Selfforce on a scalar charge in a circular orbit about a ReissnerNordström black hole
Abstract
Motivated by applications to the study of selfforce effects in scalartensor theories of gravity, we calculate the selfforce exerted on a scalar charge in a circular orbit about a ReissnerNordström black hole. We obtain the selfforce via a modesum calculation, and find that our results differ from recent postNewtonian calculations even in the slowmotion regime. We compute the radiative fluxes towards infinity and down the black hole, and verify that they are balanced by energy dissipated through the local selfforce – in contrast to the reported postNewtonian results. The selfforce and radiative fluxes depend solely on the black hole’s chargetomass ratio, the controlling parameter of the ReissnerNordström geometry. They both monotonically decrease as the black hole approaches extremality. With respect to an extremality parameter , the energy flux through the event horizon is found to scale as as .
pacs:
04.20.âq,04.25.Nx, 04.70.BwI Introduction
The selfforce acting on a particle moving in a curved spacetime has been a fascinating subject for some time, principally motivated by the prospect of detecting lowfrequency gravitational waves with future spacebased missions such as LISA AmaroSeoane et al. (2012). While selfforcebased gravitational waveforms remain elusive, progress in selfforce research has been steady, and has made key contributions to a fuller understanding of the stronglygravitating twobody problem Poisson et al. (2011); Barack (2009); Wardell (2015). Beyond direct applications to gravitationalwave astronomy, the selfforce has proven useful as a theoretical probe of the nonlocal features of the spacetime in which the particle moves. More specifically, the static selfforce has been shown to be sensitive to central and asymptotic structure Drivas and Gralla (2011); Taylor (2013); Kuchar et al. (2013). At the frontier of selfforce research there remains strong momentum for the calculation of fully selfconsistent gravitational waveforms from extrememassratio inspirals (to second order in the mass ratio), in addition to active research programs pushing to develop the formalism to higher dimensions Taylor and Flanagan (2015); Harte et al. (2016, 2017) and alternative theories of gravity Zimmerman (2015). For the latter, it is of strong interest to understand how the extra gravitational degrees of freedom and their coupling might impact selfforced dynamics.
The work reported in this paper began with a consideration of selfforces in the context of alternative theories of gravity, particularly in scalartensor theories, as inspired by the seminal work of Zimmerman Zimmerman (2015). One tantalizing result of this work was the possibility of “scalarization” of a compact object by the action of the selfforce. However, a close inspection of Zimmerman (2015) quickly reveals that this effect requires a nontrivial scalar field residing on a curved spacetime background. Nohair theorems for black holes in scalartensor theories then severely limit the possible realizations for scalarization via selfforce. To be sure, there are loopholes to these theorems; certain scalartensor theories do admit hairy black hole solutions Torii et al. (2001); Kanti et al. (1996); Sotiriou and Zhou (2014); Antoniou et al. (2018, 2017). But these solutions are obtained mostly with numerical integration Sotiriou and Zhou (2014), making them difficult to study as backgrounds for concrete selfforce calculations. (See, however, Babichev et al. (2017) for some simple hairy black hole solutions in Horndeski theory.)
A wellknown solution in scalartensor theory is the socalled BocharovaBronnikovMelnikovBekenstein (BBMB) solution Bocharova et al. (1970); Bekenstein (1974, 1975) in conformal scalarvacuum gravity. This theory is defined by the action
(1) 
and the BBMB solution reads
(2) 
with
(3) 
The metric in this solution is the extremal ReissnerNordström solution, and the scalar field is nontrivial, though it clearly diverges at the putative event horizon at . This divergence muddles the interpretation of as a true event horizon and, correspondingly, of the BBMB solution as a legitimate black hole solution. But this interpretational issue can be eschewed when one’s primary concern is the impact of the scalar degree of freedom on the gravitational dynamics. This is the viewpoint espoused by our research program, of which this work is an initial step. Our proposal is to use the BBMB solution as a theoretical playground for studying scalartensor selfforce effects.
Apart from this broader goal, the ReissnerNordström spacetime is in itself an interesting spacetime on which to study selfforce effects. As the unique, sphericallysymmetric and asymptotically flat solution to the EinsteinMaxwell equations, it describes the spacetime outside a charged sphericallysymmetric mass distribution. It is characterized by its mass and charge , and the spacetime is described by the metric
(4) 
where and . Note that the coordinate is connected to Q. If the object in question is a black hole, then it will have the following features:

There are 2 horizons, an inner horizon at , and an outer horizon at , which happens to be the event horizon of the black hole.

In the case where , the black hole becomes extremal, with a degenerate horizon (and thus zero temperature).
Despite these interesting properties, astrophysical considerations preclude significant charge buildup, and thus the selfforce on a ReissnerNördstrom background has been largely neglected. The notable exception is a recent work by Bini et al. (Bini et al., 2016) in which they produced a 7 postNewtonian (PN) order calculation of the scalar selfforce on a circular geodesic. In this paper, we go beyond the PN approximation and present the first modesum calculation of the full strongfield scalar selfforce on a circular geodesic. In the process of doing so, we found that our results for the selfforce and energy flux is in disagreement with the slowmotion formulas presented by Bini et al. (Bini et al., 2016). This is surprising, as one would expect a numerical calculation to agree with a PN calculation up to the order reported. We have yet to establish a reason for this discrepancy. Nevertheless, we present several consistency checks on our results, to show that this disagreement is not from an error in the numerical calculation.
The paper is organized as follows: In Section II we provide a brief review of circular geodesics in ReissnerNordström spacetime. In Section III we provide a derivation of the scalar field generated by a scalar point charge in a geodesic circular orbit, subject to ingoing wave conditions at and outgoing wave conditions at . In Section IV we briefly review the modesum regularization scheme, as well as providing a brief derivation of the regularization parameters for geodesic circular orbits in ReissnerNordström spacetime. In Section V we provide numerical results computed from a frequencydomain calculation and compare it with analytical results obtained from (Bini et al., 2016).
Ii Circular geodesics
Considered as a test particle, the scalar charge will move along a geodesic of the ReissnerNordström spacetime. Two Killing vectors of spacetime, and , provide the conserved quantities
(5)  
(6) 
where is the proper time along the orbit and is the particle’s fourvelocity.
Due to the spherical symmetry of the spacetime, the test particle will move along a fixed plane. We can always choose our coordinates so that this plane is described by . Combining these with the normalization, , we arrive at the radial equation
(7) 
where is the effective potential. For circular orbits (), the fourvelocity reads
(8) 
where
(9) 
is the angular velocity of the particle with respect to an asymptotic observer. Note that this quantity, as an observable, is invariant to coordinate transformations. Circular orbits also require , which gives the condition
(10) 
while in Eq. (7) gives
(11) 
which can be combined to give
(12) 
Putting this into Eq. (9) we finally get
(13) 
Normalization of the fourvelocity then gives
(14) 
This completes the determination of the fourvelocity for a particle in a circular geodesic.
Iii Field equations
iii.1 Multipole decomposition
We assume that the scalar field is a small perturbation of the fixed ReissnerNordström spacetime, and that it satisfies the minimally coupled scalar wave equation
(15) 
sourced by a scalar charge density . We model this scalar charge density as a function distribution on the worldline, written as
(16) 
which for a circular orbit becomes
(17)  
(18) 
Using the spherical harmonic completeness relations, we can further rewrite as
(19)  
(20)  
(21) 
A similar decomposition for the scalar field into spherical harmonics and Fourier modes yields the form
(22) 
With Eqs. (21) and (22), Eq. (15) reduces to
(23) 
We now want to impose boundary conditions.
iii.2 Boundary conditions
The wave equation can also be rewritten in terms of the socalled tortoise coordinate
(24) 
Defining (where ) we get
(25) 
where is the Laplacian on the unit twosphere. Decomposing into its sphericalharmonic components
(26) 
Eq. (25) becomes
(27) 
The homogeneous part of this equation appears like a flatspace wave equation [in (1+1) dimensions] with a potential . This potential vanishes as (or as ) and as ().
The appropriate boundary conditions are ingoing waves at the event horizon and outgoing waves at infinity. Since (where for circular orbits), we shall then impose that as and as . Correspondingly, for the boundary conditions of interest are
(28) 
and
(29) 
These boundary conditions serve as initial data in the integration of Eq. (23). In practice, the integration cannot begin exactly at the horizon because vanishes and the potential term in Eq. (23) blows up. [The potential term of Eq. (27) is regular, but the horizon in these coordinates is inaccessible at .] Instead, we then begin the integration slightly away from the horizon, at for . An asymptotic solution as can be obtained by inserting the ansatz
(30) 
into Eq. (23). This gives a recurrence relation for the coefficients which reads
(31) 
The same considerations apply to the boundary condition as . Again we work with the ansatz
(32) 
and obtain a recurrence relation for using Eq. (23). This reads
(33) 
Iv Regularization
iv.1 Modesum regularization
To obtain the scalar selfforce, we must first subject the unregularized force to a regularization procedure. In our case, we use the modesum scheme Barack and Ori (2000); Burko (2000); Barack and Ori (2003), where the selfforce is constructed from regularized spherical harmonic contributions. We start with full force derived from the retarded field
(34) 
where is the mode component (summed over ) of the full force at an arbitrary point in the neighborhood of the particle. At the particle location, each is finite, although the sided limits often produce different values (which we then label as ) and the sum over may not converge. We then obtain the selfforce using a modebymode regularization formula
(35) 
where the regularized contributions no longer have the ambiguity and the sum over is guaranteed to converge. The regularization parameters (independent) and have been obtained for generic orbits about a Schwarzschild black hole Barack et al. (2002), and a Kerr black hole Barack and Ori (2003). In the next subsection we present a derivation of and for circular orbits about a ReissnerNordström black hole.
iv.2 Regularization parameters
The procedure for deriving modesum regularization parameters is by now wellestablished Barack et al. (2002); Barack and Ori (2003); Haas and Poisson (2006); Heffernan et al. (2012, 2014); Heffernan (2012). Here, we directly follow the approach of Heffernan et al. (2012, 2014); Heffernan (2012), extending it to the case of ReissnerNordström spacetime as given by the line element, Eq. (4). Since the essential details remain the same, we refer the reader to Refs. Heffernan et al. (2012, 2014); Heffernan (2012) for an extensive discussion, and give here only the key equations and results.
We start with an expansion of the DetweilerWhiting singular field (Detweiler and Whiting, 2003) through nextfromleading order in the distance from the worldline,
(36) 
Here, we have already specialized to the case of circular, equatorial orbits, and have introduced and
(37) 
The above expressions are given in terms of the Riemann normal coordinates and , the same as are described in Heffernan et al. (2014).
It turns out that for circular orbits the and componets of the selfforce are purely dissipative, meaning that only the radial component of the selfforce requires regularization. We thus compute the contribution from the singular field to the radial component of the selfforce using . Doing so, taking , and keeping only terms which will not vanish in the limit we get
(38) 
Here, , just as in Heffernan et al. (2014).
Next, we obtain the regularization parameters by decomposing this into sphericalharmonic modes (as usual, we only need to consider the case since the other modes do not contribute). Doing so, and taking the limit , we find
(39) 
from which we can immediately read off the and regularization parameters. Here,
(40) 
are complete elliptic integrals of the first and second kind, respectively.
V Selfforce calculation
v.1 Scalar energy flux
Global energy conservation dictates that the local energy dissipation, represented by the component of the selfforce, is accounted for by the energy flux carried by scalar field radiation. We numerically calculate the energy flux to infinity and down the black hole, and verify that the result is consistent with the energy lost through the local dissipative selfforce.
We briefly review the relevant formalism used to calculate the energy flux. The stressenergy tensor of the scalar field is given by
(41) 
With , we construct the differential energy flux over the following constant hypersurfaces: , represented by , and , represented by . The differential energy flux then takes the form
(42) 
where is the unit normal vector of the hypersurface, and is the hypersurface element. We then rewrite Eq. (42) as
(43) 
Integrating over the twosphere, we then express the energy transfer as
(44) 
Substituting the multipole expansion defined by Eq. (22) into Eq. (44), we then arrive at the following expression for the energy transfer
(45) 
We present sample numerical data for , and in Tables 1, and 2 respectively. We see that as the extremality parameter approaches zero, monotonically decreases. We also note that compared to , exhibits a dramatic decrease as . We then investigate the scaling behavior of with respect to , which we present in Fig. 1 and 2. We note that while exhibits no discernible scaling behavior, exhibits power law scaling as , which in Fig. 2 corresponds to . This behavior for has been previously observed for nearextremal Kerr black holes, with a power scaling of Gralla et al. (2015).
In the same figures, we compared our numerical data with the slowmotion analytic formulas for the energy fluxes derived by Bini et al. Bini et al. (2016). While the qualitative behavior of Bini et al.’s formula is similar to our numerical results for , that cannot be said for . The qualitative behavior exhibited by Bini et al.’s formula for is opposite to that of our numerical results, and the disagreement worsens as .
v.2 Dissipative component of the selfforce
For circular orbits, the dissipative components of the scalar selfforce are , and . We note that due to in the circular orbit case, there is a simple relationship between the dissipative components of the selfforce:
(46) 
This relationship indicates that we need only one component to calculate. In this work, we choose to calculate .
For our setup, the local energy dissipation must be accounted for by the energy fluxes towards infinity, and down the black hole. This energy balance relation can be expressed in terms of the selfforce
(47) 
This allows us to test our computation of by verifying that our numerical results satisfy Eq. (47).
Sample numerical results for are presented in Table 4. As a check, we compared our Schwarzschild results with those of Warburton and Barack Warburton and Barack (2010), and we are in agreement to all significant figures presented. Looking at our results, we see that as the black hole approaches extremality , the dissipative selfforce decreases. One concludes from this that the black hole charge suppresses local energy dissipation.
We also compared our results to the slowmotion formula for the dissipative selfforce derived by Bini et al. Bini et al. (2016), presented in Fig. 3. We see that the qualitative behavior of our results completely differs from Bini et al.’s formula, which worsens as . We note that this discrepancy in qualitative behavior is also present for , as presented in Fig. 1.
Numerical Result  PN Result  

As a consistency check, we then examined the energy balance relation exhibited by our numerical results and Bini et al.’s slowmotion formulas. Sample data are presented in Table 3. While it is expected that energy balance will be better satisfied by numerical calculations compared to PN calculations, we note that the energy balance error exhibit by Bini et al.’s formulas are of relative 3 PN order, one order higher than the expected error in their formulas.
v.3 Conservative component of the selfforce
For circular orbits, the conservative component of the selfforce is contained entirely in . The calculation of this conservative selfforce is more complicated than the dissipative piece, as the modesum requires regularization. We then checked the effect of the regularization parameters on the modesum, as presented in Fig. 4. Looking at the high mode components, we see that the regularization parameters work as expected, leaving a residual field which exhibits falloff behavior.
Sample numerical data for is presented in Table 5. We compared our Schwarzschild results with those of DiazRivera et al. DiazRivera et al. (2004), and we are in agreement up to six significant figures. Looking at our results, we see that as the black hole approaches extremality , the conservative selfforce decreases. This implies that the black hole charge suppresses the entire selfforce.
We also compared our numerical results to the slowmotion formula for the conservative selfforce derived by Bini et al. Bini et al. (2016). We present this in Fig. 3, and we see that while Bini et al.’s formula follow the same qualitative behavior of our results, we begin to deviate as .
Vi Conclusion
In this work we presented the first modesum calculation of the selfforce exerted on a particle in circular orbits about a ReissnerNordström black hole. We also present in this work regularization parameters , and for circular orbits in ReissnerNordström spacetime.
We tested the validity of our results in various ways. The results for the Schwarzschild limit was found to agree with the results found in the literature Warburton and Barack (2010); DiazRivera et al. (2004). We confirmed numerically that the local energy dissipation is balanced out by the energy carried away by scalar waves towards infinity and down the event horizon. We also investigated the mode falloff of the conservative selfforce, and found that after subtracting the and regularization parameters the modes of the residual field fall off as , as expected.
Our results indicate that as the black hole’s electric charge increases, the selfforce decreases in magnitude. This dampening is notably drastic for the flux of scalar radiation towards the event horizon, where in nearextremal ReissnerNordström black holes, the scalar radiation flux scales as . This behavior is also seen for nearextremal Kerr black holes Gralla et al. (2015), thus a more detailed calculation is recommended as a future study.
We also compared our results with the slowmotion formulas obtained by Bini et al. Bini et al. (2016). While our results agree in the Schwarzschild limit, they disagree for , and as the electric charge increases, the disagreement between the results increases. We have yet to establish the reason for this disagreement.
We expect some of our results to have some bearing on future selfforce studies in black hole solutions of scalartensor theories. The BBMB solution of conformal scalarvacuum gravity is exactly the extremal ReissnerNordström geometry, so similar results might be obtained in situations where the scalar field in these alternative theories can be approximated as test fields. Other hairy black hole solutions have now been discovered in other scalartensor theories of gravity and some of these are also of ReissnerNordström form Babichev et al. (2017). Selfforce phenomenology in these theories remains completely uncharted, and so remains a promising area of future research. By exploring the scalar selfforce from a minimallycoupled scalar field in the ReissnerNordström spacetime, we hope to have provided a useful guide and some benchmark numerical results for future selfforce calculations in alternative theories of gravity.
Vii Acknowledgments
This research is supported by the University of the Philippines OVPAA through Grant No. OVPAABPhD201613 and by the Department of Science and Technology Advanced Science and Technology Human Resources Development Program  National Science Consortium (DOST ASTHRDPNSC).
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