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

TL;DR

This work analyzes the scalar-field self-force on a small mass with scalar charge in Schwarzschild spacetime, focusing on circular orbits, slightly eccentric orbits, and the ISCO. Using perturbation theory and mode-sum regularization, it separates dissipative and conservative self-force effects and computes how the scalar self-force shifts orbital parameters, including a inward ISCO shift and an increase in the ISCO angular frequency. For circular orbits the self-force is outward and reduces $\Omega_o$, $E$, and $J$; for slightly eccentric orbits the precession rate is reduced; and the ISCO moves inward by $\Delta R_{is}=-0.122701\,q^2/\mu$ with $\Delta\Omega_{is}/\Omega_{is}=0.0291657\,q^2/(\mu M)$. These scalar-field results provide essential intuition and a methodological bridge toward the gravitational self-force problem, while underscoring key differences due to the absence of scalar charge in the black hole and the resulting center-of-mass dynamics.

Abstract

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 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 q^2/mu, and it increases the angular frequency of the ISCO by the fraction 0.0291657 q^2/mu M.

Figures (8)

• Figure 1: The regularized field ${\Psi_{\text{o}}}^{\text{R}} m/q$ at the particle and the radial component of the self force $(\partial_r{\Psi_{\text{o}}}^R)m^2/q$ as a function of the radius ${r_\text{o}}$ for circular orbits close to the black hole.
• Figure 2: The same as Fig. \ref{['dpdrFigA']} except for larger radii.
• Figure 3: The fractional changes in $J$, $E$ and ${\Omega_\text{o}}$, from the self force as a function of the radius ${r_\text{o}}$ for circular orbits close to the black hole. These quantities are calculated using Eqs. (\ref{['DeltaJ']})--(\ref{['DeltaOm']}).
• Figure 4: The same as Fig. \ref{['JEOFigA']} except for larger radii.
• Figure 5: For $q^2/\mu \ll {r_\text{o}}-6m$, the change in $\Omega_r$ from the self force as a function of the radius ${r_\text{o}}$ for slightly eccentric orbits close to the black hole. The limiting behavior at $q^2/\mu \ll {r_\text{o}}-6m << m$ is given in Eq. (\ref{['DeltaOmrb']}). See Eq. (\ref{['DeltaOmrc']}) for $q^2/\mu \approx {r_\text{o}}-6m$.