Scalar field self-force effects on orbits about a Schwarzschild black hole

For a particle of mass $\mu$ and scalar charge $q$, we compute the effects of the scalar field self-force upon circular orbits, upon slightly eccentric orbits and upon the innermost stable circular orbit (ISCO) of a Schwarzschild black hole of mass $m$. For circular orbits the self-force is outward and causes the angular frequency at a given radius to decrease. For slightly eccentric orbits the self-force decreases the rate of the precession of the orbit. The effect of the self-force moves the radius of the innermost stable circular orbit inward by $0.122701\times q^2/\mu$, and it increases the angular frequency of the ISCO by the fraction $0.0291657\times q^2/\mu m$.

Figure 1

The regularized field ${\Psi }_{\mathrm{o}}^{\mathrm{R}}m/q$ at the particle and the radial component of the self-force $({\partial }_r\Psi _{\mathrm{o}}{}^R)m^2/q$ as a function of the radius $r_{\mathrm{o}}$ for circular orbits close to the black hole.

Figure 2

The same as Fig. 1 except for larger radii.

Figure 3

The fractional changes in $J$, $E$, and ${\Omega }_{\mathrm{o}}$, from the self-force as a function of the radius $r_{\mathrm{o}}$ for circular orbits close to the black hole. These quantities are calculated using Eqs. (24, 25, 26).

Figure 4

The same as Fig. 3 except for larger radii.

Figure 5

For $q^2/\mu \ll r_{\mathrm{o}}-6m$, the change in ${\Omega }_r$ from the self-force as a function of the radius $r_{\mathrm{o}}$ for slightly eccentric orbits close to the black hole. The limiting behavior at $q^2/\mu \ll r_{\mathrm{o}}-6m\ll m$ is given in Eq. (46). See Eq. (43) for $q^2/\mu \approx r_{\mathrm{o}}-6m$.

Figure 6

From numerical analysis, the fractional change in ${\Omega }_r$ from the self-force as a function of the radius $r_{\mathrm{o}}$ for slightly eccentric orbits far from the black hole.

Figure 7

For $q/m\ll r_{\mathrm{o}}-6m$, the change in ${\Omega }_{\mathrm{p}\mathrm{r}}$, from the self-force as a function of the radius $r_{\mathrm{o}}$ for slightly eccentric orbits close to the black hole.

Figure 8

For $q/m\ll r_{\mathrm{o}}-6m$, the fractional change in ${\Omega }_{\mathrm{p}\mathrm{r}}$ from the self-force as a function of the radius $r_{\mathrm{o}}$ for slightly eccentric orbits far from the black hole.