }^m$) in this multidomain computational framework.

Figure 4

This figure shows the dependence of the truncation error (${log}_{10}|a_N|$) with respect to the number of collocation points for simulations of circular motion at the LSO ($r_p=6M$) corresponding to the $\ell =m=2$ mode, in particular, for the field ${\psi }_2^2$. The data, taken from the subdomain $r^{*}-r_p^{*}\in [0,20M]$ indicates the exponential convergence of the numerical method.

Figure 5

We show here three plots of a snapshot of the evolution of the scalar charged particle in circular orbital motion at the LSO for the mode $\ell =10$ and $m=6$. Each plot shows different evolutions of the variables $\psi$ (left), $\varphi$ (center), and $\phi$ (right), for a fixed number of collocation points ($N=50$), but for different numbers of subdomains ($D=2$, 6, 10, 16). In this we can see how increasing the number of subdomains leads to a better accuracy (we can see how the waves get better resolved and the solution converges as we increase the number of subdomains) with a much less computational cost as if we would have increased the number of collocation points in a single domain computational framework.