Gravitational self-force on a particle in circular orbit around a Schwarzschild black hole

We calculate the gravitational self-force acting on a pointlike particle of mass $\mu$, set in a circular geodesic orbit around a Schwarzschild black hole. Our calculation is done in the Lorenz gauge: For given orbital radius, we first solve directly for the Lorenz-gauge metric perturbation using numerical evolution in the time domain; we then compute the (finite) backreaction force from each of the multipole modes of the perturbation; finally, we apply the “mode-sum” method to obtain the total, physical self-force. The temporal component of the self-force (which is gauge invariant) describes the dissipation of orbital energy through gravitational radiation. Our results for this component are consistent, to within the computational accuracy, with the total flux of gravitational-wave energy radiated to infinity and through the event horizon. The radial component of the self-force (which is gauge dependent) is calculated here for the first time. It describes a conservative shift in the orbital parameters away from their geodesic values. We thus obtain the $O(\mu )$ correction to the specific energy and angular momentum parameters (in the Lorenz gauge), as well as the $O(\mu )$ shift in the orbital frequency (which is gauge invariant).

Figure 1

A numerical grid cell, of dimensions $h\times h$ (see description in the text). Our 2D grid is based on characteristic (Eddington-Finkelstein) coordinates $v$ and $u$. These are related to the Schwarzschild coordinates through $v=t+r_{*}$ and $u=t-r_{*}$, where $r_{*}=r+2M\ln [r/(2M)-1]$.

Figure 2

Finite-difference scheme for terms in the field equations involving single $v$ derivatives (diagrams to go with the description in the text). The dashed line represents the worldline, with two cases shown.

Figure 3

Diagram to explain how $r$ derivatives are taken at the worldline (see description in the text). The dashed line represents the particle’s trajectory on the numerical grid. The SF is calculated at the point labeled “0.”

Figure 4

Analytic fitting for the large-$l$ tail of the SF, exemplified here for $r_0=6M$. We used the fitting formula (45) with $N=2$, and based on the modes $10\le l\le 15$. Circles (“$\odot$”) represent actual data obtained for $F_{\mathrm{reg}}^{rl}$ (calculated from $r_0^{-}$), for the various modes $3\le l\le 25$. The dashed line is the analytic fit. Left/right panels show the same data on linear/log scales. The large-$l$ tail of the mode-sum series shows the $l^{-2}$ falloff expected from theory (cf. Fig. 7).

Figure 5

Numerical convergence of the calculated SF, demonstrated here for the $r$ component, for $r_0=6M$. The left and right panels show $l=2$ and $l=15$, respectively. Plotted is the difference $F_{\mathrm{reg}}^{rl}[h_i]-F_{\mathrm{reg}}^{rl}$ between the value of the regularized mode computed with step size $h_i$, and the value extrapolated to $h\to 0$. Each panel displays both one-sided values of the force: “left” and “right” stand for $r_0^{-}$ and $r_0^{+}$ values, respectively. The reference lines (dotted) have slopes $\propto h^2$. This demonstrates the quadratic convergence of the numerical calculation.

Figure 6

Time evolution of the metric perturbation. We plot the absolute values of the various functions ${\overline{h}}^{(i)lm}$, evaluated at the location of the particle, as a function of $t$, for $l=2$ and $m=1$, 2. Top two figures are for $r_0=6M$; lower two figures are for $r_0=100M$. The time $t$ is indicated along the horizontal axis in units of the orbital period, $T_{\mathrm{orb}}$. The initial transient phase is due to the imperfection of the initial conditions; these spurious waves dissipate rapidly, clearing the stage for the physical, stationary solution. The SF is calculated at late time, when the effect of the initial spurious waves is negligible.

Figure 7

Large-$l$ behavior of the regularized modes $F_{\mathrm{reg}}^{\alpha l}$. The reference lines in the left panel are $\propto l^{-2}$; the reference line in the right panel is $\propto \exp (-1.22l)$. The left panel displays the radial component $F_{\mathrm{reg}}^{rl}$ (calculated using internal derivatives) for $3\le l\le 25$ and for $r_0=6M$, $100M$. The regularized modes appear to fall off as $\propto l^{-2}$ at large $l$, in agreement with theory. The right panel shows the $l$-modes of the $t$ component (for $r_0=6M$); these are already regularized, and show an exponential falloff at large $l$. (Note the scale for the two panels is different: log-log for the left panel, semilog for the right.)

Figure 8

The radial component of the SF. The data in the upper panel correspond to the second column in Table . The inset shows an expansion of the ISCO area. The solid line is a plot of the large-$r_0$ analytic fit given in Eq. (56). The dashed line represents the complementary near-ISCO fit (58). The lower panel shows the relative difference between the numerical data and values obtained using the analytic-fit formulas. Using the large-$r_0$ fit for $r_0\ge 8M$ and the near-ISCO fit for $6M\le r_0\le 8M$, one recovers all numerical data to within the numerical accuracy.

Figure 9

The shift in the energy (left panel) and angular momentum (right panel) parameters, due to the conservative SF effect. The plots show the relative shifts $\Delta \mathcal{E}\equiv (\mathcal{E}-{\mathcal{E}}_0)/{\mathcal{E}}_0$ and $\Delta \mathcal{L}\equiv (\mathcal{L}-{\mathcal{L}}_0)/{\mathcal{L}}_0$, where ${\mathcal{E}}_0$ and ${\mathcal{L}}_0$ are the geodesic values. Solid and dashed lines correspond to the large-$r_0$ and near-ISCO analytic fits, Eqs. (56, 58), respectively. Recall $\Delta \mathcal{E}$ and $\Delta \mathcal{L}$ are gauge dependent; the values shown here are the Lorenz-gauge values.

Figure 10

Shift in the orbital frequency due to the conservative SF effect. Shown is the quantity $\Delta ({\Omega }^2{)}_{\mathrm{GR}}\equiv ({\Omega }^2-{\Omega }_0^2)/{\Omega }_0^2-(-2\mu /M)$, describing the relative shift in ${\Omega }^2$, minus the leading-order Keplerian term. The solid line shows the large-$r_0$ (3PN) analytic fit, Eq. (60). The dashed line corresponds to the near-ISCO fit (58).