Self-force and Green function in Schwarzschild spacetime via quasinormal modes and branch cut

The motion of a small compact object in a curved background spacetime deviates from a geodesic due to the action of its own field, giving rise to a self-force. This self-force may be calculated by integrating the Green function for the wave equation over the past worldline of the small object. We compute the self-force in this way for the case of a scalar charge in Schwarzschild spacetime, making use of the semianalytic method of matched Green function expansions. Inside a local neighborhood of the compact object, this method uses the Hadamard form for the Green function in order to render regularization trivial. Outside this local neighborhood, we calculate the Green function using a spectral decomposition into poles (quasinormal modes) and a branch cut integral in the complex frequency plane. We show that both expansions overlap in a sufficiently large matching region for an accurate calculation of the self-force to be possible. The scalar case studied here is a useful and illustrative toy model for the gravitational case, which serves to model astrophysical binary systems in the extreme mass-ratio limit.

Figure 1

Green function along the innermost stable circular geodesic orbit of radius $r_0=6M$ (top) and along the eccentric geodesic orbit with $p=7.2$, $e=0.5$ at the point in the orbit where $r=6M$ and $\frac{dr(\tau )}{d\tau }>0$ (bottom). The left plots show the orbit and the first four null geodesics that start on the orbit, pass around the Schwarzschild black hole, and reintersect the orbit. The right plot illustrates the retarded Green function $G_{\mathrm{ret}}(x,x^{\prime })$ where $x^{\prime }$ is a point on the orbit at a time $\Delta t$ in the past of $x$. The singular features at points $x_1,x_2,x_3,\dots$ correspond to the arrival of the null geodesics (1), (2), (3),… shown in the left plots. The singular features follow a fourfold repeating pattern, arising from the passage of the null wavefront through caustics.

Figure 2

Schematic representation of the method of matched expansions [see Eq. (2.1)].

Figure 3

Contour deformation in the complex frequency plane. The residue theorem of complex analysis allows one to reexpress the integral over (just above) the real line of the Fourier modes $G_{\ell }^{\mathrm{ret}}$ of the GF as an integral over a high-frequency arc (HF), plus an integral around a branch cut (BC) of the Fourier modes, plus an integral over the residues at the poles (QNMs) of the Fourier modes.

Figure 4

Scalar QNM frequencies in the complex frequency plane in Schwarzschild spacetime for $n\mathrm{\text{:}}0\to 5$ and $\ell \mathrm{\text{:}}0\to 100$. The frequencies are symmetric with respect to the negative imaginary axis. In the fourth quadrant, the frequencies move down the plane with increasing $n$ and across to the right with increasing $\ell$.

Figure 5

The QNM excitation factors in the Argand plane. These plots show the real and imaginary parts of ${\mathcal{B}}_{\ell n}$ for the fundamental ($n=0$, left) and first overtones ($n=1$, right) of the scalar field on Schwarzschild spacetime, from $\ell =2\dots 50$. The data points show values determined with the MST method (see text), and the dotted lines show the asymptotic approximation for large $\ell$ [Eq. (31) in Ref. [24]].

Figure 6

QNM contribution for individual overtones $n=0$, 1, 2, 3, 4, 5 to the GF (left) and to the radial derivative of the GF (right) as a function of time $\Delta t$ along a circular geodesic of radius $r_0=6M$. Note that we use the symbol ${\mathcal{G}}_n^{\mathrm{QN}}$ (with $G_{\mathrm{QN}}=\sum _{n=0}^{\infty }{\mathcal{G}}_n^{\mathrm{QN}}$, in obvious notation) in this figure only—not to be confused with ${\mathcal{G}}_{\ell }^{\mathrm{QN}}$.

Figure 7

Dominant BC mode ${\mathcal{G}}_{\ell =0}^{\mathrm{BC}}$ (left) and ${\partial }_r{\mathcal{G}}_{\ell =0}^{\mathrm{BC}}$ (right) along a circular geodesic of radius $r_0=6M$ as a function of time. We plot the various contributions from different integration intervals in Eq. (4.4): The black curve is integration over $M\nu \mathrm{\text{:}}0\to 0.05$ (“very small” frequency regime), the dashed blue curve is over $M\nu \mathrm{\text{:}}0.05\to 0.225$ (“small” frequency regime), and the dotted purple curve is over $M\nu \mathrm{\text{:}}0.225\to 9$ (“mid” frequency regime).

Figure 8

Individual $\ell$ modes of the BC contribution to the GF (left) and its radial derivative (right) as a function of time along a circular geodesic of radius $r_0=6M$.

Figure 9

Combining QNM and BC contributions to compute the GF (left) and radial derivative of GF (right) in the DP along a circular geodesic of radius $r_0=6M$.

Figure 10

Validation of the GF (left) and its radial derivative (right) in the DP along a circular geodesic of radius $r_0=6M$. As a reference, we use a numerical time-domain approximation to the GF [57].

Figure 11

Matching between QL expansion (red dashed curve) and $\mathrm{QNM}+\mathrm{BC}$ expansion in the DP (black curve) for GF (top) and its radial derivative (bottom) as a function of time along a circular geodesic of radius $r_0=6M$ (left) and an eccentric orbit with $p=7.2$, $e=0.5$ instantaneously moving outward ar $r_0=6M$. The inset shows the relative difference $|1-G_{\mathrm{QL}}/(G_{\mathrm{QN}}+G_{\mathrm{BC}})|$ between the two expansions in the matching region, where $G_{\mathrm{QL}}$ is the GF calculated using the QL expansion.

Figure 12

Partial field (top) and partial SF (bottom) as a function of integration time, $\Delta t$, along a circular geodesic of radius $r_0=6M$ (left) and an eccentric orbit with $p=7.2$, $e=0.5$, instantaneously moving outward at $r_0=6M$. The exact value computed using a highly accurate frequency domain method [59] is given by the black dashed line. The insets show the sensitivity of the integration to the matching time. For these, we consider the full field and self-force by integrating all the way to $t^{\prime }=-\infty$, using large-$t$ asymptotics where they are valid.

Figure 13

Partial self-force as a function of integration time, $\Delta t$, along a range of circular geodesic orbits of radius $r_0$. The times when null geodesics intersect the circular timelike geodesics are indicated by black lines. The light gray lines are contours of constant orbital phase, $\gamma =\pi ,2\pi ,3\pi ,\dots$.