Regularized calculation of the retarded Green function in a Schwarzschild spacetime

The retarded Green function for linear field perturbations of black hole spacetimes is notoriously difficult to calculate. One of the difficulties is due to a Dirac-$\delta$ divergence that the Green function possesses when the two spacetime points are connected by a “direct” null geodesic. We present a procedure which notably aids its calculation in the case of Schwarzschild spacetime by separating this direct $\delta$ divergence from the remainder of the retarded Green function. More precisely, the method consists of calculating the multipolar $\ell$ modes of the direct $\delta$ divergence and subtracting them from the corresponding modes of the retarded Green function. We illustrate the usefulness of the method with some specific calculations in the case of the scalar Green function and self-field for a point scalar charge in Schwarzschild spacetime.

Figure 1

Green function $\ell$ modes for a scalar field on Schwarzschild spacetime as a function of time. The point $x$ is fixed at $r=6M$, and we vary the points $x^{\prime }$ along a circular geodesic at $r=6M$. Left: The red curve is $G_{\ell }^{\mathrm{ret}}$ via the method in Ref. [10], and the blue curve is $G_{\ell }^{\mathrm{d}}$ from Eq. (20) for $\ell =3$. Right: $G_{\ell }^{\mathrm{ret}}-G_{\ell }^{\mathrm{d}}$ for various $\ell$ values.

Figure 2

Retarded Green function $G_{\mathrm{ret}}(x,x^{\prime })$ for a scalar field on Schwarzschild spacetime as a function of time. The point $x$ is fixed at $r=6M$, and we vary the points $x^{\prime }$ along a circular geodesic at $r=6M$. The solid blue line shows $G_{\mathrm{ret}}$, Eq. (3), computed using a truncated and smoothed multipolar mode sum. The solid orange line shows $G_{\mathrm{nd}}(x,x^{\prime })$, Eq. (22), also computed using a truncated and smoothed multipolar mode sum. In both cases, smooth-sum parameters of ${\ell }_{\max }=100$ and ${\ell }_{\mathrm{cut}}=20$ were used. The dashed red line shows the Hadamard bitensor $V(x,x^{\prime })$ computed from a short-distance Padé-resummed Taylor series [32].

Figure 3

Plot of the regularized self-field ${\mathrm{\Phi }}_R$ [calculated via Eq. (29)] as a function of matching coordinate time $t_m$ for a scalar charge on a circular geodesic at $r=6M$. The solid blue line is obtained using Eq. (29) as it is (i.e., with $G_{\ell }^{\mathrm{ret}}$ in the summand); the solid orange line is obtained using Eq. (29) with $G_{\ell }^{\mathrm{ret}}$ replaced by $G_{\ell }^{\mathrm{ret}}-G_{\ell }^{\mathrm{d}}$ in the summand. In both cases, the numerical integration was truncated at $t=200M$, with the contribution for $t>200M$ accounted for by a late-time approximation which assumes the branch-cut contribution to the GF dominates (see, e.g., Refs. [9, 10]). The dashed black line is a highly accurate reference value computed using the mode sum regularization method [33].

Figure 4

The regularized self-field ${\mathrm{\Phi }}_R$ for a scalar charge on a circular geodesic at $r=6M$ as a function of the parameter ${\ell }_{\mathrm{cut}}$ in the smoothing factor in the $\ell$ sum (see Sec. 2). Blue curve: ${\mathrm{\Phi }}_R$ is calculated via Eq. (29) with ${\tau }_m=18M$. Red curve: ${\mathrm{\Phi }}_R$ is calculated via Eq. (29) with ${\tau }_m=18M$ and with $G_{\ell }^{\mathrm{ret}}$ replaced by $G_{\ell }^{\mathrm{ret}}-G_{\ell }^{\mathrm{d}}$. Green curve: ${\mathrm{\Phi }}_R$ is calculated via Eq. (29) with ${\tau }_m=0$ and with $G_{\ell }^{\mathrm{ret}}$ replaced by $G_{\ell }^{\mathrm{ret}}-G_{\ell }^{\mathrm{d}}$.