Green functions of the Regge-Wheeler and Teukolsky equations in Schwarzschild spacetime

Abstract

We present a calculation of the full retarded Green functions of the Regge-Wheeler and Teukolsky equations obeyed by gravitational field perturbations of Schwarzschild spacetime. We perform the calculations for spacetime points along: (i) a timelike circular geodesic (where null-separated points are not at caustics); and (ii) a static worldline (where null-separated points are at caustics). These Green functions show a 4-fold singularity structure away from caustics, and 2-fold at caustics (similarly to the case of scalar field perturbations, which we also reproduce). Physical oscillations near the singularities appear in the gravitational case, which were not present in the scalar case. We obtain our results by developing various numerical and analytical methods.

Figures (21)

• Figure 1: Past-directed null geodesics emanating from a field point $x$ on a timelike worldline, orbiting around the equator of a Schwarzschild black hole and re-intersecting the worldline at various base points $x'$ (light crossings). Top: first four orbiting null geodesics in the setting of a timelike circular geodesic at $r_0=6M$; the type of singularity in the GF at each intersection is indicated and follows the $4$-fold cycle in Eq. \ref{['eq:4-fold']}. Bottom: first four (two if taking the symmetry into account) orbiting null geodesics in the setting of a static worldline at $r_0=6M$; the GF singularity structure follows the $2$-fold cycle in Eq. \ref{['eq:2-fold,g=0']}. The dotted, inner circular curve corresponds to the photon orbit at $r=3M$; a circle of radius $r=6M$ is the solid, black curve on the top plot and the dashed, black curve on the bottom plot. The number in parenthesis in the labels indicates the number of caustic crossings by the null geodesics before re-intersecting the timelike worldline.
• Figure 2: Uniform grid distribution used to solve the characteristic initial value problem \ref{['eqn:sglCIDEqn']}--\ref{['eq:CID']} for ${}_{s} g_\ell(v,u)$. For the cell $SENW$ on the grid, the value of ${}_{s} g_\ell(v,u)$ at $N$ is determined from its previously calculated values at $S$, $E$ and $W$, together with the value of $\mathcal{Q}_s(r)$ at $O$.
• Figure 3: Modes ${}_{2}G_{\ell}$ and $G^{\textrm{dir}}_\ell$ of, respectively, the spin-2 RW GF (orange curves) and its direct Hadamard term (blue curves), as well as their difference (green curves), as functions of $\Delta t$. The points are on $r=r'=6M$ and the modes are for $\ell=2$ (top) and 5 (bottom).
• Figure 4: Fourier modes ${}_{2} G_{\ell=20,\omega}$ (continuous curves) of the RW ${}_{2} G_{\ell=20}$, together with the leading-order in the asymptotic expansion for large $\omega\in\mathbb{R}$ in Eq. \ref{['eqn:sGwlAsymtotic']} of the imaginary part (dashed black line), as functions of $M\omega$ for $r=r'=6M$.
• Figure 5: Relative difference between ${}_{2} G_\ell$ computed via CID (Eqs. \ref{['eqn:sGlAnsatz']}--\ref{['eq:CID']}) and via the Fourier integral in Eq. \ref{['eqn:sGwlFourierIntegral']}, as a function of $\Delta t$ for $r=r'=6M$ and $\ell=2$.