Self-force on a scalar charge in a circular orbit about a Reissner-Nordström black hole

Motivated by applications to the study of self-force effects in scalar-tensor theories of gravity, we calculate the self-force exerted on a scalar charge in a circular orbit about a Reissner-Nordström black hole. We obtain the self-force via a mode-sum calculation and find that our results differ from recent post-Newtonian calculations even in the slow-motion regime. We compute the radiative fluxes toward infinity and down the black hole and verify that they are balanced by energy dissipated through the local self-force—in contrast to the reported post-Newtonian results. The self-force and radiative fluxes depend solely on the black hole’s charge-to-mass ratio, the controlling parameter of the Reissner-Nordström geometry. They both monotonically decrease as the black hole approaches extremality. With respect to an extremality parameter $\epsilon$, the energy flux through the event horizon is found to scale as $\sim {\epsilon }^{5/4}$ as $\epsilon \to 0$.

Figure 1

Linear-log plot of ${\dot{E}}_{+}$ vs $\epsilon$ for $(M\mathrm{\Omega }{)}^{-2/3}=50$. The orange dots represent the numerical results, while the solid blue line represents the results obtained from Bini et al.’s slow-motion formula for the outgoing flux. Disagreement between the two results increases as $\epsilon \to 0$.

Figure 2

Log-log plot of ${\dot{E}}_{-}$ vs $\epsilon$ for $(M\mathrm{\Omega }{)}^{-2/3}=50$. The orange dots represent the numerical results, while the solid blue line represents the results obtained from Bini et. al.’s slow-motion formula for the horizon flux. For near-extremal values of $\epsilon$, ${\dot{E}}_{r=r_{+}}$ exhibits power law falloff in $\epsilon$, which in this example is found to scale as ${\dot{E}}_{r=r_{+}}\sim {\epsilon }^{5/4}$.

Figure 3

Linear-log plot of $\epsilon$ vs $F_t$, for $(M\mathrm{\Omega }{)}^{-2/3}=50$. The orange dots represent the numerical results, while the solid blue line represents Bini et al.’s slow-motion formula for the dissipative self-force. Disagreement between the two results increases significantly as $\epsilon \to 0$.

Figure 4

$l$-modes for the conservative self-force for $(M\mathrm{\Omega }{)}^{-2/3}=10$ and $\epsilon =0.5$, along with the results from the regularization by $A_r$, $B_r$, and additional regularization parameters ($D_r$, $E_r$, $F_r$) obtained from a numerical fit.

Figure 5

Linear-log plot of $\epsilon$ vs $F_r$ for $(M\mathrm{\Omega }{)}^{-2/3}=50$. The orange dots represent the numerical results, while the solid blue line represents the results obtained from Bini et al.’s slow-motion formula for the conservative self-force. Disagreement between the two results increases as $\epsilon \to 0$; however, it is not as severe as compared to the dissipative self-force.