Generic effective sources for first-order in mass-ratio gravitational self-force calculations in Schwarzschild spacetime

Abstract

The numerical calculation of gravitational self-force in extreme mass ratio inspiral systems is fundamentally challenging due to the singular nature of point-particle sources. To overcome these difficulties, the effective source method offers an innovative alternative by replacing traditional regularization techniques with a reformulation of the problem. In this paper, we present the first fully analytic framework for constructing effective sources to compute the gravitational self-force for generic orbits in Schwarzschild spacetime. By reformulating the singular field through angular modulation in terms of a tetrad decomposition, the effective source can be constructed with the linear combination of scalar modes. The derived effective source is continuous across the particle's worldline, enabling efficient numerical implementation in $1+1$ dimensions.

Figures (7)

• Figure 1: Circular case: Real components of the puncture field modes $\bar{h}^{(i)lm}$ for a particle on a circular orbit at $r_p=10~M$ and $\phi_p=\pi/6$. Top panels display even-parity components $\bar{h}^{(1-7)}$ for $(l,m)=(2,2)$, showing continuous variation across the particle's position (vertical dashed line at $\delta r=0$). Bottom panels show odd-parity components $\bar{h}^{(8-10)}$ for $(l,m)=(3,2)$, exhibiting characteristic parity-dependent behavior.
• Figure 2: Circular case: Real components of the puncture field with respect to $r$ modes $\partial_r \bar{h}^{(i)lm}$ for a particle on a circular orbit at $r_p=10~M$ and $\phi_p=\pi/6$. Top panels display even-parity components $\bar{h}^{(1-7)}$ for $(l,m)=(2,2)$ and the bottom panels show odd-parity components $\bar{h}^{(8-10)}$ for $(l,m)=(3,2)$, exhibiting characteristic parity-dependent behavior.
• Figure 3: Circular case: Real components of the effective source modes $S_{\rm eff}^{(i)lm}$ for a particle on a circular orbit at $r_p=10~M$ and $\phi_p=\pi/6$. Top panels display even-parity components $\bar{h}^{(1-7)}$ for $(l,m)=(2,2)$, showing continuous variation across the particle's position (vertical dashed line at $\delta r=0$). Bottom panels show odd-parity components $\bar{h}^{(8-10)}$ for $(l,m)=(3,2)$, exhibiting characteristic parity-dependent behavior. All components remain continuous and finite at $\delta r=0$, validating the regularization scheme.
• Figure 4: Eccentric case: Real components of the puncture field modes $\bar{h}^{(i)lm}$ for a particle on an eccentric orbit with the velocity $u^r=\frac{1}{3 \sqrt{95}}$ and $u^\phi=\frac{7}{30 \sqrt{38}}$ at $r_p=10~M$ and $\phi_p=\pi/6$. Top panels display even-parity components $\bar{h}^{(1-7)}$ for $(l,m)=(2,2)$, showing continuous variation across the particle's position (vertical dashed line at $\delta r=0$). Bottom panels show odd-parity components $\bar{h}^{(8-10)}$ for $(l,m)=(3,2)$, exhibiting characteristic parity-dependent behavior.
• Figure 5: Eccentric case: Real components of the puncture field with respect to $r$ modes $\partial_r \bar{h}^{(i)lm}$ for a particle on an eccentric orbit with the velocity $u^r=\frac{1}{3 \sqrt{95}}$ and $u^\phi=\frac{7}{30 \sqrt{38}}$ at $r_p=10~M$ and $\phi_p=\pi/6$. Top panels display even-parity components $\bar{h}^{(1-7)}$ for $(l,m)=(2,2)$ and the bottom panels show odd-parity components $\bar{h}^{(8-10)}$ for $(l,m)=(3,2)$. All components remain nondifferentiability at $\delta r=0$, validating the regularization scheme.