The point-particle-limit effective-source approach for computing gravitational self-force in the Lorenz gauge

Abstract

The traditional effective-source method is hampered by complex analytical expressions and the inherent smoothness limit, which incur high computational costs and complicate implementation. To overcome these limitations, we introduce the point-particle-limit effective source method, which analytically takes the size of the effective source to zero, thereby transforming the problem into a well-defined jump condition of retarded metric field at the particle position governed by the local singular field. This formulation naturally pairs with a discontinuous Galerkin scheme, whose inherent capacity for accommodating solution discontinuities enables highly accurate enforcement of the jump conditions. We apply both the traditional and point-particle-limit effective source method to calculate the time-domain gravitational metric perturbation and gravitational self-force in the Lorenz gauge on a point particle in a circular orbit around a Schwarzschild black hole. The comparison of numerical results shows the excellent advantage of the point-particle-limit effective source method, which validates the correctness and efficiency of the point-particle-limit effective source method and thereby establishes a numerical foundation for computing generic geodesic orbits or long-time self-consistent orbital evolution.

Figures (4)

• Figure 1: Time evolution of the real components of the regular metric field $\bar{h}^{(i)}_R=\Psi^{(i)}$ at the particle position for a circular orbit around a Schwarzschild black hole. Blue points show results from the TES method, and orange points correspond to the PPLES method. Top panels display even-parity components $\bar{h}^{(1-7)}_R$ for $(\ell,m)=(2,2)$ and the bottom panels show odd-parity components $\bar{h}^{(8-10)}_R$ for $(\ell,m)=(2,1)$.
• Figure 2: Radial profile of the real components of the retarded metric field $\bar{h}^{(i)}_{\rm ret}=\Psi^{(i)}$ at the time slice $\lambda=1500$ s. Blue points show results from the TES method, and orange points correspond to the PPLES method. The retarded metric fields are compared across the computational domain, with the particle located at $\xi=20$. The two methods show excellent agreement in the far-field region, while differences near the particle arise from distinct implementations.
• Figure 3: Time evolution of the real components of the regular metric field $\bar{h}^{(i)}_R=\Psi^{(i)}$ at the particle position for a circular orbit around a Schwarzschild black hole with damping terms. Blue points show results from the TES method, and orange points correspond to the PPLES method. Top panels display even-parity components $\bar{h}^{(1-7)}_R$ for $(\ell,m)=(2,2)$ and the bottom panels show odd-parity components $\bar{h}^{(8-10)}_R$ for $(\ell,m)=(2,1)$.
• Figure 4: Radial profile of the real components of the retarded metric field $\bar{h}^{(i)}_{\rm ret}=\Psi^{(i)}$ at the time slice $\lambda=1500$ s with damping terms. Blue points show results from the TES method, and orange points correspond to the PPLES method. The retarded metric fields are compared across the computational domain, with the particle located at $\xi=20$. The two methods show excellent agreement in the far-field region, together with the region near the particle.